Chapter 5xqd339/DarkenChapter_05A.doc · Web viewChapter 5 Multiplication and Division I: Meaning 5.1 Multiplication as Repeated Addition Multiplication is not really a basic operation

Chapter 5

Multiplication and Division I Meaning

51 Multiplication as Repeated Addition

Multiplication is not really a basic operation As the problems in the following activity show it is possible to solve many ldquomultiplicationrdquo problems by using a simpler operation

Activity 51A

A Solve the following problems using addition and appropriate units Draw pictures if it is helpful to do so

1 Three children are playing a game Each child gets four cards How many cards are in use

2 A rectangular baby quilt is made of four strips each containing six squares How many squares are in this quilt

3 Rachel has two pairs of shorts and three T-shirts Assuming she is indifferent to color coordination how many outfits does she have

4 A water bottle has a capacity of 112 liters of water How many liters of water can five of these bottles hold

B Answer the following

1a Each of the problems in part A involved repeated ___________________

b Each of the problems in part A could have been solved more efficiently using what operation

_____________

c Thus multiplication can be defined as __________________________________________________________

2 Consider the following sets hearts hearts hearts hearts hearts hearts

a There are _____ sets with ______hearts in each set The union of these sets includes six ____________

b In other words 3 ∙ (2 __________) = 6 _______________

c In this problem 3 refers to the ___________________ of sets and 2 refers to the ___________ of a set

3 Reconsider problem 4 in part A Five referred to the _______________ of bottles and three quarters of a liter referred to the _______________ of a bottle

4 In these situations it seems that one of the numbers in a multiplication refers to the _____________________

and the other refers to the _______________________________

In all of the above problems answers can be found by using repeated addition There are so many situations involving repeated addition that this process is called multiplication (Be warned however that repeated addition is not the only meaning of multiplication We will study another meaning in a later section of this chapter)

Basic Definition of Multiplication as Repeated Addition

For m a whole number the product m bull B is the total number of objects in m disjoint sets eachcontaining B elements m is called the multiplier and B is called the multiplicand

m bull B = B + B + B + + B

m times

The two numbers m and B play two very different roles in this basic meaning of multiplication The multiplier m is the number of sets while the multiplicand B is the size of the set The result of a multiplication is called a product In situations in which multiplication is defined as repeated addition the multiplicand can be any type of number but the multiplier must be a whole number

Total = (Number of sets) bull (Size of the set) darr darr darr

Product = Multiplier bull Multiplicand

Example 1 Melissa invited all of her running friends over for a morning run followed by brunch She bought three dozen eggs for the occasion How many eggs did she buyTotal number of eggs = 3 sets of 12 eggs = 12 eggs + 12 eggs + 12 eggs = 3 12 eggs = 36 eggs

ldquoOfrdquo and ldquoTimesrdquo

Notice that ldquoofrdquo is the word we often use to describe the size of a set For instance we might say that a platoon includes three squads of 10 soldiers This phrasing indicates that the total number can be found by repeated addition aka multiplication IThus the use of the word ldquoofrdquo can be a signal to multiply Conversely ldquotimesrdquo can often be translated as ldquoofrdquo For example ldquo3 times 5rdquo can be interpreted to mean ldquo3 sets of fiverdquo or 3 fives

Teaching Tip Sometimes children are told that ldquoofrdquo meansldquotimesrdquo This is a misleading overgeneralization ldquoOfrdquo is one of the most common words in the English language and often does not mean ldquotimesrdquo For example in the following sentence ldquoNine of the 12 students in the class passed the testrdquo it would be nonsensical to multiply 9 by 12 It actually makes more sense to say that ldquotimesrdquo often means ldquoofrdquo

Factors and Multiples

The multiplier and multiplicand are also called factors A whole number product is called a multiple of each factor Example 2 Consider 3 2 = 2 + 2 + 2 = 6

a 3 is the multiplier 2 is the multiplicand and 6 is the product b 2 is the size of the set and 3 is the number of setsc 3 and 2 are factors of 6 while 6 is a multiple of 3 and 2

Every whole number except 0 has a finite number of whole number factor but all whole numbers have an infinite number of whole number multiples

Example 3 Set of factors of 6 = 1 2 3 6 set of multiples of 6 = 0 6 12 18

Teaching Tip Students often confuse factors with multiples For instance a student might say that 3 is a multiple of 6 or that 12 is a factor of 6 Since these are important vocabulary words teachers need to spend time making sure students learn which is which Mnemonic devices such as ldquoFactors are firstrdquo or ldquoMultiples multiply monotonouslyrdquo may be helpful to some students

As the next examples indicate many different notations are used to indicate multiplication Example 4 (a) Product of 2 and 3 = 2 times 3 = 2 threes = 2 3 = (2)(3) = 2(3) = 2 3 = 2 bull 3

(b) Product of x and y = xy = x bull y

Units in Repeated Addition

A sum has the same unit as its terms For example 3 feet + 3 feet is equal to 6 feet Similarly since the basic meaning of a product is the repeated sum of multiplicands the product has the same unit as the multiplicand

Example 5 Five yardsticks are placed end to end How many feet is it from one end to another5 bull 3 feet = 3 feet + 3 feet + 3 feet + 3 feet + 3 feet = 15 feet

Activity 51B

A Fill in the blanks representing the total as a repeated addition Include units

Multiplier Multiplicand Total

Ex Three days a week Heidi walks 134 miles 3 134 mi 134 mi + 134 mi + 134 mi = 514 mi How many miles does she walk every week

1 Sara has two classes of 20 students How _____ ________ _______________________________many students does she have altogether

2 Peter buys three frac12-gallon bottles of milk _____ ________ _______________________________How many gallons of milk has he bought

B Answer the following questions

1a Find the area of the shaded shape on the centimeter grid to the right _________

b What is the shape of the standard unit for measuring area __________________

2a Suppose each cube to the right measures 1 cm by 1 cm by 1 cm What is the total volume of this set of cubes _________

b What is the shape of the standard unit for measuring volume _______________

Four Major Situations Involving Repeated Addition

1 Distinct Repeated Sets

Example 6 Consider the problem in which each of three children has four cards How many cards are there altogether

We have three sets of four 3 bull 4 cards = 4 cards + 4 cards + 4 cards = 12 cards

The most obvious case of repeated sets occurs with a repeating set of physical objects This physical set may be a hand of cards a platoon of soldiers a case of soft drinks and so on

2 Arrays

Consider the situation in which Rachel has three T-shirts and two pairs of shorts The following diagram illustrates one way to determine that Rachel can put together a total of six different outfits

A horizontal arrangement of objects is called a row and a vertical arrangement is called a column The above diagram with 2 rows and 3 columns is an example of a 2 by 3 array An R by C array is a set of discrete objects arranged into R rows and C columns Because the rows of an array are the same size the total number of elements in an array can be found by repeatedly adding the rows Since the row size is the same as the number of columns we have the following generalization

The total number of elements in an R by C array is R bull C

This explains why an R by C array is also described as an ldquoR C arrayrdquo

Example 7 This is a 2 5 array with two rows and five columns Total number of elements = 2 bull 5 = 5 + 5 = 10

3 Area and Volume

What is the total number of squares in a baby quilt made of four strips of six squares each This is another example of a problem that can be solved by repeated addition The quilt consists of four rows each containing six squares The total number of squares is equal to the following 4 sixes = 6 squares + 6 squares + 6 squares + 6 squares = 24 squares

This quilt also illustrates why the area of a rectangle can be found by multiplying its length by its widthFinding the number of squares in a rectangle is analogous to finding the number of elements in an array

Rectangles as Arrays of Squares

Array with 8 elements Rectangle with an area of 8 squares

Generally speaking we measure the area of a two-dimensional shape using squares The squares in a rectangle form an array in which the number of rows corresponds to the length of the rectangle while the number of columns corresponds to the width Thus the area of a rectangle is the product of its length and width

BFormulas for the areas of other special shapes are derived from this basic area formula

Example 8 The area of a right triangle with legs of length B and H is frac12BH because its area is half the area of a rectangle with length B and width H

One special area is not directly derived from the area of a rectangle The area of a circle is equal to π r 2 where r is the radius of the circle

As the following example illustrates the area of many figures can be found by partitioning the figure

Example 9 To find the area of the figure given below partition it as indicated 6 cm 6 cm Area Half-circle = 05 π (38 cm)2 asymp 2268 cm2 38 cm 76 cm Area Rectangle = 6 cm middot 76 cm asymp 456 cm2 168 cm 38 60 70 Area Triangles = 2 middot (05 middot 38 cm 70 cm) = 266 cm2

Area Total = 9488 cm2

Volume 1Prime

The standard unit for measuring volume is a cube A cube that measures one unit 1Primeby one unit by one unit has a volume of one cubic unit As the following activity illustrates the volume of the three-dimensional analog of a rectangle can be found 1Primeby repeated addition of layers of cubes One Cubic Inch

Activity 51C

1 A solid box has a length of 4 cm a width of 2 cm and a height of 3 cm

________ a What is the area or the bottom (or top) of this box

________ b How many cubic centimeters are in the first layer of this box

________ c How many layers does the box have

________ d Use the above facts to determine the volume of the box

2 What is the volume of a box that is 5 high 10 long and 3 deep ______________

3 A cylindrical water tank is 20 feet high It is known that when the water is one foot deep the volume of water in the tank is about 700 cubic feet What is the capacity of the tank _____________[Hint Think about the volume of each layer]

The formal name of a typical box is a right rectangular prism It has rectangular faces at right angles to each other A right rectangular prism with length L width W and height H 1 1can be partitioned into a series of identical one unit thick layers The volume of one of these layers has the same numerical value as Lmiddot W the area of the ldquofloorrdquo or base of the prism 1Since the number of layers corresponds to the height of the solid the volume of the right rectangular prism is as follows 1 W L

Volume of a right rectangular solid = length bull width bull height

Volumes of Solids with Congruent Bases

In general a prism is any solid with two congruent and parallel polygonal bases connected by parallel lines This means that the other faces of a prism are parallelograms

Various Prisms

A prism is a special type of cylinder A cylinder is any solid with two congruent and parallel bases not necessarily polygonal that are connected by parallel lines

Various Cylinders

Like a prism a cylinder consists of a series of congruent layers Thus its volume is the repeated sum of the volume of one layer The volume of a single layer has the same numerical value as the area of the base of the cylinder the number of layers corresponds to the height of the cylinder (The height of a cylinder is the distance between its bases If the base of a cylinder is horizontal then its height is vertical) This yields the following useful formula

Volume of a Cylinder = Area of its Base bull Height

Example 10 If the base of a kidney-shaped pool has an area of 40 square feet then filling it to a depth of one foot will require 40 cubic feet of water Every additional foot of depth will require another 40 ft 3 So filling the pool to a depth of three feet will require 40 ft3 + 40 ft3 + 40 ft3 for a total of 120 ft3

Example 11 A waste basket is a cylinder that is 2prime 3Prime high Its base has parallelsides and circular ends The parallel sides are 10 inches apart andone foot long How many gallons of water will this waste baskethold There are 231 cubic inches in a gallon

Find the area of the base It consists of two half-circles and a rectangleThe area of a circle is r2 where r is the radius In this situation the diameter is 10Prime and thus the radius is 5Prime To reduce round-off error do not round until the end of the problem

12Prime Area of rectangle = 10 12 = 120 square inches Area of two half circles = 2 (frac12 r2) where radius is 5Prime 10Prime 314159hellip 52 square inches

785 square inches

Total area of the base 1985 square inches

Volume of container 1985hellip square inches 27 inches 536057hellip cubic inches 536057hellip in3 231 in3 per gallon 232 gallons

4 Cartesian Products

Recall that the number of possible combinations of Rachelrsquos shorts and T-shirts was found by pairing each T-shirt with a pair of shorts In general the set consisting of all possible ways of pairing elements of a set A with elements of another set B is called a Cartesian product A Cartesian product can always be illustrated as an array The number of rows in this array corresponds to the number of elements in set A designated as NA and the number of columns corresponds to the number of elements in set B designated as NB Thus we have the following

If C is the Cartesian Product of A and B then NC = NA bull NB

Example 12 The license plate of a very small state consists of a letter followed by a single-digit number How many distinct license plates of this description are possible

The license plates form an array partially indicated below

0 1 2 3 4 5 6 7 8 9

A A0 A1 A2 A3 A4 A5 A6 A7 A8 A9

B B0 B1 B2 B3 B4 B5 B6 B7 B8 B9

Z Z0 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

There are 26 rows with each row containing 10 plates The total number of plates is 26 bull 10 or 260

A Cartesian product can also be described using a tree diagram as shown below

Example 13 Let S represent a pair of Rachelrsquos shorts and T represent a T-shirt The following tree diagram shows the six outfits that result from using these clothes

T1 T2 T3 T1 T2 T3

S1T1 S1T2 S1T3 S2T1 S2T2 S2T3

As the next activity demonstrates the idea of a Cartesian product can be extended to more than two sets

Activity 51D

1 Find the volume of a prism that is one foot long with a right triangular base _________________The three sides of the base measure 3 4 and 5

2 Suppose license plates consist of a letter followed by two digits

a List one license plate meeting this description _________________

b How many license plates meeting this description start with A _________________

c What is the total number of license plates _________________

3 In Tennessee license plates consist of three letters followed by three digits a How many license plates are possible in Tennessee _________________

b Suppose Tennessee deletes 38 three-letter words from use on license plates ________________How many license plates are now possible in Tennessee

4 Summarize the pattern that occurs when a fraction is multiplied by a whole number in the following

a 4 middot 12 = 12 + 12 + 12 + 12 = 42 b 3 middot 45 = 3 middot 4 fifths = 12 fifths = 125 c 2 middot 73 = 73 + 73 = 143

5 Use the pattern you observed in the previous problem to find the answer to the following word problem A chocolate nougat weighs 23 ounce How much do 5 of these nougats weigh

The set of all possible Tennessee license plates is an example of a general Cartesian product Just as a license plate is created by choosing letters and digits an element in a general Cartesian product is formed by choosing elements one at a time from several sets

N1 N2 N3 N4 Nk elements elements elements elements elements

Set 1 Set 2 Set 3 Set 4 Set k

General Cartesian Product

Each element in this Cartesian product contains one element from Set 1 one element from Set 2 and so on The total number of such elements is found as follows

Total number of elements in the Cartesian product = N1 bull N2 bull bull Nk

Example 14 How many different kinds of pizza can be made if there are five possible toppings from which tochoose

For each topping there are two choices to use the topping or not to use it Thus there are a total of five sets each containing 2 choices So the total number of pizzas is equal to 2 bull 2 bull 2 bull 2 bull 2 or 32

The next example illustrates a situation in which several sets need to be reconsidered as a single set in order to determine the appropriate number of possibilities

Example 15 Suppose Tennessee license plates consist of three letters followed by three digits with 38 three-letter words deleted from use How many license plates are possibleTotal number of allowable ldquowordsrdquo = 263 - 38 = 17538 For each word there are 103 or 1000 numbers This yields 17538 bull 1000 = 17538000 license plates

Repeated Addition with Rational Numbers as Multiplicands

When the size of a set is not a whole number using the unit fraction as the main unit leads to an easy process for computing the product

Example 16 A small measuring cup has a capacity of 38 of a liter How much water will two of these cups

2 bull 38 liter = 3 eighths of a liter + 3 eighths of a liter = 6 eighths of a liter = 68 L (or 34 L)

Example 17 I bought three half-gallons of milk today How many gallons of milk did I buy

3 bull 12 gallon = 12 gallon + 12 gallon + 12 gallon = 32 gallons = 112 gallons

As these examples illustrate we can find the product of a whole number and a rational number by multiplying the number of unit fractions ie the numerator m bull N = m bull N

If a multiplication problem contains mixed numbers change these mixed numbers to improper fractions to make use of the above property

Example 18 It takes 123 yards of ribbon to make a bow How much ribbon is needed for four bows

4 bull (123 yards) = 4 bull 5 thirds of a yard = 20 thirds of a yard = 203 yd or 623 yardsCompare this to using feet as a unit 4 bull 5 thirds of a yard = 4 bull 5 feet = 20 feet

51 Homework Problems

A Answer the following

1a State the basic definition of multiplication b In situations involving repeated addition the total can be found by multiplying the of sets by the of a

2 Define the following (a) multiplicand (b) multiplier (c) row (d) Cartesian product

3a List the four general situations leading to repeated addition b Invent and solve your own example for each situation Do not use the examples given in the text

4 Show how the area of a 3 by 5 rectangle can be found by repeated addition Use a well-labeled diagram

5 Show how the number of elements in a 3 by 5 array can be found by repeated addition Use a labeled diagram

6 Fill in the blanks (a) 4 bull 35 = 4 bull fifths = 12 (b) 3 bull 54 = 3 bull 5 = 15

7a Draw a picture to show why 2 bull 35 = 65 b Use repeated addition to find 2 bull 35 = 65

8 Explain why in situations involving repeated addition the multiplicand and the product have the same units Include an example

9 Which of the following are arrays a diams diams diams diams b 1 45 48 c d clubs spades clubs

diams diams diams 0 15 32 spades spades spades

10 State the number of rows and columns and the total number of elements in each of the arrays in the previous problem

11 Ron purchases three boxes of light bulbs Each box contains 6 packages of bulbs and each package contains two bulbs Find the total number of light bulbs purchased by using

a a series of repeated additions b multiplication c a picture d a tree diagram

12 Use a tree diagram to find the number of different pizzas if there are three types of crusts (thin medium or thick) two types of dough (white or whole wheat) and four kinds of topping combinations (plain pepperoni super and vegetarian)

13 The screen on a calculator contains pixels arranged in 62 columns and 48 rows How many pixels occupy the screen (A pixel is a single position on the screen It is either lighted or unlighted) Draw the beginnings of an array and solve this problem

14 Ryan now has only 62 toy soldiers after losing 48 in the woods yesterday a How many toy soldiers did Ryan have before playing with them in the woods b Identify the type of this problem

15 An auditorium has 100 rows The first row contains 20 chairs and each succeeding row contains one more chair than the previous row

a How many chairs are in the 100th row Solve this problem by using an organized table containing at least three rows and finding the pattern

b How many chairs are there altogether in the auditorium [Hint What is the sum of the chairs in the 1 st and 100th row What is the sum of the chairs in the 2nd and 99th row]

16 License plates for a certain state contain 4 letters followed by 3 digits a State one possible license plate for this state b How many different license plates are possible c How many license plates starting with LOVE are possible d If 18 four-letter words are eliminated from the possible choices of four-letter combinations and the use of

ldquo000rdquo is eliminated how many different license plates are possible

17 Some lottery tickets consist of six digits What are your chances of winning the lottery if there is only one winning combination of digits

18 A large bag of mulch is labeled as containing 2 cubic feet of mulch How many cubic inches of mulch is this [Hint One cubic foot is 12 by 12 by 12]

19 A 10prime by 8prime patio is to be made with cement It will be 2 thick How much cement is needed

20 Explain how the area of a right triangle is related to the area of a rectangle with the same base and height Include a diagram

21 A clay brick measures 8 long 4 deep and 3 high It is hollow in the middle with sides and bottom that are 1 thick A cubic inch of clay weighs about two ounces How heavy is this brick

22 Find the volume of the wedge to the right 8 cm

23 A 20 by 30 rectangular swimming pool is 3 4 deep at one end and steadily increases to 8 deep at the other end 30 away How many gallons of water does it hold (There are about 7frac12 gallons of water in one cubic foot)

_______________________________________________ _______________________________

_________________________________________________________ ____________________

52 Division in the Context of Repeated Addition

Like multiplication division is a derived operation It is possible to solve many division problems by using more basic operations as illustrated in the next activity

Activity 52A

A Show how to solve the following problems using counting addition or subtraction Use pictures or diagrams as appropriate

1 A kindergarten teacher has one of her children distribute 10 lollipops equally to five children The child gives

one to each child then another and another until they are all gone How many lollipops does each child get

2 A class contains 24 children seated at tables in groups of four How many tables are there

3 I cut 3 apples in half and gave away all the half-apples one to each child in the room How many children are in the room

B Travis Zack and Chad are playing with toy soldiers Travis has eight toy soldiers Zack has six and Chad hasfourteen All three boys organize their soldiers into pairs Then Travis and Zack team up against Chad

1 Compare the pairs in each ldquoarmyrdquo This situation illustrates that (8 2) + (6 2) is the same as (___ + __)

2 Make a generalization using fraction form A + B =______________________________C C _________________

A The Basic Definition of Division

Just as subtraction is the inverse of addition division is the inverse of multiplication

BASIC DEFINITION OF DIVISION

Division is the Inverse of MultiplicationA divide B = is equivalent to B = A for B ne 0

The first number in a division is called the dividend the second is the divisor and the result is the quotient

Dividend divide Divisor = Quotient

Example 1 Consider 12 divide 3 = 412 is the dividend 3 is the divisor and 4 is the quotient 12 divide 3 = 4 because 12 = 3 bull 4

In other words if we can formulate a problem into the multiplication sentence A bull = C then we can find the unknown factor by reformulating the sentence into a division sentence = C divide A Notice that the product in the multiplication sentence corresponds to the dividend in the corresponding division sentence

Example 2 The floor of a right rectangular solid measures 3 m by 2 m and the solid has a volume of 30 m3 What is the height of the solid

V = LWH =gt 30 = 3 bull 2 bull H =gt 30 = 6 bull H So H = 30 m3 divide 6 m2 = 5 m

B Two Major Interpretations of Division

All situations involving division are equivalent to multiplication problems with a missing factor However two quite different situations give rise to division 1 Division as Partitioning Total divide Number of Parts = Size of the Part

The total is known the number of sets (multiplier) is known but the size of the set (multiplicand) is unknown

Example 3 Ten candies were distributed equally to five children How many candies did each child get

Solution A The problem is to determine the size of the set given the number of sets The solution can be found by partitioning Ten partitioned into five equal parts yields two candies per part

sect sect sect sect sect sect sect sect sect sect

Solution B We have an unknown multiplicand namely the number of candies given to each child Thus we have 5 bull B = 10 By the definition of division B = 10 divide 5

Teaching Tip Young children can partition a set by dealing out the elements in the set like cards in a card game Later on such experiences with partitioning can help children understand this basic meaning of division

Example 4 A pizza has been cut into eight equal pieces and Anne eats two pieces If two people share the remaining pizza equally how much of a pizza will each person eat

If six pieces are split evenly between two people each person will get three pieces

As these examples illustrate division can be used to find the size of a part given the original quantity and the number of parts into which it is partitioned This is called the partitioning interpretation of division

Partitioning Interpretation of Division

For B a natural number A divide m can be interpreted to mean the size of a part when A is partitioned into m equal parts

m parts

Units in Partitioning Problems

In situations involving partitioning the quotient is the size of a part when the dividend is partitioned into the number

of parts specified by the divisor Hence the quotient as part of the dividend has the same unit as the dividend

Example 5 Sixty feet of rope is cut into 12 pieces of equal length How long is each piece

60 feet divide 12 = 5 feet

2 Division as Repeated Subtraction Total divide Size of the Part = Number of Parts

Example 6 A class contains 24 children seated at tables in groups of four How many tables are there

lt---------- How many tables ------------gt

Solution A Add fours until we reach 24 4 + 4 = 8 8 + 4 = 12 12 + 4 = 16 16 + 4 = 20 20 + 4 = 24We added 6 fours to get 24 so the answer is 6 tables

Solution B Subtract 4 repeatedly from 24 until we reach 0 24 - 4 - 4 ndash 4 - 4 - 4 - 4 = 0 We had to subtractsix fours so there are six tables

Solution C Find a missing multiplier m so that m bull 4 = 24 That is find m such that m = 24 divide 4

Division as repeated subtraction occurs in situations where a known quantity has been partitioned into equal parts of a known size The problem is to determine the number of parts

Repeated Subtraction Interpretation of Division

For B ne 0 A divide B can be interpreted to mean the number of Brsquos contained in A or the number of times B can be subtracted from A

B B B B B B

A divide B Number of parts of size B in set A

Stated another way we have A - B - B - B - B = 0

Example 7 Since 36 - 9 - 9 - 9 - 9 = 0 we have 36 divide 9 = 4

Units in Repeated Subtraction

In situations involving repeated subtraction the quotient is the number of divisors in the dividend Hence the quotient does not have a reference unit For this reason we say that the units of the dividend and divisor ldquodivide outrdquo just as common factors divide out

Example 8 How many 200rsquos are in 600

There are 3 sets of 200rsquos in 600 Thus we can say that in the division of 6 hundred by 2 hundred the hundreds units divide out

Example 9 A child arranges six toy soldiers into sets of two soldiers each How many sets are there

6 toy soldiers divide 2 toy soldiers = 3 =gt There are 3 sets of two soldiers in the set of six soldiers

C Rational Numbers in Division

1 Quotients as Rational Numbers

Partitioning whole numbers can lead to parts with fractional sizes Such problems reveal a surprising connection between quotients and fractions

Activity 52B

A Three pizzas are to be shared equally among four people How much pizza does each person get

1 The problem is to partition 3 pizzas into 4 equal parts and determine the size of the part That is we want to find _____________ divide ___

2a Draw a diagram that shows how to solve this problem by cutting each pizza into four pieces Shade the pieces to be claimed by the first person

b We have 3 pizzas divide 4 = 12 _______ of a pizza divide 4 = 3 ___________

3 Thus 3 divide 4 is equivalent to the rational number _______

B Use diagrams to solve the following problems

1 Adrienrsquos will states that one ninth of his estate is to be given to a certain charity and the remaining part of the estate is to be partitioned equally between his two children What fraction of the estate will each child inherit

2 Annarsquos will stated that one quarter of her estate was to be given to a certain charity and the remaining part of the estate was to be partitioned equally between her two children What fraction of the estate did each child inherit

The above activity illustrates the following relationship between quotients and fractions

The Connection Between Quotients and Fractions

For any real numbers A and B with B ne 0 A divide B is the same as AB

The relationship between AB and A B is not obvious For instance consider 3 divide 5 and 35 We can interpret 3 divide 5 to mean the size of a part when three units are partitioned into five equal parts we can interpret 35 to mean three of five equal parts of one unit On the face of it these seem to be very different problems They are certainly different processes Yet as the following example illustrates they yield the same result

Example 10 To partition 3 acres into 5 equal parts 1 acre 1 acre 1 acre

a Convert 3 acres into 15 fifths of an acre b 15 fifths of an acre divide 5 = 3 fifths of an acre = 35 acre

Thus we have three interpretations for a fraction AB

1 AB can refer to A parts of a unit that has been partitioned into B equal partsExample ldquo35 of an acrerdquo refers to three parts of an acre that has been partitioned into five equal parts

2 AB can refer to the ratio of two quantities where for every A elements in the first quantity there are B elements in the second quantityExample ldquoThe ratio of girls to boys in our class is 35rdquo means that there are three girls for every five boys

3 AB can refer to A divided by B This interpretation has multiple meanings including partitioning and repeated subtraction Example If three acres of land are to be shared equally by five heirs to an estate then each heir receives 3 acres divide 5 or 35 of an acre

2 Rational Number Dividends and Divisors

What is the meaning of an expression like 34 divide 2 This division of a fraction by a whole number can be interpreted as partitioning Just as with whole numbers the key to partitioning a fraction into two equal parts is to convert the fraction into a form that includes a multiple of two

Example 11 Partition 34 of a pizza equally between two people

Cut each of the fourths into two parts That is convert 34 to 68 Now we have6 eighths of a pizza divide 2 = 3 eighths of a pizza = 38 pizza

What is the meaning of an expression like 3 divide 34 or 34 divide18 These divisions can be interpreted in the context of repeated subtractions as the next activity illustrates

Activity 52C

A Mary is going to make giant three-quarter-pound hamburgers How many hamburgers can she make with three pounds of hamburger meat

1 Solve this problem using repeated subtraction

2 The problem is to find out how many quarter-pounds are in 3 pounds

a The division associated with this problem is 3 lbs divide _____ lb

b Convert 3 lbs to quarter-pounds

c 3 lbs divide 34 lb = ___ quarter-pounds divide ___ quarter-pounds = _____ (Note that the units cancel out)

d So Mary can make ____ hamburgers

B Solve the following problems without using standard algorithms

1 If a strip of metal that is 15 sixteenths of an inch long is cut into pieces that are 3 sixteenths of an inch long how many of these smaller pieces will there be

2 If a strip of metal that is 34 of a foot long is cut into pieces that are 18 of a foot long how many of these smaller pieces will there be

3 Suppose a restaurant serves a glass of wine that is 18 of a liter If a box of wine holds 156 liters how many glasses of wine can be poured from this box Include the fractional part of a glass in your answer [Hint Convert to twenty-fourths]

Understanding the process of dividing a fraction by a fraction is not straightforward To make sense of these types of division it is helpful to use the repeated subtraction interpretation of division and a common unit As the following examples illustrate this boils down to finding a common denominator

Example 12 Suppose six acres are divided into three-quarter-acre lots How many lots will there be

6 acres = 24 quarter-acres =gt 6 acres 34 acre = 24 quarter-acres 3 quarter-acres = 8

Example 13 If 212 tons of gravel are to be poured into bins each holding half of a ton how many bins areneeded

Convert to half-tons 212 tons 12 ton = 5 half-tons 1 half-ton = 5

Fortunately a relatively simple pattern occurs Following is the explanation for this pattern

1 Use the Fundamental Property of Fractions to generate equivalent AB CD= ADBD BCBDfractions with the same denominator

2 Since AD and BC have the same unit namely the unit fraction 1BD ADBD BCBD = AD divide BCthis division can be interpreted to mean ldquoHow many BCrsquos are in ADrdquo

3 As we shall see a quotient can be interpreted as a fraction AD divide BC =BCBD

4 The Shortcut AB divide CD = ADBC

Teaching Tip Sometimes this shortcut is called ldquocross-multiplyingrdquo This is a very bad idea ldquoCross-multiplyingrdquo more commonly refers to a shortcut used to solve proportions For instance the proportion 3x = 85 can be solved by ldquocross-multiplyingrdquo to obtain the equivalent equation 3 5 = 8x In contrast the result of ldquocross-multiplyingrdquo when dividing fractions is a fraction not an equation When different processes are referred to by the same name students often confuse the results Thus it is better not to refer to the division of fractions as ldquocross-multiplyingrdquo A pedagogically better way of computing the quotient of two fractions which involves inverting the divisor will be discussed later in this chapter

Example 14 Finding 112 14 using a variety of methods

(a) Repeated subtraction as visualization In your mindrsquos eye visualize the number of quarter pieces of pizza in 112 pizzas There are six such pieces

(b) Formal repeated subtraction 112 - 14 - 14 - 14 - 14 - 14 - 14 = 0 =gt 112 14 = 6

(c) Common unit 112 14 = 6 fourths 1 fourth = 6

(d) Shortcut 112 14 = 32 divide 14 = (3 middot 4)(2 middot 1) = 6

D Remainders and Two Useful Theorems

It is a curious fact that inverse operations are often not as well behaved as the original operations Here is a case in point multiplying two whole numbers yields a whole number but dividing two whole numbers can result in a remainder

Activity 52D

1 It takes 15 inches of ribbon to make a certain kind of bow a Suppose Mary has 50 inches of ribbon How many bows can she make with this ribbon and how much

ribbon will be left over

b Specify a length of ribbon that can be used to make bows without having any ribbon left over

c Give a general description of the lengths of ribbon that can be used to make bows without having any ribbon left over

d Use your calculator to determine how much ribbon will be left over if Mary makes as many ribbons as possible from a roll containing 88 feet of ribbon Report your answer in inches

2 The maximum class size for kindergartners in one state is 18 A school has 50 kindergartners What is the smallest number of kindergarten classes that this school must have

3 At a practice a coach divides his team into groups of four girls each He assigns any remaining players to be referees If 23 players show up how many will be referees

4 Three children steal into the kitchen late one night and find their motherrsquos secret cache of 11 chocolate bars

a If the children decide to split the chocolate bars evenly how many chocolate bars _____________does each child get

b In the context of this problem explain the meaning of the remainder of 2 in the equation 11 3 = 3 R 2

c Explain what happened to this whole number remainder in this problem

Division will lead to a ldquoleft-overrdquo when the dividend is not a whole number multiple of the divisor

Example 15 Twenty-six grapefruits are being packed into boxes that hold six grapefruits each How many boxes will be filled and how many grapefruits will be left over 26 is not a multiple of 6 Instead 26 = 4 bull 6 + 2 So there will be four full boxes with two grapefruits left over

26 grapefruits

6 grapefruits 6 grapefruits 6 grapefruits 6 grapefruits 2 gf

In general if A and B are whole numbers then either (a) A is a whole-number multiple of B or (b) A is the sum of a whole-number multiple of B and a remainder The remainder has the same unit as the dividend The relationship between the dividend A the divisor B the whole number quotient q and the remainder r is summarized as follows

The Division Theorem

For any whole numbers A and B with B ne 0 A can be written as qB + rwhere q and r are unique whole numbers with 0 le r lt B

q Brsquos r

This theorem is called the Division Theorem because of the connection between A divided by B and A written as q middot B + r If A consists of q Brsquos plus a remainder r then A divide B is equal to q with a remainder of r

Example 16 The following statements convey the same informationa 242 = 5 bull 43 + 27 b 242 contains 5 forty-threes with 27 left over c 242 divide 43 is equal to 5 with a remainder of 27

It is common (at least in elementary school) to indicate a whole-number quotient and remainder using the ldquoRrdquo notation as illustrated in the next example Note that ldquoRrdquo does not indicate addition

Example 17 ldquo14 divide 5 = 2 R 4rdquo means that 14 = (2 bull 5) + 4 In other words 14 contains 2 fives with 4 left over

Another useful theorem related to division is illustrated in the following example

Example 18 Bridge is a card game involving exactly four players Marge is organizing a bridge party at her retirement community First eight people sign up so Marge prepares two tables for four Then another 12 people sign up so Marge prepares three more tables for a total of five tables Obviously if all 20 people had signed up at the same time Marge would also have prepared five tables This illustrates the following fact 20 = 12 + 8 = 12 + 8

4 4 4 4In general we have the following result

Quotient of a Sum Property

If A B and C are real numbers with C 0 then A + B = A + B C C C

This is called the Quotient of a Sum Property because it states that the quotient of a sum (A + B) is the same as the sum of the quotients AC and BC

Teaching Tip Many students find the Quotient of a Sum Property rather strange when it is read from left to right Just ask them to read the property from right to leftmdashin this direction the property should be very familiar See how the Quotient of a Sum Property plays a role in the next example

Example 19 Forty-one acres are to be divided into eight lots of equal size What will be the size of each lot

Since 41 acres = 8 middot 5 acres + 1 acre each lot will include 5 acres If the remaining acre is partitioned equally among the eight lots each lot will increase by an eighth of an acre Thus the total size of each lot will be 518 acres

Summary 41 acres8 = 40 acres8 + 1 acre8 = 5 acres + 18 acre = 518 acres

As this example shows a quotient can be expressed as a non-whole number that includes the remainder as a fractional part of the divisor

If A = qB + r then A B = qB + R = qB + r = q + r B B B B

Example 20 387 8 = (48 middot 8 + 3) 8 = 48middot 8 + 3 = 48 middot 8 + 3 = 48 + 3 = 48⅜ 8 8 8 8

The concept of whole number quotients also applies to problems involving fractional dividends and divisors In such cases be careful to interpret the remainder correctly

Example 21 Suppose three and a quarter liters of acid is being poured into half-liter containers

a How many containers will be filled Include fractional parts

Compute the answer using the shortcut 314 liters 12 liters = 134 21 = 132 = 612

This means that 612 containers will be filled

b How many full containers will there be and how much acid will be left over

Since 314 12 = 612 there will be six full containers The left-over acid would fill 12 of a half liter container so there is 14 of a liter of left-over acid

Remember that the fractional part of a quotient is equal to the remainder divided by the divisor To find the remainder in terms of original units multiply the fractional part of the quotient by the divisor

Finding Whole Number Remainders from Quotients in Decimal Form

If a calculator is used to find a quotient the answer is usually expressed in decimal form The whole number quotient q is clearly identifiable as the whole number part of this decimal One way to find the whole number remainder is to use the relationship between A B q and r A = qB + r Solving this for r yields the following equation r = A ndash qB In other words find r by subtracting q Brsquos from A

Example 22 242 divide 43 = 56279069hellip =gt 242 = 5 middot 43 + r =gt r = 242 ndash 5 middot 43 = 27

Described in another way When we compute 242 divide 43 as 562hellip we have determined that there are five 43rsquos in 242 plus a remainder To find the remainder subtract the five 43rsquos from 242

Another way to find the whole number remainder r is to recognize that the fractional part of the decimal represents the ratio of r to the divisor Thus r can be found by multiplying this fractional part by the divisor Avoid rounding errors by using all the digits provided by your calculator for the fractional part

Example 23 242 divide 43 = 56279069hellip =gt r = 43 middot 06278069hellip = 27

Situations Involving Whole Number Quotients and Remainders

While there are many division situations in which the answer is a non-whole number quotient there are many division situations in which the answer must be a whole number These situations usually involve units that are indivisible ie units that cannot be partitioned into smaller units

Example 24 The organizer of the schoolrsquos May Day event decides to form six rows of chairs for the audience She wants the same number of chairs in each row There are eighty-seven chairs available Howmany chairs should be in each row

Find 87 divide 6 = 14 r 3 This means that 87 = 14 bull 6 + 3 Put 14 chairs in each row with three chairs left over

Example 25 The sixth grade is scheduled to see the play ldquoThe Lion Kingrdquo but the bus has broken down Parents with minivans are being recruited to take all 87 sixth graders to the play If each minivan carries sixpassengers (not including the driver) how many parents with minivans need to be recruited

Since 87 = 14 bull 6 + 3 we can fill up 14 vans and part of another van This means we need 15 vans to take all 87 sixth graders to the play (Alternately line up 14 parents with minivans and one parent with a sedan)

As the above examples illustrate sometimes the quotient is rounded up and sometimes it is rounded down to find the appropriate answer to a question Use common sense to decide which way to round

Sometimes the remainder plays the starring role in a division problem That is sometimes the relevant part of a division is not the quotient but the remainder Consider the next examples

Example 26 January 1 2002 fell on a Tuesday On what day did January 31 2005 fall

Starting with January 1 every seven days there will be another Tuesday January 29 will fall on a Tuesday because it is 28 days after January 1 Thus January 31 will fall on a Thursday

Example 27 December 25 2005 falls on a Sunday On what day will December 25 2009 fall

There are 365 days in most years and 365 = 52 bull 7 + 1 This means that a year consists of 52 full weeks plus a day That extra day the remainder in the division 365 7 means that from one 365-day year to the next every date moves forward one day So December 25 2006 will fall on a Monday and December 25 2007 will fall on a Tuesday The year 2008 is a leap year with 366 days the extra day occurring on February 29 This means that all dates after February 29 move forward two days from the previous year Thus December 25 2008 will fall on Thursday December 25 2009 will fall on a Friday

Teaching Tip An efficient way to identify leap years which normally occur when the year is divisible by four is to use the following property a whole number is divisible by four if and only if the last two digits are divisible by four For example 2036 will be a leap year because 36 is divisible by 4

Various examples in this section have illustrated four effects of the remainder These are summarized below

Four Possible Effects of the Remainder

1 Eliminate the remainder Round the quotient down to the nearest whole number 2 Round the quotient up to the next whole number 3 Retain the remainder as the answer 4 Include the remainder in the answer as a fractional part of the divisor

Teaching Tip Students have been known to lose track of the existence of whole number quotients and remainders in later grades because they become so accustomed to using calculators that yield only decimal quotients Their memories can be jogged by working problems that require whole number answers not decimal answers

Summary

Division is defined as the inverse of multiplication From an understanding of multiplication as finding a total given a number of repeated sets there arise two understandings of division The first is to find the size of the repeated set The second is to determine the number of these repeated sets Complications occur because of the backwards nature of division especially as it relates to the existence of remainders and the behavior of rational numbers

A Concepts

1 Definitions Properties and Vocabulary a State the basic definition of division b Use the basic definition of division to rewrite A ⅜ = as a multiplication sentence c Rewrite the following multiplication sentence as a division sentence 4 = 23

2a Use the basic definition of division to rewrite 8 0 = as a multiplication sentence b Explain why this multiplication sentence and hence the division sentence has no solution

3 Identify the divisor dividend and quotient in the following division sentence 6 13 = 18

4 List three numbers in each of the following sets a Multiples of 12 b Factors of 12 c Numbers divisible by 12

5 Justify your answers to the following a Is 24 a multiple of 8 b Is 24 divisible by 8 c Is 24 a factor of 8 d Is 0 a multiple of 8 e Is 0 divisible by 8 f Is 0 a factor of 8

6 Why can division always be interpreted as the process of finding an unknown factor

7 Which of the following can be interpreted as A B for B 0 a AB b A B c Number of Brsquos in A d where A = B

8 Explain the meaning of 56 using a the basic definition of an elementary fraction b division interpreted as partitioning c division interpreted as repeated subtraction with a whole number quotient and remainder

9 The Division Theorem a For any two whole numbers A and B A can be written as a of Brsquos plus a b Show this relationship for A = 17 and B = 3 c Show this relationship for A = 6 and B = 17 d If A = cB + d describe A B e Fill in the blanks 37893 = 87 + and 37893 87 = R

10 Fill in the blanks a If 27 divide 4 = 634 then 27 = bull 4 + b If 473 = 8 bull 56 + 25 then 473 divide = 8 + 25

11 Which of the following are equivalent to 56 = 9 bull 6 + 2 a 56 divide 9 = 6 R 2 b 56 divide 6 = 9 R 2 c 56 divide 9 = 6256 d 56 divide 9 = 629 e 56 divide 6 = 9 + 2

12 The Quotient of a Sum Theorem a State the sum that is the same as (x + y)z b According to the Quotient of a Sum Theorem 963 is the same as 903 + c Determining the number of threes in 96 is the same as determining the number of threes in 90 and adding this

to the number of threes in d The Quotient of a Sum Theorem states that first adding A and B and then dividing the sum by C is the same

as first dividing A by C and dividing B by C and then

B Division as Partitioning

1 Describe the meaning of 6 2 in terms of partitioning

2 Identify which of the following three quantities is unknown in a partitioning problemtotal size of repeated set number of repeated sets

3 Write and solve a story problem that involves partitioning for each of the following conditions a The dividend is three fifths b The quotient is three fifths c The dividend is 0 d The divisor is 0

4a Identify which of the following three quantities have the same units in a partitioning problem total size of repeated set number of repeated sets

b Explain why these two quantities have the same unit Include an example

5 Use the partitioning interpretation of division to explain why A A = 1 for A 0

6a For division interpreted as partitioning (total) divide (number of parts) = b What type of number occurs as the divisor in a partitioning problem and why

7a A divide B can be interpreted as the process of partitioning a set of size A into B parts and finding b Using this interpretation we have 8 people divide 2 = Justify your answer

C Division as Repeated Subtraction

1 Describe the meaning of 6 2 in terms of repeated subtraction

2 Identify which of the following three quantities is unknown in a repeated subtraction problemtotal size of repeated set number of repeated sets

3 Write and solve a story problem that involves repeated subtraction for each of the following conditions a The dividend is three fifths b The quotient is three c The dividend is 0 d The divisor is 0 e The divisor is 13

4a Identify which of the following three quantities have the same units in a repeated subtraction problem total size of repeated set number of repeated sets

b Explain why these two quantities have the same unit Include a word problem as an illustration

5 Use the repeated subtraction interpretation of division to explain why A A = 1 for A 0

6a A divide B can be interpreted as the process of finding how many times B must be subtracted from A to get

b Using this interpretation we have 6 feet divide 3 feet = because

7a Use the repeated subtraction interpretation of division to explain why 8 tenths divide 2 tenths = 4 b Explain why AB CB = A C in terms of repeated subtraction and the common unit of the dividend and

divisor

8 Invent a story for each of the following and find the answers a 18 lbs divide 3 lbs = b 18 lbs divide 3 =

9 Which of the following can be computed by determining M 2 a What number should I multiply 2 by to get M b What is the size of a part if M is partitioned into two parts of equal size c How many twos are in M d If M is partitioned into halves how many halves will there be

D Rational Numbers and Division

1 Rational Divisors a Invent a story that can be solved by finding 313 divide 23 b Draw a labeled diagram that illustrates how to find the solution

2 Rational Dividends a Invent a story that can be solved by finding 412 divide 3 b Draw a labeled diagram that illustrates how to find the solution

3 Explain why 158 divide 38 is the same as 15 divide 3 using the repeated subtraction interpretation of division and unit fractions

4 Rational Quotients a Use a diagram to illustrate how to divide two pizzas evenly among three people b Fill in the blanks with appropriate unit fractions 5 divide 6 = 30 divide 6 = 5 c Suppose 4 units are partitioned into M equal parts Describe the size of a part

5 Find 112 divide 38 by the following methods a repeated subtraction b common denominators c a third method of your own choosing

E Remainders

1 Basics a Under what circumstances will division of whole numbers include a nonzero remainder b When the remainder is 0 the dividend must be a (multiplefactortermproduct) of the divisor c A remainder in a division problem can be considered as a fractional part of the

2 Find the whole number quotient and remainder for the division 4379 35

3a List the four possible effects of a remainder on the answer of a division problem b Invent a word problem for each of these four effects

F Problem Solving

1 The teacher decides to organize his class of 22 students into teams of four children each with the ldquoleftoverrdquo children working with her How many teams will there be and how many children will be working with the teacher

2 If a 735 acre lot is to be divided equally into 6 lots what will be the size of each lot

3 I cut oranges into fourths and gave a piece to each of 22 children How many whole oranges did I use

4 Twenty-five children are going on a field trip in vans holding 7 children each How many vans are needed

5 January 1 2004 falls on a Thursday Determine the day of the week for January 1 2012

6 The 15th day of a certain year falls on a Thursday On what day of the week will the 327 th day of the year fall

7 A construction company is paving a 214 mile stretch of freeway at the rate of 200 yards a day How long will it take to complete the job

8 The Martian year is almost exactly 687 days Suppose Martians have seven-day weeks like we do If the Martian year of 2005 started on a Monday on what day of the week would the Martian year of 2006 fall

9 On Venus the year is a little over 224 days Suppose Venutians have five-day weeks (Monday through Friday) with leap years that occur every three years and contain two extra days The Venutian year of 2005 started on a Monday and is a leap year

a On what day of the week will the Venutian year of 2006 start b On what day of the week will the Venutian year of 2009 start

10 The water in a tank weighs 66875 pounds One cubic foot of water weights 625 pounds How many cubic feet of water does the tank hold

11 A manufacturer had a roll of 750 yards of linen goods that he cut into pieces 27 inches long to make dish towels He sold the towels at $480 a dozen

a If he sold all the towels what was his revenue [Hint Revenue is the amount of money taken in] b If the cost of producing and cutting the roll of linen goods was $380 what was the profit per towel

12 A chemistry professor is preparing for a lab with 18 students Each pair of students will need a tenth of a liter of a 40 nitric acid solution for the dayrsquos experiment How much of this acid must the professor prepare

13 A 314 yard strip of steel is to be used to make pieces that are a half foot long How many pieces can be made and how much steel will be left over

14 An estate worth one and a half million dollars is to be shared equally among five heirs How much does each heir inherit

15 Eight and two thirds miles of interstate are to be paved in 20 days How much road should be paved each day on average Report your answer in feet

16 How many nails weighing 38 of an ounce can be made from a third of a pound of metal____________________________________________________________________________________________

______________________________________________________________________________________ ______

53 Multiplication as a Means of Comparison

Besides repeated addition multiplication has a second major meaning This is illustrated in the following activity

Activity 53A

A Jerry Nick and Melissa went running one Saturday morning Melissa ran twice as far as Jerry Nick ran two thirds as far as Jerry Let Jerryrsquos Nickrsquos and Melissarsquos distances be represented by J N and M respectively

1 Write an equation expressing the relationship between J and M ______________

2 Suppose Jerry ran 12 miles a Use a diagram to determine how far Nick ran

b Write an equation expressing the relationship between J and N ______________

3 In the last thirty years there has been a 200 increase in the price of bread

a ____________________________________ is 200 of ___________________________________________

b Label three sets in the following diagram the old price the increase and the new price

c If a loaf of bread cost 50cent thirty years ago how much does it cost now Label the diagram _____________appropriately to find the answer

4 Suppose an employee gets one tenth off the sticker price

a ____________________________________ is 110 of _____________________________________________

b How much will an employee pay for an item with a sticker price of $60 Label the following diagram using the information in this problem and figure out the answer Label three sets the sticker price the discount and the discounted price

Multiplication in Comparison Situations

In the above problems multiplication is used to describe the relationship between two quantities In such situations the product is not a total but an amount that is described relative to a base of comparison The multiplier indicates how many or how much of the base is necessary to generate the described amount

Described Amount = m bull Base of Comparison

Example 1 Melissa ran twice as far as Jerry

Let J = Jerryrsquos distance and M = Melissarsquos distance J bull______________bull

We have M = 2 bull J M bull______________bull_______________bull

Example 2 A 200 increase means that the increase is two times the original price If the original price was 50cent then the increase is 2 bull 50cent or 100cent The new price will be 50cent + 100cent or $150

Teaching Tip Especially when an increase is over 100 of the original value students may forget to add the increase to the original price to find the final value Warn them to be extra careful when they are working with these types of problems

Rational Number Multipliers

If the multiplier is a whole number multiplication in comparison situations is similar to repeated addition In the above example for instance 2 bull J still means J + J Unlike repeated addition however multipliers in comparison situations can be non-whole rational numbers As the next example illustrates the meaning of these multipliers is directly based on the meaning of elementary fractions

Example 3 Nick ran two thirds as far as Jerry This means that Nickrsquos distance N is two thirds of Jerryrsquos distance J or two of three equal parts of Jerryrsquos distance J bull_____bull_____bull____bull

N = 23 of J N bull_____bull_____bull

Since 23 plays exactly the same role in this example as 2 the multiplier 2 did in the previous example it seems reasonable to interpret ldquo23 of Jrdquo as multiplication For instance if Jerry ran 12 miles then 23 middot J means to partition 12 into three equal parts and select two of these equal parts 23 middot 12 = (12 divide 3) middot 2 = 8

In general for any positive rational number ND ND middot B means ND of B where ND is interpreted as an elementary fraction That is ND middot B means N of D equal parts of B ND middot B = (B divide D) middot N

Example 4 If Y = 7 middot X then Y is 7100 of X or seven of one hundred equal parts of X

Example 5 The guests ate two thirds of a box of 24 candies How many candies did they eat

Solution A To find 23 of 24 first partition 24 into three equal parts This yields 8 candies in each part with 16 candies in two parts The guests ate 16 candies

Solution B 23 middot 24 candies = 23 of 24 candies = 2 middot (24 candies divide 3) = 2 middot 8 candies = 16 candies

As the next activity illustrates this process does not always yield a whole number

Activity 53B

A John ordered eight pizzas for a party His guests ate two-thirds of all the pizza How much pizza did they

1 Solve this problem by partitioning the pizzas below and shading the pizzas that were eaten

2 23 of 8 pizzas = 23 of ____ thirds of a pizza = 16 ___________________________ = 513 ____________

B Four fifths of the City Council of Newton voted in favor of a tax increase Of those in favor of a tax increase two thirds indicated that the increase should be more than 1 What fraction of the city council was in favor of a tax increase over 1

1 Suppose the large rectangle to the right represents the Newton City Council

a Shade the area representing those who voted in favor of a tax increase

b Stripe the area representing those who favored an increase of more than 1

c Use this diagram to find the answer to the question ______________

2 Symbolically

(1) The problem is to find _____ of _____ of the city council (2) Convert the base so that its numerator is a multiple of 3 45 = 12____

3a Solve the following problem by using fifteenths as the unit23 middot 45 = 23 of 1215 = 23 of 12 _______________ = 8 ________________ or 8____

b The pattern that occurs indicates the following shortcut 23 middot 45 = (2 middot 4)(___ middot ___)

Teaching Tip Fractions such as 45 can be written as either ldquofour-fifthsrdquo or ldquofour fifthsrdquo The use of two separate words emphasizes ldquofifthsrdquo as the primary unit the use of a hyphenated word emphasizes 45 as a single unit

Parts of Parts

As the last problem in the above activity illustrates it is common to describe parts of parts using multiplicative comparisons This leads to expressions such as ldquo23 of 45 of the City Councilrdquo How much is 23 of 45 The following example shows several ways of determining the answer all involving the identification of fifteenths as the key unit

Example 6 Four fifths of the class passed the test Of those who passed two thirds made at least a B Whatfraction of the class made at least a B

Students making at least a B = 23 of those who passed

= 23 of 4 5 of the class

= (23 middot 45) of the class

Solution A Use the Fundamental Property of Fractions to convert 45 to an equivalent fraction with a numerator that is a multiple of three 23 middot 45 = 23 of 45 = 23 of 1215 = 23 of 12 fifteenths = 8 fifteenths

Solution B Use a one-dimensional line segment partitioned into five equal parts Partition each of these parts into three parts and identify 23 of the small parts within 45 of class

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

Solution C Use a two-dimensional area diagram Use vertical lines to partition the rectangle into five equal parts and then use horizontal lines to partition 45 into thirds Extend the horizontal lines to partition the entire rectangle into thirds in order to determine the size of the smallest part relative to the whole

45 of the whole

the whole 23 of 45 of the whole = 815 of the whole

Partitioning a quantity into five parts and then partitioning each of these five parts into three parts

creates a total of 15 parts As the diagram illustrates 23 middotof 45 includes 8 of these 15 parts or 815

The above example indicates that there is a surprisingly simple way to compute the product of two fractions simply multiply the numerators and multiply the denominators A C = A middot C B D B middot D

Thus for example we can compute 23 middot 45 as follows 23 middot 45 = (2 middot 4)(3 middot 5) = 815 The justification for this easy shortcut is not the least bit obvious The two-dimensional area diagram may be the most straightforward way to verify that this shortcut works but the most fundamental explanation is based on understanding that 23 middot 45 means 23 of 45 and recognizing that 23 of 45 is the same as 23 of 12 fifteenths

Teaching Tip A good algorithm for computing the quotient of rational numbers can be obtained by combining two patterns We have just noted that AB bull DC = ADBC Previously we found that AB divide CD = ADBC So we have

A divide C = A D = AD B D B C BC

Since DC is the inverse of CD this rule can be summarized as follows ldquoTo divide fractions invert the second fraction and multiplyrdquo Be sure to stress to students that the second fraction not the first is to be inverted Multiplication with Decimals and Percents

If the multiplier m is between 0 and 1 m is often expressed in percent form While the form of the multiplier has no effect on the meaning of the comparison the use of percent (whichmeans hundredths) as a unit makes the use of grid paper almost a necessity for drawing an illustrative diagram

Example 7 A is 34 of B =gt A = 34 middot B =gt A = 75 middot B

To compute answers convert percents to decimal form and use the rules for decimal multiplication (Justifications for these rules will be discussed later)

Example 8 Becky invested 60 of her bonus in bonds and put the rest in her savings account If her bonus was $2500 how much money did she put in her savings account

Amount invested in bonds = 60 of B where B is the bonus B=gt Amount left in savings = 40 of bonus

= 04 middot $2500 bonds savings = $1000 60 of B 40 of B

Identifying the Components of Multiplicative Comparisons

To understand a multiplicative comparison it is very important to identify the described amount and the base of comparison As the next activity illustrates this is not as easy to do as one might think

Activity 53C

1 State the amount being referred to by the number in the following situations

a Alexandriarsquos salary now is three times what it was at her part-time position

__________________________

b One-third of my salary is used to pay my rent ___________________________

c Hamilton County has a 925 sales tax ___________________________

2 For each of the above situations describe the base to which the described amount is being compared

a ___________________________ b ___________________________ c ___________________________

3 Suppose a real estate agent earns a 10 commission for selling a house Fill in the following blanks

______________________________________ is 10 of __________________________________________

4 Suppose you buy an item at a 14 off sale Fill in the following boxes and blanks with either ldquooriginal pricerdquo ldquosale pricerdquo or ldquodiscountrdquo

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

5 The newspaper reported that the price of gasoline jumped 9 from August 1 to August 2 a Identify each of the three amounts F G and H in the following diagram as either ldquoprice on August 1rdquo ldquoprice

on August 2rdquo or ldquoprice increaserdquo

F _______________________ F G

G _______________________ H

H __________________________

b Fill in the following blanks with either ldquoprice on August 1rdquo ldquoprice on August 2rdquo ldquoprice increaserdquo or anappropriate percent

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

Here are some pointers for identifying the components of a multiplicative relationship

1 Described Amount is (___) of Base of Comparison =gt A = m bull B

A multiplicative relationship can always be phrased in the above form which corresponds directly to the equation A = m bull B

Example 9 Garyrsquos commission is one tenth of the selling price Selling Price

=gt commission = 110 middot selling price C

All StudentsExample 10 Forty percent of the students are women

=gt The number of women is 40 of the students=gt number of women = 40 of the students Women Students

2 ldquo (Amount)rdquo or ldquo(Amount) raterdquo

In many situations the described amount is stated before or after the multiplier with the multiplier expressed in percent form The base often unspecified is usually a total or the original amount

Example 11 The state has an 8 sales tax If the sticker price is $30 how much is the taxSales tax = 8 of sticker price = 008 middot $30 = $240

Example 12 The store gives a 15 employee discount employee discount = 15 middot original price

3 Part-Whole Part = m middot Whole

a Described Part

A part of a set is often described relative to the size of the set (the whole)

Example 13 One fourth of 40 students were sick How many students were sick

Number of sick students = 14 of total number of students= 14 of 40 10 10 10 10= 10

Total Number of Students

It is particularly common to describe a decrease relative to the original amount Decrease

Example 14 The size of the class decreased by a third when the instructor enforced the prerequisites Remaining Students Decrease = 13 of Original Original Class

It is common to describe decreases using percents without stating the base of comparison The original amount is always the base of comparison for a percent decrease

Example 15 ldquoAn 8 decrease in the price of gasolinerdquo means that the decrease is 8 of the old price

b The Other Part

With the part-whole model we get ldquotwo for the price of onerdquo For example if we know that 14 of the students are sick then we also know that (1 - 14) or 34 of the students are not sick If the multiplier is in percent form we find the multiplier for the other part by subtracting from 100 (100 is equal to 1)

Describing the Other Part of a Set

If A = 25 of B then the other part = 75 of B

A Other Part

25 of B 75 of B

100 of B

Example 16 At a 25 off sale what is the sale price of an item originally priced at $3495

Let P represent the original price Note that P is 100 of itselfSale price = Original Price - Discount

= 100 of P - 25 of P = 75 of P 25 middot P 75 middot P

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

In a situation in which the size of a set increases the increase is often described relative to the original

amount

Example 17 The value of a stock increases by 150 If it used to be worth $6 a share how much was the increase and how much is the stock worth now

Increase = 150 of old value= 15 bull $600 old value increase = $900

New Value = $6 + $9 = $15 New Value

The original amount is always the base of comparison for a percent increase

Teaching Tip Some students are disconcerted by the possibility that a percent may be larger than 100 This may be due to associating percents exclusively with the part-whole type of comparison When a part is compared to a whole the percent certainly cannot exceed 100 However there are many types of comparisons in which the described amount can be larger than the base of comparison For instance an increase can exceed the original amount In these situations the multiplier is larger than 100

b The New Amount

We also get ldquotwo for the price of onerdquo in increase situations because the new amount is the union of the old amount and the increase This means that the new amount can be described in terms of the old amount by adding the percent increase to 100

The Relationship Between the New Amount N and the Original Amount B

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

Example 18 Tuition has increased by 15 If the tuition was $4000 what is the new tuition

Tuition increase = 15 bull old tuition (T) Old Tuition

IncreaseNew Tuition = old tuition + increase = 100 middot T + 15 middot T 100 T 15 T

= 115 middot T = 115 middot $4000 115 T = $4600

Reporting Sensible Answers

There are some situations in which non-whole numbers do not make sense as answers In such situations round theanswer to the nearest whole number

Example 19 A teacher reported that two thirds of her class had done well on the year-end standardized tests This teacher has 25 students How many of her students did well on the testsNumber of students who did well = 23 of 25 = 16666 About 17 students did well on the tests

A Basic Concepts

1 Invent a word problem for the expression ldquo3 6rdquo based on the following meanings of multiplication a Repeated addition b Means of comparison

2 Consider the expressions ldquo23 6rdquo and ldquo23 of 6rdquo a What is the relationship between these two expressions b Explain the meaning of ldquo23 of 6rdquo Include a labeled diagram c Invent and solve a comparison word problem that is solved by computing 23 6

3 Which of the following are true in situations involving multiplicative comparisons a The described amount is never more than the base of comparison b The described amount must be a part of the base of comparison c The described amount can be a whole number multiple of the base of comparison d If one part of a set is 10 of the set then the other part must be 90 of the set e If a set increases in size by 10 then the original set is 90 of the enlarged set f If a set decreases in size by 10 then the shrunken set is 90 of the original set g In comparison situations the amount is always described explicitly h In comparison situations the base of comparison is always described explicitly

4 Fill in the blanks a If A is 23 of B and B is 14 of C then A is of C b If A is 20 of B and B is 150 of C then A is of C c If A = 04 middot B and B = 08 middot C then A is middot C

5 In the following diagrams the base of comparison B is represented by heavily outlined rectangles Describe the shaded area in terms of B in each of these situations

a b c d 66⅔B

6 In the following diagrams there are two bases of comparison B1 and B2 B2 is represented by the largest rectangle and B1 is represented by the next largest rectangle A is represented by the shaded area For each diagram estimate the multiplier for the following multiplicative relationships (a) A and B1 (b) B1 and B2 (c) A and B2 [Hint Extend lines and draw extra lines to make good estimates]

Example (a) A is 12 of B1 (B1 is striped) (b) B1 is 13 of B2

(c) A is 16 of B2

a b c d

7 Suppose Y has the following length If possible accurately draw the following lengths a a length that is twice the length of Y b a length that is 2 units longer than Y c a length that is one fourth the length of Y d a length that is a fourth of a unit less than Y e a length that is 50 more than Y f a length that is 25 less than Y

8a Explain the meaning of 35 of a number M without making reference to multiplication b What is the meaning of AB bull M where AB is a positive rational number c AB bull 23 can be computed by dividing 23 by and multiply the result by

9 Explain why 15 of 3 is the same as 3 divide 5 with the latter interpreted as partitioning

10 Which of the following are equivalent to 35 bull B a 3 of 5 equal parts of B b 3 bull (B divide 5) c B divide 35 d Partitioning B into 5 equal parts and selecting three parts

11 Find the following products of rational numbers using unit fractions and the definition of elementary fractions

a 2 bull 65 = 2 bull fifths = fifths b 13 of 7 feet = 13 of 21 of a foot = c 15 bull 1011 = 15 of ___ elevenths = d 16 bull 53 = 16 of 30 =

12 Develop examples to show that ldquoofrdquo does not necessarily mean ldquotimesrdquo while ldquotimesrdquo usually means ldquoofrdquo

13 Use each of the following methods to find 14 bull 13 a Creating an equivalent fraction with a numerator that is a multiple of 4 b Partitioning a one-dimensional line segment c Partitioning a two-dimensional rectangle

14 Write a word problem for which it makes no sense to report 13 bull 53 as 1723

15 Show how to find 35 of 10 sevenths using discrete sets

16 Six long distance runners get a take-out order of six pizzas for dinner When they get home they find that they were shortchanged one pizza They divide these five pizzas equally among themselves Which of the following expressions can be used to determine how much pizza each runner gets

a 6 5 b 15 of 6 c 5 6 d 30 sixths 6 e 16 of 5

B For each of the following(a) Identify all described amounts A and their bases of comparison B(b) Write the corresponding multiplication equations of the form A = m bull B(c) Draw and label a picture illustrating the situation(d) Write multiplication equations for ldquothe other partrdquo or ldquothe new quantityrdquo

1 The sales tax rate in Hamilton County Tennessee is 9252 A shirt is on sale for 14 off3 Two fifths of the class are women4 The price of gas went up 10 this week5 The price of gas went down 10 last week6 Three quarters of the students at the university are undergraduates Of these one third are Asian7 In 1997 234 of all pregnancies ended in abortion with 554 of these abortions occurring within the first

eight weeks of pregnancy

C Solve the following problems

1 Adrian ran three fourths as far as Paula Paula ran 24 miles How far did Adrian run

2 Alison makes $60000 more than Larry and her salary is three times his What is their combined salary

3 An employee gets a 10 discount on merchandise a What is the discount for an item marked $7995 b Determine the price the employee will pay for an item marked $14799 by doing a single multiplication

4 A companyrsquos stock lost 910 of its value when the company went bankrupt a If the stock used to be worth $20 per share how much is it worth now b If the stock is now worth $20 per share how much was it worth before

5 The cost of a certain type of computer decreased by 15 this year It used to cost two thousand dollars How much does it cost now

6 The cost of gas increased by 10 this past week a Last week gas cost two dollars a gallon How much does it cost now b The cost of gas is about to increase by another 20 What will be the new cost of gas

7 In 1999 426 of accidental deaths in the United States were caused by motor vehicles Of these 237 were people between the ages of 15 and 24 If possible answer the following questions If the question cannot be answered describe the information that would need to be known to answer the question

a What percent of accidental deaths were people between the ages of 15 and 24 who died in a motor vehicle accident

b How many people between the ages of 15 and 24 died in a motor vehicle accident in 1999 c What percent of accidental deaths in the US in 1999 were not caused by motor vehicles d What percent of accidental deaths caused by motor vehicles were not people between the ages of 15 and 24 e What percent of accidental deaths were not people between the ages of 15 and 24 whose accidental deaths

were caused by motor vehicles f What percent of people between the ages of 15 and 24 died in motor vehicle accidents

8 There were two thirds of a pizza left after a pizza party a Suppose the tired host sat down and ate half of a pizza How much pizza is now left b Suppose the tired host sat down and ate half of what was left How much pizza is now left

9 One third of the expenses for a certain business is the employee payroll One quarter of the employee payroll is for managers

a What fraction of the entire budget is for managerial employee wages b What fraction of the employee budget is for non-managerial employee wages c What fraction of the entire budget is for non-managerial employee wages

10 Seventy percent of the students at a university are women Of the latter 40 are 21 years old or older a What percent of the women are less than 21 years old b What percent of the university students are women less than 21 years old c What percent of the students are men d What percent of the students are at least 21 years of age

11 In 1992 heart disease accounted for 3310 of the 2177000 deaths in the US while suicide accounted for 137 of the deaths Of those who committed suicide 2267 were women

a Write multiplication sentences for each of the percents in this problem State the described amounts and their bases using English phrases not numbers

b Write multiplication sentences for the ldquoother partsrdquo related to each percent State the other parts and their bases using English phrases not numbers

c How many men committed suicide in the US in 1992 d What percent of the US deaths in 1992 were not due to heart disease or suicide____________________________________________________________________________________________

54 Division in the Context of Comparisons

In this section we investigate two more interpretations of division Just as there are two interpretations of division related to the basic meaning of multiplication as repeated addition there are two interpretations of division related to multiplication used as a means of comparison This multiplicative relationship is summarized as follows

Described Amount = Multiplier middot Base of Comparison If the multiplier and the base of comparison are known we use multiplication to find the described amount In contrast if the described amount is known and either the multiplier or the base of comparison is unknown we have a situation with an unknown factor That is we have a division problem

1 Unknown Multiplier Division as a Ratio

Described Amount = bull Base of Comparison

In the following activity we will investigate the connection between multipliers and ratios

Activity 54A

1 Jerry ran 12 miles Nick ran twice as far as Jerry

a Write the multiplicative relationship between Nickrsquos distance N and Jerryrsquos distance J N = ____________

b How far did Nick run ________________

c What is the ratio of Nickrsquos distance to Jerryrsquos distance Write this ratio in reduced form ________________

2 Maryrsquos salary M is three fourths of Edrsquos salary E

a Write the multiplication sentence expressing the relationship between M and E M = ____________

b If Edrsquos salary is $40000 what is Maryrsquos salary ________________

c What is the ratio of Maryrsquos salary to Edrsquos salary Write this ratio in reduced form ________________

3 Charlie bought a shirt on sale for $30 It originally cost $40

a State the ratio of the discount to the original price in percent form (ie the discount rate)

________________

b Fill in the blank discount = ______ of the original price

4 A class has 8 girls and 16 boys

a What is the ratio of girls to boys ________________

b Fill in the blank using a reduced fraction Number of girls = ____ bull number of boys

5 In light of your above work state the relationship between (a) the multiplier in the multiplicative comparison and (b) the ratio of the amount to the base ________________

6 At Superior Tech the tuition in 1999 was $18500 In 2000 it was $20000 What ________________was the percent increase in tuition

According to the basic definition of division as the inverse of multiplication A = m middot B implies that m = A divide B The problems in the above activity also indicate that the multiplier m is equal to the ratio of A to B This connection between division and ratios is the third major interpretation of division Since the ratio of A to B is also the same as AB we have the following string of equivalences

Ratio Interpretation of Division

For B ne 0 the following are equivalent for computational purposes

A divide B = A B = AB

Teaching Tip Teachers should not assume that students will immediately recognize that the multiplier in the multiplicative relationship between A and B is the same as the ratio of A to B This is a surprise to many people

Example 1 Jerry ran 12 miles and Nick ran twice as far as Jerry What is the ratio of Nickrsquos distance to Jerryrsquosdistance

Solution A The first sentence indicates that Nickrsquos distance is two times Jerryrsquos distance Since the multiplier inthis multiplicative relationship is 2 the ratio of Nickrsquos distance to Jerryrsquos distance is 2 to 1

Solution B Since Jerry ran 12 miles Nick must have run 24 miles The ratio of Nickrsquos distance to Jerryrsquos distance is 24 to 12 or 2 to 1

We have already examined a number of situations in which the ratio of two quantities is of great interest In situations involving multiplicative relationships the ratio of interest is the ratio of the described amount to the base of comparison The ratio of A to B is often called a rate if the ratio is described as a single number For instance the rate of ldquo60 miles per hourrdquo is the ratio of 60 miles to 1 hour A rate is thus a ratio in which the second quantity is expressed in terms of a single unit A noun or adjective appearing immediately before the word ldquoraterdquo is usually a reference to the described amount Below are some examples

Example 2 (a) Discount Rate = DiscountOriginal Price

(b) Sales Tax Rate = Sales TaxSticker Price

(c) Rate of Increase (or Decrease) = Increase (or Decrease)Original Amount

If a ratio or rate is to be determined the key is to identify the described amount and the base

Example 3 Peter bought a sofa on sale for $600 It originally cost $800 Find the discount rate

The discount rate is the ratio of the discount to the original price The discount is$800 - $600 or $200 so the discount rate = $200$800 = 25

Example 4 Joanne paid $540 for an item with a sticker price of $500 What was the tax rate

The tax rate is the ratio of tax to sticker price $040$500 = 8100 = 8

As the next example illustrates we often get ldquotwo for the price of onerdquo in situations involving ratios

Example 5 There are 18 girls and 6 boys in Johnrsquos class

(a) The ratio of girls to boys is 18 to 6 or 3 1

(b) The ratio of girls to the entire class 18 to 24 = 18 divide 24 = 1824 = 34 or 3 to 4

Mixed numbers usually need to be changed to improper fractions in order to compute simpler forms of ratios

Example 6 A stock that was worth 234 points fell by half a point What was the percent decrease

Ratio of decrease to original value = 12 234 = 12 divide 114 = 12 bull 411 = 422 asymp 18

ldquoSpeedrdquo is the special name given to ratios such as distance to time or words per minute

Example 7 Mark drove 200 miles in 4 hours What was his speed

Markrsquos speed = 200 mi4 hour = 50 mi1 hr = 50 miles per hour

2 Division as Finding the Unknown Base of Comparison

Described Amount = Multiplier bull

The fourth interpretation of division occurs when the base of comparison is unknown These are probably the most difficult types of division problems It is often easier to solve such problems by setting up the multiplicative relationship with the base of comparison as an unknown factor The use of diagrams the definition of multiplication and algebraic techniques are helpful in finding an unknown base

Activity 54B

A Solve the following problems

1 Peter earns three times as much money as Jim If Peter earns $60000 how much ______________money does Jim earn

2 Maria ran one third as far as Jan If Maria ran 6 miles how far did Jan run ______________

B Connie swam two thirds as far as Jan Let C and J represent Conniersquos and Janrsquos distances

1 State the multiplicative relationship between C and J ______________

2 Suppose the following line segment represents C Underneath this line segment draw a line segment that accurately represents J

3 Suppose Connie swam 600 yards Use your diagram to determine Janrsquos distance

4 Explain why Janrsquos distance can be found by dividing 600 yards by 2 and multiplying the result by 3

5 Rewrite the following as a division sentence using the basic definition of division as the inverse of multiplication 600 = 23 middot

6 Explain how to solve the following equation by multiplying both sides of the equation by a particular fraction 600 = 23 B

C Mandy bought a blouse at a 25 off sale

1 Label the parts of the diagram to the right with ldquooriginal pricerdquo ldquosale pricerdquo and ldquodiscountrdquo

2 If Mandy paid $24 for the blouse how much money did she save by buying it on sale

Finding an unknown base of comparison is a matter of working backward from the described amount

Example 8 The new church hall with an area of 4800 square feet has three times the floor space as the old church hall What was the area of the old church hall

New Church Hall

Old Church Hall

Area of new church hall = 3 middot Area of old church hall =gt Area of old church hall = One of three equal parts of 4800 square feet =gt Area of old church hall = 13 of 4800 = 4800 square feet divide 3 = 1600 square feet

In other words since the described amount is three times the base then the base will be one third of the described amount Note that 13 is the reciprocal of 3

Example 9 Bobby spent two thirds of his money to rent a DVD The rental cost $8 How much money didBobby have before renting the DVD

$4 $4 $4

Since $8 is two thirds of the original amount then $8 divided by 2 must be one third of the original amount The original amount is three of these thirds Original = 3 middot ($8 divide 2) = $12

Note that 3 middot (8 divide 2) is the same as 8 middot 32 Once again we have found the base by multiplying the amount by the reciprocal of the multiplier

As these examples illustrate an unknown base can be reconstructed by multiplying the described amount by the reciprocal of the multiplier

Finding an Unknown Base

If A = c B then B = d A d c

Algebraically this relationship is derived as follows

A = m middot B =gt A = m middot B =gt A = B =gt B = 1 middot A m m m m

When the multiplier m is in fraction form with m = cd then1m is equal to dc So we have B = dc middot A

Teaching Tip Unfortunately this division relationship between the base the described amount and the multiplier is not intuitively obvious to most people While it can be laborious to reconstruct the base using the technique demonstrated in the above examples students who do such reconstructions (with small numbers) may be more likely to solve unknown base problems correctly Students may also be more likely to solve such problems correctly by setting up the algebraic equation A = m middot B and algebraically solving for B

The relationship between the base and the described amount is directly connected to the fact that division is the inverse of multiplication as illustrated by the following diagram

Base of Comparison Described Amount

Multiply by m

Base Amount

Divide by m

It is interesting that the actual process of reconstructing the base from the described amount is related more directly to multiplying by the reciprocal of m than dividing by m This may be one of the reasons why finding a missing base is one of the most difficult problems in the standard school curriculum

Example 10 Jack owns a two-acre lot in a subdivision It is three fourths as large as the largest lot in thesubdivision How large is the largest lot

Solution A Let represent the size of the largest lot 2 acres = 34 middot =gt = 43 middot 2 acres = 223 acres

Solution B Think this through with a diagram Since two acres consists of three parts of the basewe need to partition these acres into three equal parts Do this by partitioning each acre into

thirds Two Acres Partitioned into Three Equal Parts

One part = ⅔ acre

Largest Lot = 4 parts = 4 middot (⅔ acre) = 2⅔ acres

Indirect Amounts

A complication associated with finding unknown bases is that the available information is not necessarily the amount described by the multiplier

Example 11 Daisy paid $80 for a suit on sale for 20 off How much money did she save

Let P be the original price Use the fact that 20 is equal to 15 to draw a diagram

Solution A ldquo20 offrdquo =gt discount = 20 of P

=gt sale price = 80 of P Sale Price Discount

=gt $80 = 08 P Original Price

=gt P = $80 divide 08 = $100

=gt discount = $20

Solution B $80 is 4 fifths of the original price Therefore 14 of $80 or $20 is one fifth of the original price and also the discount

Example 12 The population of Catoosa County rose by 2 in the last year The population is now 48400 Whatwas the population a year ago Let P represent last yearrsquos population

2 Increase =gt Increase in population = 2 middot P =gt Current population = 102 middot P

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

Teaching Tip Some students have a tendency to ldquosolverdquo percent problems by blindly multiplying or dividing numbers in the problem They hope to be lucky and stumble across the right answer Unfortunately luck is often in scant supply especially for two-step problems involving indirect amounts In such problems it is impossible to find the right answer by multiplying or dividing the given numbers Teachers must help students come to understand multiplicative relationships if students are to become competent with these very common and important problems

A Basic Concepts

1 Basic Relationships a State the basic multiplicative relationship between the described amount and the base of comparison b State the basic definition of division c State the definition of a ratio

2 List the four interpretations of division discussed in this chapter

3 Which of the following are correct interpretations of X divide Y for Y 0 a The size of a part when X is partitioned into Y equal parts b The number of Yrsquos in X c The unknown factor in the equation X = Y middot d The number of Xrsquos in Y e The unknown factor in the equation Y = X middot f The ratio of X to Y g The unknown base for an amount X and multiplier Y h XY

4 Which of the following are equivalent to the multiplier m in the multiplicative relationship between the described amount A and the base of comparison B

a A B b the ratio of A to B c the ratio of B to A d the reciprocal of A

5 Which of the following are equivalent to the base of comparison B in the multiplicative relationship A = 35 middot B

a A 35 b the product of A and 53 c five parts each the size of one of the three equal parts of A

d 35 A e three of five equal parts of A f 53 A

6 Draw diagrams for each of the following and determine the missing numbers a If X is four times as large as Y then Y will be of X b If X is three fourths as large as Y then Y will be as large as X c If Y increases by 20 then the result will be of Y d If Y decreases by 20 then the result will be of Y e If X is 50 of Y then Y will be of X f If X is 25 of Y then Y will be of X

7 Consider the multiplicative comparison described by A = m bull B a Solve this equation for m b Solve this equation for B c m is the ratio of to d is the base of comparison

8 For the multiplicative comparison A = m bull B decide whether the following statements are true or false a A is always less than B b m is always a percent between 0 and 100 c B must be a whole number d m is the ratio of B to A

9 Identify the bases and amounts for the fractions or percents in the following statements a 25 off b There will be a 10 tuition increase next year c One fifth of the students failed the test

10 For each of the statements in the previous problem write a multiplication equation that includes the other part or the new amount

11 What is the typical base of comparison in decrease and increase problems

12 Which of the following can be answered by computing 57 divide 25 a What is the ratio of 57 to 25 b How many times can 25 be subtracted from 57 c What is 25 of 57 d Find x if 57 bull x = 25 e Find x if 25 bull x = 57 f If 57 is 25 of another number what is that number

13 Invent and solve a word problem of the indicated type for each of the following a 2 divide 14 (missing base) b 12 divide 14 (ratio) c 14 divide 2 = 18 (missing base) d 2 divide 12 (repeated subtraction)

B Problem Solving

1 Seventy-five percent of the graduating seniors came to graduation a Fill in the blanks is 75 of b If 1200 graduating seniors were at graduation how many did not come to graduation

2 A realtor sold a house for $125000 and earned a commission of $10000 What was her percent commission

3 Karen bought a suit on sale for 25 off a is 25 of b is 75 of c If the discount was $13499 what was the original price of the suit d If the original price was $13499 what was the sale price of the suit e If the sale price was $13499 what was the original price of the suit

4 Alice saved $1895 by using her 10 employee discount to buy a VCR How much did she pay for the VCR

5 The sales tax rate is 734 a If the tax on an item is $3042 what is the sticker price b If the sticker price of an item is $3042 what is the tax c If the final price of an item is $3042 what is the tax

6 Blair paid $84799 for a sofa The sales tax rate was 6 What was the sticker price

7 At a sale Margaret bought a blouse for $2759 that had been originally priced at $4599 What was the discount rate

8 Mary makes 34 as much money as John Johnrsquos salary is $46000 a What is the ratio of Maryrsquos salary to Johnrsquos salary b What is Maryrsquos salary

9 Seth had to pay a 10 penalty when he made a late payment The penalty was $15 How much was the final bill

10 Ben invested three fourths of an inheritance He bought a boat with the remaining money If the boat cost $6000 how much money did he inherit

11 Rachel has 18 feet of string and cuts it into half-foot lengths for a project a How many pieces of string does she now have b State the division sentence that yields the answer to this question

12 April has 18 feet of string and cuts it in half for a project a How many pieces of string does she now have and how long are they b State the division sentence that yields the answer to this question

13 A half acre of land is sectioned off into 40 garden plots of equal size How big is each plot

14 A square mile is equal to 640 acres How many square feet are in an acre [Hint A square mile is 5280 feet by 5280 feet]

15 One third of the crew of a ship got seasick during a storm a If there were 6 crewmen how many got sick b If there were 6 sick crewmen how many crewmen did not get sick c If there were 6 crewmen who did not get sick how many crewmen were there altogether

16 Twenty percent of a class made Arsquos a If 40 students made Arsquos how many students did not make Arsquos b If 40 students did not make Arsquos how many students were in the class c If there were 40 students in the class how many did not make Arsquos

17 A stock lost one tenth of its value in 2000 and one quarter of its remaining value in 2001 What was the stock worth after these changes relative to its value at the beginning of 2000

18 The price of a computer dropped 10 in 1998 and another 15 in 1999 a If the computer cost $2449 in 1997 how much did it cost in 1999 b If the decrease in price was about $150 in 1998 what was the decrease in price in 1999 c What was the overall percent change in the price of computers in these two years

[Percent change is the ratio of the change in price to the original price]

19 Berta paid $3147 for a pair of pants on sale for 30 off How much money did she save by buying the pants on sale

20 Hakeem paid $140724 for a bedroom suite including an 825 sales tax How much sales tax did he pay

21 After a 7 increase full-time tuition is now $1349 What was the old tuition

22 In 1991 the United States consumed about ten times as much energy as India even though India has more than three times as many people as the United States The US consumed about 80 quadrillion Btu (ldquoBturdquo is an abbreviation for British thermal unit a measure of energy)

a How much energy did India consume b How much energy did an average American consume compared to an average Indian

23 Sarah inherited two thirds of her motherrsquos estate She decided to give one tenth of her inheritance to charity If she gave $1500 to charity how much money did she inherit

24 A teacher sent 15 students to the library This was three fourths of her class How many students are still in the classroom

25 John inherits 57 of his motherrsquos estate He invests 25 of his inheritance and spends the rest on a trip to Alaska

a What fraction of the entire estate did he invest b What fraction of his inheritance did he spend on his trip to Alaska

26 John is in charge of 57 of his motherrsquos estate He invests 25 of the entire estate in Company X and the rest of the estate for which he is responsible in mutual funds What fraction of his motherrsquos estate are in mutual funds

27 John inherits 25 of a small parcel of land His inheritance amounts to 27 of an acre What is the total acreage of the small parcel of land

28 In 1992 the world record for the 1500 meter run was 3 min 4012 sec The world record for the 1500 meter freestyle swim was 14 min 4348 sec How much faster is the world record in running compared to the world record in swimming

a Estimate answers using (1) subtraction and (2) division b Find exact answers using (1) subtraction and (2) division

____________________________________________________________________________________________

55 Proportional Reasoning

In this section we explore constant ratios in greater depth

Activity 55A

A An ad in the produce section of the supermarket reads ldquoTwo watermelons for $300rdquo

1 Answer the following questions supporting your answers with appropriate diagrams

a How much will six watermelons cost ________ b How much will five watermelons cost _________

2 Let C = cost of watermelons and W = number of watermelons

a Complete the following table d Graph your ordered pairs

W 0 1 2 5 6 10

b Express the relationship between C and W using multiplication

c Express the relationship between C and W using ratios e Find the slope of the line defined by these points

B Answer the following Assume this is a one centimeter grid

1 Identify two sets of rectangles with the same shapes Set 1 ______________ Set 2 ______________ A B C

2 Complete the following tables for each set including theratios of corresponding sides of rectangles in each set Use fraction form for your ratios

Set 1 Rectangle Short Side Long Side D E

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

Set 2 Rectangle Short Side Long Side

_______ ________ ________ _______ ________ ________

Ratio ________ ________

3 Look for a pattern and make a generalization about the ratios of the corresponding sides of ldquolook alikerdquo rectangles

4 For each of the above sets of two rectangles find the ratio of the larger area to the Set 1 _________smaller area Use fraction form [Hint These ratios are not what you might expect]

Set 2 _________

5 Fill in the following table Assume the smaller cube is 1 cm by 1 cm by 1 cm and the larger cube is 2 cm by 2 cm by 2 cm Include units

Smaller Cube Larger Cube RatioLength of a sideArea of a face

Volume of cube

Proportional Relationships and Their Connection with Multiplicative Relationships

If the ratio of two related variable quantities A and B remains constant even as the two quantities change then A and B are said to be proportional For example the ratio of the cost to the number of watermelons at a supermarket probably remains constant even as the cost and number change the ratio of sales tax to sticker price remains constant for different prices and the ratio of the velocity of a free falling object to the time it has been falling is a constant

Example 1 If a pound of asparagus costs $300 then 2 pounds will cost $600 half a pound will cost $150 a third of a pound will cost $100 and so on The constant in these situations is the ratio of weight to cost $3001 lb = $6002 lb = $15005 lb = $100(⅓ lb) All of these are ratios of 3 to 1

The equation YX = AB is equivalent to the equation Y = AB bull X Thus two quantities are proportional if and only if one quantity is a constant multiple of the other This multiplicative relationship is exactly the type of relationship we studied in previous sections In other words quantities with a multiplicative relationship also have a proportional relationship and vice versa As we shall see some problems are easier to solve using a proportion while others are easier to solve using the multiplicative relationship

Example 2 Reconsider the asparagus that cost $300 a pound Let W represent the weight of the asparagus you are buying and C represent the cost of this asparagus Presumably the ratio of price to weight is constant so we have $3001 pound = CW The equivalent multiplicative relationship is C = 3 W

Proportionality and Similar Figures

Proportional relationships are common in geometry Similar figures were defined earlier as figures that have the same shape but not necessarily the same size Now we can state more precisely that similar figures are such that their corresponding sides are proportional and their corresponding angles are congruent

Example 3 The following two right rectangular solids are similar 2 This means that the ratio of the corresponding heights 6 of these solids is the same as the ratios of the corres- 4 4

ponding lengths and the corresponding widths

Example 4 The ratio of the circumference to the diameter of a circle is constant regardless of the size of the circle d

CD = cd D

This ratio is the irrational number π c CD = π =gt C = πD C

Proportionality in One Two and Three Dimensions

Areas and volumes of similar shapes have predictable relationships 2nd

Example 5 In a little league baseball diamond it is 60 feet from home plate to first base In the major leagues this distance is 90 feet Find the ratio of these distances and the ratio of the areas of these infields (The infield is the square area bounded by the baselines) 3rd 1st

Ratio of distances = 90 ft60 ft = 32 = 15

Ratio of infield areas = 902 ft2602 ft2 = 81003600 = 94 = 225 home plate

Thus a major league base runner has to run one and a half times as far as a little leaguer to get to first base a major league infielder also has to cover over twice as much area as a little leaguer

Example 6 A small nougat of chocolate candy measures 1 cm by 1 cm by 3 cm and weighs about half an ounce A larger nougat has dimensions that are double the dimensions of the smaller nougat How much does the larger nougat weigh

As the diagram illustrates the larger nougat has a volume that is 8 times the volume of the smallernougat so it weighs 8 times as much as the frac12 oznougat or about four ounces 1 cm by 1 cm by 3 cm 2 cm by 2 cm by 6 cm

These examples illustrate the following relationships among ratios in one two and three dimensions

Dimension Type Ratio Example1-dimensional Length k 1 3 12-dimensional Area k2 1 9 13-dimensional Volume Weight k3 1 27 1

Example 7 Suppose a 5-foot tall woman weighs 100 pounds How much would a 6-foot tall woman with the same shape as the shorter woman weigh

The ratio of one-dimensional heights is 6 to 5 or 65 Since weight is associated with volume the corresponding ratio of three-dimensional volumes will be 6353 or about 173 to 1 Thus the weight of the taller woman with the same shape is about 173 middot 100 pounds or 173 pounds

Teaching Tip Most students are amazed by the above relationships among length area and volume Apparently our intuitions are working against us here Thus students should be given lots of experiences comparing one- two- and three-dimensional characteristics of similar figures and shapes It is a good idea to use manipulatives such as grid paper and building blocks for this purpose

Within and Between Ratios

Situations involving constant ratios involve four quantities There are two major ways to arrange these quantities

Example 8 The cost of 16 ounces of tomatoes is $179 If the ratio of cost to weight is constant what is the cost of 12 ounces of tomatoes Let C represent the cost of 12 ounces of tomatoes

a Use the ratios of cost to weight $17916 ounces = C12 ounces

b Use the ratios of corresponding quantities $179C = 16 ounces12 ounces

A ratio of two quantities within the same situation is a within ratio For example the above ratios of cost to weight are within ratios The ratio of weight to cost is also a within ratio A ratio of corresponding quantities in different situations is a between ratio In the above example the ratio of the first cost to the second cost is a between ratio so is the ratio of the first weight to the second weight

Solving Proportions

An equation of the form AB = CD in which two ratios are set equal to each other is called a proportion In situations involving constant ratios we often know three of the four numbers in a proportion and are interested in figuring out the fourth Below are three common ways of doing so

1 The Unit Rate Method

Example 9 A 15-oz can of clams costs $300 If the unit price is constant how much should a 22-oz can cost

The unit price is the cost per ounce For the first can of clams the unit price is $300 divide 15 oz = 20cent per ounce So 22 oz bull 20cent per oz = $440

In general the unit rate for two proportional quantities is the amount of the first quantity A per one unit of the second quantity B It is simply the reduced ratio of A to B found by calculating A divide B This corresponds to the multiplier m in the multiplicative relationship A = m bull B

Teaching Tip Send your students off to supermarkets that list unit prices to compare the unit prices of items packaged in varying sizes (eg cans of clams)

Constant ratios are the basis for creating and using scale models such as maps and model airplanes The unit rate method of determining corresponding values is particularly useful in these situations because multiple values often need to be calculated

Example 10 On a backpackerrsquos map every two inches represents five miles On the map the distances from the start to the end of two trails are 7 and 412 How long is each trail

If two inches represents five miles then one inch represents 212 miles Length of first trail = 7 inches bull 212 miles per inch = 1712 miles Length of second trail = 412 inches bull 212 miles per inch asymp 11miles

2 The Scale Factor Factor of Change Divisor of Change Method

Example 11 Cantaloupes are three for five dollars How much will six cantaloupes cost 2

3 cantaloupes = 6 cantaloupes =gt Six cantaloupes will cost $10

$5 2 bull $5

Solving the cantaloupe problem is a matter of observing that 35 is the same as 610 This is an application of the Fundamental Property of Fractions AB = nAnB for any nonzero number n The number n is referred to as the scale

factor or factor of change Since the FPF also states that AB = AdividenBdividen proportions can also be solved using a divisor of change This method is very handy if the factor or divisor of change is a small whole number

As the following example shows sometimes two factors of change can be used to find an answer

Example 12 Right triangles A and B are similar What is x x 10

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

The scale factor method works well only if the numbers are compatible that is when one number is a whole number multiple of another such as 9 and 3

Teaching Tip Students become familiar with scale factors when they are learning to add fractions with different denominatorsmdashalthough they usually do not know the process by this name For instance they find the numerator in 54 = 12 by identifying the scale factor as three Later on teachers rewrite ldquo54 = 12rdquo as ldquo54 = x12rdquo and call it a proportion Rather than insisting that students solve this equation using some other technique teachers should build on what students already know and encourage them to apply the scale factor method when appropriate 3 The Cross Products Algorithm

A third way to solve proportions is to use the following theorem and a little algebra

Cross Products Theorem The equation AB = CD is equivalent to AD = BC for B 0 and D 0

Proof A = C =gt BD A = BD C =gt B D A = B D C =gt AD = BCB D 1 B 1 D 1 B 1 D

AD and BC are called cross products Sometimes the process of converting AB = CD to the equivalent equation AD = BC is called ldquocross-multiplyingrdquo Teaching Tip Unfortunately the phrase ldquocross-multiplyingrdquo is also used to describe the shortcut for dividing a fraction by a fraction W hen different processes are referred to by the same name students often confuse the results With a proportion the result of ldquocross-multiplyingrdquo is another equation with division of fractions the result is another fraction 23 7x is equal to 2x21 but 23 = 7x is equivalent to the equation 2x = 21 To avoid confusion it is better not to refer to the division of fractions as ldquocross-multiplyingrdquo

As the next example illustrates this algebraic approach to solving proportions is useful when dealing with more difficult numbers

Example 13 A nurse knows that the dosage of a certain antibiotic is 30 ml for an 80-pound child What should the dosage be for a 105-pound child

80 lb needs 30 ml 105 lb needs x ml

Solution A Using cross products

(1) Within Ratios Proportion (2) Between Ratios Proportion

30 ml = x ml x ml = 105 lb 80 lb 105 lb 30 ml 80 lb

Both of these proportions lead to the following equation

80x = 30 bull 105 =gt x = 30 ml bull 105 lb asymp 39 ml 80 lb

Solution B Using standard equation solving techniques (and one less step than cross-multiplying)

x ml = 30 ml =gt x = 105 bull 30 asymp 39 ml 105 lb 80 lb 80

Solution C Using unit rates30 ml80 lb = 0375 ml per pound =gt 105 pounds 0375 ml asymp 39 ml

As the above example illustrates a variety of methods can be used to find an unknown in a proportional relationship The main challenge is setting up the ratios correctly This is greatly facilitated by paying attention to units and using within ratios For instance if the ratio on one side is milliliters to pounds then the ratio on the other side must also be milliliters to pounds Between ratios can also be used but care must be taken so that the quantities in the two numerators (and the two denominators) come from the same situation Another way to guarantee correct results is to set up operations so that units divide out correctly We will explore unit cancellations in the next section

Teaching Tip Proportional reasoning is far more than the ability to follow procedures for solving proportions It is important to develop studentsrsquo conceptual understanding of proportional relationships in a wide variety of settings rather than simply focusing on procedures for solving proportions

Proportions and Multiplicative Relationships

As we have already discussed quantities that are proportional also have a multiplicative relationship This means that problems can often be solved two ways either with a proportion or a multiplication sentence

Example 14 Matt paid only $240 for a refrigerator at a 40 off sale What was the original price Pldquo40 offrdquo =gt discount = 40 original price 100

=gt sale price = 60 P 100 =gt $240 = 60 =gt P = $240 100 = $400

P 100 60

Teaching Tip While some problems involving percents lend themselves to solutions using proportions not all problems do so It is important for students to know how to describe proportional relationships both multiplicatively and with ratios

Activity 55B

1 A school had a 20 increase in enrollment and now has 425 students How many more students are enrolled at the school now than before Solve this problem two ways

a Using a proportion b Using a multiplication sentence

2 The photocopying machine is set so that the new dimensions will be 300 of the original dimensions The original figure is a 2 by 3 rectangle

a Find the dimensions of the enlarged image _____________________

b Find the ratio of the area of the enlarged image to the area of the original figure

3 A woman who is five feet tall weights 100 pounds Another woman who has the same general build is 5rsquo6rdquo tall About how much does the second woman weigh [Hint Weight is related to volume]

4 A ranger wants to estimate the number of fish in a small lake Her first step is to catch and tag 20 fish Then she returns these fish to the lake Later she catches 40 fish She finds that five of these fish are tagged If she assumes that the proportion of tagged fish in the lakersquos fish population is about the same as in her second catch about how many fish are in the lake

A Basic Concepts

1 Suppose A and B are proportional quantities Which of the following must be true statements a A and B remain constant b The ratio of A to B remains constant for corresponding values of A and B c A is a constant multiple of B d B is a constant multiple of A e A and B have a multiplicative relationship f If A increases by 2 units so will B g If A doubles so will B

2 Similarity a Similar figures have the same but not necessarily the same b Two figures are similar if their sides are

3 An 18-ounce can of tomatoes costs $189 a If the price per ounce is constant how much will a 12-ounce can of tomatoes cost Solve this problem using

a proportion containing within ratios b Find and use the unit price (cost per can) to find the cost of the 12-ounce can

4 Ears of corn are advertised as ldquo10 for $2rdquo a Find and use the unit price to determine the cost of 8 ears of corn b Use the Scale FactorDivisor Method to find the cost of 15 ears of corn

5 Solve the following using the Scale Factor Method

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

6 Explain how the Scale Factor Method of solving proportions is based on the Fundamental Property of Fractions Include an example

7 Informally stated the basic principle of equation solving is as follows ldquoDoing the same thing to both sides of an equation produces an equation with the same solutions as the original equationrdquo For instance if 3 is subtracted from both sides of x + 3 = 5 the resulting equation will have the same solution as the original equation What must be done to both sides of the proportion AB = CD to produce the equivalent equation AD = BC

8 Suppose a child is having a hard time grasping the idea of constant ratios She thinks that the ratio of 8 to 5 is the same as the ratio of 9 to 6 and that the ratio of 4 to 1 is the same as the ratio of 6 to 3

a What is this childrsquos misperception b Use the basic definition of a ratio and pictures to help this child see that 4 1 is not the same as 6 3

9a If Y = 3 middot X what is the ratio of Y to X b If P and Q are proportional quantities with PQ equal to 43 what is the value of the multiplier in the equivalent

multiplicative relationship P = m middot Q

10aState the definition of π b Using a measuring tape or a ruler and string measure to the nearest millimeter the diameter and

circumference of a handy large circular item (a wastebasket the rim of a bowl a flower pot etc) Then find the ratio of the diameter to the circumference

c Find the difference between your ratio and π to three decimal places d Find your percent error the ratio of the error (from part c) to the actual value 11 Suppose the lengths of all sides of a square are tripled Use a labeled and carefully drawn illustration to show

the effect on the area of the square It may be helpful to use grid paper

12 Higher Dimensional Relationships a Carefully draw representations of two cubes one with an edge of length 1 cm and the other with an edge of

length 3 cm b What is the ratio of the areas of the front faces of these cubes c What is the ratio of the volumes of these cubes 13 If the ratios of the edges of two cubes is p q state the following a Ratio of the areas of the faces of these cubes b Ratio of the volumes of these cubes

B Suppose cans of beans are advertised at ldquo5 for $4rdquo Assume the ratio of cans to cost remains constant Let N represent the number of cans and C the cost of N cans

1 Find the unit rate2 Make a table of six pairs of values for N and C 3 Graph your ordered pairs on graph paper4 Find the slope of the line formed by your graph and compare it to the unit rate Explain any similarities5 State the relationship between N and C in two ways a Using ratios b Using multiplication

C Problem Solving

1 Avocados are advertised as ldquo4 for $3rdquo Find the cost of six avocados in three ways a Unit rate method b Factor divisor of change method c Setting up a proportion and cross-multiplying

2 The prescribed dosage of a certain antibiotic is 30 ml for a 50-pound child Answer the following questions using the method stated in parentheses

a How much antibiotic should be given to a 75-pound child (divisor factor of change) b How much antibiotic should be given to an 87-pound child (within ratios proportion) c How much antibiotic should be given to an 113-pound child (between ratios proportion)

3 The two rectangles to the right are similar a Construct a proportion using within ratios b Construct a proportion using between ratios c Use cross products to find x 8 24 d Use the factor of change method to find x x e Find the ratio of the areas of these rectangles 45

4 Grocery store 1 advertises 15-oz cans of pork and beans at ldquo4 for $1rdquo Grocery store 2 advertises a 28-ounce can of pork and beans for 59cent Determine the better deal using (a) unit rates and (b) a factor of change

5 On a map two inches represent 9 miles If two points are 35 inches apart on the map how far apart are they in actuality

6 Grocery store 1 advertises ldquo12 gallon Gatorade 3 for $5rdquo Grocery store 2 advertises ldquo64-ounce Gatorade 2$3rdquo Determine the better deal by using (a) unit rates and (b) a factor of change

7 Justify your answer for the following using labeled diagrams a One right triangle has legs of length 9 and 12 Another right triangle has legs of length 6 cm and 8 cm Are

these triangles proportional b One triangle has sides of length 9 and 12 Another triangle has sides of length 6 cm and 8 cm Are these

triangles proportional

8 Two boxes are similar The shortest side of the larger box is three times the shortest side of the smaller box a What is the ratio of the longest side of the larger box to the longest side of the smaller box b What is the ratio of the bases of the two boxes c What is the ratio of the volumes of the two boxes

9 The pitch of a roof is a measure of the roofrsquos steepness It is the ratio of the length of the vertical to the horizontal leg in the right triangle N formed underneath the roof Construction workers describe the pitch of a roof in the form ldquoN and 12rdquo which means the ratio of N to 12 12

Draw diagrams on grid paper for each of the following problems a Draw a roof with a pitch of 8 and 12 b The pitch of a roof is to be 5 and 12 If the vertical beam is to be 8 feet how long should the horizontal beam

of the truss be (The truss is the roof support represented by the isosceles triangle in the above diagram)

c A rectangular house is to be built 40 feet wide and 60 feet long Find the dimensions of the trusses needed for this house if the pitch is to be 5 and 12

10 The grade of a road refers to the ratio VH of the vertical to the horizontal change from one point on the road to another It is V often expressed in percent form because it is usually a small Hfraction

a As I-24 comes off the Cumberland Plateau in southeastern Tennessee there are large signs warning truckers of an upcoming 7 grade Explain the meaning of this number

b If one leg of a right triangle is very small compared to the other then the hypotenuse of the triangle has almost the same length as the longer leg Use this fact to estimate the height (in feet) of the Cumberland Plateau above the valley if it takes about three miles to drive down the 7 grade to the bottom of the mountain

_______________________________________________________________________ ______

Chapter 5

Activity 51A

For m a whole number the product m bull B is the total number of objects in m disjoint sets each

Total = (Number of sets) bull (Size of the set)

darr darr darr

Example 1 Melissa invited all of her running friends over for a morning run followed by brunch She bought three dozen eggs for the occasion How many eggs did she buy

Total number of eggs = 3 sets of 12 eggs = 12 eggs + 12 eggs + 12 eggs = 3 12 eggs = 36 eggs

The multiplier and multiplicand are also called factors A whole number product is called a multiple of each factor

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

3 A cylindrical water tank is 20 feet high It is known that when the water is one foot deep

the volume of water in the tank is about 700 cubic feet What is the capacity of the tank _____________

Various Cylinders

Activity 51D

1 Find the volume of a prism that is one foot long with a right triangular base _________________

The three sides of the base measure 3 4 and 5

3 In Tennessee license plates consist of three letters followed by three digits

Activity 52A

Division is the Inverse of Multiplication

For B a natural number A divide m can be interpreted to mean

Activity 52B

1 The problem is to partition 3 pizzas into 4 equal parts and determine the size of the part

That is we want to find _____________ divide ___

Activity 52C

Activity 52D

This theorem is called the Division Theorem because of the connection between A divided by B and A written as

q middot B + r If A consists of q Brsquos plus a remainder r then A divide B is equal to q with a remainder of r

Activity 53A

a ____________________________________ is 200 of ___________________________________________

b Label three sets in the following diagram the old price

the increase and the new price

c If a loaf of bread cost 50cent thirty years ago how much does it cost now Label the diagram _____________

appropriately to find the answer

a ____________________________________ is 110 of _____________________________________________

Activity 53B

A John ordered eight pizzas for a party His guests ate two-thirds of all the pizza How much pizza did they eat

B D B middot D

Thus for example we can compute 23 middot 45 as follows 23 middot 45 = (2 middot 4)(3 middot 5) = 815 The justification for this easy shortcut is not the least bit obvious The two-dimensional area diagram may be the most straightforward way to verify that this shortcut works but the most fundamental explanation is based on understanding that 23 middot 45 means

23 of 45 and recognizing that 23 of 45 is the same as 23 of 12 fifteenths

A divide C = A D = AD

Since DC is the inverse of CD this rule can be summarized as follows ldquoTo divide fractions invert the second fraction and multiplyrdquo Be sure to stress to students that the second fraction not the first is to be inverted

Multiplication with Decimals and Percents

Activity 53C

Example 11 The state has an 8 sales tax If the sticker price is $30 how much is the tax

b The New Amount

A Basic Concepts

1 Invent a word problem for the expression ldquo3 6rdquo based on the following meanings of multiplication

a Repeated addition

b Means of comparison

2 Consider the expressions ldquo23 6rdquo and ldquo23 of 6rdquo

a What is the relationship between these two expressions

b Explain the meaning of ldquo23 of 6rdquo Include a labeled diagram

c Invent and solve a comparison word problem that is solved by computing 23 6

3 Which of the following are true in situations involving multiplicative comparisons

a The described amount is never more than the base of comparison

b The described amount must be a part of the base of comparison

c The described amount can be a whole number multiple of the base of comparison

d If one part of a set is 10 of the set then the other part must be 90 of the set

e If a set increases in size by 10 then the original set is 90 of the enlarged set

f If a set decreases in size by 10 then the shrunken set is 90 of the original set

g In comparison situations the amount is always described explicitly

h In comparison situations the base of comparison is always described explicitly

4 Fill in the blanks

a If A is 23 of B and B is 14 of C then A is of C

b If A is 20 of B and B is 150 of C then A is of C

c If A = 04 middot B and B = 08 middot C then A is middot C

a b c d

66⅔B

6 In the following diagrams there are two bases of comparison B1 and B2 B2 is represented by the largest rectangle and B1 is represented by the next largest rectangle A is represented by the shaded area For each diagram estimate the multiplier for the following multiplicative relationships (a) A and B1 (b) B1 and B2

(c) A and B2 [Hint Extend lines and draw extra lines to make good estimates]

Example (a) A is 12 of B1 (B1 is striped)

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

B For each of the following

Activity 54A

5 In light of your above work state the relationship between (a) the multiplier in

the multiplicative comparison and (b) the ratio of the amount to the base ________________

Activity 54B

1 Peter earns three times as much money as Jim If Peter earns $60000 how much ______________

money does Jim earn

A Basic Concepts

1 Basic Relationships

a State the basic multiplicative relationship between the described amount and the base of comparison

b State the basic definition of division

c State the definition of a ratio

3 Which of the following are correct interpretations of X divide Y for Y 0

a The size of a part when X is partitioned into Y equal parts b The number of Yrsquos in X

c The unknown factor in the equation X = Y middot d The number of Xrsquos in Y

e The unknown factor in the equation Y = X middot f The ratio of X to Y

g The unknown base for an amount X and multiplier Y h XY

6 Draw diagrams for each of the following and determine the missing numbers

a If X is four times as large as Y then Y will be of X

b If X is three fourths as large as Y then Y will be as large as X

c If Y increases by 20 then the result will be of Y

d If Y decreases by 20 then the result will be of Y

e If X is 50 of Y then Y will be of X

f If X is 25 of Y then Y will be of X

5 The sales tax rate is 734

a If the tax on an item is $3042 what is the sticker price

b If the sticker price of an item is $3042 what is the tax

c If the final price of an item is $3042 what is the tax

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Example 3 The following two right rectangular solids are similar 2

This means that the ratio of the corresponding heights 6

of these solids is the same as the ratios of the corres- 4 4

Solving Proportions

3 The Cross Products Algorithm

Activity 55B

In all of the above problems answers can be found by using repeated addition There are so many situations involving repeated addition that this process is called multiplication (Be warned however that repeated addition is not the only meaning of multiplication We will study another meaning in a later section of this chapter)

For m a whole number the product m bull B is the total number of objects in m disjoint sets eachcontaining B elements m is called the multiplier and B is called the multiplicand

m bull B = B + B + B + + B

m times

The two numbers m and B play two very different roles in this basic meaning of multiplication The multiplier m is the number of sets while the multiplicand B is the size of the set The result of a multiplication is called a product In situations in which multiplication is defined as repeated addition the multiplicand can be any type of number but the multiplier must be a whole number

Total = (Number of sets) bull (Size of the set) darr darr darr

Product = Multiplier bull Multiplicand

Example 1 Melissa invited all of her running friends over for a morning run followed by brunch She bought three dozen eggs for the occasion How many eggs did she buyTotal number of eggs = 3 sets of 12 eggs = 12 eggs + 12 eggs + 12 eggs = 3 12 eggs = 36 eggs

The multiplier and multiplicand are also called factors A whole number product is called a multiple of each factor Example 2 Consider 3 2 = 2 + 2 + 2 = 6

a 3 is the multiplier 2 is the multiplicand and 6 is the product b 2 is the size of the set and 3 is the number of setsc 3 and 2 are factors of 6 while 6 is a multiple of 3 and 2

Every whole number except 0 has a finite number of whole number factor but all whole numbers have an infinite number of whole number multiples

Example 3 Set of factors of 6 = 1 2 3 6 set of multiples of 6 = 0 6 12 18

Activity 51B

2 Arrays

3 Area and Volume

Volume 1Prime

Activity 51C

Various Prisms

Various Cylinders

785 square inches

0 1 2 3 4 5 6 7 8 9

A A0 A1 A2 A3 A4 A5 A6 A7 A8 A9

B B0 B1 B2 B3 B4 B5 B6 B7 B8 B9

Z Z0 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

T1 T2 T3 T1 T2 T3

Activity 51D

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

Activity 51B

2 Arrays

3 Area and Volume

Volume 1Prime

Activity 51C

Various Prisms

Various Cylinders

785 square inches

0 1 2 3 4 5 6 7 8 9

A A0 A1 A2 A3 A4 A5 A6 A7 A8 A9

B B0 B1 B2 B3 B4 B5 B6 B7 B8 B9

Z Z0 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

T1 T2 T3 T1 T2 T3

Activity 51D

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

2 Arrays

3 Area and Volume

Volume 1Prime

Activity 51C

Various Prisms

Various Cylinders

785 square inches

0 1 2 3 4 5 6 7 8 9

A A0 A1 A2 A3 A4 A5 A6 A7 A8 A9

B B0 B1 B2 B3 B4 B5 B6 B7 B8 B9

Z Z0 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

T1 T2 T3 T1 T2 T3

Activity 51D

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

Volume 1Prime

Activity 51C

Various Prisms

Various Cylinders

785 square inches

0 1 2 3 4 5 6 7 8 9

A A0 A1 A2 A3 A4 A5 A6 A7 A8 A9

B B0 B1 B2 B3 B4 B5 B6 B7 B8 B9

Z Z0 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

T1 T2 T3 T1 T2 T3

Activity 51D

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

Volume 1Prime

Activity 51C

Various Prisms

Various Cylinders

785 square inches

0 1 2 3 4 5 6 7 8 9

A A0 A1 A2 A3 A4 A5 A6 A7 A8 A9

B B0 B1 B2 B3 B4 B5 B6 B7 B8 B9

Z Z0 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

T1 T2 T3 T1 T2 T3

Activity 51D

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

Various Prisms

Various Cylinders

785 square inches

0 1 2 3 4 5 6 7 8 9

A A0 A1 A2 A3 A4 A5 A6 A7 A8 A9

B B0 B1 B2 B3 B4 B5 B6 B7 B8 B9

Z Z0 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

T1 T2 T3 T1 T2 T3

Activity 51D

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

Various Prisms

Various Cylinders

785 square inches

0 1 2 3 4 5 6 7 8 9

A A0 A1 A2 A3 A4 A5 A6 A7 A8 A9

B B0 B1 B2 B3 B4 B5 B6 B7 B8 B9

Z Z0 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

T1 T2 T3 T1 T2 T3

Activity 51D

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

0 1 2 3 4 5 6 7 8 9

A A0 A1 A2 A3 A4 A5 A6 A7 A8 A9

B B0 B1 B2 B3 B4 B5 B6 B7 B8 B9

Z Z0 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

T1 T2 T3 T1 T2 T3

Activity 51D

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

_______________________________________________ _______________________________

_________________________________________________________ ____________________

Activity 52A

m parts

B B B B B B

Activity 52B

Activity 52C

Activity 52D

26 grapefruits

q Brsquos r

Summary

A Concepts

divisor

E Remainders

F Problem Solving

______________________________________________________________________________________ ______

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

2 Symbolically

Parts of Parts

45 of class 45 = 1215

|_ __|_ __|__ _|_ __| |

|__ _|_ _ _|_ _ _|_ _ _| |

23 of 1215 = 815

45 of the whole

Activity 53C

__________________________

a ___________________________ b ___________________________ c ___________________________

______________________________________ is 10 of __________________________________________

a b ________________ = 14 middot _____________________

c _________________ = 34 middot ______________________

F _______________________ F G

G _______________________ H

H __________________________

(1) is 100 of F

(2) is 9 of

(3) is ___________ of ____________________________________

a Described Part

b The Other Part

A Other Part

25 of B 75 of B

100 of B

= 075 middot $3495

= $2621 100 P

5 Expanding Amounts

a The Increase

amount

b The New Amount

B Increase

100 of B X of B

New Amount

N = (100 + X) of B

= 115 middot T = 115 middot $4000 115 T = $4600

A Basic Concepts

a b c d 66⅔B

(c) A is 16 of B2

a b c d

Activity 54A

________________

Activity 54B

New Church Hall

Old Church Hall

$4 $4 $4

Multiply by m

Base Amount

Divide by m

One part = ⅔ acre

Indirect Amounts

=gt P = $80 divide 08 = $100

=gt discount = $20

=gt 48400 = 102P

=gt P = 48400 divide 102 asymp 47500

A Basic Concepts

B Problem Solving

____________________________________________________________________________________________

Activity 55A

W 0 1 2 5 6 10

_______ ________ ________ _______ ________ ________

Ratio ________ ________ F G H

_______ ________ ________ _______ ________ ________

Ratio ________ ________

Set 2 _________

Volume of cube

CD = cd D

Solving Proportions

$5 2 bull $5

10 = 5 and 5 = x =gt x = 15 inches 6 3 3 9 6 9

P 100 60

Activity 55B

A Basic Concepts

a 4 = 12 b 6 = x c 48 = 24 d 15 = 10 5 x 9 3 150 x 6 x

C Problem Solving

_______________________________________________________________________ ______

Chapter 5

Activity 51A

darr darr darr

Activity 51B

2 Arrays

3 Area and Volume

Activity 51C

Various Cylinders

Activity 51D

Activity 52A

Activity 52B

Activity 52C

Activity 52D

Activity 53A

a ____________________________________ is 200 of ___________________________________________

a ____________________________________ is 110 of _____________________________________________

Activity 53B

B D B middot D

Activity 53C

b The New Amount

A Basic Concepts

a Repeated addition

a b c d

66⅔B

(b) B1 is 13 of B2 (c) A is 16 of B2

a b c d

Activity 54A

Activity 54B

money does Jim earn

A Basic Concepts

Activity 55A

Ratio ________ ________ F G H

Ratio ________ ________

Solving Proportions

Activity 55B

Documents

Chapter 5xqd339/DarkenChapter_05A.doc · Web viewChapter 5 Multiplication and Division I: Meaning 5.1 Multiplication as Repeated Addition Multiplication is not really a basic operation