Math3 5 Extras

8/3/2019 Math3 5 Extras

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Topics 1 - 6

1) Multiplication I dont want to add that over and over again.

2) Division Hey, I need to share these

3) Remainders How many 4-passenger cars to get all of youto The Avenue?

4) Factors, Multiples and Divisibility How many socks shouldbe in the dryer?

5) Fractions and Equivalent Fractions How much of thatchocolate bar did I get?

6) Fractions/Decimals/Percentages A dollar can do that!


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Multiplication


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Applications of multiplication

Quickly find the answer to adding thesame number repeatedly

Finding the area of a rectangle Calculate the total for a bill, where

multiples of several items are purchased

Scaling up proportionally, such asdoubling a recipe

Converting between units


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Sample Problem

What is the area of a room that is 10ftlong and 13 ft wide?

Area A= length * widthA=10ft x 13 ftA=130 ft2


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Important Characteristics

Multiplication by Zero

Multiplication by 1

Commutative Property

Associative Property

Distributive Property

N*0=0 3*0=0

X*1=X 3*1=3

A*B=B*A 3*2=2*3

A*(B*C)=(A*B)*C 3*(2*4)=(3*2)*4

F*(G+H)=(F*G)+(F*H)

5*(17)=5*(10+7)=(5*10)+(5*7)=50+35


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Special Application: Exponents

Finding the area of a squareWhen we multiply an number by itself,

we can use exponents the smallnumber tells how many time the numberis multiplied.The area of a square with side 7m is:7m*7m = 72m2 = 49m2


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Division

(no Remainder)


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Applications of Division

Instead of subtracting repeatedly, we divide

Distribute/Share/Sub-divide into equal portions

Determine how much each person gets whenthings are divided in proportion

Scaling down proportionally, such asrepresenting a country on map

Converting between units


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Division Example Using

Proportions

Abe, Ron and Ben share 4 pizzas during the superbowl. Eachpizza has 6 slices. Each pizza has Each time they fill their plates

Abe takes 1 slice, Ron takes 3 slices and Ben takes 4 slices, untilthe pizza is done. How many slices does each person take?

The pizza is consumed in proportion 1:3:4.Each time that they fill their plates they take a total of 1+3+4=8slices.There are 4*6=24 slices available.Number of times they can fill their plates: 24/8=3

Abe takes 3 *1 = 3 slices

Ron takes 3*3 = 9 slicesBen takes 3*4 = 12 slices(Check : 3+9+12 = 24 slices of pizza)


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Division

(with Remainder)


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Division with Remainders: Uses

Ignore (or recycle) the remainder if itdoesnt affect the outcome

Account for the need for an extra item, ifit does affect the outcome (fitting theremaining people in a golf cart after allthe others are full)

Finding the item in a certain place in arepeating pattern


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Division, With Remainder

Example

The swim team wears a different swim cap each practice.The order is always: red, orange, yellow, green, blue, indigoand violet. What color will the caps be on the 26 th day?

The pattern repeats every 7th practice:ROYGBIVROYGBIVROY.So find the remainder when 26 is divided by 7: 267 = 3 R 5

Count to the 5th place in the pattern and thats the answer,so the Blue cap!

(Note that if the remainder is zero, then the answer is thelast item in the pattern).


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Primes, Factors, Multiples

and Divisibility


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Primes, Factors and Multiples

A number that can only be divided by 1 anditself is called a prime number. In other words,a prime number has only 1 and itself as

factors. The first prime (and only even prime) istherefore 2. When two factors are multiplied, they produce

a multiple All multiples can be expressed as the product

of prime numbers and 1. Every whole number has 1 and itself as

factors.


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Factors and Primes - Application

There are 36 peoplecoming to the familyreunion. How many tablesshould I order?It depends. The factors of36 are 1,2,3,4,6,9,12,18,36.We probably dont want 36tables with a single personseated at each table, but alloff the other numbers mightwork. For example 2 tableswith 18 people each or 4tables with 9 each, etc.

Graphic: National Library of Virtual Manipulatives (Factor Tree) http://nlvm.usu.edu


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Divisibility Rules All multiples of 2 are even numbers. All multiples of 3 have digits that add up to a multiple of 3. All multiples of 4 have multiples of 4 in the last 2 digits (including 00). All multiples of 5 end in 0 or 5. All multiples of 6 are multiples of both 2 AND 3. All multiples of 7: take each digit from the units (ones) place and going

from right to left, multiply by 1,3,2,6,4,5 (repeat the pattern as often asnecessary). Add the products together the number is a multiple of 7,if this sum leaves no remainder when divided by 7. Long division maybe quicker.

All multiples of 8 have multiples of 8 in the last 3 digits (long division

sometimes faster than this test). All multiples of 9 have digits that add up to a multiple of 9. All multiples of 10 end in 0.


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Greatest Common Factor (GCF)

The biggest factor that 2 or more numbers havein commonI have 35 roses and 20 carnations, what is thelargest number of bouquets that I can make, ifeach buoquet must have the same number ofeach kind of flower?I need to find the GCF of 35 and 20:20=4x5=22x5 and 35=5x7The largest number that 20 and 35 have incommon is 5. I can make 5 buoquets, each with7 roses and 4 carnations.


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Least Common Multiple (LCM)

The smallest number that is divisible by the numbers forwhich I must find the LCM.

A package has 12 hotdogs and each bag has 8 buns. How

many packages and bags must I buy in order to have abun for each hotdog and spend as little as possible?

I must find the LCM of 8 and 12. First I must factorize eachnumber: 8=2x2x2 and 12=2x2x3. Find the maximumnumber each prime occurs in each of the two factors, so3 twos (2x2x2) and 1 three (3). Multiply these together to

get the LCM: 2x2x2x3 = 24. I need 24 hotdogs (2packages of 12 hotdogs) and 24 buns (3 packages of 8buns).


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Fractions and

Equivalent Fractions


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Fractions and Equivalent

Fractions

Answers are typically given in the simplest or smallestfraction.

Equivalent Fractions: Whatever you do to the numerator,you must do to the denominator.

When ordering, adding, or subtracting fractions, we needthem to have the same denominator, so we must findequivalent fractions

When multiplying or dividing fractions its often easiest totry to simplify or find an equivalent fraction that makes thecalculations easier.

Use the LCM of the denominators to find the newdenominator for all the fractions. We then call this theLeast Common Denominator (LCD).


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Equivalent Fractions example

Sammy the slug traveled 3/4m, 1/3m, 5/8m, 2/9m and6/7m to find his meals today. How far did he traveltoday?

3 and 7 are prime, 4=2x2, 9=3x3, 8=2x2x2LCM=2x2x2x3x3x7 = 504

Total distance =

504

378

7332

7332

22

3

4

3

504

168

73222

73222

3

1

3

1

504

315

733

733

222

5

8

5

504

432

332*2*2

332*2*2

7

6

7

6

504

112

7222

7222

3*3

2

9

2

m5043972

504

1405

504

432

504

112

504

315

504

168

504

378


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Fractions, Decimals and

Percentages

Hint: $1.00


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Applications

Simple interest

Sales tax

Sales price markup and discount(percent increase and percent decrease)


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A Dollar - Decimals

Were used to adding money line up thedecimal point and add the numbers that fall in

the same column. Keep the decimal point inthe same place in the answer. A tip that helpswith addition and subtraction is to fill in theempty spaces with zero.

Find the sum of 103.275, 7.09 and 0.9876.


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A Dollar - Percentages A dollar has 100 pennies. When we express the number of pennies that we

have as a fraction we are actually writing a percentage. If we have 65 cents,we have 65/100 pennies or 65% of a dollar. When we express any fraction asan equivalent fraction with 100 in the denominator, we are writing a percentage.

A whole is 100% (100 cents of the dollar), 200% would be 2 wholes.

When finding a percentage ofa number, write the percentage as a fraction,then multiply the number by percentage. Simplify the fractions before findingthe final answer.

Write 1/5, 3/10, 7/40 and 230/1000 as percentages.

.

%23100

23

101000

10230

1000

230

%5.17100

5.17

5.2

5.2*

40

7

%30100

30

10

10*

10

3

%20100

20

20

20*

5

1


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A Dollar - Fractions A quarter (25cents) and a half dollar (50cents) are fractions that we

use all the time. With fractions, the keyword of tells us to multiply. Convert whole

numbers to fractions, by dividing them by 1.

When multiplying fractions, first simplify, by canceling any number inthe denominator by any number in the numerator.What is 5/6 of 30? What is 1/7 of $21.35?

.

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Math3 5 Extras