Sampler - Mathematics in Context - Britannica

FormulasPerimeterandArea_6.epsDesigned to be used with Britannica’s
Mathematics in Context® or any
middle school mathematics curriculum as
• A supplement to your regular math program
• A program for extended time classes
• A tool for remediation and review
Problems for extra practice, further exploration,
and reinforcement of skills!
middle school mathematics curriculum:
middle school mathematics curriculum:
Middle School MathematicsCompanion
Designed to be used with Britannica’s Mathematics in Context® or any middle school mathematics curriculum: Problems for:
• Extra practice • Further exploration• Reinforcement of skills
Encyclopædia Britannica, Inc. 331 N. LaSalle Street
Chicago, IL 60654
This workbook contains samples from all three grade level workbooks
The perfect Companion for every mathematics student in the middle grades!
• One write-in student workbook each for grade levels 6, 7, and 8
• Organized by math topic to insure comprehensive coverage at each grade
• Multiple choice, extended response, and open response questions for every topic
• Spiral review questions in each section
• Special “Focus On” selected mathematics topics:
Absolute value, order of operations, comparing rational numbers, inequalities, formulas and equations, area, perimeter, and volume
• Correlated to state standards on request
Grade 6 ISBN: 978-1-60835-058-2 Grade 7 ISBN: 978-1-60835-059-9 Grade 8 ISBN: 978-1-60835-060-5
Middle School Mathematics
Companion Practice Workbook
Sampler
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No part of this work may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage or retrieval system, without permission in writing from the publisher.
ISBN 978-1-60835-087-2
1 2 3 4 5 C 13 12 11 10 09
Table of Contents Grade 6: Mathematical Models 1 Ratio Tables 2
Bar Models 4
Number Lines 6
Grade 7: Integers 15 Introduction to Integers 16
Locating Integers on the Number Line 18
Adding and Subtracting Integers 20
Multiplying Integers 22
Coordinate Pairs 24
Focus On: Division of Integers 26
Division of Integers Practice Problems 27
Grade 8: Linear Functions, Quadratics, and Factoring 29 Operating with Sequences 30
Slope 32
Focus On: Solving Equations 40
Solving Equations Practice Problems 41
For an answer key go to: http://info.eb.com/html/print_math_in_context.html. Scroll to the bottom of page and click on Sampler Companion Workbook Answer Key.
Grade 6 Mathematic al Models
1
Name Date
Marcia is baking banana muffins to sell at her basketball team’s fundraiser. This is her favorite banana muffin recipe.
1. If she wants to make 120 banana muffins, how can Marcia find out what amount of each ingredient she needs?
2. Marcia decides to make 96 banana muffins. How many eggs will she use?
A. 2 C. 8
B. 6 D. 12
3. The table below shows the cost for three different types of plants at a garden center.
Find the cost of the following orders:
a. 18 pepper plants
b. 24 tomato plants
c. 12 marigold plants
Banana Muffins (makes 24)
cup margarine, softened 1 cups mashed bananas 1 cup sugar 1 teaspoon baking soda 2 eggs 1 cups flour
Preheat the oven to 375° F. Cream margarine and sugar until smooth. Beat in eggs, then bananas. Add flour and baking soda, stirring until mixed. Fill muffin paper liners. Bake for 30 minutes.
1 2
1 2
1 2
Plant Cost
Pepper–6 plants $2.50
Tomato–3 plants $3.00
Marigold–24 plants $5.00
4. A grocery store buys cereal by the case. Last week’s delivery had 23 cases of cereal. Use the ratio table to calculate the number of boxes of cereal that the store received.
A. 156 boxes C. 240 boxes
B. 264 boxes D. 276 boxes
5. The ratio table below shows the price of different numbers of pizzas from Pizza Pizzazz.
Locate the price, in the table, that is incorrect. What is the correct price for that number of pizzas?
$12Price
Pizzas
Number of Boxes
Number of Cases
12 24 120
1 2 10
Math Content Students will use tables to find equivalent ratios and calculate ratio values.
Ratio Tables
6. A contractor is building a patio behind a house. For the floor, he uses the recipe below to mix concrete.
a. A cubic foot of sand weighs 90 pounds. Use the ratio table below to find the weight of sand needed to make 15 cubic yards of concrete.
b. Complete the ratio table for the number of cubic yards of concrete indicated.
Concrete (makes 1 cubic yard)
6 cubic feet cement 15 cubic feet sand 12 cubic feet gravel 3 cubic feet water
1 2
2 5 10
7. A store parking lot has 25 rows of parking spaces. Each row has the same number of spaces. There are 375 total parking spaces. Complete the table below to determine the number of spaces in each row.
Review 8. The pie chart below shows which field
trip idea students liked best. If 200 students were surveyed, estimate the number who chose the museum trip.
9. The bar graph below shows the number of cars sold at a dealership in one week. How many cars were sold on Wednesday?
Spaces
Ca rs
So ld
SatFriThuWedTueMonSun
Museum
Nature Center
Name Date
1. The bar model below is divided into equal parts. Label each part with a fraction.
2. What fraction is represented by the shaded portion of the bar model below? Express your answer in simplest form.
3. In which bar model does the shaded portion represent ?
A.
B.
C.
D.
4. Kim had a strip of 8 stamps. She used 6 of the stamps to mail letters.
a. Draw a bar model that shows the stamps she used as the shaded part of the model and the remaining stamps as the unshaded part.
b. What fraction of the strip of stamps does Kim have left?
1 4
5. A water storage tank at a factory has a gauge on the outside so that employees can estimate the amount of water in the tank.
a. What fraction of the tank is filled?
b. If a full tank holds 2,000 liters of water, how much water is in the tank?
6. A bar gauge on a large coffee maker shows the amount of coffee remaining. A full container holds 60 cups of coffee. Fill in the gauges to show the amounts indicated.
a. 45 cups b. 10 cups c. 30 cups
Math Content Students will use bar models to represent fractions, percentages, and decimals.
Bar Models
7. Kai is downloading a program onto his computer. He sees the following bar that shows the progress of the downloading.
If the total size of the program is 4.4 MB, what is the best estimate of the amount that has been downloaded?
A. 0.4 MB C. 1.6 MB
B. 1.1 MB D. 2.2 MB
8. Alana is downloading a program. After 20 seconds, she sees the screen below.
a. How can Alana figure out the size of the program?
b. Draw the bar model that will be on the screen after 40 seconds, if the program downloads at a constant rate.
Saving:
9. Use the bar model to calculate a 20% tip.
Review 10. The ratio table shows the number of
pencils in a given number of boxes.
If each box contains 15 pencils, which value in the ratio table is incorrect and what is the correct value?
11. Calculate the values and fill in the blank spaces in the ratio table.
$29.00
100%
$2.90
10%
Eyes
30 6015 120 200
Bar Models Name Date
Name Date
, , , ,
The sign at the beginning of a hiking trail shows the following distances. Use the sign to answer questions 2 and 3.
2. The number line below represents the trail. Label each of the locations on the number line.
3. If the trail is exactly 1 mi long, what fraction of the trail remains when you reach the stream crossing?
1 4
3 4
1 3
2 3
1 2
Y Z
4. In a track and field competition, four athletes throw a heavy iron shot. Their distances are shown in the table below.
a. Show the four throwing distances on the number line below. Label the distances with the first initial of the athlete and the distance.
b. Which two athletes had distances that were closest to each other?
c. What was the distance between the shortest throw and the longest throw?
12 13 14 15
Walters 12.2
Sanchez 13.2
Chen 12.8
Thomas 14.2
Math Content Students will use number lines to compare and order fractions and decimals.
Number Lines
5. Mark a point on this number line and label its value.
For questions 6 through 8, use jumps of 0.1 and 1 to “jump” between the points.
6. Jump from 1.2 to 2.1 in two jumps.
7. Jump from 0.8 to 3.6 in five jumps.
8. Jump from 1.2 to 2.9 in the fewest possible number of jumps.
10 2 3
10 2 3
10 2 3
10 2 3
Review 9. In the bar model below, what fraction is
represented by the shaded portion? Express your answer in simplest form.
10. A bar gauge on the computer shows that 1.5 MB of a program have been downloaded. The size of the program is 6.0 MB. Fill in the bar gauge to show the progress of the download.
Number Lines Name Date
Name Date
Use the map scale below to answer questions 1 and 2.
1. Julie estimated that the distance from the town where she lives to the town where her grandmother lives is 25 miles. Estimate the distance between the towns measured in kilometers.
2. About how long is 5 miles?
A. 3 km C. 8 km
B. 5 km D. 15 km
3. The downtown section of Hilldale is arranged in a regular grid. Each city block is mile long.
Ms. Casey’s class is going on a field trip to the museum. How many blocks will they have to walk from the school, which is at 1st Avenue and Main Street, to the museum at 6th Avenue and Lee Street? How many miles is that?
Museum
School
km
4. On a long bus trip, Jamal recorded the time the bus traveled to reach certain distances. He placed the data on the double number line below.
a. How could Jamal use his graph to find out how much time the bus will need to travel 400 miles?
b. What assumption does Jamal have to make in his calculation?
c. How much time will Jamal calculate for the 400-mile trip?
50 150100 250200
mi
Math Content Students will use double number lines to estimate and calculate ratios and to develop algorithms for fractions.
Double Number Lines
5. At the local market, mixed nuts cost $2.50 per kilogram. Make a double number line to show the cost of , 1, 1 , and 2 kilograms of nuts.
6. The double number line below compares centimeters to inches.
About how many inches is the same length as 20 centimeters?
A. 4 inches
B. 8 inches
C. 10 inches
D. 20 inches
7. This double number line compares centimeters and millimeters.
There are 100 centimeters in a meter. How many millimeters are there in a meter?
10 3020 50 6040
mm
in
1 2
1 2
8. At a hardware store, small nails cost $2.40 per kilogram. The double number line below shows the scale reading for one bag of nails.
a. Fill in the prices for full kilogram measures of nails on the double number line.
b. Use the double number line to find the cost of the bag of nails.
Review 9. The track team trains each day after
school for 1 hours. Fill in the ratio table below to show the number of hours after each number of days of training.
10. What fraction is represented by the shaded portion of the bar model below? Express your answer in simplest form.
1 2
Training Hours
Companion Practice Workbook, Grade 6 Mathematical Models 9
10 Mathematical Models Companion Practice Workbook, Grade 6
Name Date
1. Wendy is collecting eggs at her aunt’s farm. She places 12 eggs in each carton. Use the ratio table to determine the number of cartons she will need if there are 288 eggs collected.
2. In an archery class, four students shot arrows at a target. They measured the distance from the center of the target to each arrow. The results are shown in the data table.
a. On the number line below, show the distance from the center for each archer.
b. How much farther from center was Leanne’s arrow than Paolo’s arrow?
1 2 3 54 6 7
Eggs
Cartons
Amy 4.8
Paolo 5.9
Thomas 5.1
Leanne 6.5
3. On a camping trip, the nature club uses tents that can sleep 4 people. On the next weekend trip, there will be 52 campers. Use the double number line to determine how many tents will be needed.
4. Leta is making trail mix to take on the camping trip. She uses this recipe for her mix.
Complete the ratio table to determine the amount of ingredients for the number of bags shown in the table.
Raisins (cups)
2 cups raisins 3 cups nuts
cup pretzels 1 cups cereal1
2
tents0
0
campers
Math Content Students will use models to represent mathematical concepts.
Applications of Models
5. The centimeter ruler below can be used to model jumps between points.
a. Using jumps of 1 cm and 1 mm, what is the fewest number of jumps needed to go from 1.2 cm to 3.1 cm?
b. How many 1 cm jumps would be needed to go from 0 to 0.5 m?
6. Use the bar graph below to calculate a 15% tip on a $75 restaurant bill.
a. Calculate the values for 5% and 10%. Write the values on the bar model.
b. How can you use those values to calculate the 15% tip?
100%
$75.00
0.0
cm
7. At a school supply store, spiral notebooks cost $2.50 each.
a. Make a ratio table that can be used to calculate the cost of 24 notebooks.
b. Make a double number line that can be used to calculate the cost of 24 notebooks.
c. What is the cost of 24 notebooks?
Review 8. The table below shows the total number
of bolts for a given number of boxes of bolts.
How many bolts are there in 14 boxes?
9. A bar gauge on the computer shows that 3.5 MB of a program have been downloaded. The size of the program is 14.0 MB. Fill in the bar gauge to show the progress of the download.
Number of Bolts
Number of Boxes
Companion Practice Workbook, Grade 6 Mathematical Models 11
Comparing and Ordering Like Fractions Two fractions that have the same denominator are called like fractions. For example and are both fourths. To compare and order like fractions, compare the numerators as shown in the examples:
Comparing and Ordering Unlike Fractions If two fractions have different denominators, they must be converted to like fractions before they can be compared. This can be done by multiplying both the numerator and the denominator by the same value. For example, to convert to sixths, multiply by (which is equal to 1): × = .
Comparing and Ordering Decimals To compare and order decimal numbers, first compare the whole number portion: 4.5 > 3.4 > 2.4.
If the whole numbers are the same, compare the first number after the decimal point (tenths place). If that digit is the same, compare the second digit after the decimal point (hundreths place). Continue comparing place value until you reach a digit that has a higher number.
1 3
2 2
1 4
6 6
1 3
2 2
3 4
12 Mathematical Models Companion Practice Workbook, Grade 6
Name Date
Compare and
× = which is < ,
× = × = × =
< <
< < because 1 < 5 < 1111 12
5 12
1 12
11 12
1 12
5 12
Compare 3.675 and 3.657
The whole number and the first decimal digit are the same.
The second decimal digit of 3.675 is 7.
The second decimal digit of 3.657 is 5.
3.675 > 3.657
The first decimal digit of 4.449 is 4.
The first decimal digit of 4.542 and 4.548 is 5.
The second decimal digit of 4.542 and 4.548 is 4.
The third decimal digit of 4.548 is 8, while the third decimal digit of 4.542 is 2.
4.548 > 4.542 > 4.449
, , 1 , 1
2. Four swimmers finished a race in the times shown in the table below.
List the swimmers in order of their times, from shortest time to longest time.
3. Which of these numbers has the greatest value?
A. 2.619 B. 2.568 C. 2.564 D. 2.618
Swimmer Time (seconds)
4. Compare these pairs of numbers using the symbols <, =, or >.
a.
56.352, 56.061, 58.998, 56.115
6. Compare these pairs of numbers using the symbols <, =, or >.
a. 7.359 7.539
b. 45.23 46.08
c. 2.357 2.351
d. 0.056 0.23
, , ,
Math Content Students will compare and order rational numbers: fractions, decimals, and whole numbers.
Ordering Rational Numbers
8. After a party, Don compared the amount of pizza that was left over. The cheese pizza was cut into 10 slices and 4 pieces were left. The mushroom pizza was cut into 12 slices and 5 pieces were left over. Both pizzas were the same size.
a. Explain how Don can determine which pizza has the greater amount remaining.
b. Express the amount of each pizza remaining as a fraction of the whole pizza. Then simplify each fraction, if possible.
c. Express both fractions in an equivalent form using the same denominator.
d. Which pizza has the greater amount left over?
9. Order the following numbers from least to greatest:
, 2 , 2, ,
10. Which two decimals in the list below have the same value?
2.10 2.20 2.1 2.01
11. Which numbers in the list below have the same value?
6, 5 , 5 , 2 3
3 3
36 6
2 3
1 3
2 5
2 2
Name Date
Name Date
1. Miguel lives in San Francisco, California. His friend Lola lives in New York, New York. When he calls Lola at 4 P.M. his time, it is 7 P.M. her time.
a. What is the time difference between the two cities?
b. How do you know what time it is in San Francisco if you are given the time in New York?
c. What time is it in New York when it is 11 A.M. in San Francisco?
2. The surface of the Dead Sea has an elevation of 530 m below sea level. The elevation of sea level is written as 0, and a location with an elevation of 400 m above sea level is written as +400. How would you write the elevation of the Dead Sea?
A. 0
B. –400
C. –530
D. 530
3. In the Fahrenheit temperature scale, the freezing point of water is 32°F. Which of the following is true about the Fahrenheit temperature scale?
A. All positive temperatures are above freezing.
B. All negative temperatures are below freezing.
C. Some negative temperatures are warmer than 32°F.
D. Some negative temperatures are warmer than some positive temperatures.
4. Weather forecasters use temperature to help predict whether it will rain or snow. At temperatures below freezing, water turns to ice, and snow can form. At temperatures above freezing, water is a liquid, and it comes down as rain. In the Celsius temperature scale, the freezing point of water is 0°C. What can positive and negative Celsius temperatures tell you about the weather?
Math Content Students will understand and use positive and negative numbers in various situations and problems.
Introduction to Integers
5. Keilani works at a comic book store. She keeps track of the total number of comics in the store by noting changes in a chart. She uses positive numbers to note a delivery of new comics. She uses negative numbers to note the sale of comics from the store. The chart below shows her chart for Monday.
a. How many comics in all were deliv- ered on Monday? How many comics in all were sold? How did you come to these answers?
b. Were there more or fewer comics in the store at the end of the day than there were at the beginning of the day? How do you know?
c. Would Keilani use a positive or a negative number to describe the overall change in the number of comic books in the store over the whole day? Explain.
Time Number of Comics
10:20 A.M. +20
11:45 A.M. –7
1:35 P.M. –15
2:59 P.M. +120
3:17 P.M. –20
3:31 P.M. –10
4:52 P.M. –4
6. A deposit is when you add money to a savings account. A withdrawal is when you take away money from a savings account. How might you use positive and negative numbers to describe deposits and withdrawals?
Review 7. 3 × 1,000 + 2 × 100 + 3 × 1 + 5 × =
A. 3,235
B. 3,203.5
C. 3,231.5
D. 32,315
8. In 1998, chickens in the United States laid almost 80 billion eggs. How many dozen is that?
1 10
Companion Practice Workbook, Grade 7 Integers 17
18 Integers Companion Practice Workbook, Grade 7
Name Date
1. Make true statements using <, =, or > and write each statement in words.
a. –35 15
b. 200 –300
c. –43 –47
a.
b.
c.
d.
3. Below is a part of a number line with numbers ranging from –40 to 40. Fill in the boxes.
–40 40
ADD 40–20
4. Below is a part of a number line with numbers ranging from –20 to 20.
Which two points on the number line have a difference of 20?
A. A and B
B. A and C
C. B and C
D. C and D
5. A robot is located at point X on the number line below. The robot is given the following instructions: subtract 3, add 2, add –1, and subtract –1. After following the instructions in order, at what point on the number line is the robot located?
A. –4
B. –2
C. –1
D. 2
–4 –3 –2 –1
0 1 2 3 4
Math Content Students will compare, order, and solve problems using positive and negative numbers on number lines.
Locating Integers on the Number Line
6. A building has a ground floor called Level 0. There are 12 floors of offices above the ground floor that are called Levels 1–12. There are 3 floors of parking below the ground floor that are called (from bottom to top) Level –3, Level –2, and Level –1.
a. Draw a vertical number line to represent the building.
b. A delivery person parks on Level –2 and takes an elevator up 6 floors to make a delivery. At what level did the delivery person make the delivery? Show on your number line where the delivery was made.
c. Write the delivery scenario as an arithmetic problem using positive and negative integers.
7. For each statement below, say whether it is “always true,” “sometimes true,” or “never true.” Then, for each statement, give two examples that support your answer.
a. “A positive number is greater than another positive number.”
b. “A negative number is greater than a positive number.”
c. “A positive number is greater than a negative number.”
Review 8. Raul’s business experienced a net loss of
$30 on Monday, a net gain of $40 on Tuesday, and a net loss of $10 on Wednesday. What can you conclude about the total amount of money his business earned during the three days?
A. It experienced a net loss of $0.
B. It experienced a net loss of $10.
C. It experienced a net gain of $10.
D. It experienced a net gain of $80.
9. What does a negative number represent on the Celsius temperature scale?
Locating Integers on the Number Line Name Date
Name Date
a.
b.
a.
b.
+
+
+ +
5 – –4 =
–1 – 2 =
–2 + –5 =
8 + –3 =
5. Complete the arithmetic tree. If the sign is –, subtract the number above the sign on the right from the number above the sign on the left.
6. Suppose that it is currently 5°C outside. Which of these changes in the weather would result in a negative temperature?
A. The temperature gets 5 degrees colder.
B. The temperature gets 10 degrees colder.
C. The temperature gets 5 degrees warmer.
D. The temperature gets 10 degrees warmer.
+
–1014 –3
Math Content Students will solve problems involving addition and subtraction of positive and negative numbers.
Adding and Subtracting Integers
7. Look at the number line below.
Which kind of calculation would involve moving to the left on the number line?
A. adding zero
B. adding a positive number
C. subtracting a positive number
D. subtracting a negative number
8. Why is subtracting 5 the same as adding –5 on a number line? Why is subtracting –5 the same as adding 5? Write out each calculation in your answer.
9. In accounting, losses of money are often written down using red ink, while money earned is written down using black ink. This way, a business owner can tell just by glancing at a balance sheet whether the business is “in the red” (has a net loss) or if it is “in the black” (has a net profit). Imagine that a business has lost more money than it has earned in a week. Would the total sum for that week be written in red ink or black ink?
Does this total sum represent a positive or negative number? Explain.
–4 –3 –2 –1 0 1 2 3 4
10. A submarine rises and sinks to different depths underwater. Rising in depth is recorded as a positive number. Sinking in depth is recorded as a negative number. The chart below shows the movements of a submarine over the course of two hours.
If the submarine started out at a depth of –500 ft, what was its final depth after the two hours? Show your work.
Review 11. Is it possible for a number to not be
negative or positive? Explain.
12. Below is a part of a number line with numbers ranging from –8 to 8.
Which two points on the number line have a difference of 6?
A. A and B
B. A and D
C. B and C
D. B and D
Time Movement (in ft)
Name Date
1. Solve by rewriting each problem as an addition problem.
a. 100 × 4 =
b. –17 × 3 =
c. –30 × 6 =
3. Find each product.
a. 5 × –10 × 2 =
b. –3 × –1 × 8 =
c. –10 × –20 × –10 =
4. A number is multiplied by –1. The product is then subtracted from the original number. What can you conclude about the final answer?
A. It is zero.
D. It is double the original number.
5. Look at the double number line below.
Which multiplication statement corresponds to –2 on this number line?
A. –2 × 3 = –6
B. –2 × –3 = –6
C. –2 × –3 = 6
D. –2 × 6 = –12
6. A company says that it serves 47 million people every day. How many people is that every week? Write out the problem as an addition problem and as a multiplication problem.
–4 –3 –2 –1 0 1 2 3 4
12 9 6 3 0 –3 –6 –9 –12
Math Content Students will apply the rules for multiplying integers to solve problems involving multiplication of positive and negative numbers.
Multiplying Integers
7. Complete the multiplication tree.
8. a. Three negative numbers are multiplied together. Is the final answer positive or negative? Explain your reasoning.
b. Four negative numbers are multiplied together. Is the final answer positive or negative? Explain your reasoning.
×
×
× ×
3
9. A mountain climber starts his day at the top of a mountain at an altitude of 4,000 m. During his descent, his change in altitude averages about –150 m per hour. If he hikes for 8 hours, what is his total change in altitude? At what altitude is he located at the end of the 8-hour hike? Show your work.
Review 10. The diameter of the sun is about
1,391,000 km. What is this number in scientific notation?
A. 1.391 × 103
B. 1.391 × 104
C. 1.391 × 106
D. 1.391 × 107
+
+
+ +
Name Date
Use the coordinate plane below to answer questions 1 and 2.
1. Identify the coordinates for each point.
a. Point A:
b. Point B:
c. Point C:
d. Point D:
2. a. What is the name for the point at the very center of the coordinate plane, where the two number lines meet?
b. What are the coordinates for this point?
D
C
A
B
5
4
3
2
0
1
–1
–1 1 2 3 4 5–2–3–4–5
–2
–3
–4
–5
Use the coordinate plane below to answer questions 3 and 4.
3. Plot each point on the coordinate plane.
a. Point D: (1, 5)
b. Point E: (–2, 1)
c. Point F: (3, –3)
d. Point G: (–3, 3)
4. Add –2 to the first coordinate of each point and plot this new set of points. What do you observe?
5
4
3
2
0
1
–1
–1 1 2 3 4 5–2–3–4–5
–2
–3
–4
–5
Math Content Students will use standard notation for describing (x, y) coordinates, plot and label points on a coordinate system, and perform transformations on shapes in coordinate space.
Coordinate Pairs
5. Which of the following operations would cause a plotted drawing to shrink?
A. Add 3 to both coordinates of each point.
B. Subtract 3 from both coordinates of each point.
C. Multiply both coordinates of each point by 3.
D. Divide both coordinates of each point by 3.
6. Which of the following operations would cause a plotted drawing to rotate?
A. Add –1 to both coordinates of each point.
B. Add to both coordinates of each point.
C. Multiply both coordinates of each point by –1.
D. Multiply both coordinates of each point by .
7. a. What do the coordinates for all of the points along the y-axis have in common?
b. What does multiplying the first coordinate of each point by zero do to a plotted drawing?
1 2
1 2
between two adjacent hash marks 5?
A.
B.
C.
D.
9. How is a thermometer like a number line? What do the two objects have in common?
10. Complete the multiplication tree.
×
×
××
Name Date
Using a Picture You can use a number line or a grid to help you visualize a division problem.
Restating as a Multiplication Problem Multiplication and division are inverse operations. One way to look at a division problem is to rewrite it as a multiplication problem. Similarly, a multiplication problem can be rewritten as an addition problem.
Rules for Dividing Integers Two rules can tell you whether the answer to a division problem is positive or negative:
1. If the numbers have the same sign, then the answer is positive. 2. If the numbers have different signs, then the answer is negative.
Notice that these rules are exactly the same as the rules for multiplying integers.
Focus on: Division of Integers
Find –45 ÷ 9
–45 ÷ 9 = –5
The signs are the same, so the answer is positive.
= 11
Find the quotient 16 ÷ –2
It takes 8 arrows that are each 2 points long to cover 16 points on the number line moving left, negative.
16 ÷ –2 = –8
Find
A grid of 39 squares can be divided into 3 groups of 13.
= 339 13
39 13
–75 = (–25) + (–25) + (–25)
Find
136 = –1(–17 + –17 + –17 + –17 + –17 + –17 + –17 + –17)
–1 multiplied by the sum of eight –17s is 136.
136 = –17 × –8 = –8136 –17
136 –17
136 –17
Name Date
1. Solve by rewriting each problem as a multiplication problem.
a. –27 ÷ 9 =
b. –1 ÷ –1 =
a. =
b. =
–24 –8
75 –15
4. What can you conclude about the quotient of a negative number divided by a positive number?
A. It is zero.
B. It is an even number.
C. It is a positive number.
D. It is a negative number.
5. A number is divided by –1. The quotient is then added to the original number. What can you conclude about the final answer?
A. It is zero.
B. It is an even number.
C. It is a positive number.
D. It is a negative number.
6. A negative number is divided by a positive number, and the quotient is then divided by a negative number. Is the final answer positive or negative? Explain your reasoning.
Math Content Students will apply the rules for dividing integers to solve problems involving division of positive and negative numbers.
Division of Integers
7. Zelda has a booth at a craft fair. When her sales are greater than her expenses, she has a positive daily profit. When her expenses are greater than her sales, she has a negative daily profit. The table below shows her daily profit each day of the fair.
a. If her daily profit on Sunday is three times her daily profit on Thursday, what will be her total profits over the four days of the fair? Show your work.
b. What will be her average daily profit over the four days? Show your work.
8. The fraction is equal to the fraction Use the rul es for dividing integers to explain how this is possible.
–3 4
3 –4
Sunday ?
9. At 8 P.M., it is 0°C. The temperature drops by the same amount every hour for 8 hours, such that the temperature at 4 A.M., is –24°C. By how much did the temperature change each hour? Show your work by setting up a problem using division of integers.
10. The table below shows the low temperature for each day during one week in January.
What was the average daily low temperature that week? Show your work.
Day Low Temperature
Monday –14°C
Tuesday –8°C
Wednesday 2°C
Thursday 8°C
Friday 0°C
Saturday –10°C
Sunday –13°C
Division of Integers
Name Date
Name Date
1. How many smiley faces will the 100th figure have?

B. n+2 D. All of the above
3. The steps are equal. Fill in the missing numbers and expressions.
3
13
4. A sequence is represented by the expression –3n + 4.
a. What are the first three terms of the sequence?
b. What part of the expression makes the sequence decrease?
5. a. Fill in the missing numbers for the arithmetic sequence.
1, , 5, , , 11, …
b. Write the expression for the sequence.
c. Use diamonds () to make a visual pat- tern that corresponds to this sequence.
6. What is the sum of –4n – 3 and 6n + 8?
A. 9n
Operating with Sequences
a. (–9 + 6h) + (–4 – 2h) =
b. (4 – 2c) + = (–2 + 5c)
8. Fill in the missing numbers and expressions.
9. Rewrite the following expression to be as short as possible.
(8 + b) + b + (–2+ b) + (1 + 2b)
10. The American Civil War began in 1861, and World War II ended in 1945. How many years are between 1861 and 1945?
A. 84 C. 116
B. 106 D. 124
6 + 3x
18 180
4 – x
11. Let n be the year the United States entered World War II. The year the war started was (n – 2). The year the war ended was (n + 4). How many years long was the war?
12. What is the missing expression?
a. 6(–1 + 2x) =
Review 13. There are 20 students in Mrs. Xavier’s
class. She needed two helpers, so she ran- domly drew a name out of a hat and picked Michiko. Then, without replacing the name, she drew a second name. What is the probability that Shawn will be a helper?
14. A baseball player calculates that the probability of his hitting a ball when he is up to bat is 29%. About how many balls does he expect to hit if, during the season, he bats 42 times?
n-3
n-2
n-1
n
Companion Practice Workbook, Grade 8 Linear Functions, Quadratics, and Factoring 31
32 Linear Functions, Quadratics, and Factoring Companion Practice Workbook, Grade 8
Name Date
1. A swimming pool is 3 ft deep at the shal- low end. For each step Juanita takes towards the other end, the pool is about 0.25 ft deeper. Juanita records this information as the following equation.
D = 3 + 0.25S
a. What does the D in the formula stand for?
b. What does the S in the formula stand for?
c. Complete the following table that fits the formula D = 3 + 0.25S.
d. Use the table to draw a graph that fits the formula D = 3 + 0.25S.
S 0 1 2 3 6
D (in ft) 6.5
a. y = 6 – 2x
b. y = 2(3 – 2x)
c. If you graphed both equations on a coordinate system, how would the graphs compare?
3. Which of the following equations will not have a graph that is a straight line?
A. y = 8x C. y = 8x – 2
B. y = x D. y = 8x21 8
x y
Slope
4.
b. Why does one graph appear steeper than the other?
5. Find an equation of the straight line with x-intercept 3 and y-intercept 4.
1
1
b. What is the y-intercept?
c. What is the x-intercept?
d. Write the equation of the line.
Review 7. Which of the following expressions is
equivalent to 7(9 – 2d)?
A. 63 – 2d C. 7 × 9 + 7 × 2 + 7 × d
B. 16 – 9d D. 63 – 14d
8. What is (5f + 4) – (2f – 8)?
1 –1
Slope Name Date
Name Date
1. The chart shows the number of milks and orange juices bought during a 7-day fundraiser in Mr. Jackson’s class.
a. In the last column of the chart, complete the values of M + J.
b. Use a line graph to show the number of milks sold, and label the graph M.
c. Use a line graph to show the number of orange juices sold, and label the graph J.
d. Use a line graph to show M + J.
Day Milk (M) Orange Juice (J) M + J
1 5 10
2 2 9
3 3 6
4 8 2
5 9 4
6 1 8
7 4 8
10 2 3 4 5 6 7 8
2. You are on a boat at the lake. The boat is traveling at 36 km/hr pulling a skier. You walk from the back of the boat to the front of the boat at 6 km/hr.
The graph, B, of y = 36x represents the distance the boat travels, and the graph, P, of y = 6x represents the distance you travel each hour.
a. What is the equation for B + P?
b. What is the slope of the graph of B + P?
c. What does this slope represent?
3. Which equation would represent 2G?
A. y = 2x + 1 C. y = 4x + 1
B. y = 4x - 2 D. y = 2x – 2
G
2 3 4 5 6 7 8
Math Content Students will translate among different mathematical representations and make and interpret graphs in a coordinate system.
Adding Graphs
4. a. Using the graph of lines A and B below, draw the sum graph, A + B.
b. What points did the graphs of A and B have in common?
c. Does A + B have the same points? Why or why not?
5. Graphs F and M intersect at point (4, 5). Explain why the graph of (F + M) intersects F and M at (4, 5), too.
1 2
0 1
6. Graph A corresponds to y = x – 5.
Graph B corresponds to y = x + 3.
Which equation represents the graph of A + B?
A. y = 4x C. y = x + 8
B. y = x – 2 D. y = 4x – 2
7. Graph W corresponds to y = 6 – 9x. Write an equation that corresponds to W.
Review 8. Which equation represents the following
graph?
9. What is (–6d + 3) + (d – 10)?
3 4
1 3
1 4
7 8
Adding Graphs Name Date
Name Date
1. When using the cover method to solve the equation 5(x + 2) = 20, what is the value of x + 2?
A. 2 C. 5
B. 4 D. 20
2. Tariku babysits and calculates her fee by using the formula F = 5 + 8H.
a. What do you think F and H mean?
b. What is the meaning of each number in the formula?
c. Hosea also babysits, and he simply charges $10 per hour. Write an equation for Hosea’s fee.
d. Draw the graphs from both formulas. Label them A and H.
e. Your mom says that Hosea is more expensive than Tariku. What is your comment?
3. Cell phone company S charges $25 a month. Cell phone company T charges $20 a month plus $0.50 per call. The graph represents the charges for each company. What does the intersection point of the graphs represent?
4. Use the specified method to solve the equation.
48 + 6n = 24 – 2n
a. Balance Method
b. Difference-is-0 Method
c. Why do you get the same solution using either method?
0
5
10
15
20
25
30
35
2 4 6 8 10 12 14 16 18 20
T S
Solving Equations
4 + 3x = 3x + 10
b. What does the solution tell you about the graph?
Review 6. The table corresponds to a linear graph.
What is the slope of the graph?
A. 2
B. 3
C. 5
D. 10
x y
–3 –18
–1 –8
1 2
3 12
5 22
7. Let graph A be represented by the equation y = –2x + 6, and graph B be represented by the equation y = 3x – 4.
a. Write the equation that represents A + B.
b. Graph A, B, and A + B. Be sure to label each graph.
Solving Equations Name Date
Name Date
1. Three triangles are shown below.
a. For the perimeter P of the first trian- gle, the formula is P = 3a. Explain this formula.
b. What is the formula for perimeter Q?
c. What is the formula for perimeter R?
a a
b c
Perimeter = R
2. This is a cross figure. The sum of the lengths x and y is 10 feet.
What is the perimeter of the figure?
3. Which is the formula for the area of the figure?
A. A = w + 2z + xy
B. A = 2z + 2w + 2x
C. A = zw – xy
D. A = zw + xy
z
z
Math Content Students will write expressions and find area and perimeter.
Formulas for Perimeters and Areas
4. Use the picture to find the equivalent expressions.
A. (a + j)(m + n) = am + jm + an + jn
B. (a + j)(m + n) = am + jn
C. (a + j)(m + n) = a2 + jn + jm
D. (a + j)(m + n) = am2 + jn2 + an + jm
5. a. Draw a picture to show r(s + t).
b. Draw a picture to show rs + rt.
c. Explain why these expressions are equivalent.
d. Calculate rs + rt if r = 15 and s + t = 21.
a
m
n
j
Review 6.
Which part of the difference graph shows the point of intersection for A and B?
A. slope
C. x-intercept
D. y-intercept
7. A line has slope –5 and y-intercept of 120. What is the x-intercept?
20
40
20
Solving One-Step Equations To solve an equation, isolate the variable on one side of the equation. The Addition Property of Equality and the Multiplication Property of Equality state that you can add (or subtract) and multiply (or divide) each side of the equation by the same number or expression without chang- ing the solution. Always check your solution by substituting it into the original equation.
Solving Multi-Step Equations Some equations require more than one step to solve. For these equations, follow the steps below.
Step 1 Simplify Each Side If there are parentheses, use the Distribution Rule to write an equivalent expression. Rewrite the expressions on each side of the equation to be as short as possible.
Step 2 Gather All Variable Terms on One Side If there are variable terms on both sides of the equation, move one of the terms to the other side of the equation by adding or subtracting it from both sides. Rewrite the expressions on each side of the equation to be as short as possible.
Step 3 Isolate the Variable Add or subtract numeric terms so that the variable term is by itself on one side. Multiply or divide by the coefficient of the variable term to get an equation of the form “x = a number.” Simplify the resulting number, if necessary.
Step 4 Check the Answer Substitute the solution into the original equation and see if it works.
Focus On: Solving Equations
Name Date
x = –10.3
6(k – 4) – 2k = k + 9
6k – 24 – 2k = k + 9 (Step 1) 4k – 24 = k + 9
4k – k – 24 = k – k + 9 (Step 2) 3k – 24 = 9
3k – 24 + 24 = 9 + 24 (Step 3) 3k = 33
=
k = 11
6(11 – 4) – 2(11) = 11 + 9 (Step 4) 66 – 24 – 22 = 20
20 = 20 (check)
1. Solve x = –12.
2. Which step should you take to solve the equation x – 5.6 = 1.02?
A. Add 5.6 to each side.
B. Subtract 5.6 from each side.
C. Multiply each side by –5.6.
D. Divide each side by –5.6.
3. James has 6 times as many stamps as Bryah. Together they have 224 stamps.
a. Choose a variable to represent the number of stamps that Bryah has.
b. Write an expression for the number of stamps that James has. Use the same variable from part (a).
c. Write an equation for the total number of stamps that the boys have. Then solve the equation.
d. How many stamps does Bryah have?
e. How many stamps does James have?
3 4
–3(5p + 24) + 9 = 2(3 – 2p) – 12
5. What is the solution to the following equation?
16.3 – 7.2b = –8.18
B. b = 3.4 D. b = – 812 720
812 720
Solving Equations
6. A student completes several steps and comes up with the equation 5x = 2x. The student then divides each side by x, get- ting 5 = 2. He says that there is no solution. Solve the equation to show why the student was incorrect.
7. a. Solve 2(x + 3) + 4 = 2(x + 5)
b. What does the solution tell you? For what values of x is the equation true?
8. A friend tells you that the simplest way to solve the equation below is to multiply each side by 100.
0.05(q + 2) + 0.1q = 2
a. Show the equation that results from multiplying by 100.
b. Why is this a mathematically accept- able step?
c. Why might some see this strategy as helpful?
9. Which equation is not a step in solving the following equation?
19 – (2x + 3) = 2(x + 3) + x
A. 16 = 5x + 6
Solving Equations
Name Date

Documents

Sampler - Mathematics in Context - Britannica