Sampler - Encyclopædia Britannica Mathematics in Context

Transcription

S
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School M
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Practice Workbook
ica’s
tann
Bri dle School Mathematic
s
Mid
an
Comp ion k
boo
Practice Work
Grade 6
nnica’s
used with Brita
Designed to be
any
in Context® or
Mathematics
curriculum:
mathematics
middle school
Problems for:
• Extra practice
oration
• Further expl
t of skills
• Reinforcemen
annica’s
Brit School Ma
dle
Mid
them
atic
s
Companion
Practice Workbook
Grade 7
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
nnica’s
Brita
le School Ma
them
Midd
atic
s
Companion
Practice Work
book
Grade 8
Designed to be
used with Brita
nnica’s
Mathematics
in Context® or
any
middle school
mathematics
curriculum:
Problems for:
• Extra practice
• Further explorat
ion
• Reinforcemen
t of skills
Designed 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!
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
Encyclopædia Britannica, Inc.
331 N. LaSalle Street
Chicago, IL 60654
To learn more, call 1-800-344-9629
or visit mathincontext.eb.com
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
Encyclopædia Britannica, Inc.
Chicago • London • New Delhi • Paris • Seoul • Sydney • Taipei • Tokyo
© 2010 Encyclopædia Britannica, Inc. Britannica, Encyclopædia Britannica,
and the thistle logo are registered trademarks of Encyclopædia Britannica, Inc.
All rights reserved.
No part of this work may be reproduced or utilized in any form or by any
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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
Bar Models
2
4
Number Lines
6
Double Number Lines
8
Applications of Models
10
Focus On: Ordering Rational Numbers
12
Ordering Rational Numbers Practice Problems
13
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
Adding Graphs
34
Solving Equations
36
Formulas for Perimeters and Areas
38
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
Mathematical Models
1
Name
Date
Ratio Tables
Math Content
Students will use tables to find equivalent ratios and calculate ratio values.
Marcia is baking banana muffins to sell at
her basketball team’s fundraiser. This is her
favorite banana muffin recipe.
Banana Muffins (makes 24)
1
2
cup margarine, softened
1 cup sugar
2 eggs
1 12 cups mashed bananas
1 teaspoon baking soda
1 12 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.
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.
Number of Cases
1
2
10
Number of Boxes
12
24
120
A. 156 boxes
B. 264 boxes
C. 240 boxes
D. 276 boxes
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
B. 6
C. 8
D. 12
3. The table below shows the cost for three
different types of plants at a garden center.
Plant
Cost
Pepper–6 plants
$2.50
Tomato–3 plants
$3.00
Marigold–24 plants
$5.00
5. The ratio table below shows the price of
different numbers of pizzas from Pizza
Pizzazz.
Pizzas
1
2
5
10
12
Price
$12
$24
$72
$120
$144
Locate the price, in the table, that is
incorrect. What is the correct price for
that number of pizzas?
Find the cost of the following orders:
a. 18 pepper plants
b. 24 tomato plants
c. 12 marigold plants
2 Mathematical Models
Companion Practice Workbook, Grade 6
Name
Date
Ratio Tables
6. A contractor is building a patio behind a
house. For the floor, he uses the recipe
below to mix concrete.
Concrete (makes 1 cubic yard)
6 cubic feet cement
15 cubic feet sand
12 12 cubic feet gravel
3 cubic feet water
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.
Rows
1
5
10
25
Spaces
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.
Sand (cu ft)
1
2
5
10
Weight (lb)
90
180
450
900
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.
Where to Take a Field Trip
Museum
Theater
Nature
Center
Concert
b. Complete the ratio table for the number
of cubic yards of concrete indicated.
Concrete (cu yd)
1
Cement (cu ft)
2
3
4
9. The bar graph below shows the number of
cars sold at a dealership in one week.
How many cars were sold on Wednesday?
Sand (cu ft)
Cars Sold in One Week
Water (cu ft)
Cars Sold
Gravel (cu ft)
18
16
14
12
10
8
6
4
2
0
Sun Mon Tue Wed Thu Fri Sat
Day
Mathematical Models
3
Name
Date
Bar Models
Math Content
Students will use bar models to represent fractions, percentages, and decimals.
1. The bar model below is divided into equal
parts. Label each part with a fraction.
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.
2. What fraction is represented by the
shaded portion of the bar model below?
Express your answer in simplest form.
a. What fraction of the tank is filled?
3. In which bar model does the shaded
portion represent 14 ?
b. If a full tank holds 2,000 liters of
water, how much water is in the tank?
A.
B.
C.
D.
4. Kim had a strip of 8 stamps. She used
6 of the stamps to mail letters.
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. 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?
a. 45 cups
b. 10 cups
c. 30 cups
Name
Date
Bar Models
7. Kai is downloading a program onto his
computer. He sees the following bar that
shows the progress of the downloading.
9. Use the bar model to calculate a 20% tip.
$2.90
$29.00
10%
100%
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
B. 1.1 MB
C. 1.6 MB
D. 2.2 MB
Review
10. The ratio table shows the number of
pencils in a given number of boxes.
8. Alana is downloading a program. After
20 seconds, she sees the screen below.
Boxes
1
2
4
8
16
Pencils
15
30
60
120
200
If each box contains 15 pencils, which
value in the ratio table is incorrect and
what is the correct value?
Saving:
30%
3.0 MB downloaded
a. How can Alana figure out the size of
the program?
11. Calculate the values and fill in the blank
spaces in the ratio table.
b. Draw the bar model that will be on the
screen after 40 seconds, if the program
downloads at a constant rate.
Number of Cows
1
Eyes
2
Ears
2
Feet
4
Stomach chamber
4
10
30
50
Mathematical Models
5
Name
Date
Number Lines
Math Content
Students will use number lines to compare and order fractions and decimals.
1. Use the fraction strips below to make a
number line that shows these fractions:
1,
4
3,
4
1,
3
2,
3
1
2
4. In a track and field competition, four
athletes throw a heavy iron shot. Their
distances are shown in the table below.
Shot Put Distances
The sign at the beginning of a hiking trail
shows the following distances. Use the sign
to answer questions 2 and 3.
Old Cabin
1
2 mi
Overlook
1
3
mi
Stream Crossing
3
4
mi
X
Y
Z
beginning of trail
Distance (m)
Walters
12.2
Sanchez
13.2
Chen
12.8
Thomas
14.2
a. Show the four throwing distances on
the number line below. Label the
distances with the first initial of the
athlete and the distance.
12
2. The number line below represents the
trail. Label each of the locations on the
number line.
Athlete
13
14
15
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?
3. If the trail is exactly 1 mi long, what fraction of the trail remains when you reach
the stream crossing?
Name
Date
Number Lines
5. Mark a point on this number line and
label its value.
0
1
2
Review
9. In the bar model below, what fraction is
represented by the shaded portion?
3
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.
0
1
2
3
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.
7. Jump from 0.8 to 3.6 in five jumps.
0
1
2
3
8. Jump from 1.2 to 2.9 in the fewest
possible number of jumps.
0
1
2
3
Mathematical Models
7
Name
Date
Double Number Lines
Math Content
Students will use double number lines to estimate and calculate ratios and to develop algorithms
for fractions.
Use the map scale below to answer questions
1 and 2.
0
10
10
20
20
30
40
30
mi
50
km
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.
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.
0
1
2
3
4
5
hr
0
50
100
150
200
250
mi
a. How could Jamal use his graph to find
out how much time the bus will need
to travel 400 miles?
2. About how long is 5 miles?
A. 3 km
B. 5 km
C. 8 km
D. 15 km
3. The downtown section of Hilldale is
arranged in a regular grid. Each city
block is 101 mile long.
b. What assumption does Jamal have to
make in his calculation?
Museum
Lee St.
Main St.
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
3rd Ave
2nd Ave
1st Ave
School
c. How much time will Jamal calculate
for the 400-mile trip?
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?
Name
Date
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 12 , 1, 1 12 , and
2 kilograms of nuts.
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.
0
6. The double number line below compares
centimeters to inches.
1
2
3
4
1
5
6
8
7
2
2
4 kg
3
a. Fill in the prices for full kilogram
measures of nails on the double
number line.
cm
in
3
1
About how many inches is the same
length as 20 centimeters?
b. Use the double number line to find the
cost of the bag of nails.
A. 4 inches
B. 8 inches
C. 10 inches
D. 20 inches
Review
9. The track team trains each day after
school for 112 hours. Fill in the ratio table
below to show the number of hours after
each number of days of training.
7. This double number line compares
centimeters and millimeters.
Days
0
1
2
3
4
5
6
cm
0
10
20
30
40
50
60
mm
2
3
5
10
12
Training Hours
There are 100 centimeters in a meter.
How many millimeters are there in a
meter?
10. What fraction is represented by the
shaded portion of the bar model below?
Mathematical Models
9
Name
Date
Math Content
Students will use models to represent mathematical concepts.
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.
Cartons
1
2
4
10
Eggs
12
24
48
120
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.
0
0
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.
2
10
Distance (cm)
Amy
4.8
Paolo
5.9
Thomas
5.1
Leanne
6.5
6
20
8
30
10
12
40
14
50
tents
60
campers
4. Leta is making trail mix to take on the
camping trip. She uses this recipe for her
mix.
Trail Mix (makes 8 bags)
2 cups raisins
3 cups nuts
3
4 cup pretzels
1 12 cups cereal
Distance from Center
Archer
4
Complete the ratio table to determine the
amount of ingredients for the number of
bags shown in the table.
a. On the number line below, show the
distance from the center for each archer.
Bags of Trail Mix
8
16
24
40
Raisins (cups)
1
2
3
4
5
6
b. How much farther from center was
Leanne’s arrow than Paolo’s arrow?
7
Nuts (cups)
Pretzels (cups)
Cereal (cups)
Name
Date
5. The centimeter ruler below can be used
to model jumps between points.
a. Make a ratio table that can be used to
calculate the cost of 24 notebooks.
0.0
cm
7. At a school supply store, spiral notebooks
cost $2.50 each.
1
2
3
4
5
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. Make a double number line that
can be used to calculate the cost of
24 notebooks.
b. How many 1 cm jumps would be
needed to go from 0 to 0.5 m?
c. What is the cost of 24 notebooks?
6. Use the bar graph below to calculate a
15% tip on a $75 restaurant bill.
Review
8. The table below shows the total number
of bolts for a given number of boxes of
bolts.
$75.00
Number of Boxes
Number of Bolts
5% 10%
12
480
14
?
100%
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?
2
80
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.
Mathematical Models
11
Name
Date
Focus On: Ordering Rational Numbers
Comparing and Ordering Like Fractions
Two fractions that have the same denominator are called like fractions. For example 14 and 34 are
both fourths. To compare and order like fractions, compare the numerators as shown in the
examples:
Compare 3 and 7
8
8
3 < 7 because 3 < 7
8 8
Order 5 , 1 , and 11 from least to greatest
12 12
12
1 < 5 < 11 because 1 < 5 < 11
12 12 12
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 13 to sixths, multiply by 22 (which is equal to 1): 13 × 22 = 66 .
Compare 1 and 7
2
12
1 × 6 = 6 which is < 7 ,
2 6 12
12
so 1 < 7
2
12
Order 3 , 2 , and 7 from least to greatest
4 5
10
3 × 5 = 15
7 × 2 = 14
2 × 4 = 8
4
5
20
5
4
20 10
2
20
2 < 7 < 3
5
10
4
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.
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
Order 4.542, 4.548, and 4.449, from greatest to least.
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
Name
Date
Ordering Rational Numbers
Math Content
Students will compare and order rational numbers: fractions, decimals, and whole numbers.
1. Order these numbers from least to
greatest:
5, 3, 11, 1
8
8
8
2. Four swimmers finished a race in the
times shown in the table below.
Swimmer
Time (seconds)
Brandi
41.56
Athena
40.87
Jin
41.23
Frances
41.28
List the swimmers in order of their times,
from shortest time to longest time.
4. Compare these pairs of numbers using
the symbols <, =, or >.
a. 7
8
b. 1 1
3
c. 5
4
d. 24
19
13
16
2
4
5
15
19
5. Order the following numbers from
greatest to least:
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
7. Locate the following fractions on the
number line below:
3. Which of these numbers has the greatest
value?
A.
B.
C.
D.
3,
8
3,
4
1,
2
1
4
2.619
2.568
2.564
2.618
0
1
Mathematical Models
13
Name
Date
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.
9. Order the following numbers from least
to greatest:
2,
3
21,
3
2,
2
2
2,
5
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.
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,
52,
3
53,
3
36
6
d. Which pizza has the greater amount
left over?
Grade 7
Integers
15
Name
Date
Math Content
Students will understand and use positive and negative numbers in various situations and problems.
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.
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. What is the time difference between
the two cities?
A. All positive temperatures are above
freezing.
B. All negative temperatures are below
freezing.
b. How do you know what time it is in
San Francisco if you are given the
time in New York?
C. Some negative temperatures are
warmer than 32°F.
D. Some negative temperatures are
warmer than some positive
temperatures.
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?
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?
A. 0
B. –400
C. –530
D. 530
16 Integers
Name
Date
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.
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
a. How many comics in all were delivered on Monday? How many comics
in all were sold? How did you come to
these answers?
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 ×
1
10
=
A. 3,235
B. 3,203.5
C. 3,231.5
D. 32,315
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?
8. In 1998, chickens in the United States
laid almost 80 billion eggs. How many
dozen is that?
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.
Integers
17
Name
Date
Math Content
Students will compare, order, and solve problems using positive and negative numbers on
number lines.
1. Make true statements using <, =, or >
and write each statement in words.
a. –35
15
b. 200
–300
4. Below is a part of a number line with
numbers ranging from –20 to 20.
–20
A
B
0
C
D
20
Which two points on the number line
have a difference of 20?
A. A and B
c. –43
B. A and C
–47
C. B and C
D. C and D
2. Complete the following lines.
a.
–20
ADD 40
30
SUBTRACT 45
–20
ADD _
0
ADD 60
–30
b.
c.
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?
–4
–3
–2
–1
0
1
2
3
4
x
numbers ranging from –40 to 40. Fill in
the boxes.
–40
18 Integers
40
A. –4
B. –2
C. –1
D. 2
Name
Date
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.
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.”
a. Draw a vertical number line to represent the building.
b. “A negative number is greater than a
positive number.”
c. “A positive number is greater than a
negative number.”
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.
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.
c. Write the delivery scenario as an
arithmetic problem using positive
and negative integers.
D. It experienced a net gain of $80.
9. What does a negative number represent
on the Celsius temperature scale?
Integers
19
Name
Date
Math Content
Students will solve problems involving addition and subtraction of positive and negative numbers.
1. Complete each addition calculation.
a.
b.
8
+
–3
=
–2
+
–5
=
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.
–3
14
–10
+
−
2. Complete each subtraction calculation.
a.
–1
–
=
2
−
–2
b.
–
5
–4
=
21
+
−
3. Complete each calculation.
a. 0 – 3 =
−
b. 6 – (–10) =
c. –9 + 8 =
d. –1 – (–1) =
4. Complete the arithmetic tree.
–4
6. Suppose that it is currently 5°C outside.
Which of these changes in the weather
would result in a negative temperature?
3
+
A. The temperature gets 5 degrees colder.
–2
6
+
+
B. The temperature gets 10 degrees
colder.
C. The temperature gets 5 degrees warmer.
D. The temperature gets 10 degrees
warmer.
+
20 Integers
Name
Date
7. Look at the number line below.
–4
–3
–2
–1
0
1
2
3
4
Which kind of calculation would involve
moving to the left on the number line?
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.
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.
Time
Movement (in ft)
10:00 A.M.
–100
10:15 A.M.
+25
10:18 A.M.
+200
10:45 A.M.
–150
10:59 A.M.
–75
11:07 A.M.
–100
11:52 A.M.
+120
If the submarine started out at a depth of
–500 ft, what was its final depth after the
two hours? Show your work.
Review
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.
11. Is it possible for a number to not be
negative or positive? Explain.
numbers ranging from –8 to 8.
–8
A
B
0
C
D
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
Integers
21
Name
Date
Math Content
Students will apply the rules for multiplying integers to solve problems involving multiplication
of positive and negative numbers.
1. Solve by rewriting each problem as an
addition problem.
a. 100 × 4 =
b. –17 × 3 =
c. –30 × 6 =
5. Look at the double number line below.
–4
–3
–2
–1
0
1
2
3
4
12
9
6
3
0
–3
–6
–9
–12
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
2. Find each product.
a. –28 × 5 =
b. –102 × –11 =
3. Find each product.
a. 5 × –10 × 2 =
b. –3 × –1 × 8 =
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.
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.
B. It is a positive number.
C. It is a negative number.
D. It is double the original number.
22 Integers
Name
Date
7. Complete the multiplication tree.
–1
3
×
–1
–4
×
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.
×
×
8. a. Three negative numbers are multiplied
together. Is the final answer positive
or negative? Explain your reasoning.
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
b. Four negative numbers are multiplied
together. Is the final answer positive or
negative? Explain your reasoning.
D. 1.391 × 107
11. Complete the arithmetic tree.
–3
c. Use your answers to the questions
above to develop a rule for multiplying negative numbers. Is the product
of an even number of negative numbers positive or negative? How about
the product of an odd number of
negative numbers?
8
+
–12
10
+
+
+
Integers
23
Name
Date
Coordinate Pairs
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.
Use the coordinate plane below to answer
questions 1 and 2.
C
5
5
4
4
A
3
–5
–4
–3
–2
–1
3
2
2
1
1
0
–1
D
Use the coordinate plane below to answer
questions 3 and 4.
1
2
3
4
5
–5
–4
–3
–2
–1
0
–2
–2
–3
–3
–4
–4
–5
–5
3
4
5
3. Plot each point on the coordinate plane.
a. Point A:
a. Point D: (1, 5)
b. Point B:
b. Point E: (–2, 1)
c. Point C:
c. Point F: (3, –3)
d. Point D:
d. Point G: (–3, 3)
2. a. What is the name for the point at the
very center of the coordinate plane,
where the two number lines meet?
2
–1
B
1. Identify the coordinates for each point.
1
4. Add –2 to the first coordinate of each
point and plot this new set of points.
What do you observe?
b. What are the coordinates for this
point?
24 Integers
Name
Date
Coordinate Pairs
5. Which of the following operations would
cause a plotted drawing to shrink?
A. Add 3 to both coordinates of each
point.
Review
8. For which number line is the distance
between two adjacent hash marks 5?
A.
–25
B. Subtract 3 from both coordinates of
each point.
B.
–25
C. Multiply both coordinates of each
point by 3.
C.
–20
D. Divide both coordinates of each point
by 3.
D.
–20
6. Which of the following operations would
cause a plotted drawing to rotate?
0
0
25
0
0
20
9. How is a thermometer like a number
line? What do the two objects have in
common?
A. Add –1 to both coordinates of each
point.
B. Add 12 to both coordinates of each
point.
C. Multiply both coordinates of each
point by –1.
D. Multiply both coordinates of each
point by 12 .
10. Complete the multiplication tree.
7. a. What do the coordinates for all of
the points along the y-axis have in
common?
-3
-2
×
-1
3
b. What does multiplying the first
coordinate of each point by zero do
to a plotted drawing?
×
×
×
Integers
25
Name
Date
Focus on: Division of Integers
Using a Picture
You can use a number line or a grid to help you visualize a division problem.
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 39
13
A grid of 39 squares can be divided into 3 groups of 13.
39 = 3
13
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.
What is –75 ÷ –25?
–75 ÷ –25 = ? is the same as
–75 = –25 × ?
–75 = (–25) + (–25) + (–25)
There are three (–25)s in –75.
–75 = –25 × 3
–75 ÷ –25 = 3
Find 136
–17
136 = ? is the same as 136 = –17 × ?
–17
136 = –1(–17 + –17 + –17 + –17 + –17 + –17 + –17 + –17)
–1 multiplied by the sum of eight –17s is 136.
136 = –17 × –8
136 = –8
–17
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.
Find –45 ÷ 9
The signs differ, so the answer is negative.
–45 ÷ 9 = –5
26 Integers
Simplify the fraction by finding the quotient –825
–75
The signs are the same, so the answer is positive.
–825 = 11
–75
Name
Date
Division of Integers
Math Content
Students will apply the rules for dividing integers to solve problems involving division of
positive and negative numbers.
1. Solve by rewriting each problem as a
multiplication problem.
a. –27 ÷ 9 =
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.
b. –1 ÷ –1 =
C. It is a positive number.
D. It is a negative number.
2. Find each quotient.
a. –510 ÷ –17 =
b. 42 ÷ –7 =
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.
3. Simplify each fraction by finding the
quotient.
C. It is a positive number.
D. It is a negative number.
a. 75 =
–15
b. –24 =
–8
c.
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.
–6,000
=
125
Integers
27
Name
Date
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.
Day
Daily Profit
Thursday
–$10.00
Friday
$35.00
Saturday
$17.00
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.
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.
10. The table below shows the low temperature for each day during one week in
January.
b. What will be her average daily profit
over the four days? Show your work.
–3
4
8. The fraction is equal to the fraction
Use the rules for dividing integers to
explain how this is possible.
28 Integers
3
–4
Day
Low Temperature
Monday
–14°C
Tuesday
–8°C
Wednesday
2°C
Thursday
8°C
Friday
0°C
Saturday
–10°C
Sunday
–13°C
What was the average daily low temperature that week? Show your work.
Grade 8
Linear Functions,
Quadratics, and Factoring
29
Name
Date
Math Content
Students will translate among different mathematical representations, write expressions, and
combine like terms.
1. How many smiley faces will the 100th
figure have?
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?
2. Which expression could be used to
calculate the number of triangles in the
next term?
△△
A. 2n
B. n+2
△△△△
n
C. 2
D. All of the above
5. a. Fill in the missing numbers for the
arithmetic sequence.
1,
, 5,
,
, 11, …
b. Write the expression for the sequence.
3. The steps are equal. Fill in the missing
numbers and expressions.
A
B
C
c. Use diamonds (◇) to make a visual pattern that corresponds to this sequence.
36
–2
3
6. What is the sum of –4n – 3 and 6n + 8?
9
A. 9n
B. 10n + 11
13
18
–9
42
C. 2n + 5
D. 2n – 5
30 Linear Functions, Quadratics, and Factoring
Name
Date
7. What is the missing expression?
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?
a. (–9 + 6h) + (–4 – 2h) =
b. (4 – 2c) +
= (–2 + 5c)
-
=
0
10. The American Civil War began in 1861,
and World War II ended in 1945. How
many years are between 1861 and 1945?
1870
A. 84
B. 106
1900
1890
n+2
n+5
n+4
n+6
Review
(8 + b) + b + (–2+ b) + (1 + 2b)
1880
n
n+3
b. 4(–1 + 2x) + 2(1 – 5x) =
18
9. Rewrite the following expression to be as
short as possible.
1860
n-2
n+1
a. 6(–1 + 2x) =
4–x
6 + 3x
n-1
12. What is the missing expression?
8. Fill in the missing numbers and
expressions.
18
n-3
1920
1910
1940
1930
1950
13. There are 20 students in Mrs. Xavier’s
class. She needed two helpers, so she randomly 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?
C. 116
D. 124
Linear Functions, Quadratics, and Factoring
31
Name
Date
Slope
Math Content
Students will translate among different mathematical representations, define slope, and identify
intercepts.
1. A swimming pool is 3 ft deep at the shallow 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.
2. Complete the table for each equation.
a. y = 6 – 2x
x
D = 3 + 0.25S
–2
a. What does the D in the formula stand
for?
–1
y
0
1
b. What does the S in the formula stand
for?
2
b. y = 2(3 – 2x)
c. Complete the following table that fits
the formula D = 3 + 0.25S.
x
y
–2
S
0
1
2
3
6
D (in ft)
–1
6.5
0
1
d. Use the table to draw a graph that fits
the formula D = 3 + 0.25S.
2
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
B. y = 1 x
8
C. y = 8x – 2
D. y = 8x2
Name
Date
Slope
4.
6.
4
4
3
3
2
2
1
1
1
2
3
4
1
2
3
4
5
6
–1
–2
Graph A
7
6
5
(2, 4)
4
3
2
1 (1, 1)
1 2 3 4 5 6 7
a. What is the slope of the line?
b. What is the y-intercept?
c. What is the x-intercept?
d. Write the equation of the line.
Graph B
a. What is the slope of each graph?
Review
b. Why does one graph appear steeper
than the other?
7. Which of the following expressions is
equivalent to 7(9 – 2d)?
A. 63 – 2d
B. 16 – 9d
5. Find an equation of the straight line with
x-intercept 3 and y-intercept 4.
C. 7 × 9 + 7 × 2 + 7 × d
D. 63 – 14d
8. What is (5f + 4) – (2f – 8)?
33
Name
Date
Adding Graphs
Math Content
Students will translate among different mathematical representations and make and interpret
graphs in a coordinate system.
1. The chart shows the number of milks and
orange juices bought during a 7-day
fundraiser in Mr. Jackson’s class.
Day Milk (M) Orange Juice (J)
1
5
10
2
2
9
3
3
6
4
8
2
5
9
4
6
1
8
7
4
8
M+J
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?
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.
3. Which equation would represent 2G?
8
7
6
5
4
3
2
1
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.
16
G
1 2 3 4 5
14
6 7 8
12
10
8
6
4
2
0
1
2
3
4
5
6
7
8
A. y = 2x + 1
B. y = 4x - 2
C. y = 4x + 1
D. y = 2x – 2
Name
Date
Adding Graphs
4. a. Using the graph of lines A and B
below, draw the sum graph, A + B.
11
10
9
6. Graph A corresponds to y = 34 x – 5.
Graph B corresponds to y = 14 x + 3.
Which equation represents the graph of
A + B?
A
8
7
6
B
A. y = 4x
B. y = x – 2
C. y = x + 8
D. y = 4x – 2
5
4
3
2
1
0
7. Graph W corresponds to y = 6 – 9x.
Write an equation that corresponds to 31 W.
1
2
3
4
5
6
b. What points did the graphs of A and B
have in common?
Review
8. Which equation represents the following
graph?
c. Does A + B have the same points?
Why or why not?
8
7
6
5
4
3
2
1
1 2 3 4 5
5. Graphs F and M intersect at point (4, 5).
Explain why the graph of 12 (F + M)
intersects F and M at (4, 5), too.
A. y = 5 + x
B. y = 5 – x
6 7 8
C. y = –5 – x
D. y = –5 + x
9. What is (–6d + 3) + (d – 10)?
35
Name
Date
Solving Equations
Math Content
Students will translate among different mathematical representations, solve equations, and use
different solution methods.
1. When using the cover method to solve
the equation 5(x + 2) = 20, what is the
value of x + 2?
A. 2
B. 4
C. 5
D. 20
3. Cell phone company S charges $25 a
month. Cell phone company T charges
$20 a month plus $0.50 per call. The
graph represents the charges for each
company. What does the intersection
point of the graphs represent?
2. Tariku babysits and calculates her fee by
using the formula F = 5 + 8H.
35
a. What do you think F and H mean?
25
Cost ($)
30
b. What is the meaning of each number
in the formula?
20
15
10
5
0
c. Hosea also babysits, and he simply
charges $10 per hour. Write an equation for Hosea’s fee.
S
T
2
4
6
8 10 12 14 16 18 20
Number of Calls
d. Draw the graphs from both formulas.
Label them A and H.
4. Use the specified method to solve the
equation.
48 + 6n = 24 – 2n
a. Balance Method
b. Difference-is-0 Method
e. Your mom says that Hosea is more
expensive than Tariku. What is your
comment?
c. Why do you get the same solution
using either method?
Name
Date
Solving Equations
5. a. Solve the equation.
4 + 3x = 3x + 10
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. What does the solution tell you about
the graph?
b. Graph A, B, and A + B. Be sure to
label each graph.
Review
6. The table corresponds to a linear graph.
x
y
–3
–18
–1
–8
1
2
3
12
5
22
What is the slope of the graph?
A. 2
B. 3
C. 5
D. 10
37
Name
Date
Math Content
Students will write expressions and find area and perimeter.
1. Three triangles are shown below.
a
a Perimeter = P
x
y
a
c
2. This is a cross figure. The sum of the
lengths x and y is 10 feet.
c
a
Perimeter = Q
b
c
x yy
y x
x y
y x
x
y
What is the perimeter of the figure?
a
Perimeter = R
a. For the perimeter P of the first triangle, the formula is P = 3a. Explain
this formula.
3. Which is the formula for the area of the
figure?
z
x
y
w
z
b. What is the formula for perimeter Q?
A. A = w + 2z + xy
B. A = 2z + 2w + 2x
C. A = zw – xy
D. A = zw + xy
c. What is the formula for perimeter R?
Name
Date
4. Use the picture to find the equivalent
expressions.
a
j
m
Review
6.
80
A
60
B
40
n
20
20
40
A. (a + j)(m + n) = am + jm + an + jn
B. (a + j)(m + n) = am + jn
C. (a + j)(m + n) = a + jn + jm
2
D. (a + j)(m + n) = am2 + jn2 + an + jm
60
80
100
120
A–B
140
Which part of the difference graph shows
the point of intersection for A and B?
A. slope
B. distance from A
C. x-intercept
D. y-intercept
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.
7. A line has slope –5 and y-intercept of
120. What is the x-intercept?
d. Calculate rs + rt if r = 15 and
s + t = 21.
39
Name
Date
Focus On: Solving Equations
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 changing the solution. Always check your solution by substituting it into the original equation.
2 b = 54
3
3 × 2 b = 3 × 54
2 3
2
b = 81
check:
x + 6.2 = –4.1
x + 6.2 – 6.2 = – 4.1 – 6.2
x = –10.3
check:
–10.3 + 6.2 = –4.1
–x = 49
–x = 49
–1 –1
x = –49
check:
–(–49) = 49
2 (81) = 54
3
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.
6(k – 4) – 2k = k + 9
6k – 24 – 2k = k + 9
4k – 24 = k + 9
4k – k – 24 = k – k + 9
3k – 24 = 9
(Step 1)
(Step 2)
3k – 24 + 24 = 9 + 24
3k = 33
3k = 33
33
3
k = 11
6(11 – 4) – 2(11) = 11 + 9
66 – 24 – 22 = 20
20 = 20
(Step 3)
(Step 4)
(check)
Name
Date
Solving Equations
Math Content
Students will solve equations.
1. Solve 34 x = –12.
4. What is the solution to the following
equation?
–3(5p + 24) + 9 = 2(3 – 2p) – 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.
5. What is the solution to the following
equation?
16.3 – 7.2b = –8.18
b. Write an expression for the number of
stamps that James has. Use the same
variable from part (a).
A. b = –3.4
B. b = 3.4
C. b = 812
720
D. b = – 812
720
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?
41
Name
Date
Solving Equations
6. A student completes several steps and
comes up with the equation 5x = 2x. The
student then divides each side by x, getting 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)
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 acceptable step?
c. Why might some see this strategy as
helpful?
b. What does the solution tell you? For
what values of x is the equation true?
9. Which equation is not a step in solving
the following equation?
19 – (2x + 3) = 2(x + 3) + x
A. 16 = 5x + 6
B. 2 = x
C. 10 = 5x
D. 22 – 2x = 3x + 6

Sampler - Encyclopædia Britannica Mathematics in Context

Transcription

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