Question 1

Use diagrams to explain how one could use white (positive) and dark (negative) chips to model the following:
A) 4 + (-6)
B) 5 - (-2)

Accepted Answer

A) B) Adding two white and two dark chips does not change the value in the first box. Remove two darks chips to represent subtracting -2. This leaves seven whites, or ⁺7.

Question 2

Which property does each of the following involve?
A) (-2 + 3) + -5 = (3 + -2) + -5
B) (-2 + 3) + -5 = -5 + (3 + -2)
C) (-2 + 3) + -5 = -2 +(3 + -5)

Accepted Answer

A) commutativity of addition
B) commutativity of addition
C) associativity of addition

Question 3

Reorder these numbers from largest to smallest without using decimals: 
$- \frac { 5 } { 4 }$ $\quad $$- \frac { 4 } { 5 }$$\quad $ $- 2 \frac { 1 } { 3 }$ $\quad $$- \frac { 7 } { 10 }$

Accepted Answer

$- \frac { 7 } { 10 }$  $\quad $$- \frac { 4 } { 5 }$  $\quad $$- \frac { 5 } { 4 }$ $\quad $ $- 2 \frac { 1 } { 3 }$

Question 4

Consider clock arithmetic using a clock with four numbers: 0, 1, 2, and 3.
A) Complete these tables   B) Do you think the set of numbers-0, 1, 2, and 3-is closed under addition? If not, provide an example that shows it is not.
C) Do you think the set of numbers-0, 1, 2, and 3-is closed under multiplication? If not, provide an example that shows it is not.
D) Is there an additive identity? If so, what is it?
E) Is there a multiplicative identity? If so, what is it?
F) Does 3 have an additive inverse? If so, what is it?
G) Does 2 have a multiplicative inverse? If so, what is it?
H) Do you think addition is commutative? If so, how do we know that as we look at the table?
I) Do you think multiplication is commutative? If so, how do we know that as we look at the table?
J) Do you think addition is associative? If so, provide at least three examples.
K) Do you think multiplication is associative? If so, provide at least three examples.
L) Do you think multiplication is distributive over addition? If so, provide at least three examples.

Accepted Answer

The answer of Consider clock arithmetic using a clock with...

Question 5

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.)
-Associative property of multiplication
_____ A) -3 × (2 + 5) = -3 × (5 + 2)
_____ B) 3 × (2 + 5) = (3 × 2) + (3 × 5)
_____ C) 4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2)
_____ D) 5 + (8 + 0) = 5 + 8
_____ E) -4 × 1 = -4
_____ F) 6 + (4 + -4) = 6 + 0
_____ G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$ 
_____H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$ 
_____ I) 3 + (2 + -5) = (3 + 2) + -5
A:Commutative property of multiplication
B : Distributive property
C : Associative property of addition
D : Identity property of addition
E : Identity property of multiplication
F : Inverse property of addition
G : Inverse property of multiplication
H : Associative property of multiplication
I : Associative property of addition

Accepted Answer

A) -3 × (2 + 5) = -3 × (5 + 2)
This op

Question 6

Give the exact answer to each of the following, taking advantage of properties. Then tell which property or properties enabled you to answer them so easily.
A) If 213 × 142 = 41,401, then 142 × 213 = _____.
B) $\frac { 179 } { 196 } + \frac { 0 } { 343 } =$ _____ 
C) 84.96 × 100% = _____
D) (548 + -967) + -548 = _____

Accepted Answer

Question 7

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.)
-Additive inverse property

Accepted Answer

A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5 
A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5

Question 8

Is the statement a + b = |a| - |b| always true, sometimes true, or never true? Explain your answer.

Accepted Answer

The answer of Is the statement a + b =...

Question 9

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.)
-Associative property of addition

Accepted Answer

A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5 
A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5

Question 10

Illustrate how to use motion on the number line to find the difference.
A) -5 - _____ = -10
B) 3 - 8 = _____

Accepted Answer

A) 5; start at -5 and move lef

Question 11

Complete the following.
A) -7 × -8 =
B) 3 × -5 =
C) 16 × -2 =
D) -21 × -2 =
E) 18 ÷ -3 =
F) -2 ÷ -5 =
G) -18 ÷ 9 =
H) 4 ÷ -16 =
I) 
 $\frac { 5 } { 4 } \div \frac { 1 } { - 2 } =$ 
J) 
 $\frac { 11 } { 17 } \times \left( - \frac { 11 } { 17 } \right) =$ 
K) -8.6 ÷ 4.3 =
L) -1.2 ÷ -2.4 =

Accepted Answer

A) 56 B) ^-15 C) ^-32 D)

Question 12

A) Continue the following pattern of multiplication of integers with six more products in the pattern.
4 × 4 = 16
3 × -4 = -12
2 × -4 = -8 . . .
B) The multipliers decrease by _____ each time. The multiplicand is always -4. The product _____ by _____ each time. For this pattern to continue working, the product of two negative integers must be _____.

Accepted Answer

Question 13

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.)
-Additive identity property

Accepted Answer

A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5 
A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5

Question 14

Place the following on a number line:

$- \frac { 4 } { 5 }$ $\quad $ $- 2 \frac { 1 } { 3 }$ $\quad $- $\frac { 7 } { 10 }$ $\quad $- $\frac { 2 } { 3 }$

Accepted Answer

See that @#LAT-DLM& is placed to the lef

Question 15

Reorder these numbers from smallest to largest: $\begin{array}{lllllllll}
7.4 \% & \frac{21}{7} & \frac{5}{4} & \frac{-11}{6} & 0.2 & -1.5 & 2.8 & -0.2 & -1 \frac{5}{4}
\end{array}$

Accepted Answer

The answer of Reorder these numbers from smallest to largest:...

Question 16

Illustrate how to use the missing-addend view of subtraction on the number line to find the difference.
A) 3 - -4 = _____
B) -2 - -5 = _____

Accepted Answer

A) What is added to -4 to get to 3? Illu

Question 17

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.)
-Distributive property of × over +

Accepted Answer

A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5 
A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5

Question 18

Does zero have a multiplicative inverse? Explain.

Accepted Answer

The answer of Does zero have a multiplicative inverse? Explain....

Question 19

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.)
-Commutative property of multiplication

Accepted Answer

A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5 
A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5

Question 20

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.)
-Commutative property of addition

Accepted Answer

A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5 
A)  -3 × (2 + 5) = -3 × (5 + 2) 
B)  3 × (2 + 5) = (3 × 2) + (3 × 5) 
C)  4 + (-3 + -1) + 2 = (4 + -3) + (-1 + 2) 
D)  5 + (8 + 0) = 5 + 9 
E)  -4 × 1 = -4
F) 6 + (4 + -4) = 6 + 0
G) $\frac { 3 } { 2 } \times \frac { 2 } { 3 } = 1$
H) $3 \times ( 5 \times 0 ) = ( 5 \times 0 ) \times 3$
I) 3 + (2 + -5) = (3 + 2) + -5

Exam 10: Integers and Other Number Systems

Use diagrams to explain how one could use white (positive) and dark (negative) chips to model the following: A) 4 + (-6) B) 5 - (-2)

Which property does each of the following involve? A) (-2 + 3) + -5 = (3 + -2) + -5 B) (-2 + 3) + -5 = -5 + (3 + -2) C) (-2 + 3) + -5 = -2 +(3 + -5)

Reorder these numbers from largest to smallest without using decimals: −54- \frac { 5 } { 4 }−45​ \quad −45- \frac { 4 } { 5 }−54​\quad −213- 2 \frac { 1 } { 3 }−231​ \quad −710- \frac { 7 } { 10 }−107​

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.) -Additive inverse property

Is the statement a + b = |a| - |b| always true, sometimes true, or never true? Explain your answer.

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.) -Associative property of addition

Illustrate how to use motion on the number line to find the difference. A) -5 - _____ = -10 B) 3 - 8 = _____

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.) -Additive identity property

Place the following on a number line: −45- \frac { 4 } { 5 }−54​ \quad −213- 2 \frac { 1 } { 3 }−231​ \quad - 710\frac { 7 } { 10 }107​ \quad - 23\frac { 2 } { 3 }32​

Reorder these numbers from smallest to largest: 7.4\% 0.2 -1.5 2.8 -0.2 -1

Illustrate how to use the missing-addend view of subtraction on the number line to find the difference. A) 3 - -4 = _____ B) -2 - -5 = _____

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.) -Distributive property of × over +

Does zero have a multiplicative inverse? Explain.

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.) -Commutative property of multiplication

Match the operations and the names of the properties by placing the correct number to the left of the letters A-I. (Not all properties on the right will necessarily be used; some may be used more than once.) -Commutative property of addition

Reasoning About Quantities

Numeration Systems

Understanding Whole Number Operations

Some Conventional Ways of Computing

Using Numbers in Sensible Ways

Meanings for Fractions

Computing With Fractions

Multiplicative Comparisons and Multiplicative Reasoning

Ratios, Rates, Proportions, and Percents

Number Theory

What Is Algebra

A Quantitative Approach to Algebra and Graphing

Understanding Change: Relationships Among Time, Distance, and Rate

Further Topics in Algebra and Change

Polygons

Polyhedra

Symmetry

Tessellations

Similarity

Curves, Constructions, and Curved Surfaces

Transformation Geometry

Measurement Basics

Area, Surface Area, and Volume

Counting Units Fast: Measurement Formulas

Special Topics in Measurement

Quantifying Uncertainty

Determining More Complicated Probabilities

Introduction to Statistics and Sampling

Representing and Interpreting Data With One Variable

Dealing With Multiple Data Sets or With Multiple Variables

Variability in Samples

Special Topics in Probability

Filters

Use diagrams to explain how one could use white (positive) and dark (negative) chips to model the following: A) 4 + (^-6) B) 5 - (^-2)

Which property does each of the following involve? A) (^-2 + 3) + ^-5 = (3 + ^-2) + ^-5 B) (^-2 + 3) + ^-5 = ^-5 + (3 + ^-2) C) (^-2 + 3) + ^-5 = ^-2 +(3 + ^-5)

Reorder these numbers from largest to smallest without using decimals: $- \frac { 5 } { 4 }$ $\quad$ $- \frac { 4 } { 5 }$ $\quad$ $- 2 \frac { 1 } { 3 }$ $\quad$ $- \frac { 7 } { 10 }$

Illustrate how to use motion on the number line to find the difference. A) -5 - _ = -10 B) 3 - 8 = _

Place the following on a number line: $- \frac { 4 } { 5 }$ $\quad$ _{$- 2 \frac { 1 } { 3 }$ $\quad$}_- $\frac { 7 } { 10 }$ $\quad$ _- $\frac { 2 } { 3 }$

Illustrate how to use the missing-addend view of subtraction on the number line to find the difference. A) 3 - -4 = _ B) -2 - -5 = _