Deck 10: Integers and Other Number Systems

Question 1

Complete the following. For each subtraction problem, first rewrite it as an addition problem.

A) ^-7 + ^-8 =
B) 3 + ^-5 =
C) 16 + ^-14 =
D) ^-21 + ^-2 =
E) 17 - ^-3 =
F) ^-2 - ^-5 =
G) ^-13 - 9 =
H) 14 - ^-16 =
I)
J)
K) ^-2.6 - 4.5 =
L) ^-3.17 - 2.4 =

Answer 1

A) ^-15
B) ^-2
C) 2
D) ^-23
E) 17 - ^-3 = 17 + 3 = 20
F) ^-2 - ^-5 = ^-2 + 5 = 3
G) ^-13 - 9 = ^-13 + ^-9 = ^-22
H) 14 - ^-16 = 14 + 16 = 30
I)
J) 0
K) ^-2.6 + ^-4.5 = ^-7.1
L) ^-3.17 + ^-2.4 = ^-5.57

Question 2

Reorder these numbers from smallest to largest:

Answer 2

Question 3

Complete this "fact family" table:

Answer 3

Question 4

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

Question 5

Use diagrams to explain how one could use white (positive) and dark (negative) chips to model the following:

A) 4 + (^-6)
B) 5 - (^-2)

Question 6

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)

Question 7

Does zero have a multiplicative inverse? Explain.

Question 8

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 =

Question 9

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 _____.

Question 10

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) _____
C) 84.96 × 100% = _____
D) (548 + ^-967) + ^-548 = _____

Question 11

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

Question 12

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

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 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.)

-Multiplicative inverse property

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

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

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 15

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

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 16

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

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 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 +

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

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.)

-Multiplicative identity property
?

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 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

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

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.

Question 21

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 }$

Question 22

Place the following on a number line:

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

Question 23

Illustrate how to use motion on the number line to find the difference.

A) -5 - _____ = -10
B) 3 - 8 = _____

Question 24

Illustrate how to use the missing-addend view of subtraction on the number line to find the difference.

A) 3 - -4 = _____
B) -2 - -5 = _____