Download Packet #1 Rational Numbers File

Math 7
Packet #1
Module 2 – Rational Numbers
Ms. Pricola, Mr. Unson
2016/17
Table of Contents
Page 3 - 5………………………………………………………………………….. Review of fraction and decimal operations
Page 6 - 8 …………………………………………………………………………. Understanding Negative integers and rational
numbers – opposites, absolute value
Page 9 - 11………………………………………………………………………... Properties of numbers
Page 12 – 15 …………………………………………………………………….. Addition of integers and rational numbers
Page 16 - 22……………………………………………………………………… Subtraction of integers and rational numbers
Page 20 - 23……………………………………………………………………… Multiplication/Division of integers and rational
numbers
Page 27…………………………………………………………………………….. Integer Rules
Page 28 - 30……………………………………………………………………… Pemdas
Page 31 – 32……………………………………………………………………… Mixed Practice Signed Numbers
Page 33 - 51……………………………………………………………………… Word Problems and Mixed Review
Page 52 - 55……………………………………………………………………… Mixed Review of The Rational Number Domain
Page 56 - 82……………………………………………………………………… Review for Quizzes and Tests
Page 83 - 85……………………………………………………………………… Enrichment
Page 86…………………………………………………………………………….. Perfect Squares and Rational/Irrational Numbers
Page 2
Previous learning; adding and subtracting fractions:
Recall that when adding or subtracting fractions it is necessary to have common denominators. Refer to the
examples below:
When adding these two fractions, we notice the denominators are the same.
Nothing other than adding the numerators (top numbers) need be done.
The same rule applies to subtraction, simply subtract the numerators.
Complete the following problems:
A]
C]
B]
D]
E]
G]
F]
H]
Page 3
In this example, the denominators are uncommon, or not alike. We must
find a common denominator. This can be done using several methods:
We can find the least common multiple through listing;
# x2 x3 x4
12 | 24 | 36 | 48
4 | 8 | 12 | 16
We can see that 12 is the smallest number in both rows
We can also use prime factorization:
12 = 2 x 2 x 3 or
4=2x2
= 4
x3
Since 2 squared is the largest power of 2
and 3 is the largest power of 3, we use those
values and multiply them to find our least
common denominator.
3
= 12
Remember, whichever method, to multiply
the numerator by the same factor as the
denominator to achieve an equivalent
fraction.
Try these:
I]
K]
J]
L]
Page 4
Previous learning; adding and subtracting decimals:
Recall that when adding or subtracting decimals, it is necessary to “line up” the decimal points. Also, while
subtracting, you may need to “borrow”.
Example:
This problem can also be written vertically
Notice the decimal points are lined up
14.615
Example:
In this example we will need to “borrow”. Since we can not take 9
away from 7 we take a value of 10 from the next place, the “1”, and
leave that as “0”.
Now we subtract 9 from 17, but are left needing to “borrow” again.
This time we borrow a value of 10 from the 4, leaving it 3.
The process will continue when we get to the 4 in order to subtract 6
Complete the following:
M]
N]
O]
Page 5
Notes on negative numbers and opposites:
Opposites are numbers that are the same distance from zero on the number line but are in the opposite direction
which means they have opposite signs. Opposites also imply direction along with the sign. For example north and
south, above sea level and below sea level, temperature above zero and below zero.
-2 and 2 are opposites
1.
The opposite of A __________
2.
The opposite of B __________
3.
The opposite of E – D __________
Page 6
Absolute Value:
Absolute value is the distance of a number from zero. This number will always be positive. There can be no
negative distance.
Look at point a on the number line below. What is the distance from zero to point a?
a
Point a is 5 units from zero. So the absolute value of 5 is 5. This is written mathematically as |5| = 5
Now look at point b.
b
Point b has a distance which is also 5 units from zero. Hence, the absolute value of -5 is also 5.
Mathematically this is written as |-5| = 5
In general, it can be said that any absolute value by itself will be a non-negative value.
Look at these examples:
|7| = 7
|-99| = 99
|0| = 0
Try these problems:
P] |10| =
Q] |-12| =
R] |-501| =
Page 7
Do the following 3 problems without your calculator first and show the work. Then use your calculator to
check your answer:
1.
3
+ __________________
2.
3.
12 – 8.067
–________________
My Answer____________
My Answer______________
My Answer __________
Calculator ____________
Calculator________________
Calculator___________
Give the value of each of the following:
4. The opposite of 2/3 ____________
5. The opposite of –3.5 ____________
6. The absolute value of 9 _________
7. |-4.6| __________
For each of the following, insert the appropriate symbol: < , > , or =
8. –5 _______ –8
11.
100 _______–100
9.
12.
–6 ______|-6|
–|-12| _______|12|
Put the following in order from least to greatest:
10. |-7| _______|7|
13.
|-4| ______|3|
14. –6 , 9 , 0, –10 , -3 4/5 , 2.7 ____________________________________________________
15. |-2| , –5 , –|3| , |1| , the opposite of 4 ________________________________________
16. Graph the following on the number line:
A. –2 B. |3| C. |-1|
D. 2 1/4
Page 8
Properties of Numbers
Commutative Property – the order in which numbers are added has no effect on their sum. This is also true for
multiplication; numbers may be multiplied in any order and the resulting product will be
the same.
Examples:
2+3+5= 3+5+2
4x5= 5x4
x●y= y●x
and xy and yx are the same thing
x+y=y+x
Associative Property – the way in which numbers are grouped will have no effect on their sum. This is also true
for multiplication; numbers may be grouped in any order and the resulting product will be
the same. The order of the values does not change.
Examples:
(2 + 3) + 5 = 2 + (3 + 5)
4 x (5 x 2) = (4 x 5) x 2
The order of the numbers does not change
Distributive Property – When multiplying a number by a sum, you can multiply the number by the first value in
parenthesis, then add that product to the product of the number by the second value in
parenthesis. The resulting sum will be the same as the sum generated by adding the values
in the parenthesis first, and then multiplying by the number.
Examples:
4(2 + 3) = 4(2) + 4(3)
5(10 - 4) = 5(10) - 5(4)
(9 + 7)6 = (9)6 + (7)6
(9 - 7)6 = (9)6 - (7)6
Zero Property of Multiplication – any number times zero has a product of zero. Also, zero times any number has
a product of zero.
Examples:
3x0=0
0 x 14 = 0
0●y=0
Page 9
Identity Property of Addition (or Additive Identity) – when zero is added to any value, the resulting sum is te
original value.
Examples:
85 + 0 = 85
0 + 93 = 93
0+n=n
Identity Property of Multiplication (or Multiplicative Identity) – when one is multiplied by any number, the
resulting product is that number.
Examples:
24 x 1 = 24
1 x 47 = 47
1● y = y
1 ● (-6) = -6
Additive Inverse Property – When a number and its inverse (opposite) are added, the sum is zero.
Examples:
5 + -5 = 0
-125 + 125 = 0
a+ -a=0
Multiplicative Inverse Property – Two numbers are said to be multiplicative inverses when they have a product
of 1. (These numbers whose product is one are called reciprocals)
×
Examples:
=1
5×
=1
d∙
=1
=1
Page 10
Name the property being used to progress to the next line of work:
(
)
(
(
(
(
)
)
)
(
(
(
)
)
)
(
)
(
5 x (7 x 18) – 6 ÷ 3
______________________________
)
)
Create examples showing each of the properties from the previous page. Label each property.
Page 11
Integers
Integers are commonly referred to as “the set of whole numbers and their opposites”. Keep in mind that zero has
no opposite. Whole numbers are values beginning with zero and increasing which have no fractions or decimals.
Page 09
So the set of whole numbers can be represented as { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , , , }
While the set of integers can be represented as { , , , -4 , -3 , -2 , -1 , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , }
When adding integers (positive and negative whole numbers) it is often helpful to think of pairs of opposites
cancelling out. Thus, adding a -3 with a +3 would yield a sum of 0. -3 + +3 = 0 This is also known as a zero pair.
Let’s try a few:
9 + -9 = _____
-7 + _____ = 0
______ + 12 = 0
14 + -14 = ________
________ + _________ = 0
It can also be helpful to use a number line while starting to add integers. In example, if adding +7 and -3, you could
start at the +7 on the number line, and then move 3 places left (since the 3 is negative). This would leave you at +4.
-3
END
START
Use the number lines to show the movement of the following examples.
10 + -7 = 3
-7 + 10 = 3
1 + -9 = -8
-9 + 1 = -8
You may have noticed that the order of the integers does not alter the sum (commutative property). You
may
Page
12 have
also noticed that you can subtract the two absolute values to achieve the number, then apply the appropriate sign.
Look at the following:
-14 + 8
|-14| - |8|
14 – 8
6
Original addition problem
Subtract absolute values
8 + -14
Set up the same way
answer needs the appropriate sign (positive or negative)
So how do we determine the sign? Which is farther from zero, -14 or +8? (which absolute value is more)
Since the -14 is farther from zero, we assign the negative (-) sign to the final answer:
-14 + 8 = -6
Complete these examples using the absolute value subtraction method:
-12 + 8
7 + -15
10 + - -83
| 12 | - |
|
| | - | |
12 – 8
4 (Final answer is -4 because -12 is further away from zero than 8)
-15 + 24
What if the two signs are the same: (both positive or both negative?)
If you are adding two positive integers, say 6 + 9, then nothing is new. You simply add these as normal.
6 + 9 = 15
If the integers are both negatives, then you will see that you simply add them together, and keep the negative sign.
-3 + -6
|3| + |6|
3 + 6 = 9
-9
On a number line it would look like this:
-6
Try these:
9 + 8 =
-9 + -8 =
12 + 15 =
-12 + -15 =
You can see, the sums are identical with the exception of the sign.
Page 13
Commutative Property for Addition – change order of numbers that are being added
Commutative Property for Multiplication – change order of numbers being multiplied
Associative Property for Addition – change the grouping, ( ), of numbers being added.
Associative Property for Multiplication – change the grouping, ( ), of numbers being multiplied.
Distributive Property – Multiply the number outside the parenthesis times all numbers inside.
Additive Identity (Identity Property of Addition) – 0 + any number equals the number
Multiplicative Identity (Identity Property of Multiplication) - 1 x any number = the number
Zero Property of Multiplication – Zero times any number equals 0
For each of the following name the property that justifies the next line:
1. 3(a + 6)
3a + 18 ______________
2. 4 + (6 + 8)
(4 + 6) + 8 ________________
3. (4)(5)(10)
(4)(10)(5)___________
4. x + 0 + 9
x + 9 _________________
5. (4 ● 1) + 9
4 + 9 _________________
6. 8 + 7 ● 0
8 + 0 ____________
8 ______________
Illustrate each of the following problems using the number line to arrive at your answer:
7. –3 + 4 = _________
8. –2 + –3 = _________
9.
5 + –6 = __________
Page 14
10. –1 + –5 + 2 = __________
11. 4 + (–2) + 1 = __________
12. (-1) + (-1) + (-1) = ________
13. Rule #1 for Addition of Integers - SAME SIGN you ______________(pick either the word
add or subtract) and ___________(pick either keep or
change) the sign of the numbers.
14. Rule #2 for Addition of Integers – DIFFERENT SIGNS you ____________(pick either the
word add or subtract) and keep the sign of the number
with the larger absolute value.
Page 15
To subtract integers, you must “Add the opposite”.
In general terms, that means keeping the first
value the same, changing the subtraction symbol to addition, and then switching the sign of the last value to its
opposite. Now you are using the same rules as addition.
Examples:
10 - ( -4)
10 + 4
14
5 - 18
5 + -18
-13
-8 - ( -12)
-8 + 12
4
-70 - 30
-70 + -30
-100
Try these on your own:
A.
25 - ( -9)
B. 3 - 47
C. -12 - ( -62 )
D.
-78 - 41
Do you get the same results using your calculator? Try these problems. Next to each problem, write the exact
buttons you pressed after you’ve written the answer.
answer
calculator button pressed
14 + -20 =
-14 + -20 =
14 - 20 =
-14 - ( -20) =
Does the same thing happen for decimals? (Hint: estimate your answer by using integers first)
6.34 - 10.25 =
-87.01 + 23.478 =
Page 16
-14.74 - ( -5.3254) =
Practice adding and subtracting integers.
A]
B]
C]
D]
E]
F]
G]
(
)
H]
(
)
I]
(
)
J]
K]
L]
M]
N]
O]
P]
(
)
Q]
(
)
(
)
R]
Page 17
Now let’s use our calculator to do the same thing using fractions. Again, write down the exact buttons pressed to
find your answer.
answer
calculator buttons pressed
Now let’s try a few word problems:
Jackie brought home 3/4 of a birthday cake from school. Her dad proceeded to eat 1/8 of that. How much of the
original cake was left?
A football team was on the 20 yard line. They gained 4 yards on the first down, lost 6 yards on the second down,
and lost another 2 yards on the third down. On what yard line did the team have to punt from? Expressed as an
integer, where were they on the field after three downs in relation to their starting point (line of scrimmage)?
20 + 4 - 6 - 2 =
What integer describes their location in regards to the line of scrimmage (starting point)? _________
Would the answer have been different if they lost the 6 yards on the first down, then
gained the 4 yards on the second down, and still lost the other 2 yards on the third?
Meaning, does the order of gains and losses change the final result?
Page 18
Practice adding and subtracting.
A]
B]
C]
D]
E]
F]
G]
H]
I]
J]
K]
L]
M]
N]
O]
P]
Q]
R]
Page 19
Mixed Number and Decimal/Fraction Combination Practice
A.
4
+
D.
4
–
G.
8 –
B.
E.
H.
7
+
C.
12
+
14
–
F.
10
–
I.
–20
+
–
Page 20
J.
M.
14
–
–7.3 +
K.
N. –
9
+
L. 2.9 – 1
+ –8.8
0. 2.05 – 3
Page 21
For each of the following use the Integer Rules to find the answer:
1.
−15 + 10 = ______
(
4.
) = ________
Work:
6.
2. −21 + (−42) = ________
5. (
(
)
Work:
8. 075 + (−6.4) = __________
Work:
)
3. −16 + 20 = _____
7. 2.5 + (−3.25) + (−4.1) = __________
Work:
For each of the following fill in the reason part of the chart by giving the property that takes you from the line
before:
Statements
1. 3(x + 4) + (−3x)
2. 3x + 12 + (−3x)
3. 3x + (−3x) + 12
4. 0 + 12
5. 12
Reasons
1. Given
2.
3.
4.
5.
Page 22
Statements
1. 2 + 3 + (−2 + −3)
2. 2 + 3 + (−3 + −2)
3. 2 + (3 + −3) + −2
4. 2 + 0 + −2
5. 2 + −2
6. 0
Reasons
1. Given
2.
3.
4.
5.
6.
When it comes to multiplying and dividing integers , there is only one rule to remember:
If the signs are the same, the answer is positive.
(n x n = p just like p x p = p)
**So n x p = n
Written as a chart it would look like this:
Sign of first value
+
+
-
Sign of second value
+
+
-
Sign of product/quotient
+
+
Here are some examples:
A.
4( 5 ) = 20
3( -4 ) = -12
-2( 5 ) = -10
-7( -3 ) = 21
Give these a try:
B.
3( 7 ) =
-3( 7 ) =
3( -7 ) =
-3( -7 ) =
Use your calculator to try the following:
C.
14( -23 ) =
( )
D.
-9( 17 ) =
E.
(
)
-47( -52 ) =
F.
9.245( -5.816 ) =
G.
-0.025( -58.2 ) =
(
)
Page 23
Division works the same way as multiplication; same signs yield a positive quotient.
Different signs the quotient is negative
Try these:
A.
B.
C.
D.
E.
Practice multiplying and dividing signed numbers.
A]
B]
C]
D]
E]
F]
G]
H]
I]
J]
M]
(
)
K]
N]
(
)
L]
(
)
O]
Page 25
Rational Numbers are numbers that can be written as fractions. Integers and whole numbers can be written as
fractions so they are rational numbers
[This page intentionally left blank]
Page 26
Integer Rules (Important to Study)
Additon
1.
Same Sign – Add and keep the sign of the numbers
Ex. -6 + (-4) = -10
2.
Different Signs – Subtract and keep the sign of the number with the greater absolute value
Ex. -8 + 2 = -6
Subtraction
-4 + 9 = 5
You have to change the problem!!!! You must change it to “add the opposite” then use
the two rules for addition
Ex. -8 – 5 change to -8 + -5 and your answer is -13
-5 – (-2) change to -5 + 2 and your answer is -3
6 – (-4) change to 6 + 4 and your answer is 10
Multiplication and Division – are the same two rules. The multiplication sign and the division sign are
interchangeable
Rule #1 negative x negative = positive
(n x n = p)
Rule #2 negative x positive = negative
(n x p = n)
Examples: -8 x -2 = 16
-9 ÷ -3 = 3
-4 x 3 = -12
-8 ÷2 = -4
Page 27
The Order of Operations [ PEMDAS ]
The order of operations is often misunderstood; many people believe that you should always add before you
subtract, this is not the case! In fact, when it comes to adding and subtracting, the order matters most; whichever
comes first from left to right is the operation that should be completed first. The same holds true for multiplication
and division; again, whichever appears first in the expression should be completed first.
Examples:
Use your calculator to try each of those examples given above. Make certain you achieve the same results.
Now simplify these expressions: Remember to show only 1 change per line. Use your calculator as needed.
(
)
(
(
)
)
Page 28
Let’s examine these:
Notes:
Points to ponder:
Is -5 + 3 the same thing as 3 + -5 ?
Does 50 divided by -2 yield the same quotient as -2 divided by 50?
When you gain $4, then gain another $10, then lose $5, is that the same result as if you were to gain $10, then lose
$5, then gain another $4?
Does 6 x 93 = 6 x (100 – 7)?
Page 29
Order of Operations
A]
Show only 1 change per line.
B]
D]
(
(
C]
)
(
)
)
E]
(
(
)
)
F]
G]
Page 30
Mixed practice with signed numbers.
A]
B]
C]
D]
E]
F]
G]
H]
I]
J]
K]
L]
Page 31
More practice. . . No Calculator!!!!!!
A.
–5 + –7 =
B. 28 – (–7) =
C. 10 + (–48) =
D.
3(–5) + (–7) =
E. (–3)(–5) + (–7) =
F. |
H. – |
I. – |
G. 20 ÷ (–2)(5) + 1 =
J.
– | =
– | =
–
Give an example of each property
Property
Commutative Property
Associative Property
Distributive Property
Example
| =
Identity for Addition
Additive Inverse Property
Identity for Multiplication
Multiplicative Inverse Property
Zero Property for Multiplication
K. Jessica has of a bag of popcorn.
L. Nathan spends
hours a day studying.
She wants to split this evenly among
If he spends of an hour on science,
4 people (herself and 3 others). How
on social studies, and
much of the bag will each person receive?
much time does he spend on math?
hours
hours on ELA, how
Page 32
Word Problems:
A] Sally took a ride in a hot air balloon. Starting at sea level (0 feet) she began to rise until she reached 120 feet in
the air. Following this, the pilot dropped 45 feet, then dropped another 30 feet, then rose 10 feet to clear some
trees. At this point the trip was nearly complete. How far did the pilot need to descend from this point to land at
sea level again?
B] Rose brought in a bowl of Ambrosia to share with the class. The first two students took
of the bowl and
of
the entire bowl. The third student then took of the original bowl amount. After these three students had taken
their parts of the Ambrosia, how much of the original bowl was left for the rest of the class?
C] A street vendor charges $8.99 per pound bag of jelly beans. If Suzie bought 3 bags, and ate only 1/3 of a bag,
how much was the rest of her jelly beans worth?
D] While walking the treadmill during gym class, Jaron noticed he was of the way there. Did Jaron have more
than ¼ or less than ¼ of his walk left?
Page 33
E] Elise promised all 18 kids in her class that she would spend 2/3 of an hour with each of them to help with their
projects. How much time did Elise promise to give in total spending on the project?
F] A dessert recipe calls for
cups of flour to serve 6 people. How much flour will each person be eating if the
dessert is split evenly by all 6 people?
G] The temperature was 51° at 6am. By 10am, the temperature had risen by ⁄ rds. By 5pm, the temperature had
risen another ⁄
th.
To the nearest degree, what was the temperature at 5pm?
H] Rodney’s dad said he would give him of of $100 for mowing the lawn. How much money would that be?
Page 34
Problems for The Rational Numbers Module 7.NS
1. (Taken from NYS Testing Draft 7.NS.1d and 7.NS.2c Question #8)
The numerical expression
–
(6 –
) +
is equal to
A. –
B. –
C.
D.
7.NS.1
2. Use a number line to add –5 + 7 (Teacher Notes: Questions 2 - 5 Taken from North Carolina CCSS)
3. Use a number line to subtract –6 – (–4)
4. Use a number line to illustrate:



p – q
ie. 7 – 4
p + (–q)
ie. 7 + (–4)
Is this equation true p – q = p + (–q) ?
5. Morgan has $4 and she needs to pay a friend $3. How much will Morgan have after paying
her friend? Write an equation using integers and an addition sign to illustrate your answer.
Use a number line to illustrate the problem.
6. Describe a situation in which opposite quantities combine to make 0. For example, a
hydrogen atom has 0 charge because its two constituents are oppositely charged.
(Teacher Notes: Taken from the CCSS outline)
Page 35
7. (Teacher Note: Questions 7- 14 Taken from Utah CCSS)
Compute:
– +
8. Compute:
9.
(–
)
Compute:
–3 + 7
10. Compute:
–3 – 7
11. Compute:
–3 – 7 + (–5)
12. Compute:
1.35 + (–3.57)
13. Compute: 4.5 – (–7.9)
14. Write a story that would result in the problem: (–3) + 6 + 5.7 – 8
Model the solution in two different ways.
The following 15-17 are from http://illustrativemathematics.org/standards/k8
15. Ocean water freezes at about
° C . Fresh water freezes at 0 ° C . Antifreeze, a liquid used to cool
most car engines, freezes at −64 ∘ C .
Imagine that the temperature is exactly at the freezing point for ocean water. How many degrees
must the temperature drop for the antifreeze to turn to ice?
Page 36
16.
On the number line above, the numbers a and b are the same distance from 0 . What is a+b ? Explain how you know.
17. A number line is shown below. The numbers 0 and 1 are marked on the line, as are two other numbers a and b .
Which of the following numbers is negative? Choose all that apply. Explain your reasoning.
1. a − 1
2. a − 2
3. −b
4. a + b
5. a − b
6. ab + 1
18. Use the method of using a number line and the method of using the rules for adding
integers to find the following sums:
A. –5 + (–3)
B. 10 + (–4)
C. 6 + (–9)
Page 37
D. –2 + 2
19. Use a number line, use the rules of integers, draw a model to find the following differences:
A. –8 – (–2)
B. 5 – (–1)
C. –4 – 3
20. The high temperature in Alaska one day was 3.4⁰ C. The low temperature was –2.7⁰ C.
What was the temperature range that day?
21. Find the additive inverse or opposite of each of the following:
A.
5
B. –6
C. 4.38
D.
22. Rewrite each subtraction expression as an addition expression and solve
A. –12.8 – 4.6
B. –4.5 – (–2.1)
C.
– (–
)
23. Chris’ bank balance yesterday was $500. He used his debit card and made purchases of
$27.82, $321.40, $137.59, $62.95. Lynn paid him back money that she owed him and
he deposited $140.75. What was his bank balance at the end of the day?
24. Mrs. Nuffer told Mrs. Pricola that a helium atom has two protons and that protons each have a
charge of +1. She then went on to tell Mrs. Pricola that the helium atom itself has no charge. Mrs.
Pricola was confused. Mrs. Nuffer explained that besides the protons, the helium atom is also made up of
electrons and electrons each have a charge of –1. So how many electrons must this atom
have?
25. Using thermometers to find temperature ranges where the high is a positive number and
the low is a negative or both are negative, write a story describing a situation.
26. Illustrate a sum x + y on a number line where a condition is given such as x + y is a
distance of |x| units from y and show numbers that work.
27. Find each of the following sums. Can use the properties of numbers such as
associative, additive inverse, additive identity, commutative to rearrange. Explain
what properties you use.
A. –8 + (–2.5) + 8 + 0
B. –5.24 + 6.1 + 5.24 + (–3.8)
C. –
+
– (–
)–
28. For any values of x and y what has to be true about the difference between x – y and
y – x ? Explain why this is always going to be true. Pick values for x and y to help you.
Page 38
29. Use properties of numbers to explain a quick way to find the answer to the problem:
Ken likes to ride elevators. He started on the 20th floor and rode up and down. The
following integers show his elevator trip. What floor did he end up on?
–8
+4
–6
+8
–4
7.NS.2
1. Which of the following fractions is equivalent to
? Explain your reasoning. (Teacher
Notes Questions 1 - 2 Taken from North Carolina CCSS)
A
B.
C.
Page 39
2. Examine the family of equations in the table below. What patterns are evident? Create a
model and context for each of the products. Write and model the family of equations
related to 3 x 4 = 12.
Equation
Number Line Model
2●3=6
Context
Selling two packages of
apples at $3.00 per pack
0
3
6
2 ● –3 = –6
Spending 3 dollars each on
2 packages of apples
–2 ● 3 = –6
Owing 2 dollars t o each of
your three friends
–2 ● –3 = 6
Forgiving 3 debts of
$2.00 each
3. (Teacher Note: Questions 3-5 Taken from Utah CCSS)
Compute:
4. Convert
x(
)
to a decimal using long division
5. Write a story that would result in the problem:
–1.25 ÷ 2
6. Tell whether the answer would be positive, negative, zero, or undefined if you did the
indicated operations on the following signed numbers together:
A.
(+)●(–)●(–)●(+)
B. (+) ÷ (–) ● (+)
C. (+) ÷ 0
D. 0 ÷ (–)
7. Find each of the following products:
Page 40
A.
● 20 ●
B. 2.4 ● (–0.5)
8. If appropriate, use the multiplicative inverse property, multiplicative identity, or the
distributive property to help find the answer. Explain.
(
)●(
)●1●(
)
9. The Student Leadership Club is going to sell cupcakes and juice boxes. They spent $30.25 on
cupcake mixes, $6.29 on cupcake liners, and $28.90 on juice boxes. They sold 78 cupcakes at
$0.75 each and 62 juice boxes at $1 each. How much profit did they make?
10. A rectangular garden has length 4 ft and a width of
ft. Which expression does not
represent the perimeter of the garden?
A. 2(
+
)
B. (2 ● 4 ) + (2 ●
C. 2 + (
D.
+
)
+2 )
+
+
11. Katie bought supplies for school. She bought 18 pencils at $0.20 each, 12 pens at $1.25
each, and 6 notebooks at $1.25 each. She wrote the expression $1.25(18 + 12 + 6) to
find the total that she spent. Will her expression give her the correct amount? If not, write
an expression that will and find the total cost.
12. Write each quotient as a fraction:
A. –20 ÷ 2
B. –18 ÷ –11
C. (–30.5) ÷ 0.5
13. Find each quotient:
A.
(–0.004) ÷ (–0.004)
B. –
÷
c.
÷ (
●
) ÷ (–1)
Page 41
14. Mary and her four friends bought a birthday gift for another friend. The cost of the gift
was $48.20. One of the friends had a coupon for the store worth $15. After the coupon
was applied, the friends split the cost of the gift equally. How much did each of them pay?
15. A 12 foot piece of wood is cut into 3rds. Each of those pieces are then cut into 5 congruent
pieces of wood resulting in 15 pieces of wood. What is the length of each of the final 15
pieces of wood?
16. Mr. Welsh has stock in a company that says it earned a profit of –$2,350.21 since it was
started. What does that mean? If the company has existed for 2.5 years what is its
average yearly profit?
17. John has 10 feet of wood to make steps for his deck that are each 1 feet wide. How
many steps can he make with the piece of wood that he has and how much wood will be
left over?
7.NS.3
1. Calculate: [–10(–0.9)] – [(–10) ● 0.11]
(Teacher Note: Questions 1 - 4 Taken from North Carolina CCSS)
2. Jim’s cell phone bill is automatically deducting $32 from his bank account every month. How
much will the deductions total for the year? Explain your process and answer in terms of
negative numbers.
Page 42
3. It took a submarine 20 seconds to drop to 100 feet below sea level from the surface.
What was the rate of descent?
4. A newspaper reports these changes in the price of a stock over four days:
What is the average daily change?
,
,
,
.
5. (Teacher Note: Questions 5 – 7 Taken from Utah CCSS)
A hot air balloon rises 2,150.825 feet then falls
feet. What is the final height of the balloon?
6. Create three word problems arising from situations at home that require negative
numbers to solve. Write the stories and the math problems and find the solutions. Explain
what the solution means in context.
7. Write a story problem that uses the word “sum”, but does not require addition to solve.
8. The three seventh grade classes at Sunview Middle School collected the most boxtops for a school fundraiser, and so
they won a $600 prize to share among them. Mr. Aceves’ class collected 3,760 box tops, Mrs. Baca’s class collected
2,301, and Mr. Canyon’s class collected 1,855. How should they divide the money so that each class gets the same
fraction of the prize money as the fraction of the box tops that they collected? (Teacher Note: Taken from
http://illustrativemathematics.org/standards/k8
Page 43
9. Karen’s backyard is 80 feet by
feet. What is the area of her yard?
10. Steve buys 7 bags of sand that each weigh
pounds. He uses
of the sand from one
of the bags. How much does the sand that he has left weigh?
11. The tax on one of the popular novels that Barnes & Nobles sells is $3.35. Today, they
collected $87.10 in taxes on the sale of those novels. How many of the novels did they
sell?
12. Mrs. Wilson’s fish tank had a leak. The water level in the tank fell
inches per hour for the
hours she was gone from home. What was the level of water in the tank when Mrs. Wilson found it if
there had been 42 inches of water in the tank when she left in the morning?
13. Jennifer bought pound of ham that sells for $8 a pound and
pound of Swiss cheese
that sells for $9 a pound. How much did Jennifer spend altogether?
Page 44
Mixed Practice of the Entire Module
NO CALCULATOR
1. (2 – 5 )
(–
)
2. –0.5 – (– )
3. (
–
4. 0.4 –
)
(– )
(–
)
(– )
Page 45
5. Change each of the following fractions to decimals by showing the long
division. Tell whether the decimals are terminating or repeating.
A.
6.
B.
C.
D.
E.
F.
Without a calculator you could
A. Multiply
by –2
B. Multiply
by –
C. Divide
D. Multiply
by –2
by –1 and divide by 2
Page 46
7.
Without a calculator you could
A. Multiply –
by 3 and divide by 8
B. Multiply –
by 8 and divide by –3
C. Divide –
by 3 and divide by 8
D. Divide –
by 3 and multipy by 8
8.
–0.8 ÷ (– )
9.
–
10. –
Page 47
11.
12.
(
(–
– )
– )
13. Mrs. Wilson bought 6.5 lbs of fish food at $2.75 per lb. She paid with a 50 dollar bill. How
much change did she receive?
14. Mrs DeMarco is shopping for the Boston Trip. She wants to buy 7 videos for $18.99 each and
12 disposable cameras for $8.95 each. She has $200 to spend. Does she have enough money?
If so how much money would she have left? If she does not have enough money, she wants to
buy the videos and as many cameras as she can afford. How many cameras can she afford to buy?
Page 48
15. Mr. Welsh’s children want to have a lemonade stand. It cost Mr. Welsh $24 for the
materials to make the stand, $7.99 for cups, $10.49 for lemonade mix, and $3.99 for a
plastic pitcher. The children are selling the lemonade for $1.25 a cup. The sell 41 cups of
lemonade. How much profit or loss did they have?
16. Mrs. Pricola’s plant was 8
inches when she planted it. The plant grew each week by
inches. How many inches was the plant at the end of the year? (52 weeks in a year)
How many feet was it?
17. Mrs. Caluri’s plant was
grew
inches and grew to twice its size in the first 25 weeks. It then
inch each week for the remaining weeks in the year How many inches was it at
the end of the year? (52 weeks in a year). How many feet was it?
18. The temperature at 8 AM on a February day was –2 degrees. By noon it increased by 20
degrees. By 4 PM it increased by another 16 degrees. From 4 PM to midnight it decreased
5 degrees each hour. What was the temperature at midnight?
Page 49
19. John earns $375.25 each week for 33 weeks. Kim earns $45 each day for the same
length of time. Who earns more money and how much more do they earn?
20. Mrs. Pricola’s daughter Katie ran
miles each day for 8 days. Katie’s husband Corey ran
miles each day for 10 days. What is the total number of miles ran by Katie and Corey?
21. Gracie has
of the 100 Magna Tiles that came in her set. Benji has
of the 100 Magna Tiles
that were in his set. They want to combine what they have into sets of 20. What’s the
greatest number of sets they can make?
22. Guess my fractional number. I multiplied the fraction by –
What fraction is my number?
and the product was –0.8.
Page 50
23. Guess my fractional number. I divided the fraction by
and the quotient was –0.16.
What fraction is my number?
24. How fast is Steve driving if he travels 366.6 miles in 6.5 hours at a constant rate of speed?
25. How long will it take Jennifer to drive 450 miles if she sets her cruise control at 60 mph?
Page 51
Mixed Review of The Rational Number Domain
1. Think of the sign that would represent a number that is above sea level and the sign of a number that is below
sea level. I hiked this summer at Bear Mountain State Park. I stopped when I reached an altitude of 0 meters.
Which of the following could possibly represent my hike?
A. I started my hike at 5 meters above sea level and by hiking I increased my altitude by 5 meters
B. I started my hike at sea level and by hiking I decreased my altitude by 5 meters
C. I started at 5 meters below sea level and then hiked and increased my altitude by 5 meters
D. I started at 5 meters below sea level and then hiked and decreased my altitude by 5 meters
2. During a 10 hour period the temperature changed by
degrees per hour.
During that 10 hour period did the temperature rise or fall and by how much?
A. Temperature fell by 3 degrees
B. Temperature fell by
degrees
C. Temperature rose by 9
degrees
D. Temperature rose by 10 degrees
3. On a cold day in January the temperature according to my outdoor thermometer at 4 PM was 28.6 degrees.
When the dog woke me up at 1 AM the next morning to be walked I checked the thermometer and the temperature
had decreased by 30.9 degrees. What was the temperature at 1 AM?
4. On another cold day in January, from 4 PM to midnight the temperature dropped each hour by an average of
1.08 degrees each hour. If the temperature at midnight was 18 degrees, what was the temperature at 4 PM?
Page 52
5. Mr. Unson gave his nephew a Visa gift card with a balance of $200. Which of the following would give
Mr. Unson’s nephew a balance that is the same as what he started with?
A. Depositing another $120 on the card and making a purchase of $220
B. Withdrawing $175 from the card by making an online purchase and then adding $75 to the card
C. Depositing another $85 on the card and then making a purchase of $85
D. Making a purchase of $150 and then adding $50 to the card
6. Do each of the following problems without the use of a calculator.
A.
2
+ (–2 )
B.
2
– (–2 )
C.
2
● (–2 )
D.
–
7. Do each of the following without the use of a calculator
A.
(–
)
B.
–
(–
–
)
C. –
D. –
+ (– )
(– )
Page 53
8. How would I describe, on a number line, the opposite of 5?
A. Move 5 to the left of 0
C. Move 5 to the left of 5
B. Move 5 to the right of 0
D. Move 5 to the right of 5
9. How would I describe, on a number line, the opposite of –5?
A. Move 5 to the left of 0
C. Move 5 to the left of –5
B. Move 5 to the right of 0
D. Move 5 to the right of –5
10. The wedding planner for Mrs. Pricola’s daughter’s wedding needs
yds of fabric
for each table at the wedding. They are planning on 30 tables at the wedding. The
fabric is only sold in yards. How many yards of fabric does the wedding planner need
to order?
11. Mrs. Pricola’s daughter found a fabric on clearance that coordinates well with the original
fabric. There is only 20 yards of the fabric available. They still need
yds of fabric
for each table. How many tables will they be able to cover with this new fabric?
Page 54
12. Mrs. Caluri is having a party for her son at a local arcade. She wants to buy 12 tokens
for each child to play the arcade games. She is also planning on getting a slice of pizza
for each child, a soda for each, and an ice cream sundae for each. She has a coupon for
off the cost of the tokens. According to the sign in the arcade, how much will it cost Mrs.
Caluri for each child before tax?
Happy Days Arcade Price List
Each Arcade Token
Burger
Cheese Burger
Hot Dog
Pizza
French Fries
Ice Cream Cone
Ice Cream Sundae
Soda
Bottled Water
$1.00
$3.50
$4.25
$2.75
$3.25
$2.00
$1.50
$3.00
$1.25
$1.00
Page 55
Review for Quizzes and Tests in The Rational Numbers Domain
Practice for Quiz – Integers through + and –
Give the numerical value for each of the following without using a calculator:
1. the opposite of 7 = _______
2. the additive inverse of –4 = _________
3. –(–9) = ________
4. |
6. – |
7. –|
| = _________
9. –2 + (–3) = ________
10.
12. 5 – (–5) = _________
13. |
| = _________
| = _________
–9 – 4 = _________
5. | | = ___________
8. –5 + 4 = _________
11. –6 – (–2) = ________
| = _________ 14. |
(
)| = ____
15. Rule #1 for Addition says that if I add two numbers that have the SAME SIGN I will
_________________(insert add or subtract) and ___________(insert keep or change)
the sign of the numbers.
16. Rule #2 for Addition says that if add two numbers that have DIFFERENT SIGNS I will
_________________(insert add or subtract) and ___________(insert keep or change)
the sign of the number with the greater absolute value.
17. Rule for Subtraction says that I must change the problem to ________________________
and then use the two rules for addition.
For each of the following tell what property it is that brings you from the 1 st line to the 2nd line:
18.
+(
)
19.
+0
(
20.
) + ( ) ________________
–
x 1
–
22.
_____________________
_________________________
21.
20 x 3(4 + 6) x 9
20 x 9 x 3(4 + 6) ________________
20 x 9 x 3(4) + 3(6) ______________
–5 + 5
0 ________________________
Page 56
For each of the following illustrate the problem and your answer by using the number line (do not use your
calculatror):
23. –5 + 3 = ________
24. –5 – 3 = ________
25. –2 + (–1) + 4 = _________
26. 2 – (–3) = ____________
27. –2 + –1 = ____________
You can use your calculator for the following problems:
28. What is the absolute value of 2 + (3 – 8) – 8
____________
29. |-25+9| = ___________
30. –3 – 1 + 5 + (–2) = __________
31. – 4/9 + -5/6 = _____________
32.
33. –8 – (–10) = _____________
34. What is the additive inverse of –4a _________
–10 + 8.5 = ____________
Page 57
Practice for Quiz #2 – Integers through Pemdas
Give the numerical value for each of the following without using a calculator:
1. the multiplicative inverse of -2 = _______
2. the additive inverse of 8 = _________
3. –(–3) = ________
4. – |
6. – | | = _________
7. –6(2) = _________
9. –2 + (–10) = ________
10.
12. 5 – (–5) = _________
13. |
15. –2 + (–3)(2) + 6 ÷ (–3) = _______
| = _________
5. |
| = ___________
8. –9 + 4 = _________
–10 – 2 = _________ 11. –6 ÷ (–2) = ________
| = _________ 14. |
(
)(
)| = ____
16. –4 ÷ 2 x (–2) = __________
17. Rule #1 for Addition says that if I add two numbers that have the SAME SIGN I will
_________________(insert add or subtract) and ___________(insert keep or change)
the sign of the numbers.
18. Rule #2 for Addition says that if add two numbers that have DIFFERENT SIGNS I will
_________________(insert add or subtract) and ___________(insert keep or change)
the sign of the number with the greater absolute value.
19. Rule for Subtraction says that I must change the problem to ________________________
and then use the two rules for addition.
20. Rule #1 Multiplication and Division - negative x positive = __________________
21. Rule #2 Multiplication and Division – negative x negative = __________________
For each of the following tell what property it is that brings you from the 1 st line to the 2nd line:
22.
x(
)
(
24.
23.
) ________________
+( )
0 ________________________
( )
1
______________________________________
Page 58
For each of the following illustrate the problem and your answer by using the number line (do not use your
calculatror):
25. –5 + –3 = ________
26. 5 – (–3) = ________
27. 2 + (–1) – 4 = _________
You can use your calculator for the following problems:
28. What is the absolute value of 4 + (5 – 8) – (–2)
____________
29. |-20+(–10)| = ___________
31. (
)(
33. | (
35. |
(
) ÷ 6 – (–3) + = _____________ 32.
)
)
30. –4 – 1 + 5(-2) + (–2) ÷ (–1) = __________
–15.82 + 8.5(–2.4) = ____________
| = _____________ 34. What is the additive inverse of
| = _________________
_________
36. What is the multiplicative inverse of –2 _______
Page 59
Math 7 – The Rational Numbers Domain – Test Practice on Entire Domain
SHOW ALL WORK
1. All wanted to put a railing next to his stairs. The railing needs to be
feet long. If he has
2 pieces of wood that are each
feet long, how much longer must the third piece be?
Write your answer in simplest form.
2. Kim is baking a batch of cookies. She needs
cups of sugar. If she only has
much more sugar does Kim need? Write your answer in simplest form.
3. Todd needs to make 3 steps for his deck. One step needs to be
cups, how
feet long and one step
needs to be
feet long and the third needs to be
feet long. If he has a board of wood
that is 20 feet long, how much will Todd have left when he is done making the steps? Write
your answer in simplest form.
4. While doing her math homework, Nicole wrote the following sentence in her notebook:
+(
) = (
)
( )
Which property did she use?
5. Shenica is studying the properties of numbers. Her math teacher wrote this expression on
the board: + 0 = which property does this expression illustrate?
6. Seliah is practicing for the long jump for the upcoming track and field meet. Her first jump
measured
feet. Her second jump was
feet. How much longer was the second
jump?
Page 60
7. Tabitha and her two best friends are making two batches of chocolate chip cookies. The
recipe for one batch of cookies is below:
cups of all purpose flour
cup of sugar
cup of packed brown sugar
1 cup (2 sticks) of butter softened
1 teaspoon baking soda
1 teaspoon salt
1 teaspoon vanilla extract
2 large eggs
2 cups (12-oz pkg.) chocolate chips
1 cup chopped nuts
What is the total amount of flour, sugar, and brown sugar needed to make two batches
of cookies?
8. Last week, Holly bought three five-pound bags of apples and a four-pound bag of cherries.
She made an apple pie and cherry pie for the bake sale. The apple pie called for
pounds
of apples, and the cherry called for
pounds of cherries. How many pounds of apples and
how many pounds of cherries were left after she made the pies?
9. Bailey’s dog food comes in two sizes:
pounds for the large bad and
for the smaller
size. How many pounds of dog food did Joan purchase if she bought two large bags and one
small bag of Bailey’s dog food?
10. Zane loves winter sports. A typical day at the slopes for him includes practicing snowboarding for
hours, downhill skiing for
hours, and the snow tubing for an hour. How much
time does Zane spend on the slopes on a typical day?
Page 61
11. Joyce is
feet tall. She is at the amusement park with her family and they are about to
ride the newest roller coaster. When they get to the front of the line, there is a sign that
says you must be at least
feet tall to ride the coaster. How much taller than the
required height is Joyce?
12. Bernie spends three hours a day on homework. If he spends of an hour on Science, of
an hour on Social Studies, and of an hour on ELA, how much time does Bernie spend on
math?
13. Use long division to express each of the following fractions as decimals:
A.
B.
C.
14. Cindy wants to make
orders of tacos. Each taco needs
ounces of cheese. How many
ounces of cheese will she need? Express your answer in simplest form.
15. There are 32 students in Ms. Jayne’s third period class.
her class. How many students received an A?
of the students received an A in
16. If there are 30 days in June and Steve only worked of the days, how many days did Steve
work? Express your answer in simplest form.
17. Andy pitched
innings a game for 14 games. How many innings did Andy pitch?
Page 62
18. The area of a rectangle is length x width. Sajad’s swimming pool is shaped like a
rectangle. The length of the pool is
feet. The width is
feet. What is the area
of Sajad’s swimming pool?
19. There are of a pound of grapes left afte a picnic. Cody wants to split them evenly among
his three friends. How much will each friend get? Express your answer in simplest form.
20. Gino has a stick of pepperoni that is
inches long. He wants to cut inch pieces to put
on his large pizza. How many pieces can Gino get from his stick of pepperoni?
21. The area of a rectangle is length x width. What is the length of a rectangular pool table that
has a width of
feet and an area of
square feet? Express your answer in simplest
form.
22. Find the numerical value for each of the following:
A. |
|
B. – |–
|
C. –(–2)
D. the additive inverse of
E. the multiplicative inverse of
F. |( (
G. | (
)
)
(– )|
)
|
Page 63
23. Use the number line to answer the question.
Which statement best models the number line?
A. | | = 3
B. |
|=3
C. | | = –3
D. –3 = 3
24. Write the following subtraction problems as addition problems and find the answer
A. –6 – 5
B. –2 – (–1)
C. 10 – (–10)
25. Ann wrote four statements using absolute value. Which of Ann’s statements is wrong?
A. | | = –8
B. | | = 5
C. |
|=3
D. – | | = –7
26. One scuba diver descended 15 meters below the surface of a lake. Another diver
descended 8 meters below the surface. At the same time, a seagull was flying 2 meters
above the lake’s surface, and another seagull was flying 10 meters above the surface.
Which situation has the greatest absolute value in relation to the surface of the lake?
(Hint: The surface of the lake is 0 feet)
A.
B.
C.
D.
The scuba diver that is 15 meters below the lake’s surface
The scuba diver that is 8 meters below the lake’s surface
The seagull that is 2 meters above the lake’s surface
The seagull that is 10 meters above the lake’s surface
Page 64
27. A model rocket was launched from the ground and shot 150 feet straight up. It then
fell back down to the ground and landed in the same place from which it was launched.
Which expression shows how far the rocket traveled?
|–|
|
|–|
|
|+|
|
A. |
B. |
C. |
You may use your calculator for this part but must SHOW ALL WORK
D. |
| + (–|
28. Sue bought a 3 pounds of flour for cupcake wars at school to share with her friends.
She used
pounds of the flour and divided the remaining equally among her 4 friends.
The expression below represents the number of pounds of flour Sue gave to each of her
friends.
(
)
A. How could this expression be rewritten using the distributive property? Explain how
this change would make the expression easier to evaluate.
_____________________________________________________________________
______________________________________________________________________
______________________________________________________________________
|)
B. Evaluate the expression you wrote in Part A. How many pounds of flour did Sue
give to each of her friends?
Answer__________
29. Evaluate:
(
)
Show your work
Answer___________
30. John has 9 feet of wood to make new stairs for his deck. He needs
step.
Page 65
feet for each
A. Set up a complex fraction that represents the number of steps John can make.
Complex Fraction________________
B. Simplify your complex fraction from Part A to solve, and express your answer as a
mixed number.
Mixed Number __________
C. Explain what the whole number and fractional parts of the mixed number answer
represent in the context of the situation described.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
31. A number line is shown below.
A. Plot the point
on the given number line
B. What is the opposite of the number
?
______________
C. Using the number line and the definition of additive inverse to explain why the
number 3 and its opposite are additive inverses.
____________________________________________________________________
_____________________________________________________________________ Page 66
32. Point A is shown on the number line below.
What is the additive inverse of the number represented by point A?
33. Evaluate ─12 + 12
Evaluate 25 – 50
34. If a, b, c, and d are non-zero integers, which of the following is equal to
A.
B.
∙
?
C.
35. A number line is shown below.
Which of the following expressions represents the distance between ─2 and 3 on
the number line?
A. |-2+3|
36. Divide:
B. |2 - 3|
÷
C. |-3 – (-2)|
D. |-2 -3|
SHOW YOUR WORK
Page 67
37. Simplify:
÷ 14 SHOW YOUR WORK
38. What is the value of
39. Evaluate
∙(
)
8 ÷ 0 ?
SHOW YOUR WORK
40. Ann simplified the expression shown below as follows.
–15 ─ 25 ─ (– 4) + (─1)
─40 ─ (–4) + (─1)
─44 + (─1)
Step 1
Step 2
─-45
Step 3
In which Step was Ann’s mistake? Correct the problem starting on the line where her
mistake occurred and complete the problem to get the correct answer.
41. On the number line shown below, point A has a value of 4
What number must be added to 4 to get a sum of 0 ?
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42. Evaluate:
–
+ (
) +
HINT: You might want to look at it first and
think about your properties.
MUST SHOW YOUR WORK
43. The energy consumption of an appliance is measured in kilowatt-hours (kWh)
and is the product of the kilowatts per hour the appliance uses and the number
of hours it uses energy. The Unson family’s washer uses 2 kilowatts per hour. If Mr.
Unson runs the washer for
MUST SHOW YOUR WORK.
hours, how much energy will the washer use?
44.. Which of the following situations could be represented by the equation
shown below?
+ (
) = –3
A. The temperature one morning was 4
degrees Celsius. It decreased by 8
degrees Celsius during the day to reach a low of
Celsius.
B. Joe studied for 4 hours, starting at 8:00 PM, so he still needs to study for
3 1/2 more hours.
C. Mary spent
dollars on a sandwich. She had 8 dollars, so she
has 3 1/2 dollars left.
D. Louis is 4
miles away from finishing a marathon. He is running at a
speed of 8 miles per hour, so he will finish in 3 1/2 hours.
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45. A scuba diver finds some interesting sea creatures at 5 feet below sea level, represented
by the number
–
, and another interesting group of sea creatures at at
feet
below sea level represented by the number
. What is the distance between these
two points below sea level? MUST SHOW YOUR WORK
Page 70
More review of The Rational Numbers Domain (#1)
1. What is the value of |–
|
2. What is the additive inverse of 16 ?
3. What is the reciprocal of
?
4. What is the reciprocal of 4 ?
5. What is the multiplicative inverse of
6. What is the reciprocal of
?
?
?
7. What is the multiplicative inverse of
8. While doing her math homework, Nicole wrote the following sentence in her notebook:
(
)
(
)
(
)
Which property did Nicole use?
A. Commutative property of addition
B. Associative property of addition
C. Distributive property
D. Identity property of addition
Page 71
9. Stephen is studying the properties of numbers. His math teacher wrote this expression on the
board:
+ 0 =
Which property does this expression illustrate?
A.
B.
C.
D.
Identity property of addition
Identity property of multiplication
Zero property of multiplication
Distributive property
10. The expression
A.
B.
C.
D.
x 1 =
is an example of the
Identity property of addition
Identity property of multiplication
Zero property of multiplication
Distributive property
11. What is the additive inverse of
?
A.
B.
C.
D.
12. What is the additive inverse of –4a ?
A.
B. 4a
C.
D.
Page 72
13. Determine the absolute value of the following expression
2(3 – 8) + 8
14. Determine the absolute value of the following expression
–2 – 3(5 + 7) + 1
15. Determine the value of the following expression
| (
)
|
16. Determine the value of the following expression to the nearest tenth
|( (
)
)
(
)|
17. Determine the value of the following expression
|
(
)
|
18. What is the multiplicative inverse of –2 ?
A. 1
B. 2
C.
D.
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19. The multiplicative inverse of
is what number?
A. 1
B.
C.
D.
20. The multiplicative inverse of
multiplied by –
is equal to what number?
A. 1
B.
C. –
D. –1
21. Use the number line to help answer the question
3
-5 -4 -3 -2 -1 0 1 2 3 4 5
Which statement best models the number line?
A. | | = 3
B. |
|=3
C. | | = –3
D. –3 = 3
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22. Use the number line to help answer the question.
-5 -4 -3 -2 -1 0 1 2 3 4 5
What has an absolute value of 3?
A. 3, only
B. –3, only
C. 3 and –3
D. 3, –3, and 0
23. Cam wrote four statements using absolute value. Which of Cam’s statements is incorrect ?
A. | | = –8
B. | | = 5
C. |
|=3
D.
| | = –7
24. One scuba diver descended 15 meters below the surface of a lake. Another diver descended 8
meters below the surface. At the same time, a seagull was flying 2 meters above the lake’s surface,
and another seagull was flying 10 meters above the surface. Which situation has the greatest
absolute value in relation to the surface of the lake?
A.
B.
C.
D.
The scuba diver that is 15 meters below the lake’s surface
The scuba diver that is 8 meters below the lake’s surface
The seagull that is 2 meters above the lake’s surface
The seagull that is 10 meters above the lake’s surface
25. A model rocket was launched from the ground and shot 150 feet straight up. It then fell back down to
the ground and landed in the same place from which it was launched. Which expression shows how
far the rocket travelled?
A. |
|
|
|
B. |
|
|
|
C. |
|
|
|
D. |
|
( |
|)
Page 75
More review of The Rational Numbers Domain (#2)
1. Dan is building a frame that is
How many supports will Dan use?
2. Which answer expresses
feet wide. The supports that he is using are put in every foot.
as a decimal?
A. 0.6
B. 0.8
C. 0.65
D. 1.6
3. Which answer expresses
as a decimal?
A. 1.8
B. 0.56
C. 0.63
D. 0.42
4. Which expresses the fraction
A. 2.3
B. 0.3
C. 0.6
D. 0.6
as a decimal?
5. Write the fraction
as a decimal.
6. Write the fraction
as a decimal.
7. Write
as a decimal.
8. Write the mixed number
as a decimal number.
9. Which number is equivalent to
A. 0.5
B.
C. 1.2
D.
10. What is the reciprocal of
11. What is the reciprocal of 4?
?
?
Page 76
12. What is the multiplicative inverse of
?
Page 77
13. What is the reciprocal of
?
14. What is the multiplicative inverse of
15. The expression
A.
B.
C.
D.
x 1 =–
?
is an example of the
Identity property of addition
Identity property of multiplication
Zero property of multiplication
Distributive property
16. What is the multiplicative inverse of –2 ?
A. 1
B. 2
C.
D.
17. The multiplicative inverse of
A. 1
B. –
is what number ?
C.
D.
Page 78
18. The multiplicative inverse of
multiplied by
is equal to what number?
A. 1
B.
C. –
D. –1
19. There are 28 students in Mrs. Stanley’s class .
of them got a “B” in math class. How many
students got a “B” ?
A. 5
B. 6
C. 3
20. There are 30 days in June, and Mike worked
D. 4
of those days. How many days did Mike work
in June?
A. 5
21. Jim spends
B. 6
B.
D. 8
days a month away from home. He does this for
is Jim away from home?
A.
C. 7
months. How many days
C.
D.
22. A factory makes sheets of metal that are 2
how many inches thick will the stack be?
Page 79
inches thick. If a worker makes a stack of 5 sheets,
A.
B. 2
C.
D. 4
23. At a restaurant,
of the dishes on the menu are vegetarian. Of the vegetarian dishes,
dishes. What fraction of the dishes are vegetarian pasta dishes?
A.
B.
C.
D.
are pasta
Page 80
Enrichment Section
1. Tell whether each of the following are rational or irrational and explain why.
A. –65
B. 32.014
C. 0.08333….
D. 1.2345….
2. Convert the repeating decimal 5.08 to a fraction
3. What two integers is √
4. Locate √
between?
on a number line
5. Approximate the value of each irrational expression.
A. 3π
B.
6. What is the best approximation for √
A. 6
B. 7
C. 2π – 3
E. √
F. √
C. 6.6
D. 6.65
7. Which would be the more accurate estimate for π ? 3.14 or
? Explain your answer.
Page 81
8. Which chair height is taller: one that is
how you would find this answer.
9. Approximate √
√ feet tall or one that is
+ √
10. Put the following in order and then plot them on a number line:
A.
√
B.
√
π
–2.1
√
–1.1
√ feet tall. Explain
11. Compare using <, > or =
A. √
and 1.44
B. 9.2 and
C. 5 and √
Page 82
ENRICHMENT
These problems taken from www.illustrativemathematics.org under the 8th grade CCSS
1. Without using the square root button on your calculator, estimate √
as accurately as possible to
2 decimal places.
(Hint: It is worth noting that
=400 and
=900 .)
2. Represent each of the following rational numbers in fraction form.
1. 0.333
2. 0.317
3. 2.16
3. Decide whether each of the following numbers is rational or irrational. If it is rational, explain how you
know.
1. 0.333
2. √
3. √ =1.414213…
4. 1.414213
5. π=3.141592…
6.
7. 11
8.
=0.142857
9. 12.3456565656
Page 83
4. For each pair of numbers, decide which is larger without using a calculator. Explain your choices.
1.
or 9
2. √
or √
3. √ or 8
4. −2π or −6
5. Without using your calculator, label approximate locations for the following numbers on the number line.
1. π
2. −(
√
4. √
3.
×π)
ENRICHMENT PROBLEMS FOR ALL OF THE TOPICS
ALL OF THE FOLLOWING ARE TAKEN FROM NORTH CAROLINA’S CCSS (They had a footnote acknowledging that some
of the problems and graphics were taken from the Arizona Department of Education)
8.NS.1
1. Change 0.4 to a fraction
2. What pattern do you notice when you change fractions that have a denominator of 9
to a fraction?
What pattern when the denominators are 99?
What pattern when the denominators are 11?
8.NS.2
1. Compare √ and √ in three different ways . Make one of them graphing and one of then
finding an approximation of each to the nearest tenths place.
2. Find an approximation for √
Page 84
Page 85
Memorize:
These are the Perfect Squares you need to know. They are Rational numbers
Page 86