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Unit 3: Rational Numbers
ANSWER KEY
The following unit includes:
Adding/Subtracting Integers on a Number Line
Adding/Subtracting Integers with Rules
Multiplying/Dividing Integers
Adding/Subtracting Decimals
Multiplying Decimals
Dividing Decimals
Adding/Subtracting Fractions
Multiplying Fractions
Dividing Fractions
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Math in Demand
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Teacher Notes: Provided in this file is the answer key for Unit 3. There
are a few things that you might find helpful:
(1) You can differentiate for your students by deciding to print the blank template or answer key. I like to use
direct instruction and have my students fill in the blank template. However, I accommodate any of my students that have difficulty writing by providing the answer key.
(2) If you feel that I left anything out, you can ALWAYS have students add any additional notes. Every teacher
likes to teach a concept differently than another.
(3) You can use colored paper when printing the cutouts. This makes it less for students to color and it still
makes the page look colorful! Also, have students color code important info with highlighters!! It looks great!
(4) You can print the interactive notebook at 90% to make it fit into a smaller composition notebook.
If you have any questions, please don’t hesitate to email me at [email protected]. Thank you so much!
Math in Demand
Unit 3 Checklist
(Rational Numbers) As you go through the unit, check off the concepts that you have mastered. Leave the other ones unchecked so that you know which concepts you need to study before a test!
The following unit includes:
□ Adding/Subtracting Integers on a Number Line
□ Adding/Subtracting Integers with Rules
□ Multiplying/Dividing Integers
□ Adding/Subtracting Decimals
□ Multiplying Decimals
□ Dividing Decimals
□ Adding/Subtracting Fractions
□ Multiplying Fractions
□ Dividing Fractions
©2016 Math in Demand
©2016 Math in Demand
Unit 3 Overview (Rational Numbers)
Concepts(s):
Essential Questions:
Vocabulary:
Important Dates: Pre-Test: Post-Test: Vocabulary Quiz:
Scores: Pre-Test: Post-Test: Vocabulary Quiz:
Rationals Integers Decimals Fractions Operations
Numerator Denominator Reciprocal Quotient
How are adding and subtracting integers similar? Why is it important to understand operations on rational
numbers? How can number lines be used to interpret real world
problems?
Adding/Subtracting Integers on a Number Line Adding/Subtracting Integers with Rules Multiplying/Dividing Integers Adding/Subtracting Decimals Multiplying Decimals Dividing Decimals Adding/Subtracting Fractions
This overview shows you important information that you will be learning throughout the
unit. Make sure to fill out important dates and scores!
Multiplying Fractions Dividing Fractions
Rational Numbers (Adding/Subtracting Integers
on a Number Line)
Follow these steps to add or subtract integers:
©2016 Math in Demand
I can ___________________
_______________________
Subtraction always comes back to addition!!!
For example, 5 – (-4) becomes ____________
and
-8 – 7 becomes ______________
Numbers get bigger!
Step 1:
Step 2:
Place a dot on the number line for the first given number
Move from the dot either left or right according to the other given number.
Move right when adding a positive integer
Move left when adding a negative integer
Your stopping point is your answer!
Example:
-30 -29 -28 -27 -26 -25 -24 -23 -22 -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6
What is -26 – (-9)? -26 + 9
Numbers get smaller! A number line is a great tool to use when
adding or subtracting integers!
5 + 4
-8 + -7
-17
add and subtract integers
on a number line.
Rational Numbers (Adding/Subtracting Integers
on a Number Line)
1.) 1 – 7 = -6
2.) -2 – (-3) = 1
3.) -2 – 9 = -11
Reflection: I learned…
©2016 Math in Demand
On a rating of 1-5, how comfortable are you with this concept?
(5 is the highest)
1 2 3 4 5
Answer
Answer
Answer
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-6
1
-11
Rational Numbers (Adding/Subtracting Integers
with Rules)
I can ___________________
_______________________
©2016 Math in Demand
Change the following to addition:
1.) -9 – 2 = 2.) 7 – (-12) =
We can add integers by using rules
Addition Rules
Positive + Negative (Different Signs)
Subtract the two numbers. Keep the sign of the largest absolute value number.
Negative + Negative OR Positive + Positive (Same Signs)
Add the two numbers. Keep the sign.
Examples:
We can also add or subtract integers by using counters:
-8 + 5
-3
-9 + -2
7 + 12
1.) -4 – (-7) = 3
-4 + 7 = 3
2.) 3 – 8 = -5
3 + -8 = -5
add and subtract integers
using the addition rules.
Intro to Algebra (Adding/Subtracting Integers
with Rules)
Reflection: I learned…
©2016 Math in Demand
On a rating of 1-5, how comfortable are you with this concept?
(5 is the highest)
1 2 3 4 5
1
2
3
4
5
6
-10 + 8 = -2
-15 – 7 = -22
19 – (-6) = 25
-21 – (-3) = -18
-12 – (-4) – 6 = -14
45 – 47 – (-21) = 19
Rational Numbers (Multiplying/Dividing Integers)
I can ___________________
_______________________
©2016 Math in Demand
Multiplying Integers Dividing Integers
Neg x Pos = Neg
Pos x Neg = Neg
Neg x Neg = Pos
Pos x Pos = Pos
Neg ÷ Pos = Neg
Pos ÷ Neg = Neg
Neg ÷ Neg = Pos
Pos ÷ Pos = Pos
Another Way to Think About Signs:
What do you notice about the table?
There is a pattern!
# of Negatives Outcome
1 Negative
2 Positive
3 Negative
4 Positive
If you have an odd number of negatives, then your answer is negative!
If you have an even number of negatives, then your answer is positive!
Examples: 1.) -12 · 4 = -48
2.) −− = 5
multiply and divide integers.
Intro to Algebra (Multiplying/Dividing Integers)
Reflection: I learned…
©2016 Math in Demand
On a rating of 1-5, how comfortable are you with this concept?
(5 is the highest)
1 2 3 4 5
1
2
3
4
5
6
-11 · -5 = 55
-200 ÷ -5 = 40
-19 · 6 = 114
21 ÷ -3 = -7
-2 · -5 · 3 = 30
-1·-1·-1·-2·-1·-1 = 2
Rational Numbers (Adding/Subtracting Decimals)
I can ___________________
_______________________
©2016 Math in Demand
What is a decimal?
Step 1:
Step 2:
Step 3:
Follow these steps to add decimals:
Why are decimals important?
Examples:
add and subtract decimals.
1.) 0.078 + 1.5 =
0.078 + 1.500 1.578
2.) 8.98 – 4.781 =
8.980 -4.781 4.199
A decimal is written in our base-ten number system. There are ten single-digit numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Decimals are important in our everyday life. We use decimals every day when we are purchasing items at
the store, counting money or making change, and reading an odometer.
Write down the numbers with one underneath the other so that the decimals are lined up.
If necessary, add in any zeros that are needed to fill in any empty “spots”.
Add the numbers using column addition. Make sure to bring down the decimal.
Rational Numbers (Adding/Subtracting Decimals)
Reflection: I learned…
1.) 0.089 + 31.356 = 31.445
2.) 93.4 – 0.0761 = 93.3239
3.) -0.456 – (-23.41) = 22.954
Answer
Answer
Answer
©2016 Math in Demand
On a rating of 1-5, how comfortable are you with this concept?
(5 is the highest)
1 2 3 4 5
31.445
93.3239
22.954
Rational Numbers (Multiplying Decimals)
Example:
0.345 x 2.45 = 0.84525
©2016 Math in Demand
I can ___________________
_______________________
Step 1:
Step 2:
Step 3:
Follow these steps to multiply decimals:
Multiplication Signs Negative x Positive = Negative Positive x Negative = Negative Negative x Negative = Positive Positive x Positive = Positive
multiply decimals.
Ignore the decimal and multiply as usual.
Count the total number of digits behind the decimal point(s).
Place the decimal according to the same number of digits you counted.
Rational Numbers (Multiplying Decimals)
Reflection: I learned…
1.) 6.7 x 3.45 = 23.115
2.) -0.8 x 0.346 = -0.2768
3.) -2.13 x -0.78 = 1.6614
Answer
Answer
Answer
©2016 Math in Demand
On a rating of 1-5, how comfortable are you with this concept?
(5 is the highest)
1 2 3 4 5
23.115
-0.2768
1.6614
Rational Numbers (Dividing Decimals)
Step 1:
Step 2:
Step 3:
Follow these steps to divide decimals:
Example:
©2016 Math in Demand
I can ___________________
_______________________
Division Signs Negative ÷ Positive = Negative Positive ÷ Negative = Negative Negative ÷ Negative = Positive Positive ÷ Positive = Positive
5.50 0.50
5.50 ÷ 0.50 = ?
550 50
50 550 11
-50 50 -50 0
11
divide decimals.
If divisor is not a whole number, move the decimal point to the right to make a whole number. Move the decimal in the dividend by the same number of places.
Divide as usual (long division).
Place the decimal point above the decimal point in the dividend.
Rational Numbers (Dividing Decimals)
Reflection: I learned…
1.) -85.5 ÷ 5 = -17.1
2.) -20.4 ÷ -6.4 = 3.1875
3.) 140 ÷ -5.5 = -25.4545…
Answer
Answer
Answer
©2016 Math in Demand
On a rating of 1-5, how comfortable are you with this concept?
(5 is the highest)
1 2 3 4 5
-17.1
3.1875
-25.45
Rational Numbers (Adding/Subtracting Fractions)
©2016 Math in Demand
I can ___________________
_______________________
What is a fraction?
Step 1:
Step 2:
Step 3:
Follow these steps to add or subtract fractions:
Before we can add or subtract fractions,
we must have:
Example:
Common Denominators
If denominators are different, find the least common denominator.
Multiply the numerators AND denominators by a number to get the LCM.
Add the numerators. Keep the denominators the same. If possible, reduce.
+ = =
What is + ?
add and subtract fractions.
A fraction is a ratio of two numbers. In other words, an integer divided by an integer.
1.) 28 – 5x + 2 + 4x a.) The expression has ____ terms. b.) Underline the variables c.) Box the coefficients d.) Circle the constants
Rational Numbers (Adding/Subtracting Fractions)
Reflection: I learned…
1.) + = + = =
2.) − = − = =
= + = =
3.) + = +
©2016 Math in Demand
On a rating of 1-5, how comfortable are you with this concept?
(5 is the highest)
1 2 3 4 5
Answer
Answer
Answer
Rational Numbers (Multiplying Fractions)
©2016 Math in Demand
I can ___________________
_______________________
Multiplication Signs Negative x Positive = Negative Positive x Negative = Negative Negative x Negative = Postive Positive x Positive = Positive
How do we multiply fractions? ∙ =
Multiply the numerators together
and multiply the denominators
together!!!
“Multiplying across” is not the same thing as “cross multiplication”.
Examples:
Make sure to SIMPLIFY your answers!
multiply fractions.
1.) ∙ = =
− = −
2.) ∙ − =
Rational Numbers (Multiplying Fractions)
Reflection: I learned…
1.) ∙ =
2.) ∙ − = ∙ − = − = −
3.) − ∙ − = − ∙ − = =
©2016 Math in Demand
On a rating of 1-5, how comfortable are you with this concept?
(5 is the highest)
1 2 3 4 5
Answer
Answer
Answer
-24
Rational Numbers (Dividing Fractions)
©2016 Math in Demand
I can ___________________
_______________________
Division Signs Negative ÷ Positive = Negative Positive ÷ Negative = Negative Negative ÷ Negative = Postive Positive ÷ Positive = Positive
How do we divide fractions? ÷ = ∙ =
Step 1: Keep the first fraction the same. Step 2: Flip the second fraction.
Step 3: Multiply the fractions. Step 4: Simplify.
Examples:
divide fractions.
∙ = =
1.) ÷ =
∙ − = − = −
2.) ÷ − =
Rational Numbers (Dividing Fractions)
Reflection: I learned…
1.) ÷ = ∙ = =
= −
2.) − ÷ = − ∙ = −
3.) − ÷ − = − ∙ − =
©2016 Math in Demand
On a rating of 1-5, how comfortable are you with this concept?
(5 is the highest)
1 2 3 4 5
Answer
Answer
Answer
−
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