Name: __________________________________________ Date: _________
Math 7 Ms. Conway
Review Packet: Unit 1 – The Number System
Key Concepts
Module 1: Adding and Subtracting Integers
7.NS.1, 7.NS.1a, 7.NS.1b, 7.NS.1c, 7.NS.1d, 7.NS.3, 7.EE.3
• To add integers with the same sign, add the absolute value of the integers and use the
sign of the integers for the sum. (Lesson 1.1)
• To add integers with different signs, subtract the smaller absolute value from the greater
absolute value. The sign of the sum will be the sign of the addend with the greater
absolute value. (Lesson 1.2)
• Subtracting one integer from another integer is the same as adding its opposite. (Lesson
1.3)
• To solve multi-step problems involving addition and subtraction of integers, use a four
step problem-solving plan. (Lesson 1.4)
Module 2: Multiplying and Dividing Integers
7.NS.2, 7.NS.2a, 7.NS.2b, 7.NS.2c, 7.NS.3
• The product of two integers with the same sign is positive. The product of two integers
with different signs is negative. (Lesson 2.1)
• The quotient of two integers with the same sign is positive. The quotient of two integers
with different signs is negative. (Lesson 2.2)
• To simplify an expression with more than one operation, use the order of operations.
(Lesson 2.3)
Module 3: Rational Numbers
7.NS.1, 7.NS.1a, 7.NS.1b, 7.NS.1c, 7.NS.1d, 7.NS.2, 7.NS.2a, 7.NS.2c, 7.NS.2d, 7.NS.3, 7.EE.3
• A number that can be written as a terminating decimal or a repeating decimal is a
rational number. (Lesson 3.1)
• Addition and subtraction of rational numbers can be demonstrated on a number line.
(Lesson 3.2)
• To subtract a number, add its opposite. (Lesson 3.3)
• The product of two numbers with different signs is negative. The product of two numbers
with the same signs is positive. (Lesson 3.4)
• The quotient of two numbers with different signs is negative. The quotient of two
numbers with the same signs is positive. (Lesson 3.5)
• Solving real-world and mathematical problems involves applying properties of
operations as well as being able to strategically convert rational numbers to any form to
better facilitate computation and estimation. (Lesson 3.6)
Adding and Subtracting Rational Numbers Flow Chart
Are there any fractions, mixed numbers, or decimals?
If not, skip down to “Subtracting Integers” or “Adding Integers”.
Fractions/Mixed Numbers Decimals
If there are mixed numbers, Put the number with the
convert mixed numbers to improper fractions. greater absolute value on top.
(see below)
If the fractions have different denominators,
find the least common denominator (LCD). Line up the decimals.
(see below)
Add zeros as place holders,
if necessary.
Decide if you have to add or subtract, and then follow the directions below.
Subtracting Integers
a number minus a negative number a number minus a positive number
(ex: 2 – – 6 OR –2 – – 6) (ex: 2 – 6 OR –2 – 6)
Use
Keep Change Change!
change signs to positive Change it to adding a negative number
(ex: 2 + 6 OR –2 + 6) by putting a + in front of the –.
Adding Integers
same signs different signs
find the sum (+); find the difference (–);
keep the original sign use the sign of the number
with the larger absolute value
If your fraction needs to be simplified:
Convert the improper fraction to a mixed number, if necessary:
Simplify:
Divide by the same number in the numerator and denominator
HELPFUL REMINDERS:
Subtracting Integers Hint:
Keep Change Change
Keep the first number Change the operational sign Change the second number to its additive inverse
Example:
K CC KCC
–3–5 11 – – 2
–3+–5 11 + + 2
Same sign: Add, keep original sign 3+5=8! –8 11 + 2 = 13 ! + 13 Same sign: Add, keep original sign
Converting Mixed Numbers to Improper Fractions
Finding the Least Common Denominator (LCD)
1) Write the multiples of both denominators by counting by each
number.
2) Circle the numbers that are in common (the same) between both sets
of multiples.
3) Choose the least (smallest) of the common numbers. This is the
LCD.
4) Change both original fractions into equivalent fractions using the
LCD. Do this by multiplying. (THINK: What do you have to multiply
to get from the original denominator to the LCD? Multiply that
number by the numerator to get the new numerator.)
!
Multiplying Rational Numbers Flow Chart
Multiplying Multiplying Multiplying
Whole Numbers Fractions Decimals
& Mixed Numbers
Convert Write numbers one
mixed numbers to on top of the other.
improper fractions. IGNORE DECIMAL
(see above) POINTS!
Multiply: Multiply the way you
numerator " numerator normally would.
Multiply the absolute denomin ator " denomin ator
value of both numbers.
Move the decimal in the
! Simplify the fraction. product the correct
number of spaces
(based on factors).
Convert to a mixed
number, if necessary.
(See flow chart above.)
Decide the sign of the answer.
(see chart below)
Sign of Factor p Sign of Factor q Sign of Product pq
+ - -
- + -
+ + +
- - +
factor = a number that is multiplied by another number to get a product
product = the answer in a multiplication problem
Dividing Rational Numbers Flow Chart
Dividing Dividing Fractions Dividing Decimals
Whole Numbers & Mixed Numbers
Convert Set up a long division
mixed numbers to problem, using the absolute
improper fractions. values of both numbers.
(see above)
Set up and use KCF. Move the decimal in the
(Keep Change Flip) divisor to the end.
Move the decimal in the
Divide the absolute values Multiply: dividend the same number
of both numbers. numerator " numerator of spaces, and then move it
denomin ator " denomin ator up into the quotient too.
Simplify the fraction. Do long division
! as you normally would.
Convert to a mixed number,
if necessary.
(See flow chart above.)
Decide the sign of the answer.
(see chart below)
Sign of Dividend p Sign of Divisor q Sign of Quotient p
q
+ - -
- + -
+ + +!
- - +
The location of the negative sign does not affect the quotient.
For a fractional quotient, the negative sign can be in the numerator,
in the denominator, or in front of the fraction.
#
= undefined
0
0
= zero
#
! !
!
Writing Rational Numbers as Decimals
To convert rational numbers to decimals, use long division.
numerator
= denominator) numerator
denominator
Remember:
If the decimal comes to an end, it is a terminating decimal.
If the decimal continues forever, it is a repeating decimal.
An improper fraction is greater than 1. Therefore,
! its equivalent decimal should also be greater than 1.
--------------------------------------------
1) Divide: denomin ator) numerator
2) Remember to add decimals (in the dividend AND the quotient), and zeros.
3) Then, either:
a. Add zeros in the dividend and continue dividing until the remainder is 0.
! OR
b. Stop dividing once you discover a repeating pattern in the quotient.
i. Write the quotient with its repeating pattern and indicate that the
repeating numbers continue by putting a line over only the
repeating numbers.
Writing Mixed Numbers as Decimals
To convert mixed numbers to decimals,
rewrite the fractional part of the number as a decimal using long division.
numerator
= denominator) numerator
denominator
Remember:
Keep the whole number from the mixed number as the
whole number in the decimal, by writing it to the
LEFT of the decimal point.
!
Mixed numbers are greater than 1 so their
decimal equivalents should also be greater than 1.
--------------------------------------------
1) Turn the fractional part into a long division problem.
2) Rewrite the fractional part of the number as a decimal.
3) Rewrite the mixed number as the sum of the whole part and the decimal part.
Adding Three or More Rational Numbers
When adding more than two rational numbers,
GROUP NUMBERS WITH THE SAME SIGN
and combine them first.
Then, combine numbers with different signs,
using the rules for adding/subtracting integers.
REMEMBER:
When lining up decimals, if there are different amounts of numbers
after the place value, add zero placeholders.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
For example:
2.32 – 11.2 + 3.95
Since two numbers are positive, (2.32 + 3.95) – 11.2
group those together:
Combine those numbers: 6.27 – 11.2o
Combine numbers with different signs, –4.93
using rules for adding integers:
!
!
!
! "#$%&'$($(!
" )*+%$%&(!
# ,-&.*-./"&.+%! $ .0.(.+%!
! !
!"#$%&'!(%(#%")*+(%
% 11.&.+%! & ,2&#"/&.+%!
! !
