**Unit 1: Integers and Rational Numbers**

**Integers and Rational Numbers**

Skill 1:

Skill 2

Skill 3

Skill 4

A rational number is a number that can be written as a fraction. An irrational number is a number that cannot be shown as a simple fraction.

compare and order integers and absolute value.

In skill 1, students must demonstrate mastery comparing and ordering integers with and without absolute value, which is the distance from zero on a number line.

add and subtract integers

In skill 2, students essentially combine integers which are either negative or positive.

**UNIT 1:**

multiply and divide integers

STANDARDS PRACTICE

STANDARDS PRACTICE

STANDARDS PRACTICE

write algebraic equations

STANDARDS PRACTICE

While learning skill 4, students must show the ability to create an equation by breaking down a word problem given to them and using variables.

Skill 5

solve 1-step addition and subtraction equations

STANDARDS PRACTICE

Using their knowledge from skill 4, students demonstrate the ability to solve the one-step equations they previously formulated by using inverse operation.

Skill 6

solve 1 step multiplication and division equations

Students solve one step equations in which they use inverse operations to isolate the variable and complete the problem.

STANDARDS PRACTICE

Skill 12

comparing and ordering rational numbers

students compare and order rational numbers based on their values.

STANDARDS PRACTICE

Skill 13

multiply positive and negative fractions

STANDARDS PRACTICE

Skill 7

solve PEMDAS problems

Students solve expressions using PEMDAS, an ordering system that stands for parentheses, exponents, multiplication, division, addition and subtraction.

Skill 8

evaluate expressions using substitution

Students are able to substitute given values in the place of variables in an expression and complete the problem.

Skill 9

know and apply math properties

Students know and are able to apply math properties such as Commutative, Associative, and Identity properties.

Skill 10

write decimals as fractions

Skill 11

STANDARDS PRACTICE

STANDARDS PRACTICE

Skill 14

Skill 15

Write a fraction as a terminating or repeating decimal

!

Substitution problems are solved exactly like order of operations problems, with PEMDAS, once the values are substituted.

STEPS:

1: PARENTHESES. Always complete what's in the parentheses before anything else.

(PEMDAS)

So in a problem like this, you would do (7x8) first.

(7x8) + 6 x 2 - 4

2

2: EXPONENTS. When you're done with parentheses, move on to exponents.

Now the problem reads:

56 + 6 x 2 - 4

2

With 6 solved, we are one step closer to solving the problem.

2

56 + 36 x 2 - 4

3: MULTIPLICATION & DIVISION

Step three, multiplication and division, is done left to right. So multiplication is not done before division in the problem.

56 + 36 x 2 - 4

56 + 72 - 4

4: ADDITION AND SUBTRACTION

56 + 72 - 4

The same rules from Step 3 apply for Step 4: solve from left to right, so now we would solve 56 + 72 first.

128 - 4

124!

A property is a feature of an object that is always true. The following properties are true for all numbers.

TERMINATING OR REPEATING?

Every rational number can be written as a terminating or repeating decimal. A decimal like 0.625 is called a terminating decimal because the division ends, or terminates, with a remainder pf zero.

If the division does not end, a pattern of digits repeats. Repeating decimals have a pattern in their digits that repeats without end. A repetend is used to indicate that a digit or a group of digits repeats.

Divide: top in, bottom out.

MULTIPLY ACROSS:

the same rules for integers apply; same signs, the answer will be positive. Different signs, and the answer will be negative.

When multiplying positive and negative integers, these rules apply. If the signs of the two numbers are the same, the product or quotient will be positive. If the signs are different, the answer will be negative.

TERMINATING DECIMALS

0.25 = 1/4

0.0025 = 1/400

1.35 = 1 = 1

35/100

7/20

REPEATING DECIMALS

0.7 = 7/9

0.34 = 34/99

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STANDARDS PRACTICE

Skill 16

Skill 17

divide positive and negative fractions

add and subtract like fractions

add and subtract unlike fractions

solve equations with rational numbers

The same rules for integers apply; same signs, the answer will be positive. Different signs, and the answer will be negative.

MULTIPLICATIVE INVERSE:

A number multiplied by its multiplicative inverse is equal to 1 (one)

To solve a fraction division problem, the second dividend must be changed into its multiplicative inverse. That way, you can multiply the two numbers.

STANDARDS PRACTICE

When adding or subtracting fractions with like denominators, the denominators stay the same when the numerators are added or subtracted, unless the answer can be simplified afterwards.

STANDARDS PRACTICE

When adding and and subtracting unlike fractions, students must find common denominators of both the numbers in the equation and only once the denominators are the same can the student complete the problem. Simplify if needed.

STANDARDS PRACTICE

Solve one step equations with rational numbers (fractions or decimals) in them