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Unit 3 Reference Sheets

Standards-based reference sheets for 7th Grade Unit 3 covering expressions, one- and two-step equations, and inequalities with graphing.

Iisaac_daigle

Unit 3 Reference Sheets

Standards-based reference sheets for 7th Grade Unit 3 covering expressions, one- and two-step equations, and inequalities with graphing.

Iisaac_daigle

7th Grade Math Unit 3 Reference Sheet for Quiz

Expressions, Integer Rules, One-Step & Two-Step Equations

Quiz Reference

Writing Expressions Key Words and Phrases

ADDITION +

• Sum
• More than
• Increased by
• Total
• Plus
• All together

SUBTRACTION −

• Difference
• Less than
• Decreased by
• Subtracted from
• Minus

MULTIPLICATION •

• Times
• Product
• Twice (\(2n\))
• Double / Triple
• Of
• Per

DIVISION /

• Divided by
• Quotient
• Half / Third
• Separate

"Turn Around" Phrases: Phrases like less than and subtracted from indicate that you need to reverse the order!

"4 less than \(n\)" → \(n - 4\)

Integer Operation Rules

ADDING INTEGERS

Same Signs → Friends (Add)

Add numbers and keep the sign.

Different Signs → Fight (Subtract)

Subtract absolute values. Who wins? More \(+\) or More \(-\)?

SUBTRACTING INTEGERS

Change to Addition: KEEP CHANGE CHANGE

\(a - (-b) \rightarrow a + (+b)\)

\(a - b \rightarrow a + (-b)\)

MULTIPLYING & DIVIDING

Same signs = positive • Different signs = negative

\((+) \cdot (+) = (+)\)

\((-) \cdot (+) = (-)\)

\((-) \cdot (-) = (+)\)

\((+) \div (-) = (-)\)

\((-) \div (+) = (-)\)

\((-) \div (-) = (+)\)

Solving Equations (One-Step & Two-Step)

Steps to Solve One-Step Equations:

Locate the variable.
Determine the operation tied to the variable.
Use inverse operations on both sides of the equal sign.
Check your solution!

\(x + 2 = 7\)

\(-2 \quad -2\)

\(x = 5\)

\(x - 3 = 6\)

\(+3 \quad +3\)

\(x = 9\)

\(3x = 12\)

\(\div 3 \quad \div 3\)

\(x = 4\)

\(\frac{x}{2} = 5\)

\(\cdot 2 \quad \cdot 2\)

\(x = 10\)

Steps to Solve Two-Step Equations:

Locate the variable.
Undo addition or subtraction first.
Undo multiplication or division next.
Check your solution!

\(2x + 5 = 11\)

\(-5 \quad -5\) (undo +)

\(2x = 6\)

\(\div 2 \quad \div 2\) (undo •)

\(x = 3\)

\(\frac{x}{4} - 3 = 2\)

\(+3 \quad +3\) (undo −)

\(\frac{x}{4} = 5\)

\(\cdot 4 \quad \cdot 4\) (undo /)

\(x = 20\)

Simplifying Expressions: To simplify an algebraic expression: 1. Distribute → 2. Combine like terms. \(a(b + c) = ab + ac\)

7th Grade Math Unit 3 Reference Sheet for Quiz Keep both sides balanced by performing identical operations

Unit Test Reference Sheet

Unit 3 Reference Sheet: Expressions & Equations

Combining Like Terms, Distributive Property, Factoring & Equation Steps

Page 1 of 2

COMBINING LIKE TERMS

\(6b\) + \(4c\) − \(2b\) + \(7c\) = \(\mathbf{4b + 11c}\)

Group identical variable families including the sign in front of each term.

THE DISTRIBUTIVE PROPERTY

\(a(b + c)\)

\(= ab + ac\)

\(a(b - c)\)

\(= ab - ac\)

\(6(x + 2) = 6(\phantom{x}) + 6(\phantom{2}) = 6x + 12\)

\(\underline{\phantom{a}}(\underline{\phantom{a}} + \underline{\phantom{a}}) = \underline{\phantom{a}}\cdot\underline{\phantom{a}} + \underline{\phantom{a}}\cdot\underline{\phantom{a}}\)

\(\underline{\phantom{a}}(\underline{\phantom{a}} - \underline{\phantom{a}}) = \underline{\phantom{a}}\cdot\underline{\phantom{a}} - \underline{\phantom{a}}\cdot\underline{\phantom{a}}\)

FACTORING EXPRESSIONS

Divide each term by the Greatest Common Factor (GCF):

\(\underline{\phantom{GC}}(\underline{\phantom{a}} + \underline{\phantom{b}})\)

↑ GCF ↑ FACTORS

\(\underline{\phantom{GC}}(\underline{\phantom{a}} - \underline{\phantom{b}})\)

↑ GCF ↑ FACTORS

Simplifying Expressions:

An algebraic expression is in simplest form when it has combined like terms and no parentheses.

1. Distribute → 2. Combine like terms

INTEGER OPERATION RULES

ADDING INTEGERS

Same Signs → Friends (Add)

Add values, keep the sign.

Different Signs → Fight (Subtract)

Subtract values. Who wins? More \(+\) or More \(-\)?

SUBTRACTING INTEGERS

Change to Addition: KEEP CHANGE CHANGE

\(a - (-b) \rightarrow a + (+b)\)

\(a - b = a + (-b)\)

MULTIPLYING DIVIDING

Same signs = positive • Different signs = negative

\((+) \cdot (+) = (+)\)

\((-) \cdot (+) = (-)\)

\((-) \cdot (-) = (+)\)

\((+) \div (-) = (-)\)

\((-) \div (+) = (-)\)

\((-) \div (-) = (+)\)

Double sign rule: \(-(-b) = +b\)

SOLVING EQUATIONS

Perform inverse operations on both sides to isolate the variable

Inverse Operations:

Addition (\(+\)) Undo by Subtracting

Subtraction (\(-\)) Undo by Adding

Simplifying Expressions Practice Worksheet

Chapter 3 (3.1–3.4) Practice Worksheet

Simplifying Expressions • Show your work to receive partial credit. (32 points)

Name:

Date: Period:

Simplify each expression.

1. \(6m + 7 - 2m + 3\) (2 pts)

Answer:

2. \(9 + 5k + 2k - 4\) (2 pts)

Answer:

3. \(3y + 12 - 7y\) (2 pts)

Answer:

4. \(-\frac{1}{4}x - 5 + \frac{3}{4}x - 2\) (3 pts)

Answer:

5. \(-4.2w + 1.5 - 1.6w - 7.3\) (3 pts)

Answer:

6. \(-\frac{2}{5}x - 3 + \frac{7}{10}x - 8\) (3 pts)

Answer:

Chapter 3 Practice Worksheet Page 1 of 2

Chapter 3 (3.1–3.4) Practice Worksheet

Simplify each expression / Factor

Page 2 of 2

7. \(5(8x - 4)\) (2 pts)

Answer:

8. \(4(3x + 6)\) (2 pts)

Answer:

9. \(6x + 4(x + 3) + 3(x + 5)\) (3 pts)

Answer:

10. \(\frac{1}{4}(16x - 28) + 7x\) (3 pts)

Answer:

11. \(-3.2(p + 3) + 6.4\) (3 pts)

Answer:

12. Factor: \(7x + 42\) (2 pts)

Answer:

13. Factor: \(12x + 20y\) (2 pts)

Answer:

Chapter 3 Practice Worksheet Total Points: 32

Multiplication (\(\cdot\)) Undo by Dividing

Division (\(/\)) Undo by Multiplying

Two-Step Equations Solving Order:

Locate the variable.
Undo the addition or subtraction first (zero out constant).
Undo the multiplication or division next (isolate variable).
Check your solution by substituting into the original equation!

\(2x + 5 = 11\)

\(-5 \quad -5\)

\(2x = 6 \rightarrow \mathbf{x = 3}\)

\(\frac{x}{4} - 3 = 2\)

\(+3 \quad +3\)

\(\frac{x}{4} = 5 \rightarrow \mathbf{x = 20}\)

Unit 3 Mathematics • Page 1 of 2 Turn page for Key Words, Inequalities, and Number Line Graphing

Unit 3 Reference Sheet: Inequalities & Graphing

Translation Vocabulary, Inequality Symbols, Graphing Rules & Solving Inequalities

Page 2 of 2

WRITING EXPRESSIONS KEY WORDS AND PHRASES

ADDITION (+)

Sum, More than, Increased by, Total, Plus, All together

SUBTRACTION (−)

Difference, Less than, Decreased by, Subtracted from, Minus

MULTIPLICATION (•)

Times, Product, Twice, Double / Triple, Of, Per

DIVISION (/)

Divided by, Quotient, Half / Third, Separate

Watch out for Subtraction! Phrases like less than and subtracted from indicate you must reverse the order:

"5 less than \(n\)" → \(n - 5\)

INEQUALITY SYMBOLS

<

• "less than"

• "fewer than"

>

• "greater than"

• "more than"

• "exceeds"

≤

• "less than or equal to"

• "at most"

• "maximum"

≥

• "greater than or equal to"

• "at least"

• "no less than"

READING / WRITING INEQUALITIES

Format for writing/reading in words:

All numbers

[Variable Phrase] [Inequality Phrase] [Number]

Format for writing inequalities:

\(\underline{\text{Variable}}\) \(\underline{\text{Symbol}}\) \(\underline{\text{Number}}\)

Example: "All numbers greater than or equal to \(-3\)" → \(x \ge -3\)

GRAPHING INEQUALITIES

Step 1: Use an open or closed circle on the number line.

Step 2: Draw an arrow in the direction of numbers that make the statement true.

Open circle ○ Use with > or < (not included)

Closed circle • Use with ≥ or ≤ (included)

When variable is on the left, arrow points in direction of symbol (\(x > a \rightarrow\), \(x < a \leftarrow\)).

SOLVING INEQUALITIES (Same process as solving equations) Reversal Rule

\(x - 5 \le 3\)

\(+5 \quad +5\)

\(x \le 8\)

Closed circle •, left

\(y + 4 > 9\)

\(-4 \quad -4\)

\(y > 5\)

Open circle ○, right

\(6x \le 18\)

\(\div 6 \quad \div 6\)

\(x \le 3\)

Closed circle •, left

\(\frac{x}{5} > 4\)

\(\cdot 5 \quad \cdot 5\)

\(x > 20\)

Open circle ○, right

Multiplying or Dividing by a Negative Number: You must reverse (flip) the inequality sign!

\(-2x < 6 \rightarrow x > -3\)

Unit 3 Mathematics • Page 2 of 2 Check circle type (open vs. closed) and verify arrow direction

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