Chapter 8

This study sheet provides students and parents with the basic concepts of each chapter. Students still need to apply these skills in

context. They need to know when to apply each concept, often after working through a word problem, table, chart, or graph. Some

problems may be more challenging than the ones shown here, but students first need to understand these basic concepts. There

are usually several ways to solve a math problem, but this guide will show you the easiest way for 6th

graders. The sections are listed

in the order that I plan on teaching them, and that is subject to change. We do not use every section of the textbook.

Click on the blue links to navigate through the study guide. You can also view videos at Khan Academy and Virtual Nerd.

Section

8.1

Topic:

Solutions of Equations

Common errors to avoid:

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An equation is a number sentence with an equal sign.

A variable, like x, takes the place of a number.

To determine if a given value is a solution to an equation, just replace the variable with that value.

Example: x – 6 = 15 Is 10 a solution? 10 – 6 = 9 So, no, 10 is not a solution.

You should write down the equation with the value “plugged in”. x + 7 = 18 ; x =11 It’s much easier to understand if you write 11 + 7 = 18.

Determine whether the given value is a solution to the equation.

x = 20 ; x = 80

Answer

IXL: Choose the Solution Mathopolis: Solutions of Equations

Section

8.2

Topic:

Write Equations

Common errors to avoid:

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You can translate between words and numbers with math symbols.

Example: “Ten more than a number is 30.” is the same as n + 10 = 30

You need to know your vocabulary!

Here are the most common phrases:

Addition: sum, more than Subtraction: difference, less than Multiplication: product, sets, of Division: quotient, groups, per Equal: is

Subtraction tends to be the most difficult, because the order matters! “x less than 20 is 5” means to start with 20, then subtract x from it, to equal 5. 20 – x = 5

Write an equation for the word sentences. “The quotient of a number and 7 is 6.”

Answer

Writing Equations IXL: Write the Equation Buzz Math: Real Situations Multiple Choice

Section

8.3

Topic:

Model and Solve Addition Equations Common errors to avoid:

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You can build models or draw models to solve addition equations.

Example: The equation x + 3 = 7

can be represented by The goal is to get x by itself.

So, cross out the 3 extra circles on the left side. To keep the equation balanced, also cross off 3 circles on the right side. Now the equation reads x = 4, which is your solution.

Remember to keep your equation balanced. Whatever you do to one side, you have to also do to the other.

Model and solve the equation x + 2 = 8. Answer

IXL: Choose the Model Manipulatives Click on Manipulatives. Choose Algebra tiles. Drag the green x’s and yellow 1’s.

Section

8.4

Topic:

Solve Addition and Subtraction Equations Common errors to avoid:

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To solve an equation, you want to get the variable all by itself, so you can find its value. We use inverse operations, which means opposite operations, to work backwards. Example: n + 14 = 44 You want to get rid of the “+14”, so subtract it from both sides. n + 14 = 44 -14 -14 Now you have n = 30 You can check: 30 + 14 = 44 , so you know you’re correct!

You can and should always check your solution! Just plug the solution back into the equation to check it.

Solve the equation. 25 = x – 11 Answer

Equation Soccer Interactive Crossword Puzzle Basketball Equation Game Fast Circles

AAA Math

Section

8.5

Topic:

Model and Solve Multiplication Equations Common errors to avoid:

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You can build models or draw models to solve multiplication equations.

Example: The equation 3x = 12

can be represented by

The goal is to get x by itself. Since the inverse (opposite) of multiplication is division, divide by splitting the x’s and the objects into 3 equal groups. So the solution is x = 4

It’s easiest if you stack the X’s vertically. 2x = 8 Should look like

NOT

Model and solve the equation 5x = 15 Answer

Manipulatives Click on Manipulatives. Choose Algebra tiles. Drag green x’s and yellow 1’s.

Section

8.6

Topic: Solve Multiplication and Division Equations

Common errors to avoid:

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To solve an equation, you want to get the variable all by itself, so you can find its value. We use inverse operations, which means opposite operations, to work backwards.

Example:

= 6

You want to get rid of the “÷ 4”, so multiply by 4 on both sides.

= 6

4 4 Now you have n = 24

You can check:

= 6 , so you know you’re correct!

You need to recognize the different ways to show division or multiplication.

means x ÷ 2

2x means 2 times x

Solve the equation by showing your inverse operation work. 4x = 28 Answer

Pong! Number Balls Division Matching AAA Math Division Equations

Section

8.7

Topic:

Solve Equations with Fractions Common errors to avoid:

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To solve an equation, you want to get the variable all by itself, so you can find its value.

We use inverse operations, which means opposite operations, to work backwards.

Example:

y = 25

You want to get rid of the “times

”, so divide by

on both sides.

y = 25

÷

÷

Now you have y = 50

You can check:

(50) = 25 , so you know you’re correct!

You need to recognize the different ways to show division or multiplication.

means z ÷ 3

z means

times z

Bill ate

bag of

chips, which means he ate 24 chips. Use the equation

b = 24 to

determine the number of chips that are in a full bag. Answer

Jeopardy IXL: Fraction Equations

Section

8.8

Topic:

Solutions of Inequalities

Common errors to avoid:

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An inequality is a number sentence with >, <, ≥, or ≤ sign.

A variable, like x, takes the place of a number.

To determine if a given value is a solution to an inequality, just replace the variable with that value.

Examples: y < 0 = 15 Is 4 a solution? Since 4 < 0 is not true, then no, 4 is not a solution. n ≥ -3 Is 0 a solution? Since 0 ≥ -3 is true, then yes, 0 is a solution.

You should write down the inequality with the value “plugged in”. x < 7 ; x = 0.5 It’s much easier to understand if you write 0.5 < 7 Remember, less is to the left on a number line.

Determine whether the given value is a solution to the equation.

a < 3

; x =

Answer

Yes or No? IXL: You decide!

Section

8.9

Topic:

Write Inequalities

Common errors to avoid:

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To write an inequality, first write down these symbols, veritically:

> < =

Then, just circle the symbol or symbols that fit that situation. Example: You will receive at least $25.

> Yes, because you could get more than $25 < No, because you won’t receive less than $25

= Yes, because you could receive exactly $25

So combine the symbols to write x ≥ 25.

If you don’t write the symbols vertically and circle them, it’s often too much information to remember in your head. Make it easy and write them down!

> < =

Write an inequality for this sentence: The temperature was no more than 70⁰. Answer

Mathopolis Writing Inequalities

Section

8.10

Topic:

Graph Inequalities

Common errors to avoid:

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You can represent an inequality by graphing it on a number line. Example: Graph the inequality x > 4 Just find 4 on a number line, and mark it with a circle.

Since the greater than symbol points to the right, draw an arrow to the right. This shows that any number greater than 4 is a solution. For less than, draw an arrow to the left. For greater than or equal to, or less than or equal to, fill in the circle, and draw the arrow. Example: Graph the inequality x ≤ -3

This shows that -3 and all values less than -3 are solutions.

You have to pay attention to the direction of the symbol, and match it to the direction of your arrow. < means left > means right

Draw a number line and graph the inequality n ≥ -4 Answer

Interactive IXL: Graph the Inequality Mathopolis: Graph Inequalities IXL: Write the Inequality

Section Answer Return to Study Guide

8.1

Yes.

(80) = 20

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Section Answer Return to Study Guide

8.2

n ÷ 7 = 6 or

= 6

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8.3

x = 6

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8.4

36 = x , or x = 36

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8.5

x = 3 Model:

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8.6

x = 7 Work: 4x = 28

4 4 Both sides were 4

x = 7

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8.7

b = 36 , or There are 36 chips in a full bag.

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8.8

Yes,

< 3

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Section Answer Return to Study Guide

8.9

t ≤ 70 (any variable can be used)

work: > It is not more than 70. < It could be less than 70. = It could be equal to 70.

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8.10

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