Grade 6 Workbook — Chapter 11: Writing and Solving One-Step Equations

SOL 6.PFA.3 · Companion to Textbook Chapter 11

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/.

PAGE 1 — Chapter opener

Chapter 11 · Writing and Solving One-Step Equations

Standard 6.PFA.3

In this chapter you will:

• Use the words equation, variable, expression, term, and coefficient correctly
• Model and solve equations with balance scales, algebra tiles, and colored chips
• Solve one-step equations using properties of equality
• Confirm every solution with a model or a check
• Write an equation from a situation
• Write a situation from an equation

Words to know: variable · expression · equation · term · constant · coefficient · unit fraction · solution · zero pair

Coefficients here are integers or unit fractions. Every other number is an integer. No calculator needed.

PAGE 2 — The words of algebra

11.1 The Language of Algebra

Fill in the blanks.

A ____________ is a letter that stands for a number.

An ____________ is a mathematical phrase with no equal sign. You ____________ it.

An ____________ is a mathematical sentence with an equal sign. You ____________ it.

A ____________ is the number multiplied by a variable.

A ____________ is a term that is only a number.

Expression or equation? Circle one for each.

Name the coefficient.

PAGE 3 — Breaking an expression apart

Parts of an Expression

Complete the table.

Is it a solution? Show your substitution.

Is $x = 4$ a solution of $x + 6 = 10$? Substitute: ____________ Yes / No

Is $n = 6$ a solution of $n + 5 = 12$? Substitute: ____________ Yes / No

Build your own. Write an expression with two terms and a coefficient of $5$: ______________

Turn it into an equation: ______________

Explain. Ravi says $x$ has no coefficient. What is he missing?

PAGE 4 — Exit ticket 11.1

Exit Ticket · Lesson 11.1

Name: ________________________ Date: ____________

1. Is $5p = 30$ an expression or an equation? ______

2. Name the coefficient in $\tfrac{1}{3}n$. ______

3. How many terms are in $2y + 7$? ______ What is the constant? ______

4. What does it mean for a number to be a solution of an equation?

PAGE 5 — The balance scale

11.2 Balance Scales

FIGURE: fig1-balance-scale-equation.png (half width, right)

[A level two-pan balance scale with a block labeled $x$ and three unit blocks on the left pan and eight unit blocks on the right pan, with the equation $x + 3 = 8$ printed beneath.]

A balanced scale is the picture of an equal sign. Remove the same amount from both pans and it stays balanced.

Write the equation each scale shows, then solve it.

Explain. Jamal removes 2 weights from the left pan only. What happens, and what should he do instead?

PAGE 6 — Algebra tiles

Algebra Tiles

FIGURE: fig2-algebra-tiles-key.png (full width)

[A legend showing four tiles side by side: a long green rectangle labeled $x$, a long red rectangle labeled $-x$, a small yellow square labeled $1$, a small red square labeled $-1$.]

FIGURE: fig3-algebra-tiles-solving.png (full width)

[Three stacked rows showing $x + 3 = 8$ modeled with tiles, then three unit tiles crossed out on each side, then the result $x = 5$.]

A zero pair is one positive tile and one negative tile together. Its value is ______, so removing it changes ____________.

Write the equation each tile model shows, then solve.

Describe your moves for $x - 4 = 2$.

PAGE 7 — Confirming with a model

Confirm the Solution

Solving finds an answer. Confirming proves it. Replace the variable tile with that many unit tiles and count both mats.

Confirm or correct each proposed solution.

Apply it. A backpack and a 3-pound weight balance an 11-pound weight.

Equation: ______________ Move: ______________ Backpack weight: ______ lb

Confirm on the scale: _______________________________________________

Explain. Why is confirming a solution worth the extra time?

PAGE 8 — Exit ticket 11.2

Exit Ticket · Lesson 11.2

Name: ________________________ Date: ____________

1. A scale holds a block $x$ and 5 unit weights on the left, 12 unit weights on the right. Write the equation. ______________

2. Solve it and name your move. Solution: ______ Move: ______________

3. What is a zero pair? _______________________________________________

4. How would you use tiles to confirm $x = 7$ solves $x + 3 = 10$?

PAGE 9 — Addition and subtraction equations

11.3 Adding and Subtracting to Solve

FIGURE: fig4-inverse-operations-balance.png (full width)

[Two balance scale diagrams side by side. The first shows a block $x$ with seven weights against twelve weights, captioned $x + 7 = 12$. The second shows the block alone against five weights, captioned $x = 5$. An arrow between them reads "subtract 7 from both sides".]

Addition and subtraction are inverse operations. Whatever the equation does to the variable, do the opposite — to both sides.

Solve. Show the move and the check.

PAGE 10 — More addition and subtraction

Keep the Balance

Solve and check.

$-4 = q + 9$ $q =$ ______ Check: ______________

$r - 20 = -20$ $r =$ ______ Check: ______________

$b + 13 = 2$ $b =$ ______ Check: ______________

$m - 6 = -2$ $m =$ ______ Check: ______________

Write your own. Write an equation with solution $x = -8$ that is solved by adding a number to both sides.

Apply it. After spending $17, Elena has $28 left.

Equation: ______________ Starting amount: $______ Check: ______________

Find the error. Devin solves $x + 9 = 4$ and gets $x = 13$.

What went wrong? _______________________________________________

Correct solution: ______ How would a check have caught it? ____________________

PAGE 11 — Exit ticket 11.3

Exit Ticket · Lesson 11.3

Name: ________________________ Date: ____________

1. Solve $x + 8 = 15$. ______

2. Solve $n - 9 = -2$. ______

3. Is $y = -5$ the solution of $y + 12 = 7$? Show your check. ______________

4. Why must you do the same operation to both sides?

PAGE 12 — Multiplication and division equations

11.4 Multiplying and Dividing to Solve

FIGURE: fig5-equal-groups-tiles.png (half width, right)

[Two rows of algebra tiles showing $3x = 12$: three $x$ rectangles against twelve unit squares, then the same tiles divided by dashed lines into three equal groups pairing one $x$ with four units.]

Name the coefficient, then choose the move.

PAGE 13 — Unit fractions and negatives

Watch the Coefficient

$\tfrac{1}{4}x$ and $\tfrac{x}{4}$ mean the same thing. Both are undone by multiplying both sides by 4.

Solve and check.

$12m = -60$ $m =$ ______ Check: ______________

$\tfrac{1}{10}v = -9$ $v =$ ______ Check: ______________

$-4x = -28$ $x =$ ______ Check: ______________

$\tfrac{1}{2}w = -6$ $w =$ ______ Check: ______________

Write your own. An equation with a unit fraction coefficient whose solution is $x = 30$: ______________

Check: ______________

Apply it. A teacher divides 84 markers equally among tables, and each table gets 12 markers.

Equation: ______________ Number of tables: ______ Check: ______________

Find the error. Nina solves $\tfrac{1}{4}x = 8$ by dividing both sides by 4 and gets $x = 2$.

What went wrong? _______________________________________________

Correct solution: ______

PAGE 14 — Exit ticket 11.4

Exit Ticket · Lesson 11.4

Name: ________________________ Date: ____________

1. Solve $8x = 72$. ______

2. Solve $\tfrac{1}{7}n = 5$. ______

3. Solve $-6y = 30$. ______

4. Why do you multiply, not divide, when the coefficient is a unit fraction?

PAGE 15 — From words to equations

11.5 Writing an Equation from a Situation

The five steps. Read it all → name the ____________ → find the ____________ → find the ____________ → write and check the equation.

Write the equation, then solve.

PAGE 16 — Situations in context

Real Situations

Write and solve.

1. A submarine descended 30 meters to a depth of $-75$ meters. Find the starting depth.

   Equation: ______________ Starting depth: ______ m

2. Each pizza has 8 equal slices, and there are 96 slices in all. Find the number of pizzas, $p$.

   Equation: ______________ Pizzas: ______

   Full sentence: _______________________________________________

3. The temperature fell 14 degrees to reach $-6$°F. Find the starting temperature.

   Equation: ______________ Starting temperature: ______ °F

4. A rope is cut into 6 equal pieces, each 7 feet long. Find the original length, $L$.

   Equation: ______________ Length: ______ ft

Find the error. For "Omar gave 5 marbles away and has 14 left," Carla writes $m + 5 = 14$.

Why does that not match? _______________________________________________

Correct equation: ______________ Solution: ______

PAGE 17 — Exit ticket 11.5

Exit Ticket · Lesson 11.5

Name: ________________________ Date: ____________

1. A number plus 13 is 30. Equation: ______________ Solution: ______

2. Six equal rows hold 54 chairs. Equation: ______________ Solution: ______

3. A hiker's elevation rose 25 meters to reach 10 meters. Equation: ______________ Start: ______ m

4. How do you decide which quantity should be the variable?

PAGE 18 — From equations to words

11.6 Writing a Situation from an Equation

Write a situation for each equation, then solve.

1. $x + 3 = 12$

   Situation: _______________________________________________

   Solution: ______

2. $5n = 45$

   Situation: _______________________________________________

   Solution: ______

3. $x - 7 = 2$

   Situation: _______________________________________________

   Solution: ______

4. $\tfrac{1}{2}m = 9$

   Situation: _______________________________________________

   Solution: ______

PAGE 19 — More stories from equations

Your Turn to Write

Write a situation and solve.

5. $x + 11 = 30$: _______________________________________________ Solution: ______

6. $9k = 72$: _______________________________________________ Solution: ______

7. $\tfrac{1}{6}t = 5$: _______________________________________________ Solution: ______

8. $n - 12 = 8$: _______________________________________________ Solution: ______

9. $w + 15 = 4$ — pick a context where a negative answer makes sense.

   Situation: _______________________________________________ Solution: ______

10. $\tfrac{1}{8}c = 3$ — set it in a classroom.

    Situation: _______________________________________________ Solution: ______

Find the error. Priya writes this for $4x = 20$: "Priya had 4 dollars, earned some money, and now has 20."

Why does it not match? _______________________________________________

Fix it: _______________________________________________

PAGE 20 — Exit ticket 11.6

Exit Ticket · Lesson 11.6

Name: ________________________ Date: ____________

1. Situation for $x + 8 = 22$: _______________________________________________ Solution: ______

2. Situation for $6n = 42$: _______________________________________________ Solution: ______

3. Situation for $\tfrac{1}{3}p = 7$: _______________________________________________ Solution: ______

4. How do you decide what the variable stands for?

PAGE 21 — Chapter 11 review, part 1

Chapter 11 Review

Part A · Algebraic vocabulary

1. Expression or equation? a) $7n$ ______ b) $7n = 28$ ______ c) $x - 5$ ______ d) $x - 5 = 0$ ______

2. For $12k + 9$: variable ______ terms ______ coefficient ______ constant ______

3. Coefficients: a) $-6y$ ______ b) $m$ ______ c) $\tfrac{1}{7}d$ ______ d) $\tfrac{h}{2}$ ______

4. Your own expression with coefficient 3: ______________ As an equation: ______________ Solution: ______

Part B · Modeling

5. Scale: block $x$ + 6 weights on the left, 14 weights on the right.

   Equation: ______________ Move: ______________ Solution: ______

6. Model and solve $5x = 35$ with tiles: _______________________________________________

7. Model and solve $x - 3 = 4$ with tiles, naming the zero pairs: _______________________________________________

Part C · Solving

8. a) $x + 9 = 23$ ______ b) $n - 14 = 6$ ______ c) $y + 8 = 1$ ______ d) $m - 5 = -12$ ______

9. a) $7a = 63$ ______ b) $\tfrac{1}{4}b = 9$ ______ c) $-8c = 40$ ______ d) $\tfrac{d}{5} = -6$ ______

10. Every step for $\tfrac{1}{3}x = 12$, naming the property:

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PAGE 22 — Chapter 11 review, part 2

Chapter 11 Review (continued)

Part D · Confirming solutions

11. Confirm with tiles: is $x = 9$ the solution of $x + 6 = 15$? ______ Show it: ______________

12. Malik says $4x = 24$ gives $x = 8$. Correct? ______ True solution: ______

13. How does a balance scale show an incorrect solution is wrong?

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Part E · Writing equations

14. A number decreased by 9 is 16. Equation ______________ Solution ______

15. A club sold 132 tickets, 12 times the number of members. Equation ______________ Members ______

16. One fourth of the pencils in a box is 7. Equation ______________ Pencils ______

17. The temperature rose 11 degrees to reach 3°F. Equation ______________ Start ______ °F

Part F · Writing situations

18. $x + 7 = 25$: _______________________________________________ Solution ______

19. $8n = 96$: _______________________________________________ Solution ______

20. $\tfrac{1}{5}p = 4$: _______________________________________________ Solution ______

Part G · Application and reasoning

21. $4s = 36$ for a square's perimeter. $s =$ ______ What does the 4 represent? ____________________

22. Why do $x + 5 = 12$ and $x = 7$ have the same solution? ____________________

23. A savings account fell $40 and now holds $85. Situation and equation: ______________ Start: $______

24. Yusuf divides both sides of $\tfrac{1}{2}x = 10$ by 2. Error: ____________________ Correct solution: ______ Check: ______

Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Figure widths: fig2, fig3, and fig4 full width; fig1 and fig5 half width placed right
• Tile colors: keep the tile legend colors identical on every page so students can read a model at a glance
• Work tables: the solve-and-check tables on pages 9 and 12 need tall rows — at least 0.6 in — so students can show the operation on both sides
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow