Grade 6 Workbook — Chapter 9: Proportional Relationships and Unit Rate

SOL 6.PFA.2 · Companion to Textbook Chapter 9

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 9 · Proportional Relationships and Unit Rate

Standard 6.PFA.2

In this chapter you will:

• Find the unit rate from a table, a story, or a graph
• Fill in missing values in a ratio table
• Decide whether a relationship is proportional
• Graph a proportional relationship
• Move between words, tables, and graphs

Words to know: rate · unit rate · per · proportional relationship · ordered pair · origin

Every unit rate in this chapter is positive.

PAGE 2 — What a unit rate is

9.1 Unit Rate

FIGURE: fig9-1-unit-rate-price-tags.png (full width) [Two price tags: "4 pounds for $12.00" with an arrow labeled "divide both by 4" pointing to "1 pound for$3.00" marked UNIT RATE, with the caption "12 divided by 4 equals 3, so the unit rate is 3 dollars per pound."]

Fill in the blanks.

A rate compares two quantities with ____________ units.

A unit rate is a rate whose second quantity is ______.

To find a unit rate, ____________ the first quantity by the second.

The word "per" means ______________________.

Find each unit rate.

PAGE 3 — Unit rate from tables and graphs

Unit Rate in a Table and on a Graph

Find the unit rate for each table. Show the division for one column.

1.

Unit rate: ______ Division: ______________

2.

Unit rate: ______ Division: ______________

On a graph, the unit rate is the second coordinate of the point where the first coordinate is ______.

A graph of cost against pounds passes through $(1, 5)$. Unit rate: ______

A graph of pages against hours passes through $(1, 6)$ and $(3, 18)$. Unit rate: ______

Better buy. Cashews: 3 lb for $7.50, or 5 lb for $11.25.

Unit rate 1: ______ Unit rate 2: ______ Better buy: ______

PAGE 4 — Exit ticket 9.1

Exit Ticket · Lesson 9.1

Name: ________________________ Date: ____________

1. A train travels 240 miles in 4 hours. Unit rate: ______

2. Eight notebooks cost $24. Price per notebook: ______

3. Unit rate for the table: ______

   Bags	5	10
   Pounds	40	80

4. In your own words, what does a unit rate tell you?

PAGE 5 — Missing values

9.2 Missing Values in a Ratio Table

FIGURE: fig9-2-ratio-table-missing.png (full width) [A ratio table with Hours worked 2, 5, blank, 9 over Pay 30, blank, 90, blank, and a callout reading "Unit rate: 30 divided by 2 = 15 dollars per hour" with arrows labeled "x 15 to go down" and "divide by 15 to go up".]

Given the first quantity → multiply by the unit rate. Given the second quantity → divide by the unit rate.

Complete each table. Write the unit rate first.

1. Unit rate: ______

2. Unit rate: ______

3. Unit rate: ______

4. Unit rate: ______

PAGE 6 — Use the unit rate to solve

Solve with the Unit Rate

Show the unit rate, then the answer.

1. Five kilograms of flour cost $20. Cost of 9 kg?

   Unit rate: ______ Answer: ______

2. A car travels 220 miles in 4 hours. Distance in 7 hours?

   Unit rate: ______ Answer: ______

3. Eight muffins cost $10. Cost of 14 muffins?

   Unit rate: ______ Answer: ______

4. Three gallons of paint cover 1,050 sq ft. Coverage of 7 gallons?

   Unit rate: ______ Answer: ______

5. A car goes 250 miles on 10 gallons.

   Miles per gallon: ______ Distance on 16 gallons: ______ Gallons for 350 miles: ______

Find the error. To find the cost of 12 pounds of apples at $2 per pound, Sam computed $12 \div 2 = 6$ and answered $6.

What went wrong? _______________________________________________

Correct answer: ______ A quick check that catches it: ______________

PAGE 7 — Exit ticket 9.2

Exit Ticket · Lesson 9.2

Name: ________________________ Date: ____________

1. Six pounds of rice cost $9. Cost of 10 pounds: ______

2. Complete the table.

   Hours	3	5	9
   Miles	21

3. A student solves 15 problems in 45 minutes. Time for 25 problems: ______

4. When do you multiply by the unit rate, and when do you divide?

PAGE 8 — Proportional or not

9.3 Proportional or Not?

FIGURE: fig9-3-proportional-vs-not-graphs.png (full width) [Two coordinate grids side by side: the left, titled "Proportional", shows a straight line through the origin passing through (1,2), (2,4), (3,6), (5,10); the right, titled "Not proportional", shows a straight line crossing the vertical axis at (0,4) through (1,6), (2,8), (4,12).]

The three tests.

Table: divide the second quantity by the first in every column. Proportional if all quotients are ______________.

Context: proportional if there is a constant rate and ______ starting fee.

Graph: proportional if the graph is a ____________ line through the ____________.

Decide for each table. Write YES or NO and the unit rate when it is proportional.

PAGE 9 — Contexts and graphs

Contexts and Graphs

Write PROPORTIONAL or NOT, and give a reason.

Graphs.

A graph through $(0,0)$, $(2,6)$, $(5,15)$: proportional? ______ Unit rate: ______

A graph through $(0,4)$ and $(2,10)$: proportional? ______ Why not? ______________

Explain. Owen says, "The graph is a straight line, so it must be proportional." Why is a straight line not enough?

PAGE 10 — Exit ticket 9.3

Exit Ticket · Lesson 9.3

Name: ________________________ Date: ____________

1. Proportional? Unit rate if yes: ______

   Hours	3	6	10
   Dollars	12	24	40

2. Proportional? Items 1, 2 / Cost $3, $5 ______

3. A delivery service charges $8 plus $2 per package. Proportional? ______

4. Why must a proportional graph pass through $(0,0)$?

PAGE 11 — Graphing

9.4 Graphing Proportional Relationships

FIGURE: fig9-4-graph-proportional-line.png (full width) [A coordinate grid with "Time (hours)" across and "Distance (miles)" up, showing points (0,0), (1,4), (2,8), (3,12) joined by a straight ray from the origin, with (1,4) circled and labeled "Unit rate: 4 miles in 1 hour".]

Write the ordered pairs for each table.

1. Hours 1, 2, 3 / Miles 3, 6, 9 → ______________________________

2. Bags 2, 4, 6 / Pounds 10, 20, 30 → ______________________________ Unit rate: ______

3. Hours 1, 2, 3 / Pages 8, 16, 24 → ______________________________

Build a table, then plot. Blueberries cost $2.50 per pound.

Ordered pairs: ______________________________

BLANK GRID: first quadrant, horizontal axis 0 to 5 labeled "Pounds", vertical axis 0 to 12 labeled "Cost (dollars)", gridlines every 1 unit

PAGE 12 — Reading a graph

Reading a Graph

Fill in.

A proportional graph through $(1, 9)$ has unit rate ______.

A proportional graph through $(4, 20)$ has unit rate ______.

A proportional graph through $(3, 21)$ has unit rate ______, so at $x = 5$ the value is ______.

A proportional graph through $(8, 24)$ has unit rate ______, and the first coordinate is ______ when the second is 45.

Steepness. Line A passes through $(1, 3)$; line B passes through $(1, 5)$.

Steeper line: ______ What steeper means here: _______________________________________________

Apply it. A printer prints 15 pages per minute.

Ordered pairs: ______________________________ Minutes for 90 pages: ______

PAGE 13 — Exit ticket 9.4

Exit Ticket · Lesson 9.4

Name: ________________________ Date: ____________

1. Ordered pairs for Hours 1, 2, 3 / Pages 8, 16, 24: ______________________________

2. A proportional graph passes through $(1, 6)$. Unit rate: ______

3. For that graph, the second coordinate when the first is 7: ______

4. How can you tell from a graph that a relationship is proportional?

PAGE 14 — Three views

9.5 Words, Tables, and Graphs

FIGURE: fig9-5-three-representations.png (full width) [Three panels — Words: "A store sells grapes for 2 dollars per pound"; Table: Pounds 0–4 over Cost 0, 2, 4, 6, 8; Graph: points (0,0) through (4,8) on a straight ray from the origin — under a banner explaining that the unit rate 2 appears in all three.]

Where is the unit rate?

Complete the conversions.

1. Table Hours 1, 2 / Miles 9, 18 → verbal description: _______________________________________________

2. "A worker earns $4 per hour" → table:

   Hours	0	1	2	3	4
   Dollars

3. Graph through $(1, 12)$ → unit rate ______, verbal description: _______________________________________________

4. Table Hours 2, 4, 6 / Dollars 7, 14, 21 → unit rate ______, verbal description: _______________________________________________

5. Graph through $(2, 9)$ → unit rate ______, table:

   First	1	2	3	4
   Second

PAGE 15 — Compare across views

Compare Across Views

Convert to unit rates, then compare.

1. Car A: 150 miles in 3 hours. Car B: graph through $(1, 45)$.

   A: ______ B: ______ Faster: ______

2. Runner A: 6 km in 30 minutes. Runner B: graph through $(10, 2.5)$ (minutes across, km up).

   A: ______ B: ______ Faster: ______

   Distance in 60 minutes — A: ______ B: ______

Which describe the same relationship? Circle all that match.

(i) "3 dollars per pound" (ii) a table with 2 lb costing $6 and 5 lb costing $15 (iii) a graph through $(1, 4)$

Reason: _______________________________________________

Explain. Where do you see the unit rate in a table, and where on a graph? Why is it the same number?

PAGE 16 — Exit ticket 9.5

Exit Ticket · Lesson 9.5

Name: ________________________ Date: ____________

1. Unit rate for Pounds 3, 6 / Dollars 15, 30: ______

2. A machine makes 24 parts per hour. Point on its graph where the first coordinate is 1: ______

3. A proportional graph passes through $(1, 7)$. Write a situation it could represent.

4. How do a table and a graph of the same relationship carry the same information?

PAGE 17 — Chapter 9 review, part 1

Chapter 9 Review

Part A · Unit rate

1. Unit rate for Books 3, 6, 9 / Dollars 21, 42, 63: ______

2. 156 miles in 3 hours: ______

3. Graph through $(1, 11)$: ______

Part B · Missing values

4. Complete: First 2, 5, ______ / Second 9, ______, 36 Unit rate: ______

5. Four pounds of coffee cost $14. Cost of 10 pounds: ______

Part C · Proportional or not

6. Items 2, 3, 5 / Cost 8, 12, 20 → ______ Unit rate: ______

7. Items 1, 2 / Cost 6, 11 → ______

8. Tutoring: $12 registration plus $4 per hour → ______ Why? ______________

9. Straight line through $(0,2)$ and $(1,5)$ → ______ Why? ______________

PAGE 18 — Chapter 9 review, part 2

Chapter 9 Review (continued)

Part D · From a context to a table or graph

10. A faucet fills 7 gallons per minute. Unit rate: ______

    Minutes	0	1	2	3	4	5
    Gallons

    Ordered pairs: ______________________________

11. Three shirts cost $36. Unit rate: ______

    Shirts	1	2	3	4	5
    Cost (dollars)

Part E · Connecting representations

12. Hours 2, 4 / Miles 13, 26 → verbal: _______________________________________________ Point at first coordinate 1: ______

13. Graph through $(4, 18)$ → unit rate ______, three-column table:

    First	1	2	3
    Second

14. Same relationship? (i) "$5 per hour" (ii) 3 hours pays $15 (iii) graph through $(2, 12)$ → ______________

Part F · Application and reasoning

15. 5 pounds of rice for $12.50. Unit rate: ______ Cost of 8 lb: ______ Pounds for $20: ______

16. Plan A: 10 cents per minute. Plan B: $5 plus 5 cents per minute. Proportional plan: ______ 200 minutes on A: ______ on B: ______

17. Why is a straight line that crosses the vertical axis above the origin not proportional?

    ---

18. Worker A earns $108 in 9 hours. Worker B's graph passes through $(4, 52)$. A: ______ B: ______ Earns more: ______ by ______

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
• Blank grid: page 11 needs a first-quadrant grid, horizontal 0–5 and vertical 0–12, gridlines every 1 unit, axis labels printed
• Tables: minimum cell height 0.4 in so students can write values inside cells
• Figure widths: all Chapter 9 figures full width; fig9-3 needs enough height for two side-by-side grids to stay readable
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow