Grade 6 Mathematics · Unit 3: Ratios & RatesStandard 6.AT.2

Lesson 3.2Understand Rates and Unit Rates

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can describe a rate using its two different units and use a model to find a travel time, and I can find a unit rate to compare prices and decide the better buy.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
Rates and Unit RatesSpanish: Tasas y tasas unitarias A rate compares different units; a unit rate compares a quantity to 1 unit. 180 miles in 3 hours = 60 miles per 1 hour.
RateSpanish: Tasa A ratio comparing two amounts with different units, like miles per hour. 96 pages / 3 hours
Unit RateSpanish: Tasa unitaria A rate for just 1 of something, like cost for 1 item. 32 pages per 1 hour
PerSpanish: Por For each one. Example: 5 dollars per book. miles per hour, cost per pound
RatioSpanish: Razón A way to compare two amounts, like 3 to 2. 5 to 3 or 5:3
Unit priceSpanish: Precio unitario A unit rate that gives the cost of exactly 1 item, like $0.60 per game. $3 for 5 games → $0.60 per 1 game

How it worksWorked examplefinding a unit rate

These numbers are not on your problems. The steps are. Follow them with your own numbers.

A reader finishes 96 pages in 3 hours at a steady speed. Find the pages per hour.

  1. Write the rate as a fraction. 96 pages / 3 hours
  2. Divide to make the bottom number 1. 96 ÷ 3 = 32
  3. Say it with the word PER. 32 pages per 1 hour
  4. Use it to predict. in 5 hours: 5 × 32 = 160 pages

Answer: The unit rate is 32 pages per hour.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

A train covers 84 kilometers in 4 hours at a steady speed.

  1. Write the rate as a fraction. km / hours
  2. Divide to make the bottom number 1. 84 ÷ =
  3. Say it with PER. km per 1 hour
  4. Predict 6 hours. 6 × = km
Answer:unit rate and the distance in 6 hours

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • The rate is for every .
  • I divided by to find the unit rate.
  • The unit rate is per .
  • To find the amount for , I .

Word bank rateunit ratepereachdividemultiplyfor everyspeedpredictconstanthourpound

Watch outA common mistake

A common mistake in Rates and Unit Rates is comparing total costs instead of unit rates: a student sees Pack X at $4.00 for 10 tokens and Pack Y at $6.00 for 16 tokens and picks Pack X just because $4.00 is the smaller number, without dividing to find the cost per token ($0.40 for Pack X vs. $0.375 for Pack Y — Pack Y is actually the better buy). A second common mistake is dividing backwards — finding tokens ÷ cost instead of cost ÷ tokens — which turns 'dollars per token' into 'tokens per dollar' and gives an answer that looks reasonable but answers the wrong question. Always divide the total cost by the number of items, and check that your unit rate answers 'the cost for ONE item,' not the reverse.

Grade 6 Mathematics · Unit 3: Ratios & RatesStandard 6.AT.2

Lesson 3.2Understand Rates and Unit Rates

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can describe a rate using its two different units and use a model to find a travel time, and I can find a unit rate to compare prices and decide the better buy.

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    Find the unit rate for each arcade deal.

    DealTotal CostNumber of GamesUnit Rate
    Deal A$6.0012 games$0.50 per game
    Deal B$7.5010 games
    Deal C$4.808 games

    Hint Look at Deal A's finished row: $6.00 for 12 games became $0.50 per game. Your rows work the same way.

    Show your work
Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    A booth charges $4.50 for 9 games. What is the unit rate?

    1. A$0.45 per game
    2. B$0.50 per game
    3. C$2.00 per game
    4. D$4.50 per game

    Hint You know the total cost ($4.50) and the total games (9). A unit rate tells the cost of just ONE game.
    per 1
    1. 1LabelWrite what each column is.
    2. 2Find per 1Divide to get one.
    3. 3ScaleMultiply up to what is asked.
    4. 4CheckDoes the size make sense?
  2. 3Circle the letter of the best answer. Show how you know.

    A car travels 180 miles using 6 gallons of gas. What is the unit rate in miles per gallon?

    1. A6 miles per gallon
    2. B30 miles per gallon
    3. C36 miles per gallon
    4. D180 miles per gallon

    Hint Notice the units in the question: miles PER gallon. That tells you miles go on top and gallons on the bottom.
    per 1
    1. 1LabelWrite what each column is.
    2. 2Find per 1Divide to get one.
    3. 3ScaleMultiply up to what is asked.
    4. 4CheckDoes the size make sense?
  3. 4Circle the letter of the best answer. Show how you know.

    A pack of 8 markers costs $6.00. What is the cost per marker?

    1. A$0.60
    2. B$0.75
    3. C$0.80
    4. D$1.33

    Hint You have 8 markers costing $6.00 total. 'Cost per marker' means the price of ONE marker.
    per 1
    1. 1LabelWrite what each column is.
    2. 2Find per 1Divide to get one.
    3. 3ScaleMultiply up to what is asked.
    4. 4CheckDoes the size make sense?
  4. 5Circle the letter of the best answer. Show how you know.

    Which is a unit rate?

    1. A12 miles per 1 hour
    2. B24 miles in 2 hours
    3. C36 miles for 3 gallons
    4. D100 points in 5 games

    Hint A unit rate always compares to exactly ONE of something — look at the second number in each choice.

    Show your work
Part 3Explain and correctreasoning
  1. 6Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake

    1. 1Read the problemA booth charges $6.00 for 4 games. Find the unit rate (cost per game).
    2. 2Set up the division$6.00 ÷ 4 = $1.50, but the student writes $1.05 per game.
    3. 3State the unit rateThe student says each game costs $1.05.

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Hint Check the place value: is the answer about one dollar and a half, or one dollar and five cents?

    Correct work
    Correct answer:

Explain your thinkingHow does finding the unit rate help you compare the booths?

Sentence starter The best deal is Booth ___ because its unit rate of ___ per game is the ___.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 3: Ratios & RatesStandard 6.AT.2

Lesson 3.2Understand Rates and Unit Rates

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can describe a rate using its two different units and use a model to find a travel time, and I can find a unit rate to compare prices and decide the better buy.

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    Complete the table to find unit rates and determine the best deal.

    DealQuantityCostUnit Rate
    Tokens Pack A20 tokens$5.00
    Tokens Pack B35 tokens$7.00
    Tokens Pack C50 tokens$12.50
    Show your work
  2. 2Write the name of the correct group on each line.

    Sort each situation by whether the unit rate is 'per item' or 'per time'.

    Groups: Rate Per Item Rate Per Time
    • $3 per pound of apples
    • 60 miles per hour
    • $0.25 per token
    • 15 pages per minute
    • $1.50 per bottle
    • 8 laps per hour
Part 2Apply itreal-world problems
  1. 3Circle the letter of the best answer. Show how you know.

    Token packs: Pack A is 20 tokens for $5.00, Pack B is 35 tokens for $7.00, and Pack C is 50 tokens for $12.50. Which pack is the best deal per token?

    1. APack B
    2. BPack A
    3. CPack C
    4. DThey are all the same
    Show your work
  2. 4Circle the letter of the best answer. Show how you know.

    Runner A completes 4 laps in 12 minutes. Runner B completes 5 laps in 14 minutes. Who runs more laps per minute?

    1. ARunner B
    2. BRunner A
    3. CSame rate
    4. DCannot determine
    Show your work
Part 3Explain and correctreasoning
  1. 5Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake

    1. 1Read the problemPack X is 10 tokens for $4.00. Pack Y is 16 tokens for $6.00. Which is the better buy (lower cost per token)?
    2. 2Compare the dealsPack X costs $4.00 and Pack Y costs $6.00. The student says Pack X is the better buy because $4.00 is the lower total price.
    3. 3State the answerThe student chooses Pack X.

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:

Explain your thinkingHow does finding the unit rate help you compare the booths?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 3: Ratios & RatesStandard 6.AT.2

Lesson 3.2Understand Rates and Unit Rates

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can describe a rate using its two different units and use a model to find a travel time, and I can find a unit rate to compare prices and decide the better buy.

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    Two water fountains fill bottles at different rates. Fountain A fills 3 bottles in 5 minutes. Fountain B fills 7 bottles in 10 minutes. Compare their rates.

    FountainBottlesMinutesBottles per Minute
    Fountain A35
    Fountain B710
    Faster?
    Show your work
Part 2Explain and correctreasoning
  1. 2Find the mistake. Explain it, then show the correct work.

    Find the Rate Error

    1. 1Find the unit rate: 12 tickets for $812 tickets for $8
    2. 2Set up the division12 ÷ 8 = 1.5
    3. 3Write the unit rateThe unit rate is $1.50 per ticket

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:
  2. 3Answer in complete sentences.

    An arcade sells tokens in three packs: 25 tokens for $5, 60 tokens for $10.80, and 100 tokens for $19. A student says the 100-token pack is always the best deal because you get the most tokens. Is the student correct? Find all three unit rates and explain.

    Write your answer

Explain your thinkingHow does finding the unit rate help you compare the booths?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help