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

Lesson 3.5Compare Ratio Relationships

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can compare ratios by finding unit rates or using equivalent ratios.

Start hereWords for this lesson

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

WordWhat it meansExample
Unit rateSpanish: Tasa unitaria A rate for just 1 of something, like cost for 1 item. $3 per 1 pound → unit rate is $3/lb
Equivalent ratioSpanish: Razón equivalente Two ratios that mean the same thing. 4:6 and 2:3 are equivalent (both simplify to 2:3)
CompareSpanish: Comparar To look at ratios and see which is bigger, smaller, or equal. 3:4 vs. 2:5 — which has more of the first ingredient per unit?
SimplifySpanish: Simplificar To write a ratio with smaller numbers that still compares the same way. 6:8 and 3:4 are the same ratio. 8:12 → divide both by 4 → 2:3
Common denominatorSpanish: Denominador común A bottom number that two fractions can share so you can compare them. To compare 3/5 and 4/7, use LCD 35: 21/35 vs 20/35
Common multipleSpanish: Múltiplo común A number that both terms can be scaled to. 2 and 3 both reach 6

How it worksWorked examplecomparing two mixes fairly

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

Mix P is 2 cocoa to 5 milk. Mix Q is 3 cocoa to 7 milk. Which is more chocolatey?

  1. Pick the quantity to match — cocoa. Find the LCM of 2 and 3 → 6.
  2. Scale Mix P. ×3 → 6 cocoa : 15 milk
  3. Scale Mix Q. ×2 → 6 cocoa : 14 milk
  4. Same cocoa, so compare the milk. LESS milk = more chocolatey.

Answer: Mix Q is more chocolatey: for 6 cocoa it has only 14 milk.

Now you trySame steps, your turn

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

Juice A is 3 scoops to 8 cups water. Juice B is 4 scoops to 10 cups water. Which is stronger?

  1. Match the scoops. LCM of 3 and 4 =
  2. Scale A. × → scoops : cups
  3. Scale B. × → scoops : cups
  4. Same scoops, so compare the water. LESS water = stronger. Juice
Answer:the stronger juice

Say it and write itSentence starters

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

  • I scaled both ratios to .
  • Mix has and Mix has .
  • is stronger because .
  • You cannot compare them directly because .

Word bank common multipleleast common multiplescalematchcomparestrongerweakerlessmoresamefor everyfair

Watch outA common mistake

A common mistake in Compare Ratios is thinking the ratio with the bigger raw number automatically wins — like assuming Chef Tran's 8 tablespoons of cocoa must be more chocolatey than Chef Reyes's 6 tablespoons, without checking that the milk amounts (14 oz vs. 10 oz) are different. Never compare the raw numbers alone — always scale both ratios to the same amount or find each unit rate first.

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

Lesson 3.5Compare Ratio Relationships

Version ASupported practice

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

Learning target I can compare ratios by finding unit rates or using equivalent ratios.

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

    Find the unit rate for each option to compare them.

    OptionRatioUnit Rate
    Juice A6 oz for 2 servings3 oz per serving
    Juice B10 oz for 4 servings
    Juice C9 oz for 3 servings

    Hint Row 1 shows the move: 6 oz for 2 servings became 3 oz per serving by dividing.

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

    Chef A uses 2 cups of cheese for every 8 crackers. Chef B uses 3 cups of cheese for every 9 crackers. Who uses more cheese per cracker?

    1. AChef A
    2. BChef B
    3. CThey use the same amount
    4. DCannot determine

    Hint You can't compare 2 cups and 3 cups directly — the chefs use different numbers of crackers.

    Show your work
  2. 3Circle the letter of the best answer. Show how you know.

    Which ratio is greater: 3:4 or 5:8?

    1. A5:8
    2. B3:4
    3. CThey are equal
    4. DCannot determine

    Hint The second numbers (4 and 8) don't match, so the ratios aren't ready to compare yet.

    Show your work
  3. 4Circle the letter of the best answer. Show how you know.

    Which lemonade recipe is more lemony? Recipe A: 2 lemons for 6 cups of water. Recipe B: 3 lemons for 7 cups of water.

    1. ARecipe B
    2. BRecipe A
    3. CThey are the same
    4. DCannot determine

    Hint Recipe B has more lemons, but it also has more water — so compare rates, not raw counts.

    Show your work
  4. 5Circle the letter of the best answer. Show how you know.

    Which is the better deal: 4 apples for $3 or 6 apples for $5?

    1. A6 apples for $5
    2. B4 apples for $3
    3. CThey cost the same
    4. DCannot determine

    Hint 'Better deal' means the LOWER cost for one apple — not simply more apples.

    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 problemChef Priya makes hot chocolate with 2 tbsp cocoa for 4 oz milk. Chef Owen makes hot chocolate with 3 tbsp cocoa for 8 oz milk. Whose hot chocolate is more chocolatey?
    2. 2Compare the cocoaChef Owen has 3 tbsp cocoa. Chef Priya has 2 tbsp cocoa.
    3. 3Draw a conclusionSince 3 is greater than 2, Chef Owen's hot chocolate must be more chocolatey.
    4. 4ConclusionChef Owen's hot chocolate is more chocolatey.

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

    Hint Look at Step 3. The chef compared only the cocoa amounts (2 and 3) without checking whether the milk amounts were equal.

    Correct work
    Correct answer:

Explain your thinkingThe two recipes use different amounts, so you cannot compare them directly. How did building equivalent ratios let you compare them fairly?

Sentence starter I scaled both ratios to the same amount of ___, then compared the ___; Chef ___'s recipe is more chocolatey because ___.

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.3

Lesson 3.5Compare Ratio Relationships

Version BCore practice

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

Learning target I can compare ratios by finding unit rates or using equivalent ratios.

Part 1Understand the ideamodels and meaning
  1. 1Write the name of the correct group on each line.

    Match each scenario to the correct answer: Which ratio is greater?

    Groups: Correct Comparison Incorrect Comparison
    • 5:6 vs. 7:9 → 5:6 is greater
    • 2:3 vs. 4:6 → they are equal
    • 1:4 vs. 2:5 → 2:5 is greater
    • 3:7 vs. 2:5 → 2:5 is greater
    • 4:5 vs. 3:4 → 3:4 is greater
    • 1:2 vs. 3:4 → they are equal
  2. 2Complete the table. Show how you found each value.

    Two frosting recipes both mix drops of food coloring to make green frosting for a cake. Recipe A: 3 drops blue to 5 drops yellow. Recipe B: 4 drops blue to 7 drops yellow. Scale both to a common amount of yellow to compare which recipe is more blue.

    RecipeBlue DropsYellow Drops
    Recipe A35
    Recipe B47
    Recipe A (scaled to 35)35
    Recipe B (scaled to 35)35
    Show your work
  3. 3Mark and label each value on the number line.

    Place each unit rate on the number line to compare: A = $0.60/oz, B = $0.75/oz, C = $0.50/oz

    00.10.20.30.40.50.60.70.80.91
    Show your thinking
Part 2Apply itreal-world problems
  1. 4Circle the letter of the best answer. Show how you know.

    Store A sells 3 mangoes for $6. Store B sells 5 mangoes for $9. Which store offers a better price per mango?

    1. AStore A
    2. BSame price
    3. CStore B
    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 problemCompare Chef Reyes's dressing (4 tbsp oil : 6 tbsp vinegar) to Chef Tran's dressing (5 tbsp oil : 8 tbsp vinegar). Which dressing is oilier?
    2. 2Find a common denominatorLCD of 6 and 8 is 24.
    3. 3Scale each ratioChef Reyes: 6 × 4 = 24, so oil = 4 × 4 = 16, giving 16:24. Chef Tran: to reach 24 he also multiplies by 4: oil = 5 × 4 = 20, giving 20:24.
    4. 4ConclusionSince 20 > 16, Chef Tran's dressing is oilier.

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

    Correct work
    Correct answer:

Explain your thinkingThe two recipes use different amounts, so you cannot compare them directly. How did building equivalent ratios let you compare them fairly?

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.3

Lesson 3.5Compare Ratio Relationships

ChallengeExtension

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

Learning target I can compare ratios by finding unit rates or using equivalent ratios.

Part 1Understand the ideamodels and meaning
  1. 1Use the model to solve. Show your work.
    TotalChef A OilChef A VinegarChef B OilChef B Vinegar

    Chef A: 2:3 → with 15 tbsp vinegar, oil = 10 tbsp. Chef B: 3:5 → with 15 tbsp vinegar, oil = 9 tbsp. Chef A uses more oil.

    Show your work
    Answer:
Part 2Explain and correctreasoning
  1. 2Find the mistake. Explain it, then show the correct work.

    Find Kai's Mistake

    1. 1Read the problemCompare 3:5 and 4:7. Which ratio is greater?
    2. 2Find common denominatorLCD of 5 and 7 is 35
    3. 3Scale the ratios3:5 = 21:35 (3 × 7 = 21). For 4:7, Kai multiplies by 7 again: 4 × 7 = 28, giving 28:35.
    4. 4ConclusionSince 28 > 21, the ratio 4:7 is greater.

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

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

    Store A sells 5 notebooks for $12. Store B sells 8 notebooks for $20. Without calculating, a student says Store B must be cheaper because you get more notebooks. Is the student's reasoning valid? Find the unit rates to prove your answer.

    Write your answer

Explain your thinkingThe two recipes use different amounts, so you cannot compare them directly. How did building equivalent ratios let you compare them fairly?

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