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

Lesson 3.93.9 Small Group · Group 2

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can find and check equivalent ratios using multiplication and division, explain why the method works, and use it on a problem I have not seen before.

Why This Matters This is your challenge small group for Lesson 3.9. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.

Start hereWords for this lesson

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

WordWhat it meansExample
Equivalent RatiosSpanish: Razones equivalentes Ratios that compare quantities in the same relationship. 2:3 = 4:6 because both parts were multiplied by 2.
RatioSpanish: Razón A way to compare two amounts, like 3 to 2. 3 red apples : 5 green apples
RateSpanish: Tasa A ratio comparing two amounts with different units, like miles per hour. 60 miles per 1 hour
ProportionSpanish: Proporción A math sentence saying two ratios are equal. 2/3 = 4/6
SimplifySpanish: Simplificar To write a ratio with smaller numbers that still compares the same way. 6:8 and 3:4 are the same ratio. 12:18 → divide both by 6 → 2:3
Ratio tableSpanish: Tabla de razones A table of equal ratios that shows a pattern. Cups of juice: 2, 4, 6 | Cups of water: 3, 6, 9

How it worksWorked example

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

Chef Montoya's dressing uses 6 cups of oil for 4 cups of vinegar. Chef Lee's uses 9 cups of oil for 6 cups of vinegar. Do the two dressings have equivalent ratios?

  1. Write Chef Montoya's 6:4 in the first column.
  2. Divide both numbers by 2: 6 ÷ 2 = 3; 4 ÷ 2 = 2
  3. Multiply both numbers of 3:2 by 3: 3 × 3 = 9; 2 × 3 = 6
  4. Compare with Chef Lee's 9:6: 9:6 and 9:6
  5. Yes. 6:4 scales to 3:2 and then to 9:6, so the ratios are equivalent.

Now you trySame steps, your turn

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

A ratio is 2:5. Use a scale factor of 3 to make an equivalent ratio.

Answer:

Say it and write itSentence starters

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

  • I multiplied and by the same number, , so the ratio is equivalent to 3:2.

Word bank Equivalent RatiosRatioRateProportionSimplifyRatio tablemultiplysame numberequivalentscaletomatoesbroth

Watch out!A common mistake

A common mistake in Equivalent Ratios is ADDING the scale factor to one part instead of MULTIPLYING both parts by it. For example, scaling 2:5 by a factor of 4 should give 8:20 (2×4 and 5×4)

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

Lesson 3.93.9 Small Group · Group 2

Group 2Challenge practice

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

Learning target I can find and check equivalent ratios using multiplication and division, explain why the method works, and use it on a problem I have not seen before.

Why This Matters This is your challenge small group for Lesson 3.9. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.

  1. 1Complete each step, then write the final answer.
    Oats (cups)8?12Nuts (cups)6??÷ 2÷ 2× ?× ?
    Scale Mix A to 12 cups of oats. Compare the nuts.

    Mix A uses 8 cups of oats for 6 cups of nuts. Mix B uses 12 cups of oats for 9 cups of nuts. Do the two mixes have equivalent ratios?

    1. 1Divide both by 2. How many cups of nuts go with 4 cups of oats?
    2. 2Find the scale factor from 4 to 12 cups of oats.
    3. 3Multiply 3 cups of nuts by 3. How many cups of nuts?
    4. 4Mix B uses 12 cups of oats and 9 of nuts. Equivalent? Type yes or no.
    Final answer:
  2. 2Complete each step, then write the final answer.
    First amount6?9Second amount10??÷ 2÷ 2× ?× ?

    The ratio 6:10 is equivalent to 9 : ?. Find the missing number.

    1. 1Divide 6 and 10 by 2. What is 10 ÷ 2?
    2. 2What times 3 makes 9?
    3. 3Multiply 5 by 3. What is the missing number?
    4. 4Why did we use the middle column?
    Final answer:
  3. 3Complete each step, then write the final answer.
    First amountSecond amount________________________

    Is 20:8 equivalent to 25:10? Type yes or no.

    Show your work
    Final answer:
  4. 4Complete each step, then write the final answer.
    First amountSecond amount________________________

    Is 12:16 equivalent to 15:21? Type yes or no.

    Show your work
    Final answer:
  5. 5Complete each step, then write the final answer.
    Berries (cups)Yogurt (cups)________________________

    A smoothie uses 6 cups of berries for 8 cups of yogurt. Lin uses 15 cups of berries. How many cups of yogurt keep the same ratio?

    Show your work
    Final answer:
  6. 6Complete each step, then write the final answer.
    First amountSecond amount________________________

    Lia says 9:12 and 12:15 are equivalent, because both numbers went up by 3. Find the number that makes 12 : ? equivalent to 9:12.

    Show your work
    Final answer:

Explain your thinkingNo whole number turns 6 into 9. How can you still check whether 6:10 and 9:15 are equivalent?

ChallengeWrite your own problem

Write a new word problem that uses this lesson's idea. Then solve it and show every step.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help