How to teach equivalent ratios

Why this matters

A student who knows that 2 cups of blue paint go with 3 cups of yellow will often say that 6 cups of blue go with 7 cups of yellow. They added 4 to both numbers instead of multiplying both by 3. On paper, 7 looks as reasonable as 9, so the student keeps adding on every problem, and the teacher has to wait for quiz day to find out.

Equivalent ratios

Two ratios are equivalent when they describe the same relationship between two quantities. 2 cups of blue to 3 cups of yellow and 6 cups of blue to 9 cups of yellow make the same green, because both numbers were multiplied by 3. Multiplying or dividing both parts by the same number keeps a ratio equivalent, and adding the same number to both parts doesn't. A ratio table lists equivalent ratios side by side, and the Common Core asks students to make these tables, find missing values in them and use them to compare ratios (CCSS 6.RP.A.3a).

In the Flint activity, students mix paint toward a 2:3 green in a paint mixer that opens in Sparky's first message. A worksheet can only mark 7 cups wrong. In the mixer, 6 blue to 7 yellow comes out bluer than the target, and Sparky asks what the student did to each color instead of giving the answer. A student who fixes the mix builds a ratio table, then gets a new target such as 5:8.

Maya sees her mix come out too blue, and fixes it

Maya is a 6th grader who paints in the art club. She can double a ratio, but when the numbers don't double she adds the same amount to both parts.

Every student in the class gets the same paint mixer in Sparky's first message, with the same 2:3 target and the same first problem. The rest of the chat is Maya's own: Sparky sets the problems on her mural, and each next question depends on what she just answered. Maya is an example student, and her chat is shortened.

Sparky uses Maya's muralSparky asks a question instead of giving the answerSparky gives a harder problem after a right answer

M

Maya

Interest: Painting a muralStarting point: Can double a ratioGets stuck: Adds the same number to both parts

Sparky

Hi! This paint mixer makes green from blue and yellow paint. The target green is 2 cups of blue to 3 cups of yellow. Try a few mixes, then tell me: what would you paint if you had a big bucket of this green?

Maya

A mural on the wall outside the art room

Understanding check

Your mural needs 6 cups of blue. How many cups of yellow keep the same green as the target? Try it in the mixer and tell me how you figured it out.

Maya

7 cups of yellow. I added 4 to the blue to get 6, so I added 4 to the yellow too. It says Bluer than the target.

Sparky

So 6 blue and 7 yellow comes out too blue. What did you do to the blue to get from 2 to 6? And what did you do to the yellow to get from 3 to 7?

Maya

I added 4 to both. But the blue went from 2 to 6, that's times 3. So the yellow should be 3 times 3, which is 9 cups. Now it says Same green as the target.

Sparky

You found your own mistake. Now build a ratio table: three more batches that make the same green, one with 10 cups of blue, one with 12 cups of yellow, and one of your choice.

Maya

10 blue and 15 yellow. 8 blue and 12 yellow. 4 blue and 6 yellow. All three have a check mark.

Sparky

All three are right. Change the mixer's target to 5 blue : 8 yellow. How many cups of yellow go with 20 cups of blue?

Maya

32 cups of yellow. 20 is 5 times 4, so I did 8 times 4.

Sparky

Exactly right. Back in the 2:3 green, how much blue goes with 21 cups of yellow?

Found her own mistake, then solved a 5:8 mix with no hints

What Flint did for Maya

• The paint mixer in Sparky's first message turned Maya's 6:7 into a color next to the target, so she saw her mix was off before anyone told her.
• Sparky asked what she did to each color instead of giving the answer, and Maya found the times 3 herself.
• After her right answer, Sparky moved her straight to a ratio table and then a 5:8 target, without waiting for the rest of the class.
• Sparky set the first problem on her mural, the thing she said she would paint.
• Sparky graded her chat against the activity's rubric, which counts a mistake she fixed on her own toward Proficient.

A teacher can't stand next to 30 students at the moment each one mixes 6:7 and ask the right question. Sparky does it in every student's chat.

The activity page shows who added instead of multiplying

Activity analytics sums up what the class did across every chat. In this activity, Sparky's analysis guidelines ask it to note which students added the same number to both colors and whether they fixed it themselves.

Set up the activity for free

The paint mixer activity is in Flint's public library, with the mixer, Sparky's guidelines, the greeting and a rubric already set up. Open it, select Use activity to copy it into a free Flint account, and move it to a class. In the activity's Preview tab, select Try session myself to mix a few batches as a student before you assign it.

For more math activities, see Flint for math.