1. Routines for Reasoning
Connecting Representations –
Equivalent Ratios

2. Purpose
We are learning how to write number sentences from a picture. We are going to listen to what others say and think about what they mean. We are going to find equivalent ratios.

3. Today’s Thinking Goals
Think like mathematicians
Look at and make sense of equations
Solve equations visually

4. Thinking Questions What do I see in the picture?
What does the ratio mean?
How can I scale the picture to match the ratio?
How many thinking questions should we give kids at one time? Do we develop these as we go?
How is this situation behaving?
What kind of problem is this?
Does this problem remind me of another I’ve solved?
How can I decompose this problem to help me understand it better?

5. I notice __________, which makes me think __________.

6. “I know that ____ would be equivalent because…”
The ratio of Purple Circles to Green Circles is shown below. What would be an equivalent ratio?
“I know that ____ would be equivalent because…”

7. Which ratio is equivalent to which picture? (Purple to Green)1 a)
1:2 b)
15:9 c)
16:20 d)
24:36 e)
9:12 f)
24:40 2 3 4
“I noticed ________, so I __________.”
“They noticed ________, so they __________.” 5

8. Unused Answer: b) 15:9
Create a picture that would show an equivalent ratio to 15:9 using purple and green circles.
If you use this example – push students to discuss how they could modify the model to make it work.
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”

9. Reflection
“When thinking about equivalent ratios, I focused on ____________ because ____________.”