## Reflection: Developing a Conceptual Understanding Mistakes & Peer Instruction - Section 3: Exploration

When students make a mistake I like to use it to promote deeper understanding. This has the effect of both giving value to the mistake as well as teaching them from their level of comprehension. In this case the mistakes are made when students notice the denominators are increasing in a pattern, but ignore how the change relates to the denominator. The questions I ask here first acknowledge what is correct about the mistake and then help to facilitate student discussion that will guide them towards the multiplicative releationship between the numerator and denominator.

The 'student' in the discussion below represents multiple students. After each of my questions I gave the groups a couple of minutes to discuss with each other before responding.

• Me: "So, this student is noticing that the numerator is increasing by fives, is that true?"
• Student: "Yes, each number is 5 more, it's going up by five every time."
• Me: "What pattern is happening in the denominator? Is something similar happening there?"
• Student: "Well, it looks like it's going to go up by 6 every time, but then there's the 30 and it doesn't work anymore."
• Me: "So, it's not going up by 6 anymore? How many did it go up?"
• Student: "It went up 12."
• Student: "12 is six times two."
• Me: "Did it go up 2 sixes?"
• Student: "Well, we noticed something else in the bottom row. It goes one 6, 2 sixes, 3 sixes, then skips one and goes up to 5 sixes, ten sixes, then down to 8 sixes."

This is when I reintoduce the familiar sentence frame "For every ____ there are ____", but change it to: "For every increase of 5 in the numerator there is an increase of ______ in the denominator" and ask how this might help them. This should focus their attention on the relative change as the numerator and denominator are changing together.

Questioning to promote multiplicative thinking
Developing a Conceptual Understanding: Questioning to promote multiplicative thinking

# Mistakes & Peer Instruction

Unit 5: Writing and comparing ratios
Lesson 13 of 14

## Big Idea: Though there are multiple methods, certain strategies are more efficient for certain types of problems.

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Standards:
Subject(s):
Math, Number Sense and Operations, proportional relationships, common denominator, peer instruction, comparing ratios, error analysis, ratios, pattern, conceptual development, safety
49 minutes

### Erica Burnison

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