## Reflection: Developing a Conceptual Understanding Which is blackest the sequel - Section 2: Warm up

The first attempt my students make to express their ideas is not always the easiest to understand. At first I wasn't sure if it was worth the valuable time it would take to try to understand their thinking. It is really tempting to just move on to the next kid until you hear the ideas or words you're listening for. I took a risk, knowing it would take more of our precious classtime and found that it was well worth it.

When explaining why graph C was the blackest my kids said things like:

I had no idea what they were talking about until I asked for some clarification and elaboration:

• "what do you mean by that?"
• "show me on the graph what you mean?"
• "why does that mean it is blacker?"
• "why does that make sense?"

Only then did I understand that they were describing that the line for "C" was closer to the axis representing the black tiles, was more horizontal, and was moving more in the black direction which meant that the black tiles were increasing faster than the white tiles, making the pattern blacker.

When a student suggested that his points (3/12, 8/12, 36/12) were easier to compare, the above conversation helped my students understand why common denominators were helpful.  I asked the class to decide why his points were easier and they said "it's easiest to see where the points on the graph are at the same level". I continued probing with similar questions as above. When I asked "what does he mean by 'at the same level'?" someone asked to come to the projector and highlight the level where the white tiles on the y-axis were equal to 12. I then could follow up with "what's happening in terms of black and white tiles at this level?"

• "why is it easier to compare when the points are at the same level like this?" (it's easier to see that graph C has a greater number of black tiles for the same number of white tiles)
• "are there other places on the graph where this will be true?"
• "how could we predict other levels where it would be easy to compare?" (wherever the number of white tiles/denominator is the same)

Being able to use their explainations in conjunction with the context and the graph helped us make much more meaningful mathematical connections. I think it was through these connections that my students finally gained a better understanding of comparing fractions with common denominators.

Developing a Conceptual Understanding: Making student thinking visible

# Which is blackest the sequel

Unit 6: Proportionality on a graph
Lesson 8 of 10

## Big Idea: Common denominators are found when "points" on a graph are at the same "level" in relation to the y-axis.

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Standards:
Subject(s):
Math, modeling, Number Sense and Operations, common denominator, graph, Proportion, comparing ratios, ratios, pattern, questioning, making thinking visible
54 minutes

### Erica Burnison

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