## Reflection: Developing a Conceptual Understanding Scaling up ratios - Section 2: Warm up

The more diversity in the descriptions the better! I don't believe that when a single student shares the big idea that every student understands it completely. My students need ideas repeated, restated in different ways, and connected in order to fully understand a new idea. I can't count the number of times a student has said "Oh, I get it now" only after the same idea has been expressed many times. How many times have we said, "I taught it, but he didn't get it!" or asked "How can she not understand when I have already explained it?" Sometimes a student may express a mathematical idea that turns on a lightbulb for me and show me a new way of understanding it! Multiple explanations help deepen and solidify the concept as students integrate the new thinking with their own.

My job was to elicit their thinking and help them represent it in a mathematical way that they understood that would help them understand the scale factor.

After having several students share their thinking I asked them to think with their math family groups how they might express their thinking mathematically:

• In each figure one white and two black tiles are being repeated, so in figure 4 you will have one white and two black tiles 4 times: 4(WBB)
• It's going in columns, you are adding one white and two blacks each time, so in figure 4 you will have a column of 4 whites and two columns of 4 blacks: 4W + 2(4B)
• Each figure has that many (figure number) white tiles and twice as many black tiles: 4W + 4(2B)

If students have become accustomed to one right answer in math class they might not understand that all of these explanations can be correct and equivalent to each other.

• "what does this look like if we think about the tiles as a ratio?"
• "why does it make sense that these expressions represent the same thing?"

After our discussion we ended up with expressions that clearly emphasized the scale factor, but I wasn't convinced that they really understood it's significance.

• 4(WBB) = 4(1:2)
• 4W + 2(4B) = 4W + (4x2)B = 4(W+2B) = 4(1to2) = 4(1:2)
• 4W + 4(2B) = 4(W+2B) = 4(1to2) = 4(1:2)

I wanted to make sure that they understood what the 4 represented and where it came from. I think it is important to teach my students not to stop at final answers, but to continue to ask for and understand reasonableness and helps them recognize the structure of the math.

• "what does the 4 represent?"
• "will it always be a 4?"
• "How can we tell what the number will be if not a 4?"

Without the visual model and the time they had taken to describe how the pattern was changing in their own words, my students would not have been able to answer these questions. It was their verbal descriptions of the visual models that helped them conceptualize the scale factor in the end.

Don't skip the verbal description!
Developing a Conceptual Understanding: Don't skip the verbal description!

# Scaling up ratios

Unit 5: Writing and comparing ratios
Lesson 8 of 14

## Big Idea: In proportional relationships a constant ratio is maintained.

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Subject(s):
Math, Number Sense and Operations, proportional relationships, concept development, ratios, scaling ratios, ratio table, discussions, pattern
44 minutes

### Erica Burnison

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