Reflection: Developing a Conceptual Understanding Designing the floor pattern - Section 2: Warm up

The first time I was introduced to manipulatives as a teacher was watching a video of Marilyn Burns working with cuisenaire rods. The thing that struck me was how perfectly the manipulatives fit the content. I learned to design and measure the effectiveness of hands on lessons by how well the manipulatives helped students explore the mathematical concepts involved. My role is to ask the right questions to get students to explore and reflect on the big ideas.

In this case the tile designs turned out to be better tools than I expected for introducing ratio concepts. One of the hurdles with ratios is moving students from additive to multiplicative thinking. I have used context like girls and boys, red roses and white roses, red paint and yellow paint, etc. which are wonderful tools for exploring ratios. But for the initial intro the tile patterns did one thing that worked really well. My students thought of them as a "package deal". Because the 4 quadrant design was a single unit my students had no trouble with the relational thinking that the 1 black and 3 white tiles were a "package deal" or a "set".

Whereas, when I recently introduced ratios using a different context of separate units (For every 2 white roses there are 3 red roses) students had trouble seeing the separate units as a set. When I asked them how many roses there might be in a vase many had trouble seeing that they must be in multiples of 5. They claimed that we could have 2 white roses and 5 red roses, for example, because we hadn't yet added an entire second set of 3 red roses. In other words, the 2 white roses only get added after ALL 3 red roses had been added, so if we only add 1 or 2 we don't need to add the white roses yet. I love this problem and use it every year, but for the initial ratio intro the tile patterns worked so well to instill the idea of a ratio as a "package", because it can't be broken up.

When I co-taught this lesson with another teacher we came up with some questions that help invite and extend their multiplicative thinking. We noticed something several of our students were doing when they extended their initial pattern. They were grouping their design in sets of 4. Instead of adding blocks of 1 black and 3 white tiles they were using larger units of 4 black and 12 white tiles at a time. We thought they might be working multiplicatively and we didn't want it to go unnoticed by them or by the other students.

We went to these groups and asked those students how many black and white tiles they would have on the finished "floor" (how many would fill the page) and to explain how they figured it out and then we had them share with the whole class under the document camera:

• Since there are 4 black tiles in each of (these) large blocks and 30 large blocks fit on the page there should be 120 total black tiles. I did 4 times 30. I did the same for the white tiles, but I multiplied 30 by 12 since there are 12 in each block. So there are 360 white tiles.
• I found 120 black tiles because I multiplied 4 from each section by 30 that cover the floor. Then I multiplied that by 3 for the white tiles, since there are three times as many white tiles as black tiles.
• There are 4 black tiles and 12 white tiles in each large block. In the top row there are 5 large blocks, so thats 20 black tiles in that one row and 12x 5 is 60 white tiles. There are 6 rows total, so I did 20 x 6 to get 120 black tiles and 60 x 6 to get 360 white tiles.

We went on to ask the class questions like:

• Using one of these patterns how many white tiles might we find on a larger floor? Explain how you know.
• If there are (50) black tiles, how many white tiles would there be? Explain how you know.
• If there are (15 white tiles, how many black tiles would there be? Explain how you know.
• Describe the pattern piece we started with. (I like this one, because it reinforced the idea that everything we've talked about stems from this one little piece - the simplest ratio)

Developing a Conceptual Understanding: How to get the most out of the manipulatives

Designing the floor pattern

Unit 5: Writing and comparing ratios
Lesson 2 of 14

Big Idea: The ratio can be scaled up or down by multiplying or dividing.

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Standards:
Subject(s):
Math, modeling, Number Sense and Operations, equivalent ratios, ratio, scaling, simplifying ratio, hands on, ratios, pattern, student ownership, student led inquiry
54 minutes

Erica Burnison

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