Great Lesson Ideas: What Fraction of this Shape is Red?
With Fran Dickinson

[01:00:16;15]
Fran: Hi. My name is Fran Dickinson, and I teach fifth and sixth grade math. And uh, one of the ways I get my kids to think about fractions, and the concept of the whole, is by investigating with pattern blocks.

An abstract concept such as fractions, really does require some hands on manipulatives, and so the manipulative that we choose to use for fractions is pattern blocks.

"So what fractional part of my design was blue?"

We started off having the learners look at the fractional parts of my original design, and then from there, create their own.

"So, for example, Bryan here has, on his pattern block triangle paper, uh, what fraction of the design is green. So, when we come and we're solving Bryan's pattern, we're gonna be looking for what fractional part the green pieces are. OK?"

The learners are instructed to go ahead and solve the pattern block design. What fractional part is yellow, red, green, whatever.

Student: "Six out of 15 is in uh, rhombuses. Right here."

Student: "I'm counting, uh, how many triangles there are in this pattern because all the shapes could be measured in triangles."

Student: "So, we just counted all the triangles in this, and then, that was 103, and then the green ones were, we found 23 green triangles. So, our fraction is 23 out of 103."

Fran: And so learners were using any strategy that they could, using pattern blocks, numbers, words, to describe what they see.

Student: "I did it in hexagons, uh, trapezoids, triangles, and rhombuses."

Fran: In their work, I'm looking for the learners to, in the numerator, identify the parts that are in the color that they're being asked to find.

"Which one of these two numbers, the numerator or the denominator in Sam's answer, represents the whole? And, which one of these represents the yellow part?"

Student: "The yellow is the numerator."

Student: "The yellow is the 18, and then the whole is the 36."

Fran: "OK. Good."

In addition to asking them to investigate the patterns, I'm also instructing them to use multiple representations.

"When I ask you to represent your answer using these multiple representations, the reason I'm asking you to do that is because if you can do it in the numeric way, it has to be done in the geometric way as well. We have to be able to represent it in that way as well. Otherwise, it doesn't make sense for our story."

Student: "We're gonna build uh, this, and then we're gonna find out the different ways to solve it."

Student: "First I divided the entire design into triangles. There were 32 triangles, so the total number is 32 on the bottom. Twelve were red, so 12 is the number on the top, which makes 12/32 the answer."

Student: "It's uh, 0.375."

Student: "There are 32 triangles in this shape, and nine of them are red. So my answer is 9/32. We've, we found our answer, but we're trying to like explain it in words now."

Fran: It's really important for learners to uh, play around with the concept of one. Fifth graders have to think about the idea of the whole changing. In the big picture, this is about having a flexible mindset, being able to call something the "one" but it doesn't have to physically look like just one piece.

Student: "All of these pieces make up the whole, so that's, well all the triangles that make up these pieces make the whole, so that's 26 triangles. And, then um, 1-2-3-4-5-6-7-8-9 are red. So that would be 9/26ths."

Fran: That's kind of a big "AHA!" moment for the kids, and once they get there, I, I found that solutions came flying out of them, and they were really ready to, to push the boundary of the concept of the whole.