Math should tell a story — through the structure of units, and the structure of a year. There is no singular correct narrative; instead, I find joy in weaving different paths through different mathematical ideas depending on the group or grade. In sixth grade, one dominant story for the year is the story of ratios and proportional thinking, often told through fractions, percentages and ratios.
Mark Chubb (@markchubb3) posted the following prompt in April:
Which if the 2 triangles is greener?#mtbos pic.twitter.com/RTgNWaOeOa
— Mark Chubb (@MarkChubb3) April 2, 2017
Most #MTBoS people that responded used fractions to compare the two quantities.
@MarkChubb3 @gamesbygord Small triangle=3 mini green bits out of 9 bits total (yellow hex=6 bits)=3/9 green. Large tri=7/25 green. 3/9>7/25 so small tri=”greener”
— Dreams of Gerontius (@DGerontius) April 3, 2017
@MarkChubb3 @MrOrr_geek Small one is 75/225 green larger one is 63/225 green. So the small one is greener! @AMP_AZmath
— Mackenzie (@missmkuhl) April 3, 2017
I looked at the triangles again. Having just completed a unit on ratios and rates with my sixth graders, I saw the world through the lens of ratios. This lens colored my perspective on many middle school problems; where I might have seen percentages or fractions at another point in the year, ratios surfaced to the top of my mind.
I was also in the midst of planning out my unit on percentages. It felt like a world tinted red was turning orange, then yellow, as my brain shifted from ratios to rates to percentages, connecting the three. Mark’s “which is greener” problem felt like it beautifully transcended all three of those mathematical lenses. So I started planning out my lesson.
I wrote about using the Five Practices for Orchestrating Discussion .
- Anticipating student strategies
- Monitoring student work and thinking
- Selecting students to share
- Sequencing student work to highlight specific ideas and concepts
- Connecting the carefully sequenced work to illuminate the big picture
I knew that some students would still see the world through ratio glasses. I expected that some of them would find equivalent ratios, e.g. yellow:green or green:whole. Within these equivalent ratios, I expected that I would see some students constraining
Green : Whole
Left: 3: 9 –> 21: 63
Right: 7: 25 –> 21: 75
I also anticipated that some students would draw upon their background knowledge of percentages to determine the “greenness” of the two triangles.
I thought some students would focus on the number of blocks (e.g. 4 blocks in the smaller triangle), while others would focus on the area (there are 3 green triangles and 1 yellow hexagon in the smaller triangle, but the 1 yellow hexagon is worth 6 green triangles).
Within any given idea (e.g. ratio of green to whole based on the total area, not the number of blocks) I saw multiple pathways. For example:
I could think of more potential pathways than there are students in that section!
Launching the Task
To launch the problem, I showed the students the image, and took 1 minute to notice/wonder. Students were able to see how the larger triangle was composed of iterations of the smaller triangle. “But there’s an extra one in the middle to bring it together,” Caleb announced.
Students wanted to explore what would happen in future stages, but I steered the conversation back towards the greenness. Which triangle is greener? I took no suggestions — I was a little nervous that Caleb might point out that the extra block of green in the larger triangle made it extra green, and the conversation would end there. Instead, I asked students to think and then dive into their journals. Make their case.
Most students focused on the total area (e.g. not number of blocks) using ratios. I watched their pencils dot around the page counting the green and yellow individually. Several students used fractions. One tried to use decimals, and then switched over to fractions — perhaps because it is difficult to make sense of “0.44 repeating of a triangle.” One student jumped straight to percentages.
I had to decide which of the ratio examples to use. There were a few different ones to chose from, mostly green:whole based on the total area of the triangle. I chose one student who had ratios looking for equivalence, and another who compared based on unit rates.
I introduced this prompt the day after students took an assessment on ratios. They immediately noticed that I had changed our group’s wall in the classroom to reflect that we were not journeying into through just ratios anymore but into percentages and unit rates. I wanted this prompt to help bridge the ideas.
I decided to share the work in the following order:
- Equivalent Ratios
- Unit Rates & Ratios
This is quite deliberately different from the usual “concrete to abstract” or “least sophisticated the most sophisticated strategies.” The work students did with fractions reminds me of the thinking I might see in a 4th grade classroom, whereas the students who employed ratios to compare were decidedly using sixth grade skills. I decided that this order, from equivalent ratios to percentages, would emphasize the connections between them most effectively.
We started with Eleanor’s equivalent fraction work.
a: 1:3 = 3:9 (b)
In triangle a, the ratio of g to y is 3:1. In b, the triangle had 3y. If the ration of g to y for b was equal to a, there would be 9g. Instead, b has 7g. Therefore, a is greener.
“She used equivalent ratios.”
“She flipped her ratios from her work at the top to her explanation at the bottom. Originally she wrote 1:3 for A, and then she switched it to 3:1.”
“But that’s okay because she labeled the green and the yellow in her writing…”
We moved onto Lily’s work with unit rates and ratios.
Green : Yellow
B: 2 1/3 : 1
Green : Total
B: 7/10: 1
A is greener because in it the ratio of green triangles:total shapes is 3:4. That means that the ratio is also 3/4:1. The ratio of green to total in B is 7/10:1 and 3/4 is greater than 7/10 so A is greener.
Students noticed that Lily created unit rates centered on the whole as 1. They also noticed that she used fractions in her rates. “Eleanor only used whole numbers,” one girl remarked.
We compared and contrasted the two work samples. There were many similarities, but “I liked knowing what fraction to expect of the whole, and Lily’s (unit rate) work shows that better,” Jake said.
I couldn’t have devised a better transition into Ilya’s work with fractions:
A is 1/3 green because there is 9 triangle in it, and 3 green so it is 1/3.
B is less because 7/25 is less than 1/3. 7/21 is 1/3.
Students noticed that some of the same numbers appeared in Ilya and Eleanor’s work. I particular, the 1/3 green for A reminded them of Eleanor’s 1:3 and 3:1 ratios. We made connections to the previous work samples. “The numbers from Lily’s work don’t appear anywhere here,” one student observed, “but it’s like the same idea of what fraction of the whole.”
From there, we moved onto Caleb’s work with percentages.
A is 33.33% green and B is 28% green.
1 yellow = 6 green
A = 1 yellow + 3 green = 9 green
3 = 33.33% of 9
b = 3 yellow + 7 green = 25 green
7 = 28% of 25
A is greener.
“I don’t know how Caleb figured out the 28%,” Isabelle said. I thought this might happen, especially since we haven’t had studied percentages. They have occasionally come up, but only quickly. That was my choice; I knew we would have time to go into greater depth later, and did not want to shortchange the learning for the sake of a problem. Now, however, we were able to study percentages. How far do we go?
I asked Caleb to try to explain, knowing full well that his explanation may not reach Isabelle. It takes excellent teachers years to refine the skill of reaching a student where they are, using carefully curated words and models for that student. Caleb is 10 years old.
“Well, so, you know how Lily and Ilya were talking about the whole? My numbers have a whole, too. It’s 100%. So it’s like 28% out of 100% because it’s 7 green out of 25.” Not bad!
Isabelle paused. “Oh, is that because 28/100 is equivalent to 7/25?”
Several students nodded.
Isabelle optimistically shouted out, “oh, so I get percentages!”
There is, of course, a lot more learning we are going to do, but I was impressed with how quickly she saw the connection. I wonder if our conversation leading up to it both framed Isabelle’s thinking, and also helped Caleb root his explanation in mathematical language (ratios, fractions) that Isabelle already knows. It was one of those magical moments that makes me feel like an “amazing teacher.” Yes, I had cultivated an environment in which this happened, but I am (100%) certain that the scientific replicability of this exact moment is nil. We were lucky — not randomly lucky, but lucky all the same. As a teacher, I am constantly asking myself how I can increase my luck.
From there, we quickly wrapped up the conversation, and launched the longer task for the day. I was pleased with how the students helped me tell the start the story of percentages. (Big thank you to Mark Chubb for sharing that visual prompt!)