Ratios

Several years ago, I was working with a class of fourth & fifth graders. Their teacher had begun a unit on fractions and was interested in connecting fractions lớn real-world contexts. “No problem,” I told her.

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Our plan was that I would teach a lesson, she would observe sầu, and then we’d revisit it. I’d focus on talking with students about naming fractional parts, the standard symbolism of fractions, & equivalence.

My first real-world context: a six-paông chồng of water

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I showed the class the six-paông chồng I had brought lớn class và talked about one bottle being 1/6 of the six-paông chồng, two bottles being 2/6, three bottles being 3/6, và so on up to 6/6 being the same as the whole six-pachồng. The students seemed comfortable with this, & I wrote the fractions on the board:1/6 2/6 3/6 4/6 5/6 6/6

We also talked about three bottles being one-half of the six-pachồng, và that 3/6 và 50% were equivalent fractions because they both described the same amount of the six paông xã. I recorded this:3/6 = 1/2

I asked what fraction of the six-paông chồng would be gone after I drank four of the bottles và they answered 4/6 easily. I represented this numerically:1/6 + 1/6 + 1/6 + 1/6 = 4/6

I asked what fraction of the six-pack would be left after I drank four bottles, and they answered 2/6 easily. I represented this numerically with two equations:6/6 – 4/6 = 2/61 – 4/6 = 2/6

My second real-world context: a box of 12 pencils

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I continued with a different context—a box of 12 pencils. We talked about one pencil being 1/12 of the box, two pencils being 2/12, three pencils being 3/12, & so on. I wrote these fractions on the board:1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12 12/12

The pencil box gave us a way to lớn talk about another equivalent fraction for 1/2, this time 6/12. And I talked about 12/12 representing the pencils in the whole box:6/12 = 1/212/12 = 1

I asked, “If I give sầu a pencil to each of five sầu students, what fraction of the pencils would I have sầu given away?” They answered easily và I recorded numerically:1/12 + 1/12 + 1/12 + 1/12 + 1/12 = 5/12

I asked if 5/12 represented more or less than half the box, and they agreed that it was less than half. I recorded again:5/12 Then I hit a snagThe students in this class sat in small groups, & I next called the students’ attention to a table where two boys & one girl were seated. I asked them what fractional part of the students at the table were girls. Hands shot up và I had them say the fraction in unison in a whisper voice—one-third. I wrote 1/3 on the board.

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Brad noticed that the table next to his also had two boys & one girl sitting at it. Claudia commented, “So if you put the two tables together, then 2/6 would be girls.”

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Addison’s hand shot up. “Can I come up & write that in fractions?” he asked. I agreed. Addison came up và wrote on the board:1/3 + 1/3 = 2/6

I was stunned. Addison was correct that 2/6 of the students at the two tables were girls. But the addition equation that Addison wrote wasn’t correct. It’s every teacher’s nightmare when students combine the numerators và denominators to add fractions & think that adding 1/3 & 1/3, for example, gives an answer 2/6. But I didn’t think that Addison had applied that incorrect procedure. I wasn’t sure exactly what he was thinking.

To buy some time, I asked Addison to lớn explain what he had written. He said, “One out of three at Brad’s table is a girl, so that’s one-third. And it’s the same for Margaux’s table. So, if you put them together, then two out of all six kids are girls, & that’s two-sixths.” The rest of the students nodded.

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They were all pleased. I was a wreông xã.

So much for buying some time.

It’s hard lớn think & teach at the same time!I stood quietly and thought for a moment about what khổng lồ do next.

To fill the quiet, I said to the class, “When thinking about fractions, it’s important khổng lồ keep your attention on what the whole is.”

They nodded politely.

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After thinking some more, I returned to the context of the two tables of students. I said lớn Addison, “I see that you’re thinking about the two tables together.” He nodded. “So, the group of students at the two tables together has six students.” He nodded again. “Then Brad, Samantha, Jaông chồng, Margaux, Robbie, and Max, are each 1/6 of that group, just as each bottle of water is 1/6 of the whole six-pachồng.” Another nod. And because 1/6 + 1/6 equals 2/6, it makes sense lớn me that 2/6 of that group of six are girls.” I wrote on the board:1/6 + 1/6 = 2/6

None of the students seemed concerned that 1/3 + 1/3, as Addison had written, seemed lớn produce the same answer as 1/6 + 1/6, as I had written. Now I was breaking out inlớn a sweat.

I tried again to lớn explain“Let’s look at just one of the tables,” I suggested. “There are three students—Brad, Samantha, and Jaông xã. What fraction of the table does Brad represent?” The students answered 1/3 easily. “And what fraction does Jaông xã represent?” They answered 1/3 again. “And what fraction of the table are boys?” They answered 2/3. I wrote on the board, underneath what Addison had written:1/3 + 1/3 = 2/61/3 + 1/3 = 2/3


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