Worked today with 7th grade students on task 5.2 Repeating Decimals in the Illustrative Mathematics curriculum. Students were successful finding by long division that the first fraction has a decimal equivalent of 0.36, the second 0.36 with the 6 repeating, and the third 0.36 with the 36 repeating. They were able to identify what was the same and what was different about the three numbers. The fun began when we addressed question 3, which has the greatest value? There was significant disagreement within the class, a signal to me to dig deeper and see where the gaps and misconceptions lie.
Several students looked at the fractions and argued that the fraction with 11 in the denominator was the largest because a smaller denominator means a larger fraction. We discussed that the numerators were different and all three fractions seemed to be close to a third. I asked how we could compare fractions in general. Students were not sure about this, and through questioning realized that a common denominator would allow us to generate equivalent fractions with the same denominator whose numerators would enable us to order them. So what common denominator could we use? Since there were three numbers and their product would be pretty large, I suggested we look for the smallest common multiple that would work. I factored 25 into 5 x 5 and 30 into 5 x 2 x 3, and claimed that 5 x 5 x 2 x 3 x 11 would give us the smallest common multiple. Students were very bothered by the absence of the third 5 in the product, even though they conceded that the smallest common multiple of 6 and 8 is 24 and not 6 x 8 or 48. They struggled to understand that a multiple of a number must have as factors all the factors of the original number, and that a common multiple of several numbers does not need every factor of each number if those factors are duplicated across different numbers (for example, 6 and 8 each have a factor of 2, so their least common multiple is 3 x 8, not 6 x 8). So we agreed to find the equivalent fractions both ways, with my 5 x 5 x 2 x 3 x 11 as one common denominator and with 5 x 5 x 5 x 2 x 3 x 11 as another. They struggled with finding the numerators, not really understanding what to multiply by in each case, and we finally arrived at:
9/25 = 594/1650 = 2970/8250
11/30 = 605/1650 = 3025/8250
4/11 = 600/1650 = 3000/8250
It was clear by examining the numerators that 11/30 has the largest value. But some students were still unconvinced. So I suggested we break the decimal equivalents into a sum of the digits’ values according to place value. We agreed to go out to 4 places since that would cover all the repeating patterns we found.
9/25 = .3600 = 3/10 + 6/100 + 0/1,000 + 0/10,000
11/30 = .3666 = 3/10 + 6/100 + 6/1,000 + 6/10,000
4/11 = .3636 = 3/10 + 6/100 + 3/1,000 + 6/10,000
We noticed that all three had 3 tenths and 6 hundredths, making 11/30 the largest because it has the most thousandths.
Even when presented with all this evidence, some students were uneasy. They had internalized so many misconceptions about the number of digits in decimal numbers, factors and multiples, equivalent fractions, and place value in general.
I thought about all the K-5 standards that, if mastered, would have made this effort, as well as the work we completed with proportional relationships, much more manageable for my students. Among them:
Meaning of Operations:
Determine the unknown whole number in a multiplication or division equation relating three whole numbers.
… Recognize that a whole number is a multiple of each of its factors. …
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
Use decimal notation for fractions with denominators 10 or 100
Compare two decimals to hundredths by reasoning about their size…
Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.
Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2…
This is why it is so important that teachers at different levels understand what came before and what is yet to come. How did my students build (or not build, or build in a broken manner) their current understanding, how can I identify and help fill gaps and fix misconceptions, and what am I helping them lay the foundation for on their math journey? Only by focusing on coherence at every grade level can we hope to help each student become proficient.