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# Repeating decimal as approximation

## Task

- Use long division to find the repeating decimal that represents $\frac{29}{13}$
- Take the number obtained by including only the first two digits after the decimal point, and multiply that by $13$.
- Take the number obtained by including only the first four digits after the decimal point, and multiply that by $13$.
- Take the number obtained by including only the first six digits after the decimal point, and multiply that by $13$.
- What do you notice about the product of $13$ and decimal approximations of $\frac{29}{13}$ as more and more digits are included after the decimal point?
- How does what you observed in Part (e) help make sense of what it means for $\frac{29}{13}$ to be equal to the repeating decimal expression you found in the Part (a)?

## IM Commentary

The purpose of the task is to have students reflect on the meaning of repeating decimal representation through approximation. A formal explanation requires the idea of a limit to be made precise, but 7th graders can start to wrestle with the ideas and get a sense of what we mean by an "infinite decimal." Students can make observations which reinforce the topic at hand as well as lay groundwork for later developments.

This task allows students to engage in standard for mathematical practice 8, Look for and express regularity in repeated reasoning. Students are not asked explicitly to compare what they find in the second and third parts with 29, since by this grade they should make note of that on their own.

While the first part asks for application of long division (by hand), the multiplications could be done by calculator, at the teacher's discretion. The trade-offs in this choice are between further development of number fluency and opportunity for reflection if these calculations are done by hand versus greater likelihood of accuracy and answers provided more quickly (which can help observing the trend) if done by calculator.

## Solution

- The answer, after performing long division, is $2.\overline{230769}$.
- $2.23 \times 13 = 28.99$
- $2.2307 \times 13 = 28.9991$
- $2.230769 \times 13 = 28.999997$
- As we include more of the decimal expansion, we see a decimal which is less than $29$ whose difference with $29$ becomes smaller and smaller. In the product, we see that the number of 9's following the decimal is roughly the number of digits after the decimal point included in the approximation of $\frac{29}{13}$.
- Because $\frac{29}{13} \times 13 = 29$, we should expect that when we multiply numbers that are close to $\frac{29}{13}$ by $13$, the result should be to be close to $29$. In the product, we see that the number of 9's following the decimal is roughly the number of digits after the decimal point included in the approximation of $\frac{29}{13}$. Thus, it seems likely that if we used "all" the digits in the infinite decimal expansion, the product would be exactly equal. In other words, $2.\overline{230769}$ should be equal to $\frac{29}{13}$ because when you multiply it by $13$ you get $29$. What we see here is that if we take approximations to $\frac{29}{13}$ given by more and more of its repeated decimal expansion and multiply them by $13$, we get better and better approximations to $29$.

## Repeating decimal as approximation

- Use long division to find the repeating decimal that represents $\frac{29}{13}$
- Take the number obtained by including only the first two digits after the decimal point, and multiply that by $13$.
- Take the number obtained by including only the first four digits after the decimal point, and multiply that by $13$.
- Take the number obtained by including only the first six digits after the decimal point, and multiply that by $13$.
- What do you notice about the product of $13$ and decimal approximations of $\frac{29}{13}$ as more and more digits are included after the decimal point?
- How does what you observed in Part (e) help make sense of what it means for $\frac{29}{13}$ to be equal to the repeating decimal expression you found in the Part (a)?

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