BELL WORK Repeating and Terminating Decimals In the

BELL WORK •

Repeating and Terminating Decimals In the past, you may have looked at number patterns to answer questions or to find the next part of a sequence. Have you ever taken a moment to consider how amazing the mathematics is in those patterns? Today you will investigate number patterns and learn about equivalent ways to write the same number. As a team, you will work to justify why two representations are the same and share your reasoning. In this course, you will often be asked to explain your reasons and justify your answers. When you are reasoning and justifying, you will focus on what makes a statement convincing or how you can explain your ideas. As you look at patterns today, ask your team these questions to help guide your conversations: What can we predict about the next number in the pattern? How can we justify our answer?

1 -40. In this course, you will transition to using “x” as a variable, so this course will avoid using “x” as a multiplication sign. Instead, the symbol “·” will be used to represent multiplication. Calculate the value of each expression below: 1 · 8 + 1 12 · 8 + 2 123 · 8 + 3 1234 · 8 + 4 a. What patterns do you see in the expressions above? Discuss the patterns with your team. Be sure that when your team agrees on something, it is recorded on each person’s paper. b. Use the patterns you found to predict the next three expressions and their values. Do not calculate the answers yet. Instead, what do you think they will be? c. Check the solution for each expression you wrote in part (b). Were your predictions correct? If not, look at the pattern again and figure out how it is changing.

1 -42. Are 0. 999. . . , , and 1 equal? How do you know? Discuss this with the class and justify your response. Help others understand what you mean as you explain your thinking.

2 -22. REWRITING REPEATING DECIMALS AS FRACTIONS Jerome wants to figure out why his pattern from problem 2 -21 works. He noticed that he could eliminate the repeating digits by subtracting, as he did in this work: This gave him an idea. “What if I multiply by something before I subtract, so that I’m left with more than zero? ” he wondered. He wrote: “The repeating decimals do not make zero in this problem. But if I multiply by 100 instead, I think it will work!” He tried again:

2 -22 continued…. a. Discuss Jerome’s work with your team. Why did he multiply by 100? How did he get 99 sets of ? What happened to the repeating decimals when he subtracted? b. “I know that 99 sets of are equal to 73 from my equation, ” Jerome said. “So to find what just one set of is equal to, I will need to divide 73 into 99 equal parts. ” Represent Jerome’s idea as a fraction. c. Use Jerome’s strategy to rewrite as a fraction. Be prepared to explain your reasoning.

PRACTICE: CONVERT FROM FRACTION TO DECIMAL • •

PRACTICE