In this activity, students practice using long division to calculate the decimal form of a rational number. Students are expected to notice a repeating pattern (MP8).
Launch
Arrange students in groups of 2. Since the goal is for students to notice a repeating pattern, do not provide access to calculators. If needed, remind them of the previous activity where the decimal expansion of was shown to be using long division and repeated reasoning.
Give students 1–2 minutes of quiet work time, and follow with a whole-class discussion.
Student Lesson in Spanish
Activity
None
Student Task Statement
The first 3 digits after the decimal for the decimal expansion of have been calculated. Find the next 4 digits.
Long division calculations for decimal expansion, showing place value. The first line indicates the given place values of the quotient, 0 point 4 2 8. The second line indicates the division sentence 3 divided by 7; the number 7 is the left most number, followed by the long division symbol, and the number 3 inside; the 3 lines up vertically with the 0 above. On the third line reads as "minus twenty eight," with the 2 directly below the 3 from above. The fourth line reads "twenty," with the 2 directly below the 8 in 28. The fifth line reads "minus fourteen," with the 1 directly below the 2 in 20 and the 4 directly below the 0 in 20. The sixth line reads "sixty," with the 6 directly below the 4 in 14. The seventh line reads "minus fifty six" with the 5 directly below the 6 in 60, and the 6 directly below the 0 in 60. The eight line reads "4" with the 4 directly below the 6 in 56. A vertical line is drawn through all of the lines, falling between the 0 and the 4 in "0 point four two eight" and the 2 and 8 in twenty eight.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is to make sure students understand that all rational numbers have a decimal expansion that eventually repeats. Ask students to share the next 4 digits and record them on the long division calculation for all to see. Discuss:
“Without calculating, what number do you think will be next? Why?” (I think 2 will be the next digit because I can see the starting pattern has begun again.)
Continue the calculation and verify that 2 comes next and continue until reaching 4 again. Point out that this cycle will continue indefinitely—we can predict what will happen at each step because it is exactly like what happened 6 steps ago.
Tell students that all rational numbers have a decimal expansion that eventually repeats. Sometimes they eventually repeat 0s, like in . Sometimes they repeat several digits like in . If necessary, remind students that in overline notation, the line goes over the digits that repeat.
Be careful in the use of the word “pattern,” as it can be ambiguous. For example, there is a pattern to the digits of the number 0.12112111211112 . . . , but the number is not rational.
17.2
Activity
15 mins
Some Numbers Are Rational
Standards Alignment
Building On
8.EE.C.7.b
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
In this activity, students learn and practice a strategy for rewriting repeating decimals as a fraction. Students begin by arranging cards in an order that shows how a strategy is used to show . Next, students use the strategy to calculate the fractional representations of two other values.
The extension problem asks students to repeat the task for decimal expansions of and . It may be counter-intuitive for students to conclude that . Students might insist that must be strictly less than 1, which can prompt an interesting discussion. Invite students to make their argument more precise.
Launch
Arrange students in groups of 2. Do not provide access to calculators since the purpose of this activity is to rewrite a decimal representation of a number as a fraction.
Demonstrate the algorithm with an example such as converting .
Distribute a set of the slips cut from the blackline master to each group. Tell students to order the cards with their partner, then work on the second question individually. Give students 5 minutes of quiet work time and follow with a whole-class discussion.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students half of the cards to start with and introduce the remaining cards once students have completed their initial set of matches. Supports accessibility for: Conceptual Processing, Organization, Memory
Activity Synthesis
The goal of this discussion is to reinforce student understanding of the strategy introduced to rewrite a repeating decimal as a fraction. Begin the discussion by selecting 2–3 students to share their work for the second problem, displaying each step for all to see. Ask if anyone completed the problem in a different way and, if so, have those students also share.
If no students notice it, point out that when rewriting , we can multiply by 100, but multiplying by 10 also works since the part that repeats is only 1 digit long.
Conclude the discussion by asking students to rewrite using this strategy.
17.3
Activity
15 mins
Some Numbers Are Not Rational
Standards Alignment
Building On
Addressing
8.NS.A.1
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
In this activity, students revisit irrational numbers to emphasize the point that no fractional representation exists for these numbers. For all irrational numbers, long division doesn't work because there are no two integers to divide, and different methods to approximate the value need to be discussed.
Students will approximate the value of using successive approximations and discuss how they might figure out a value for using measurements of circles. They will also plot these values on number lines to reinforce the idea that irrational numbers are still numbers—they have a place on the number line even if they cannot be written as a positive or negative fraction.
Launch
Arrange students in groups of 2. Provide access to a calculator without a square root button. If that calculator does not have a button , tell students that a calculator will give something similar to 3.141592654 as a value for .
Discuss with students:
“In a previous activity, we used long division to find decimal representations of numbers. Why can't we do that with irrational numbers?” (There are no 2 numbers to divide since an irrational number cannot be written as a fraction.)
Then give students 3–4 minutes of quiet work time. Pause students and tell them to check in with their partner after the first set of questions. Continue work on the remaining questions, and follow with a whole-class discussion.
Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to respond to all parts of the first question and as much of the second question as they have time for. Supports accessibility for: Organization, Attention
Activity Synthesis
The purpose of this discussion is to deepen students' understanding that irrational numbers cannot be written as fractions, but they are still numbers with a location on the number line. As such, their values are approximated using different methods than for rational numbers. Discuss with students:
“How long do you think you could keep using the method in the first problem to find more digits of the decimal representation of ?” (My calculator only shows 9 digits to the right of the decimal, so that's as far as I could go.)
“Would this method work for any root?” (Yes, I could find the two perfect squares that the square of the number we are taking the square root of is between and then keep approximating the value of the root from there.)
Tell students that the strategy they used in the first problem is called “successive approximation.” It takes time, but successive approximation works for finding more and more precise approximations of irrational numbers, as long as there is a clear value to them check against. In the case of , since , there is a clear value to test whether approximations are too high or too low.
Lastly, select 2–3 students to share their reasoning to the last part of the second problem. Make sure students understand that since measuring has limitations of accuracy, any calculation of using measurement, such as with the beaker and the fuel tank, will have limited accuracy.
MLR8 Discussion Supports. For each response to the last part of the second problem that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language. Advances: Listening, Speaking
Lesson Synthesis
The main goal of this discussion is to emphasize that and are irrational numbers. Their decimal expansions never end and never repeat like rational numbers do. But they are still numbers!
Display the number lines for and for all to see.
Ask students what is the same and what is different about the two images (Both images show number lines that are each partitioned into 10 parts. Each successive number line is labeled with one additional digit of accuracy. One image has 4 number lines while the other only has 3. The image for has a repeating pattern and would go on forever. The image for would also go on forever but without a predictable pattern.)
Explain that the number has to be a number because a square with an area of 2 square units has as a side length. Similarly, has to be a number because it relates the radius of a circle to its area. The main takeaway is that there exist numbers that are real, but cannot be written as positive or negative fractions. Here are some questions for discussion:
“What are some decimals for which our method of rewriting decimals as fractions will work?” (any repeating decimal)
“We’ve seen that the decimal expansion of does not repeat. What would happen if we tried to use Noah’s method on ? On ?” (It would not work because those numbers are irrational.)
Student Lesson Summary
Not every number is rational. Earlier we tried to find a fraction whose square is equal to 2. That turns out to be impossible, although we can get pretty close (try squaring ). Since there is no fraction equal to , it is not a rational number, so we call it an "irrational number." Another well-known irrational number is .
Every number, rational or irrational, has a decimal expansion. For example, the rational number has the decimal expansion 0.181818 . . . with the 18s repeating forever. Irrational numbers also have infinite decimal expansions, but they don't end up having a repeating pattern.
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Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Your teacher will give your group a set of cards. Each card will have a calculations side and an explanation side.
The cards show Noah’s work calculating the fraction representation of . Arrange these in order to see how he figured out that without needing a calculator.
Use Noah’s method to calculate the fraction representation of:
Student Response
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Building on Student Thinking
Activity
None
Student Task Statement
Why is between 1 and 2 on the number line?
Why is between 1.4 and 1.5 on the number line?
How can you figure out an approximation for accurate to 3 decimal places?
Label all of the tick marks. Plot on all three number lines. Make sure to add arrows from the second to the third number line.
Elena notices that a beaker in science class says it has a diameter of 9 cm and measures its circumference to be 28.3 cm. Using these values and the equation for circumference, , what value do you get for ?
Diego learns that one of the space shuttle fuel tanks has a diameter of 840 cm and a circumference of 2,639 cm. Using these values and the equation for circumference, , what value do you get for ?
Label all of the tick marks on the number lines. Use a calculator to get a very accurate approximation of and plot that number on all three number lines.
How can you explain the differences between these calculations of ?
