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Investigation 4-1The diagram below is named for its creator, Theodorus of Cyrene (sy ree nee), a former Greek colony. Theodorus was a Pythagorean. The Wheel ofTheodorus begins with a triangle with legs 1 unit long and winds around counterclockwise. Each triangle is drawn using the hypotenuse of the previous triangle as one leg and a segment of length 1 unit as the other leg. To make the Wheel ofTheodorus, you need only know how to draw right angles and segments 1 unit long.

Use the Pythagorean Theorem to find the length of each hypotenuse in the Wheel of Theodorus (you should find 10 lengths). On picture above, label each hypotenuse with its length. Use the √ symbol to express lengths that are not whole numbers.

Task 1

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Use a cut-out copy of the ruler below to measure each hypotenuse on the wheel. Label the place on the ruler that represents the length of each hypotenuse. For example, the first hypotenuse length would be marked like this:

For each hypotenuse length that is not a whole number:

a) Give the two consecutive whole numbers between which the length lies. For example, √2 is between 1 and 2.

b) Use your ruler to find two decimal numbers (to the tenths place) between which the length lies. For example, √2 is between 1.4 and 1.5.

c) Use your calculator to estimate the value of each length and compare the result to the approximations you found in part b.

Task 2

Task 3

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In Task 2, you used a ruler to measure length. What is a reasonable level of accuracy for the lengths you found? Explain.

Jeziah uses his calculator to find √3. He gets 1. 732050808. Melody says this must be wrong because when she multiplies 1. 732050808 by 1. 732050808, she gets 3.000000001. Why do these students disagree?

Task 4

Task 5

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Investigation 4-2Task 1

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Use a calculator to write each fraction as a decimal. Tell whether the decimal is terminating or repeating. If the decimal is repeating, tell which digits repeat.Irvin

Irvin says he knows that the decimal representation of a fraction, such as 3/8,will be a terminating decimal if he can scale up the denominator to make a power of 10. What scale factor would Irvin need to use to rewrite 3/8 as x/1000? What is the decimal representation?

Jose says he can scale up but the decimal representation of 2/3 is a repeating decimal.

Do Irvin and Jose disagree? Explain.

Task 2

Task 3

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Predict which of the following fractions will have a terminating decimal representation:

For each decimal, find three equivalent fractions, if possible.0.3 0.3333 ... 0.13133133313333 ...

Rowan's calculator batteries are dead so she has to do the long division to find a decimal representation for 2/7

Continue the long division process until you are sure you can predict whether this decimal is terminating, repeating, or neither. Explain why you think your prediction is correct. Then, check your answer ona calculator.

Is it possible for a fraction to have a decimal equivalent that does not repeat and does not terminate? Explain.

Task 4

Task 5

Task 6

Task 7

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Write each fraction as a decimal. Tell whether each is a terminating or repeating decimal. If it is repeating, please tell which digits are repeating.

If you had a triangle with sides 5cm, 7cm and √74cm, would it be a right triangle? How do you know? Explain.

If you had a triangle with sides √2ft, √3ft and 2ft, would it be aright triangle? How do you know? Explain.

Practice

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Investigation 4-3Task 1

Task 2

Task 3

Write each fraction as a decimal.

Do you see any patterns in the table? If so, describe them here:

Use the patterns you found in above to write a decimal representation for each rational number. Then use your calculator to check your work.

a) 9/9 b) -10/9 c) 10/11 d) -12/11

Find a fraction equivalent to each decimal (if possible).

a) 1.222... b) 2.777... c) 0.818181... d) 0.27277277727777...

e) 1.99999... f) 0.99999... g) -7.0777... h) -5.50500500050000...

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Investigation 4-3Task 4

Task 5

The patterns from Task 1 can help you represent some repeating decimals as fractions. What about other repeating decimals, such as 0.121212 ... ? You need a method that will help you find an equivalent fraction for any repeating decimal.

Part 1) Suppose x = 0.121212... What is 100x? Is it still a repeating decimal?

Part 2) Complete the subtraction. 100x = 12.12121212... - x = 0.12121212...

99x =

Is the answer for 99x still a repeating decimal?

Part 3) Find a fraction form for 0.121212 ... by solving for x.

Part 4) Why do you think this method starts out by multiplying by 100? Explain.

Part 5) Use this method to write each repeating decimal as a fraction.

a. 0.151515 ... b. 0.123123123 ... c. 1.354354354 ...

Tell whether each statement is true or false.1. You can write any fraction as a terminating or repeating decimal.

2. You can write any terminating or repeating decimal as a fraction.

In the previous investigations, you saw that 10.111213141516 ... is an example of a decimal that never repeats and never terminates. Here is another example: 0.12122122212222 ...

You can never represent a nonrepeating, nonterminating decimal as a fraction, or rational number. For example, 1/10 is a close fraction representation of the decimal above, 12/100 is closer, and 121/1000 is even closer. You cannot, however, get an exact fraction representation for this decimal.

Numbers with decimal representations that are nonterminating and nonrepeating are called IRRATIONAL NUMBERS. Some irrational numbers have patterns, as above. Some have no patterns, but the decimals neverterminate and never repeat. You cannot express these numbers as ratios of integers or proportion.

You have worked with irrational numbers before. For example, the decimal representation of the number π starts with the digits 3.14159265 ... and goes forever without repeating any sequence of digits. The number π is irrational. The number √2 is also irrational. You could not find an exact terminating or repeating decimal representation for √2 because such a representation does not exist! Other irrational numbers are √3, √5, and √11. In fact, √n is an irrational number for any whole number value of n that is not a square number.

Are the following numbers rational or irrational: √7, √9, √0.25? Why or why not?

The set of irrational and rational numbers is called the set of REAL NUMBERS. An amazing fact is that there are an infinite number of irrational numbers between any two fractions!

Write three or more nonterminating, nonrepeating decimals that are greater than 0.5 but less than 0.6.

Why is there no limit to the number of nonterminating, nonrepeating decimals that are between 0.5 and 0.6?

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Investigation 4-4Task 1

Task 2

Write a rational-number estimate for √5 that is less than √5 and one that is greater than √5.

Write an irrational-number estimate for √5 that is less than √5 and one that is greater than √5.

Tell whether each number is rational or irrational. Explain your reasoning.

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Task 3

Task 4

As part of the rocket project, Sophie is making a pyramid nose cone. She starts with the net shown below, drawn on centimeter grid paper.

What is the exact value of q? Is the value of q a rational or irrational number? Explain.

What is the exact height h of the pyramid? Is the height a rational or irrational number? Explain.

Will the finished pyramid fit inside a cube-shaped box that is 6 centimeters wide, 6 centimeters long, and 6 centimeters high? Explain.

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Task 5

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Problems of the Week

What is the height of the Cone? What is the volume of the cone?

Suppose the edge length of the cube is 6 units. What is the volume of the pyramid?

Suppose the edge length of a cude is x units. What is the volume of the pyramid?

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Problems of the WeekFind the following areas of shaded regions

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Problems of the Week

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Problems of the Week