represented as a fraction? categories of numbers?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/im... · 9111. Howie and Steve are making cookies for themselves and some friends

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IM 8 Ch 9.2.4 What Kind Of Number Is It

CPM Materials modified by Mr. Deyo

Can the number be represented as a fraction?

What are the different categories of numbers?

Common Core Standard: 8.NS.1, 8.NS.2, 8.EE.2


By the end of two periods, I will distinguish rational numbers from irrational numbers as well as convert terminating or repeating decimals to fractions. I will also use rational approximations of irrational numbers to compare the size of irrational numbers and locate them on a number line.I will demonstrate this by completing Four‑Square notes and by solving problems in a pair/group activity.

Learning TargetTitle: IM8 ‑ Ch. 9.2.4 What Kind Of Number Is It? Date:


Home Work: Sec. 9.2.4Desc. Date Due

Review & Preview

Day 1: 4 Problems 9‑110, 9‑112, 9‑113, 9‑114

Day 2: 4 Problems 9‑116, 9‑117, 9‑118, 9‑119



Vocabulary1) squared

2) square root

3) rational number

4) irrational number



9.2.4 What Kind Of Number Is It? Any number that can be written as the ratio of two integers (a/b) with b ≠ 0 is called a rational number. A rational number can be matched to exactly one point on a number line. There are many other points on the number line, however, for which there is not a corresponding rational number. These numbers are called irrational numbers. Numbers such as , , − are irrational numbers. The rational numbers and the irrational numbers make up all of the numbers on the number line and together are called the real numbers.In this lesson you will learn how to identify a number as rational or irrational. You will also write decimals as fractions to show that they are rational. Then you will compare these kinds of numbers and place them on the number line.

9100 In previous courses, you worked with decimals that repeated and terminated. All of these are called rational numbers because they can be written as a ratio, like and = 5. Because = 3, is also a rational number.However, there are some numbers that do not repeat or terminate when they are written as decimals, such as . Such numbers are called irrational numbers. An irrational number cannot be written as a ratio of any two integers. In other words, an irrational number cannot be written as a fraction. Use your calculator to find the square root of the following numbers. Decide whether the decimals are rational (having decimals that terminate or repeat) or irrational.

= 1.41421356237

a) b) c)


9101. Do you think that you can decide by looking at it whether a number is rational or irrational? You will explore this idea in parts (a) through (d) below.a) Without doing any calculations, which of the numbers below do you think are rational numbers? Which do you think are irrational numbers? Discuss this with your team and make predictions.

c) What do you notice about the decimal forms of rational numbers compared to irrational numbers?

b) Now use your calculator and write the equivalent decimal for each of the numbers in the list. Were your predictions correct?

d) Is rational or irrational? Explain your answer.


9102 Every rational number can be written as a fraction, that is, as a ratio of two integers. Since 0.78 is described in words as “seventyeight hundredths,” it is not a surprise that the equivalent fraction is (78/100). Use what you know about place value to rewrite each terminating decimal as a fraction. Check your answers with a calculator.a) b)

c) d)

e) f)

0.19 0.391

0.001 0.019

0.5240.3


9103 Jessica knows that is a rational number, so she should be able to write it as a fraction. She wonders how to rewrite it, though. She started to rewrite it as ,but she is not sure if that correct. Is equal to ? Be ready to justify your answer.


9104ag. To help Jessica with her problem, find the decimal equivalents for the fractions below.

a) b) c)

d) e) f)

g) What patterns do you see between the fractions and their equivalent decimals? What connections do these fractions have with those you found in problem 9102? Be ready to share your observations with the class.

1999

391999

39

1 999

524999

19 999


9104h,i.

h) Use your pattern to predict the fraction equivalent for . Then test your guess with a calculator.

i) Use your pattern to predict the fraction equivalent for . Then test your guess with a calculator.

6599


9105a. REWRITING REPEATING DECIMALS AS FRACTIONS

a) Discuss Jessica’s work with your team. Why did she multiply by 100?

How did she get 99 sets of ?

What happened to the repeating decimals when she subtracted?

Jessica wants to figure out why the pattern from problem 9104 works. She noticed that she could eliminate the repeating digits by subtracting, as she wrote below on the left. This gave her an idea. “What if I multiply by something before I subtract, so that I’m left with more than zero?” she wondered. She wrote on the right:

“The repeating decimals don’t make zero in this problem. But if I multiply by 100 instead, I think it will work!” She tried again:


9105b,c. REWRITING REPEATING DECIMALS AS FRACTIONS

c) Use Jessica’s strategy to rewrite as a fraction. Be prepared to explain your reasoning.

b) “I know that 99 sets of are equal to 57 from my equation,”Jessica said. “So to find what just one set of is equal to , I will need to divide 57 into 99 equal parts.” Represent Jessica’s idea as a fraction.


9106. Show that the following repeating decimals are rational numbers by rewriting them as fractions.

a) b)

c) d)

= =

==


9107. Indicate the approximate location of each of the following real numbers on a number line. What can make this task easier? Try to do it without using a calculator.


9108. Without using a calculator, order the numbers below from least to greatest.


9109. Copy and complete the following sentences.

a) The set of all numbers on the number line are called the _______ ____________.

b) A number that has an equivalent terminating or repeating decimal is called a(n) __________ _____________.

c) A number that has an equivalent decimal that is nonrepeating is called a(n) ____________ ____________.

d) Any number that can be written as a fraction of integers is a(n) ______________ _____________.


9110a,b. Graph each of the pairs of points listed below and draw a line segment between them. Use the graph to help you find the length of each line segment. State whether each length is irrational or rational.

a) (−3, 0) and (0, −3) b) (2, 3) and (1, 2)

https://www.desmos.com/calculator/jxz8jpjsnc

http://homework.cpm.org/cpmhomework/homework/category/CC/textbook/CC3/chapter/Ch9/lesson/9.2.4/problem/9110


http://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CC3/chapter/Ch9/lesson/9.2.4/problem/9-110


9110c,d. Graph each of the pairs of points listed below and draw a line segment between them. Use the graph to help you find the length of each line segment. State whether each length is irrational or rational.

c) (3, 2) and (3, −3) d*) (2, 3) and (3, 3)






9111. Howie and Steve are making cookies for themselves and some friends. The recipe they are using will make 48 cookies, but they only want to make 16 cookies. They have no trouble reducing the amounts of flour and sugar, but the original recipe calls for 1 cups of butter. Help Howie and Steve determine how much butter they need. http://homework.cpm.org/cpmhomework/homework/category/CC/textbook/CC3/

chapter/Ch9/lesson/9.2.4/problem/9111

34



9112. Find the perimeter and area of the figure here. Show your work for each of the steps that you use.

Perimeter Area




9113. Write the following numbers in scientific notation.

a) b)


370,000,000 0.0000000000076



9114ad. Simplify each of the following expressions.

a) 4x3y · 3xy2 b) 6a5b2 · 3ab2

d)c) m2n · 9mn


35· 8 · 53

32· 23 · 53· 33



9114eh. Simplify each of the following expressions.

e) f)

h*)g*) m2n3 · 9m4n5


25· 8 · 53

32· 23 · 83· 55

m4nn3

9a4b2

15b



9115ad. Simplify each numerical expression.

a) b)

d*)c)

5 6 + 1


2 16

5 6 6 + 16 2 + 8 1



9116. Identify the following numbers as rational or irrational. If the number is rational, show that it can be written as a fraction.

a) b)

d*)c)




9117. Solve each system.

a) b)


y = 2x + 1y = −3x − 4

y = x + 4

y = x − 2

1312



9118. For the rule y = 6 + (−3)xa) What is the yintercept? y = ( )x + ( )

m =

b) What is the slope of the line?

c*) Graph the equation

( 0 , )




9119. Make a table and graph the rule that includes x‑values from −1 to 9.

y = x 2Input (x) Output (y)

10123456789


Graph the rule on graph paper.



9120. Dawn drove 420 miles in 6 hours on a rural interstate highway. If she maintains the same speed, how far can she go in 7.5 hours? http://homework.cpm.org/cpmhomework/homework/category/CC/textbook/CC3/




9121. The attendance at the county fair was lowest on Thursday, the opening day. On Friday, 5500 more people attended than attended Thursday. Saturday doubled Thursday’s attendance, and Sunday had 3000 more people than Saturday. The total attendance was 36,700. Write and solve an equation to find how many people attended the fair each day. http://homework.cpm.org/cpmhomework/homework/category/CC/textbook/CC3/



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represented as a fraction? categories of numbers?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/im... · 9111. Howie and Steve are making cookies for themselves and some friends