Relationship between Rational Numbers and Decimals

Relationship between Rational Numbers and Decimals 7th Grade Level

Mr. Alonzo

Tips for learning

• This presentation is designed to be a lesson that will be teaching you new things. If you rush through it you will fail.

• Do not rush through this slide show.

• Work the questions out on a seperate sheet of paper and take notes.

• Answer each question fully and in complete sentences.

• Ask Mr. Alonzo for help if you don't understand something.

• You may watch the videos provided in this slide show or look at the poster's in Mr. Alonzo's room for help.

Objective

The student applies mathematical process standards to represent and use rational numbers in a variety of forms. Extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers.

Rational Number Definition

• A rational number is a number that can be written as a ratio (fraction) of two integers a and b, where b is not zero.  $\frac{a}{b}$  So basically a fraction where the denominator can never be zero because you can't divide by zero.

•  $\frac{4}{7}$  is a rational number because it is a ratio (fraction) of two integers.

•  $0.37$  is also a rational number because it can be turned into the ratio (fraction)  $\frac{37}{100}$  

Turning Fractions into Decimals

To turn a fraction into a decimal you must use long division. The numerator goes in the "house" and the denominator goes outside the "house".  $\frac{1}{2}=1\div2\ =\ 0.5$  

Find the equivalent decimal form of each fraction. Remember that numbers that repeat can be written as 0.333… or

 $0.\overline{3}$  

Videos on how to turn a fraction into a decimal

Example 1: https://youtu.be/Y1V5mZaMfTk
Example 2: https://youtu.be/sCVyvfOLI6U

Multiple Choice

Convert the following rational number into a decimal. $\frac{2}{3}=$  

•  $0.\overline{66}$  

•  $2.3$  

•  $0.23$  

•  $0.60$  

Multiple Choice

Convert the following rational number into a decimal. $\frac{2}{9}=$  

•  $0.\overline{22}$  

•  $2.9$  

•  $0.29$  

•  $0.\overline{29}$  

Multiple Choice

Convert the following rational number into a decimal. $\frac{2}{9}=$  

•  $0.\overline{22}$  

•  $2.9$  

•  $0.29$  

•  $0.\overline{29}$  

Multiple Choice

Convert the following rational number into a decimal. $\frac{2}{9}=$  

•  $0.\overline{22}$  

•  $2.9$  

•  $0.29$  

•  $0.\overline{29}$  

Facts about Fractions

• When the Numerator is smaller than the Denominator you get a decimal smaller than 1.

• When the Numerator and Denominator are the same then the fraction converts to 1 whole.

• When you have an Improper Fraction with a bigger Numerator than Denominator your decimal is always bigger than 1.

• Sometimes it is better to do math with fractions because some numbers go on forever when they are in decimal form so you must learn how to do fractions.

   $\frac{1}{3}=0.\overline{33333333333}$  forever
  

Turning Decimals into Fractions

To turn a decimal into a Fraction you must know about place value.

 $0.81=\frac{8}{10}+\frac{1}{100}=\frac{80}{100}+\frac{1}{100}=\frac{81}{100}$  

Turning Whole numbers with decimals into Fractions

When you have a decimal larger than 1 you will end up with a mixed number.

 $17+\frac{5}{10}+\frac{9}{100}+\frac{1}{1000}=17\frac{591}{1000}$  

Mixed numbers must be simplified just like regular fractions

All you must do is simplify the fraction part of the mixed number. $\frac{145}{1000}\div\frac{5}{5}=\frac{29}{200}$  

 $3\frac{145}{1000}=3\frac{29}{200}$  
Here is a video example: https://youtu.be/NzIoHQfStoM

Poll

Consider the decimal 0.101001000100001000001…. Do you think this decimal represents a rational number?

• Yes it is a rational number

• No it is not a rational number

Open Ended

Considering your answer on the question of whether the decimal 0.101001000100001000001…. is a rational number or not. Explain why you think it is or is not a rational number.

Poll

Do you think a negative sign affects whether or not a number is a rational number? Use  $-\frac{5}{8}$   as an example.

• Yes it is a rational number

• No it is not a rational number

Open Ended

Considering your answer of whether you think  $-\frac{5}{8}$  is or is not a rational number explain why you think it is or isn't one?

Poll

Do you think a mixed number is a rational number?

• Yes it is a rational number

• No it is not a rational number

Open Ended

Considering your answer of whether or not you think a mixed number is a rational number explain why?

Writing Rational Numbers as Decimal

• When you turn a rational number into a decimal you end up with either a terminating decimal or a repeating decimal.

• Some decimals are "repeating decimals" (math vocab word) because one or more digits repeat infinitely.

• Other decimals are "terminating decimals" (math vocab word) because the decimals come to an end.

Example of a Terminating Decimal

•  $\frac{5}{16}$  

• Divide 5 by 16.

• Add a zero after the decimal point.

• Subtract 48 from 50.

• Use the grid to help you complete the long division.

• Add zeros in the dividend and continue dividing until the remainder is 0.

• The decimal equivalent of  $\frac{5}{16}$  is 0.3125.

Example of a Repeating Decimal

•  $\frac{13}{33}$  
  

• Divide 13 by 33.

• Add a zero after the decimal point.

• Subtract 99 from 130.

• Use the grid to help you complete the long division.

• You can stop dividing once you discover a repeating pattern in the quotient.

• Write the quotient with its repeating pattern and indicate that the repeating numbers continue. The decimal equivalent of  $\frac{13}{33}$   is 0.3939…,  $0.\overline{39}$  

Multiple Choice

Look at the following Rational Number and determine how many digits after the decimal are in the pattern that repeats.

 $\frac{1}{3}$  

• 1 digit repeats

• 2 digits repeat

• 3 digits repeat

• 4 digits repeat

 $\frac{1}{3}=0.\overline{3}$  

When you convert  $\frac{1}{3}$  to a decimal you get 0.3333... forever so only 1 digit repeats itself over and over.

Multiple Choice

Look at the following Rational Number and determine how many digits after the decimal are in the pattern that repeats.

 $\frac{4}{7}$  

• 1 digit repeats

• 2 digits repeat

• 3 digits repeat

• 4 digits repeat

• 6 digits repeat

 $\frac{4}{7}=0.\overline{571428}$  

When you convert  $\frac{4}{7}$  to a decimal you get 0.571428 and then the pattern repeats 571428 again and again forever so a total pattern of 6 different digits repeat.

Sometimes only part of the pattern repeats

Some rational numbers when converted to decimals only repeat part of the pattern. For example

 $\frac{9}{22}=0.4\overline{09}$  In this example there is a 4 in the tenths place but only the 09 repeat over and over forever. The four only appears this one time.

Open Ended

Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal.  $\frac{3}{5}=$  

Open Ended

Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal.  $\frac{89}{100}=$  

Open Ended

Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal.  $\frac{4}{12}=$  

Open Ended

Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal.  $\frac{25}{99}=$  

Open Ended

Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal.  $\frac{7}{9}=$  

Open Ended

Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal.  $\frac{9}{25}=$  

Open Ended

Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal.  $\frac{1}{25}=$  

Open Ended

Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal.  $\frac{25}{176}=$  

Open Ended

Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal.  $\frac{12}{1000}=$  

Shortcut to make Mixed Numbers into Decimals

• Seperate the whole number part from the fraction part.

• Use long division to figure out what the fraction part is as a decimal.

• Add the whole number part to the decimal part to get the total.

Open Ended

Write each mixed number as a decimal.  $11\frac{1}{6}=$  

Open Ended

Write each mixed number as a decimal.  $2\frac{9}{10}=$  

Open Ended

Write each mixed number as a decimal.  $8\frac{23}{100}=$  

Open Ended

Write each mixed number as a decimal.  $7\frac{3}{15}=$  

Open Ended

Write each mixed number as a decimal.  $54\frac{3}{11}=$  

Open Ended

Write each mixed number as a decimal.  $3\frac{1}{18}=$  

Open Ended

Maggie bought  $3\frac{2}{3}$   lb of apples to make some apple pies. What is the weight of the apples written as a decimal?

Open Ended

Harry’s dog weighs  $12\frac{7}{8}$   pounds. What is the weight of Harry’s dog written as a decimal?

Open Ended

Tom is trying to write  $\frac{3}{47}$   as a decimal. He used long division and divided until he got the quotient 0.0638297872, at which point he stopped. Since the decimal doesn’t seem to terminate or repeat, he concluded that  $\frac{3}{47}$   is not rational. Do you agree or disagree? Why?

Use this chart to solve the the next 6 problems

*Copy the number of player each sport has on a seperate sheet of paper or screen shot the picture.*
Write each ratio in the form  $\frac{a}{b}$   and then as a decimal. Tell whether each decimal is a terminating or a repeating decimal.

Open Ended

What is the ratio of basketball players to football players? What is the decimal equivalent of this ratio? Is this decimal a terminating decimal or a repeating decimal?

Open Ended

What is the ratio of hockey players to lacrosse players players? What is the decimal equivalent of this ratio? Is this decimal a terminating decimal or a repeating decimal?

Open Ended

What is the ratio of polo players to football players? What is the decimal equivalent of this ratio? Is this decimal a terminating decimal or a repeating decimal?

Open Ended

What is the ratio of lacrosse players to rugby players? What is the decimal equivalent of this ratio? Is this decimal a terminating decimal or a repeating decimal?

Open Ended

What is the ratio of football players to soccer players? What is the decimal equivalent of this ratio? Is this decimal a terminating decimal or a repeating decimal?

Open Ended

Look for a Pattern Beth said that the ratio of the number of players in any sport to the number of players on a lacrosse team must always be a terminating decimal. Do you agree or disagree? Why?

Open Ended

Use the following information to answer the following questions.

Yvonne bought  $4\frac{7}{8}$   yards of material to make a dress.

1. What is  $4\frac{7}{8}$   written as an improper fraction?
2. What is  $4\frac{7}{8}$   written as a decimal?
3. If Yvonne wanted to make 3 dresses that use  $4\frac{7}{8}$   yd of fabric each, explain how she could use estimation to make sure she has enough fabric for all of them?

Open Ended

*Vocabulary Question*

Read the following sentence and determine what math vocabulary words are missing from the blanks.

A rational number can be written as the ratio of one ______ to another and can be represented by a repeating or ______ decimal.

Open Ended

Marcus is  $5\frac{7}{24}$ feet tall. Ben is  $5\frac{5}{16}$  feet tall. Which of the two boys is taller? Justify your answer by explaining why you chose your answer.

Open Ended

Real-World Problems If one store is selling  $\frac{3}{4}$  of a bushel of apples for $9, and another store is selling  $\frac{2}{3}$ of a bushel of apples for $9, which store has the better deal? Explain your answer.

Open Ended

You are given a fraction in simplest form. The numerator is not zero. When you write the fraction as a decimal, it is a repeating decimal. Which numbers from 1 to 10 could be the denominator?

Open Ended

Julie got 21 of the 23 questions on her math test correct. She got 29 of the 32 questions on her science test correct. On which test did she get a higher score? Can you compare the fractions $\frac{21}{23}$   and $\frac{29}{32}$  by comparing 29 and 21? Explain. How can Julie compare her scores?

Open Ended

Look at the decimal 0.121122111222.… If the pattern continues, is this a repeating decimal? Explain.