Fractions and Decimals

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1 Fractions and Decimals Objectives To provide experience with renaming fractions as decimals and decimals as fractions; and to develop an understanding of the relationship between fractions and division. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Read and write decimals through hundredths. [Number and Numeration Goal 1] Represent a shaded region as a fraction and a decimal. [Number and Numeration Goal ] Rename fractions with and 0 in the denominator as decimals. [Number and Numeration Goal ] Use equal sharing to solve division problems. [Operations and Computation Goal ] Key Activities Students rename fractions as decimals and decimals as fractions. They also explore the relationship between fractions and division. Ongoing Assessment: Recognizing Student Achievement Use journal page 03. [Number and Numeration Goal ] Materials Math Journal, pp. 03, 3, and 33 Student Reference Book, p. Study Link 7 7 Math Masters, p. (optional) transparency of Math Masters, p. base- blocks calculator slate overhead base- blocks (optional) Advance Preparation Math Boxes 7 Math Journal, p. 0 Students practice and maintain skills through Math Box problems. Study Link 7 Math Masters, p. Students practice and maintain skills through Study Link activities. READINESS Creating Base- Block Designs Math Masters, p. base- blocks Students make a design with base- blocks, copy the design on a grid, and write a decimal and a fraction to describe what part of the grid is covered by the blocks. ENRICHMENT Finding Fractions, Decimals, and Percents on Grids Math Masters, p. 7 Students shade a -by- grid to represent fractions and find the percent and decimal equivalencies. ENRICHMENT Designing a Baseball Cap Rack Math Masters, pp. 7A and 7B Students use fractions with denominators of ; 0; or 1,000 to design a baseball cap rack. EXTRA PRACTICE Taking a 0-Facts Test Math Masters, pp. 1 and 1; p. 1 (optional) pen or colored pencil Students take a 0-facts test. They use a line graph to record individual and optional class scores. Teacher s Reference Manual, Grades pp., 3, 13, 1 Lesson 7 09

2 Getting Started Mathematical Practices SMP1, SMP, SMP3, SMP, SMP, SMP Content Standards.NF.1,.NF.,.NF. Mental Math and Reflexes Write a fraction on the board. Students write an equivalent fraction on their slates. Suggestions: Sample answers: _, 3, 3_ 1 _ 3, 3_ 9 _, 3_ 1 0 0, _ 0 3, 9_ 1 9 3, 1 1, 0_ 0 3, 9_ 1 Math Message Write the following fractions as decimals: 3_ 0 7_ 9_ 0 Study Link 7 7 Follow-Up Have students compare answers and share the name-collection boxes they created. 1 Teaching the Lesson Math Message Follow-Up (Math Masters, p. ) WHOLE-CLASS DISCUSSION Display a transparency of Math Masters, page as you discuss the answers. Remind students that the square is the whole. You can color the grid sections to show fractional parts or cover them with base - blocks. Color or cover one column of the bottom grid. What fractional part of the square is this? Base- Grids Teaching Aid Master 1, or 0.1 How would you write as a decimal? 0.1 Repeat with other fractions in tenths, including 3_ and 7_. Adjusting the Activity ELL Provide students with base- blocks and a copy of Math Masters, page, so they can model the decimal numbers at their desks. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Math Masters, p. Unit 7 Fractions and Their Uses; Chance and Probability

3 Next, color (or cover) one small square of the top grid on the transparency. What fractional part of the square is this? 0 Date 7 Whole Fractions and Decimals large square Student Page 1. Time , 0 or of the square is shaded. How many tenths? is shaded. How many tenths? , or How would you write as a decimal? Repeat with other fractions in hundredths, including 3_ 0 and 9_ 0. Also, give students practice converting decimals into fractions; for example, 0.3 and 0.. Tell students that in this lesson they will use a base - grid as a tool to help them rename fractions as decimals. 1 is shaded. How many tenths? , or is shaded. How many tenths? 0. 1 is shaded Math Journal, p is shaded Links to the Future Do not be concerned with reducing fractions to simplest form when converting between decimals and fractions. At this stage, it is enough for students simply to make the conversions. Naming fractions in simplest form is a Grade Goal. Renaming Fractions as Decimals and Decimals as Fractions (Math Journal, pp. 03, 3, and 33; Math Masters, p. ) INDEPENDENT Students complete journal page 03. Discuss answers, using a transparency of Math Masters, page. For each problem, ask by which number the numerator and denominator were multiplied to obtain the second fraction. Ask students to record the decimals in the Equivalent Names for Fractions table on journal pages 3 and 33. Ongoing Assessment: Recognizing Student Achievement Journal page 03 Problems 1, 7, and Use journal page 03, Problems 1, 7, and to assess students ability to rename tenths and hundredths as decimals with the assistance of a visual model. Students are making adequate progress if they are able to name the number of tenths or hundredths shaded on the grid as a fraction and rename the fraction as a decimal. Some students may be able to solve Problems and on journal page 03, which do not include a visual prompt. [Number and Numeration Goal ] Lesson 7 11

4 Date 7 Math Boxes 1. Complete the name-collection box. 3. Use pattern blocks to help you solve these problems. a. b. c. d = _ + _ 3 = _ - = _ - = _ 3_ % _ 3 _, 3_ 3, or 1 _, or _ 3 Sample answers: 1 7. There are pages in the book Ming is reading for his book report. He has two weeks to read the book. About how many pages should he read each day? 1 pages Student Page Time Math Journal, p. 0. A bag contains blue blocks, red blocks, 1 green block, and orange blocks. You put your hand in the bag and, without looking, pull out a block. About what fraction of the time would you expect to get a red block? 1. ART is an acute (acute or obtuse) angle. R The measure of ART is 0. A T Tell if each of these is closest to 1 inch, 1 foot, or 1 yard. a. the height of the door 1 yard b. the width of your journal 1 foot c. the length of your largest toe 1 inch d. the length of your shoe 1 foot 130 Renaming Fractions as Decimals with a Calculator (Math Journal, p. 03) WHOLE-CLASS Use Problem 7 on journal page 03 to model renaming fractions as decimals on a calculator. For the TI-1: Enter the fraction (press 1 n dd ). Then press F D. 0. For the Casio fx-: Enter the fraction (press 1 ). Then press. 0. Use Problem to model renaming a decimal as a fraction. For the TI-1: Enter the decimal 0.7, then press F D. 7_ 0 For the Casio fx-: Enter the decimal 0.7, then press. 3_ Discussing Fractions and Division (Student Reference Book, p. ) WHOLE-CLASS DISCUSSION STUDY LINK 7 Fractions and Decimals Write 3 equivalent fractions for each decimal. Example: 0_ 0. 0 _ 0_ _ 3_ 0_ _ 0_ _ 3_ 7_ Write an equivalent decimal for each fraction.. 3_. 3_ _. _ 9. Shade more than 3_ 0 of the square and less than _ of the square. Write the value of the shaded part as a decimal and a fraction. Decimal: Fraction:. Shade more than 1 0 of the square and less than of the square. Write the value of the shaded part as a decimal and a fraction. Decimal: 0. _ Fraction: Practice Sample answers: Study Link Master _ 0 Sample answer: Sample answer: 3, = = 13. = 39 Math Masters, p. 1 Read and discuss Fractions and Division on page of the Student Reference Book. Have students apply their understanding of division to equal-sharing division problems. For example: Nina and her mother baked dozen cookies for the book club meeting. The club has members. How many cookies are there for each member? Four dozen equals 1, or. The number models / =, =, and _ = fit this problem. The first and second number models suggest dealing out the cookies to the club members. Each member would get cookies. The third number model, _ =, suggests dividing each cookie into eighths and giving of every cookie to each person. Each person would end up with eighths. If the eighths were reassembled, they would be equivalent to cookies. NOTE _ is called an improper fraction because the numerator is greater than the denominator. Improper fractions have numerators that are greater than or equal to their denominators. Also discuss problems in which the divisor is greater than the dividend. For example: Adam ordered 3 pizzas for a party. There will be people at the party. How much pizza is there for each person? 1 Unit 7 Fractions and Their Uses; Chance and Probability

5 Point out that this problem and the cookie problem are both about sharing. The main difference is that in this problem, each share is less than one whole pizza. Draw 3 pizzas on the board or on the overhead transparency, and divide each one into fifths for the people. If Adam s guests are named Bob, Charles, Darryl, and Ed, the pizzas could be shared in the following way: Teaching Master 7 Designing a Baseball Cap Rack Karen plans to design and construct two identical horizontal racks to display her baseball cap collection. She has 1 different caps to hang on pegs. Karen s sister suggested that she add extra pegs for caps she may get in the future. Karen measured the width of some caps and decided that the pegs need to be decimeters ( _ meter) apart. Also, in order to fit on her wall, each rack cannot be more than 10 centimeters long. Help Karen design one of the identical racks. Use metric units. Fill in the blanks below as you create the design. Sample answers are given. 1. Each of Karen s racks will have pegs for hats. A B E D C A B E D C A B E D C. The total length of the rack will be 10 centimeters. 3. The first peg will be centimeters from the edge of the rack.. In the space below, draw a rough sketch of the rack. Include the measurements in your sketch. Help students see how the number model 3 / = 3_ fits this problem. The left side, 3 /, suggests dividing 3 pizzas among people. The right side, 3_, tells how much each person would get. Explain that in high school and beyond, the symbol is almost never used for division. Division is usually shown with a slash (/) or a fraction bar ( ). cm dm 10 cm. Write a fraction addition number sentence to show the total length of the rack. Sample answer: _ 0 + _ + _ + _ + _ + _ + _ + _ + _ = _ m, or 10 cm. Could there be 9 pegs on the rack? Explain your answer. No. Nine hats would require about 00 cm, and the rack can only be 10 cm long. Ongoing Learning & Practice Math Masters, p. 7A Math Boxes 7 (Math Journal, p. 0) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 7-. The skill in Problem previews Unit content. Writing/Reasoning Have students write a response to the following: Explain how you solved Problem. Sample answer: There are 1 blocks in the bag, and of them are red. So the chance of getting a red block is _ 1. Study Link 7 (Math Masters, p. ) INDEPENDENT Home Connection Students rename decimals as fractions and fractions as decimals. They color fractional parts of a base- grid and write the value of the shaded part as a decimal and a fraction. 7 Teaching Master Designing a Baseball Cap Rack continued Use your answers to Problems 1 and on Math Masters, page 7A to fill in the blanks in the sentence below. Sample answers: Each rack is 10 centimeters long and has pegs. At the lumberyard, Karen discovered she could spend less if she was willing to glue leftover pieces of wood together instead of using one long piece. She measured several boards and wrote down the lengths: 7_ meter _ 0 meter 3_ 0 meter 3_ meter 0_ 0 meter 9_ meter _ meter _ 0 meter 7_ 1 0 meter _ 0 meter Sample answers are given. 7. Can Karen use these pieces to create two racks of the length you planned? Explain why or why not. Show your work. Yes. Rack one: 7_ meter and 9_ meter; 7_ + 9_ = m, or 10 cm Rack two: _ 7 0 meter, _ 0 meter, and 3_ meter; 7_ 0 + _ 0 + 3_ = _ 10 0 m, or 10 cm Pegs come in two different packages: -pack for $3.79 or -pack for $1.99. Explain how Karen can purchase the pegs for her racks, spending as little money as possible. Karen needs 1 pegs. She should buy three -packs and one -pack for $13.3 because this is cheaper than buying two -packs and three -packs for $13.. Math Masters, p. 7B Lesson 7 13

6 3 Differentiation Options READINESS Creating Base- Block Designs (Math Masters, p. ) INDEPENDENT 30+ Min Decimal: Fraction: 0. 0 To explore representing fractions and decimals on a base- grid, have students make a design on a base- block flat with cubes and then copy the design onto one of the grids shown on Math Masters, page. Students determine how much of the flat is covered by their design and express this number as a decimal and a fraction. (See margin.) Students may choose to exchange as many cubes as possible for longs, which would result in a certain number of longs (tenths) and cubes (hundredths). ENRICHMENT Finding Fractions, Decimals, and Percents on Grids (Math Masters, p. 7) PARTNER 1 30 Min To further investigate fraction, decimal, and percent equivalencies, have students shade a base- grid to show, 3,, and _. Encourage students to discuss patterns they see and strategies they used. Ask: How could you have found the percent equivalent for _ without shading the grid? Sample answer: Use the answer for and multiply by. 7 Fraction, Decimal, and Percent Grids Fill in the missing numbers. Shade the grids. 1.. Teaching Master Sample answers: 1 ENRICHMENT Designing a Baseball Cap Rack (Math Masters, pp. 7A and 7B) To further investigate the relationships among fractions with denominators of ; 0; or 1,000, have students design two identical racks to display a baseball cap collection. Encourage students to work with a partner to complete the activity. PARTNER 1 30 Min 1 Fraction: = Fraction: 3 = 0 0 Decimal: 0.1 Decimal: Percent: 1. % Percent: 33 3 % EXTRA PRACTICE Taking a 0-Facts Test (Math Masters, pp. 1, 1, and 1) SMALL-GROUP 1 Min See Lesson 3- for details regarding the administration of the 0-facts test and the recording and graphing of individual and optional class results. 1 Fraction: = Fraction: _ = 0 0 Decimal: 0.1 Decimal: 0. Percent: 1 _ % Percent: _ % Math Masters, p Unit 7 Fractions and Their Uses; Chance and Probability

7 7 Designing a Baseball Cap Rack Karen plans to design and construct two identical horizontal racks to display her baseball cap collection. She has 1 different caps to hang on pegs. Karen s sister suggested that she add extra pegs for caps she may get in the future. Karen measured the width of some caps and decided that the pegs need to be decimeters ( _ meter) apart. Also, in order to fit on her wall, each rack cannot be more than 10 centimeters long. Help Karen design one of the identical racks. Use metric units. Fill in the blanks below as you create the design. 1. Each of Karen s racks will have pegs for hats.. The total length of the rack will be centimeters. 3. The first peg will be centimeters from the edge of the rack.. In the space below, draw a rough sketch of the rack. Include the measurements in your sketch.. Write a fraction addition number sentence to show the total length of the rack.. Could there be 9 pegs on the rack? Explain your answer. Copyright Wright Group/McGraw-Hill 7A

8 7 Designing a Baseball Cap Rack continued Use your answers to Problems 1 and on Math Masters, page 7A to fill in the blanks in the sentence below. Each rack is centimeters long and has pegs. At the lumberyard, Karen discovered she could spend less if she was willing to glue leftover pieces of wood together instead of using one long piece. She measured several boards and wrote down the lengths: 7_ meter _ 0 meter _ 3 0 meter _ 3 meter _ 0 0 meter _ 9 meter _ 7_ 0 meter _ 0 meter 1 meter _ 0 meter 7. Can Karen use these pieces to create two racks of the length you planned? Explain why or why not. Show your work. Copyright Wright Group/McGraw-Hill Pegs come in two different packages: -pack for $3.79 or -pack for $1.99. Explain how Karen can purchase the pegs for her racks, spending as little money as possible. 7B

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