# Translating between Fractions, Decimals and Percents

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1 CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Translating between Fractions, Decimals and Percents Mathematics Assessment Resource Service University of Nottingham & UC Berkeley For more details, visit: MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at - all other rights reserved

2 Translating between Fractions, Decimals and Percents MATHEMATICAL GOALS This lesson unit is intended to help students to: Compare, convert between and order fractions, decimals, and percents. Use area and linear models of fractions, decimals, and percents to understand equivalence. COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics: 6.NS: Apply and extend previous understandings of numbers to the system of rational numbers. This lesson also relates to all the Standards for Mathematical Practice in the Common Core State Standards for Mathematics, with a particular emphasis on Practices, 2,, 6, and 7:. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively.. Construct viable arguments and critique the reasoning of others.. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. INTRODUCTION This lesson unit is structured in the following way: Before the lesson, students work individually on an assessment task designed to reveal their current understanding. You then review their responses and create questions for students to consider when improving their work. Students work in small groups on a collaborative discussion task, placing decimal/percent, and fraction cards in order, along with area and linear diagrams that assist them in justifying and explaining their thinking. In a whole-class discussion, students discuss what they have learned. Finally, students revisit their initial work on the assessment task and work alone on a similar task to the introductory task. MATERIALS REQUIRED Each student will need a copy of the assessment tasks Fractions, Decimals, and Percents and Fractions, Decimals and, Percents (revisited). Each small group of students will need some plain paper, cut-up copies of all of the Card Sets, some poster paper, and a glue stick. TIME NEEDED 5 minutes before the lesson, an 80-minute lesson (or two shorter lessons), and 20 minutes in a follow-up lesson (or for homework). Exact timings will depend on the needs of the class. Teacher guide Translating between Fractions, Decimals and Percents T-

4 We recommend you either: write one or two questions on each student s work, or give each student a printed version of your list of questions and highlight appropriate questions for each student. If you do not have time to do this, you could select a few questions that will be of help to the majority of students, and write these on the board when you return the work to the student in the follow-up lesson. Teacher guide Translating between Fractions, Decimals and Percents T-

5 Common issues Assumes the length of a decimal determines its relative size (Q) For example: The student assumes 0. < 0.25 because < 25 ( longer decimals are greater). Or: Assumes 0. > 0.62 because 0. is in tenths and 0.62 is in hundredths and tenths are greater than hundredths ( longer decimals are smaller ). Or: Assumes that 0. > 0.62 because / > /62 ( longer decimals are smaller ). Compares numerators and denominators independently when comparing fractions (Q2) For example: The student reasons that / < 9/6 because < 9 and < 6. Ignores the numerator or denominator when comparing fractions (Q2) For example: The student reasons that 9/6 < /8 because sixteenths are smaller than eighths (ignores numerator). Assumes n% is the same as /n For example: The student appears to believe that 0% and ¼ are equivalent (Q). Assumes that fractions are always smaller/greater than percents For example: The student states ¼ is smaller than 0% because ¼ is a fraction (Q). Focuses on size of digits rather than entire number (Q) For example: The student states % is greater than 0. because is greater than (or 0.). Or: The student states 0.7 is greater than /5 (which it is) because 7 is greater than or 5. Answers all questions correctly with valid explanations Suggested questions and prompts Can you show me on this number line where you would place 0., 0.25 and 0.62? Which is greater in value: 0.5 or 0.50? Why? Is it true that you can tell which of two decimals is bigger by counting the digits? Which is greater 2/ or /6? Why? If you double the numerator and denominator does this change the size of the fraction? Why? Can you tell how big a fraction is by looking at the size of the numerator and denominator separately? Which is greater /8 or 6/6? Why? Now which is greater 9/6 or /8? Can you draw a diagram to show the meaning of ¼? Can you draw a diagram to show the meaning of 0%? What do you understand by ¼? What do you understand by 0%? What is the difference between and %? What is the difference between and 0.? Can you draw a diagram to show the meaning of 0.7? Can you draw a diagram to show the meaning of /5? For each of these pairs of numbers, can you find a number in between them in size? Can you write each of your numbers as a fraction, decimal and percentage? Teacher guide Translating between Fractions, Decimals and Percents T-

9 Now look at the new task sheet, Fractions, Decimals, and Percents (revisited). Can you use what you have learned to answer these questions? Some teachers give this as homework. Teacher guide Translating between Fractions, Decimals and Percents T-8

10 SOLUTIONS Assessment task: Fractions, Decimals, and Percents. The correct order is: 0.05 (least); 0.25; 0.; 0.62;.05 (greatest) Watch for students who use an algorithm without understanding. For example, they fill in zeros so that each decimal has an equal length, then compare as if these are whole numbers. This method may give correct answers but does not usually contribute to the understanding of place value. 2. The correct order is: 6 (least); 8 ; 9 6 ; ; 7 8 (greatest) One method is to convert them all to sixteenths: 6 ; 6 6 ; 9 6 ; 2 6 ; 6. (a) 0% is greater than. Students might explain this by converting 0% to 2 5 or to 25% or both to decimals. Alternatively they may draw pictures to illustrate. (b) 0.7 is greater than. Students might use similar methods. 5 (c) 0. is greater than %. Some students might think % is, which is approximately but not exactly true. Collaborative task The answers are given below, from smallest number on the left to largest on the right % % % % % % % %.25 25% Shaded answers indicate missing ones that students needed to create for themselves. In the case of 5 this could be 2 0 etc. and for the area representing 2 and the number line representing 8 many possibilities. Equally there are various possible area cards for.25. there are Teacher guide Translating between Fractions, Decimals and Percents T-9

11 Assessment task: Fractions, Decimals, and Percents (revisited). The correct order is: 0.0 (least); 0.258; 0.; 0.52;.25 (greatest) 2. The correct order is: (least); 5 6 ; 2 ; 5 8 ; (greatest) One method is to convert them all to sixteenths: 6 ; 5 6 ; 8 6 ; 0 6 ; 2 6. (a) 80% is greater than 8. Students might explain this by converting 80% to 5 or 8 to 2.5% or both to decimals. Alternatively they may draw pictures to illustrate. (b) (c) is greater than 0.6. Students might use similar methods. 0.7 is greater than 7%. Students might use similar methods. Teacher guide Translating between Fractions, Decimals and Percents T-0

12 Fractions, Decimals, and Percents. Put the following decimals in order of size, starting with the one with the least value: ,...,...,...,... Least Greatest Explain your method for doing this. 2. Put the following fractions in order of size, starting with the one with the least value: ,...,...,...,... Least Greatest Explain your method for doing this. Student materials Translating between Fractions, Decimals and Percents S- 205 MARS, Shell Center, University of Nottingham

13 . Put a check mark in each case to say which number is larger. Explain your answer each time on the dotted lines underneath. (a) 0% or Explain how you know. (b) 0.7 or Explain how you know. 5 (c) % or 0. Explain how you know. Student materials Translating between Fractions, Decimals and Percents S MARS, Shell Center, University of Nottingham

14 Card Set A: Decimals and Percents _. % % 80% 0.75 _ % 2.5% %.25 _.. % 50% % Student materials Translating between Fractions, Decimals and Percents S- 205 MARS, Shell Center, University of Nottingham

15 Card Set B: Areas Area A Area B Area C Area D Area E Area F Area G Area H Area I Student materials Translating between Fractions, Decimals and Percents S- 205 MARS, Shell Center, University of Nottingham

16 Card Set C: Fractions Student materials Translating between Fractions, Decimals and Percents S MARS, Shell Center, University of Nottingham

17 Card Set D: Scales Scale A Scale B Scale C Scale D Scale E Scale F Scale G Scale H Scale I Student materials Translating between Fractions, Decimals and Percents S MARS, Shell Center, University of Nottingham

18 Fractions, Decimals, and Percents (revisited). Put the following decimals in order of size, starting with the one with the least value: ,...,...,...,... Least Greatest Explain your method for doing this. 2. Put the following fractions in order of size, starting with the one with the least value: ,...,...,...,... Least Greatest Explain your method for doing this. Student materials Translating between Fractions, Decimals and Percents S MARS, Shell Center, University of Nottingham

19 . Put a check mark in each case to say which number is larger. Explain your answer each time on the dotted lines underneath. (a) 80% or Explain how you know. 8 (b) 0.6 or Explain how you know. (c) 7% or 0.7 Explain how you know. Student materials Translating between Fractions, Decimals and Percents S MARS, Shell Center, University of Nottingham

20 Working Together Take turns to:. Fill in the missing decimals and percents. 2. Place a number card where you think it goes on the table, from smallest on the left to largest on the right.. Explain your thinking.. The other members of your group must check and challenge your explanation if they disagree. 5. Continue until you have placed all the cards in order. 6. Check that you all agree about the order. Move any cards you need to, until everyone in the group is happy with the order. Projector Resources Translating between Fractions, Decimals, and Percents P-

21 Working Together 2 Take turns to:. Match each area card to a decimals/percents card. 2. Create a new card or fill in spaces on cards until all the cards have a match.. Explain your thinking to your group. The other members of your group must check and challenge your explanation if they disagree.. Place your cards in order, from smallest on the left to largest on the right. Check that you all agree about the order. Move any cards you need to, until you are all happy with the order. Projector Resources Translating between Fractions, Decimals, and Percents P-2

22 Decimals and Percents Projector Resources Translating between Fractions, Decimals, and Percents P-

23 Areas Area A Area B Area C Area D Area E Area F Area G Area H Area I Projector Resources Translating between Fractions, Decimals, and Percents P-

24 8 8 Fractions Projector Resources Translating between Fractions, Decimals, and Percents P-5

25 Scales Scale A Scale B Scale C Scale D Scale E Scale F Scale G Scale H Scale I 0 Projector Resources Translating between Fractions, Decimals, and Percents P-6

26 Mathematics Assessment Project Classroom Challenges These materials were designed and developed by the Shell Center Team at the Center for Research in Mathematical Education University of Nottingham, England: Malcolm Swan, Nichola Clarke, Clare Dawson, Sheila Evans, Colin Foster, and Marie Joubert with Hugh Burkhardt, Rita Crust, Andy Noyes, and Daniel Pead We are grateful to the many teachers and students, in the UK and the US, who took part in the classroom trials that played a critical role in developing these materials The classroom observation teams in the US were led by David Foster, Mary Bouck, and Diane Schaefer This project was conceived and directed for The Mathematics Assessment Resource Service (MARS) by Alan Schoenfeld at the University of California, Berkeley, and Hugh Burkhardt, Daniel Pead, and Malcolm Swan at the University of Nottingham Thanks also to Mat Crosier, Anne Floyde, Michael Galan, Judith Mills, Nick Orchard, and Alvaro Villanueva who contributed to the design and production of these materials This development would not have been possible without the support of Bill & Melinda Gates Foundation We are particularly grateful to Carina Wong, Melissa Chabran, and Jamie McKee The full collection of Mathematics Assessment Project materials is available from MARS, Shell Center, University of Nottingham This material may be reproduced and distributed, without modification, for non-commercial purposes, under the Creative Commons License detailed at All other rights reserved. Please contact if this license does not meet your needs.

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