Ratio, Proportion, and Percent: Meanings and Applications CHAPTER 13 Tina Rye Sloan To accompany Helping Children Learn Math10e, Reys et al. ©2012 John Wiley & Sons

Focus Questions How is a ratio different from a proportion? How can teachers use student intuition to develop students’ thinking about proportions? Why do students mistakenly use an additive method to find proportions rather than the correct multiplicative method? What models help develop the concept of percents? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Expectations from the Number and Operations Standard Grades 6-8 Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Principles and Standards for School Mathematics National Council of Teachers of Mathematics 2000 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Expectations from the Number and Operations Standard (cont’d) Grades 6-8 Develop meaning for percents greater than 100 and less than 1. Understand and use ratios and proportions to represent relationships. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Expectations from the Number and Operations Standard (cont’d) Compute fluently and make reasonable estimates Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios. 4’ 12’ Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Ratio-relationship between 2 quantities. 3 games for 50 cents 3/50 3 to 503:50 Ratio Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Ratio (cont’d) Patterns often lead to ratios. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Ratio: Real-World Examples Adjusting recipes to serve more or fewer people provides an opportunity to work with ratios involving multiple values. A can of mixed nuts with a ratio of cashews to almonds to peanuts of 3:5:10 A sack of lawn fertilizer with “8–12–20” printed on it, indicating ratio of the percentages of phosphorus, nitrogen, and potash in the fertilizer. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Real-World Examples Figure 13-3 Ratio table comparing the number of vehicles with number of wheels for bicycles, tricycles, and wagons. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Proportion-equal ratios (i.e. 2:3 is the same as 4:6) Joe will often tape 6 minutes of audio tape to obtain enough footage for a 30-second music clip. At that rate, how many minutes of tape will it take to shoot a 45-second music clip? 6 min tape = N min tape 30-second clip = 45-second clip cross multiply: 30 x N = 6 x x N = 270 N=9 minutes of tape Proportion Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Proportion (cont’d) A month ago, plant A was 12 cm and plant B was 15 cm tall. Now plan A is 18 cm tall and plant B is 21 cm tall. Which plant grew the most? You could compare the absolute amounts that the plants grew, which might lead you to conclude that the two plants grew equally, since both plants grew 6 cm. Or you could make a multiplicative, or relative, comparison by looking at how much each plant grew in relation to its original height. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Using Models to Solve Problems Double or Two-Sided Number Lines Ratio Tables Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Proportion Problems Use proportional reasoning to determine the length of each side of each triangle. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Percent- parts per 100 a special ratio N/100 25/100=.25=25% Percents Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Percents (cont’d) A dollar is made up of 100 cents; therefore, it provides a natural connection among percents, fractions, and decimals. 25 cents can be written as.25 25/100 = 25% Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Percents (cont’d) This model should be expanded to illustrate a wide range of percents, such as 50%, 90%, 5%, 100%,99%, 1%, and 200%. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Percent Models A meterstick provides another easily accessible and effective model. Cover part of the meterstick with paper and ask children to estimate the percent of the meterstick that is covered. A rope with 100 disks arranged in multiples of 10 in alternating colors shows 50% of each color. These two rectangular grids also show 50% Percent bars can be used to show percent. This bar is showing 75% Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Percent Models (cont’d) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Solving Percent Problems Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Solving Percent Problems The sale price on a coat was $40 and it was marked down 50%. What was its original price? Common sense suggests that the original price should be more than $40. Ratio method: 50% = $40 OP = $80 100% OP Equation method: 50% of OP = $40 OP = $40 = $80.50 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Percent Shortcuts A% of B = B% of A Ex: 28% of 50 = 50% of 28 (KA)% of B = K(A% of B) Ex: 20% of 35 is twice 10% of 35 (2x10)% of 35 = 2(10% of 35) Source: Glatzer, Arithmetic Teacher, Feb Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Percent Shortcuts (cont’d) (A+C)% of B = (A% of B) + (C% of B) Ex: 15% of 60 = (10% of 60) + (5% of 60) Source: Glatzer, Arithmetic Teacher, Feb Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Tangrams and Percent Find the area of each of the seven tangram pieces using a ruler or grid paper. Record your results. What is the total area of all seven tangram pieces? Find what percent of the whole area each piece is. Record your results. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Trevor: Fourth Month of Seventh Grade Student Work Samples Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Student Work Samples (cont’d) Deanna: Fourth Month of Seventh Grade Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Yang: Fourth Month of Seventh Grade Student Work Samples (cont’d) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Jacob: Fourth Month of Seventh Grade Student Work Samples (cont’d) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Jacob: Fourth Month of Seventh Grade (cont.) Student Work Samples (cont’d) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Bradley: Fourth Month of Seventh Grade Student Work Samples (cont’d) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Bradley: Fourth Month of Seventh Grade (cont.) Student Work Samples (cont’d) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Allan: Fourth Month of Seventh Grade Student Work Samples (cont’d) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Allan: Fourth Month of Seventh Grade (cont.) Student Work Samples (cont’d) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Jessica: Fourth Month of Seventh Grade Student Work Samples (cont’d) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012

Jessica: Fourth Month of Seventh Grade (cont.) Student Work Samples (cont’d) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10 th Edition, © 2012