VISIBLE THINKING IN MATH: USING REPRESENTATIONAL MODELS For PRopoRTIoNAL REASoNING
The CCSSM (NGA Center and CCSSO, 2010) recommend the use of representations to illustrate the concept of ratio and rate reasoning.
CCSS.Math.Content.6.RP.A.3: Use ratio and rate reasoning to solve real-world and mathematical problems, for example, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Here are some examples of the use of tables of equivalent ratios, tape or bar diagrams, double number line diagrams, or equations.
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The cost of 3 notebooks is S2.40. At the same price, how much will 10 notebooks cost?
Students might think that means each notebook is $0.80 since 2.40/3 = .80 and .80 * 10 is $8.00. Or some might consider, 3 * $2.40 = $7.20 which gives me 9 notebooks and I need one more, which is +$0.80 that totals to $8.00.
Mary’s best time for running 100 yards is 15 seconds. How long will it take Mary to run 500 yards?
A potential way to approach this is to reason up to go from 100 yards to 500 yards directly and notice the answer is five times the time taken for 15 seconds. It may also be easier to reason up to 1000 yards as students are comfortable multiplying by 10 which yields 150 seconds and then noticing 500 yards is just half of 1000 yards suggest reasoning down or halving 150 seconds.
Sue and Julie were running at the same speed around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run? (See Table 9.1, Cramer, Post, & Currier, 1993)
Notice the similarity to the last problem. Most students attempt this by writing an equation that is obtained by “cross-multiplication.” In reality, noticing that Julie completed 15 laps that is five times should immediately tell us how many laps Sue completed. Of course, there is something implicit in the problem which is not stated directly that is the two runners are running at uniform rates. This is important to solve the problem via proportional reasoning.
In grades 6-8, proportional reasoning problems may be broadly classified into three modeling approaches: (a) Quantitative Proportional Reasoning (QPR); (b) Algebraic Proportional Reasoning (APR) and; (c) Spatial Proportional Reasoning (SPR). Next, we describe the specific mathematics content through benchmark examples developed as a part each of the three modules (QPR, APR, SPR) mentioned earlier.
QPR Module: This module refers to the content knowledge needed to compare and order rational numbers presented in multiple representations including integers, percentages, positive and negative fractions, and decimals. Topics in QPR often focus on fractions and divisions; additions and subtraction of like and unlike fractions; addition and subtraction of mixed numbers; multiplying fractions by whole numbers; fraction of a set; product of fractions; dividing fractions by a whole number; dividing by a fraction. This module also helps to choose and employ appropriate operations to solve real-world applications involving rational numbers. The content developed through QPR can then be applied to concepts in probability to make predictions and decisions. Consider the following benchmark example of Sam and his wife on the next page.
The pictorial technique presented in this example is one of the many ways to become comfortable reasoning and talking about parts of discrete quantities. Note that the problem not only brings out the importance of the concept of “a unit” but they also help guide proportional reasoning. Once this concept is mastered such effective pictorial techniques provide an opportunity to apply them to a variety of related questions which teachers can later supplement in their traditional classroom.
Table 9.1 Number of laps around a track
For example, consider the following multiple-choice question from a grade 8 classroom:
Apple juice concentrate is mixed with water to make apple juice. Which final mixture has the highest percentage of apple juice concentrate?
F. 400 mL apple juice concentrate mixed with 600 mL water
G. 400 mL apple juice concentrate mixed with 400 mL water
H. 300 mL apple juice concentrate mixed with 600 mL water
I. 300 mL apple juice concentrate mixed with 400 mL water
APR Module: This module involves employing strategies to compare and contrast proportional and nonproportional linear relationships. Topics in APR also estimate and determine solutions to application problems involving percent, decimals, and other proportional relationships such as similarity, ratios, and rates. One important focus, herein, includes making connections among various representations of a numerical relationship such as tabular, graphical, pictorial, verbal, and algebraic equations. In particular, the APR module gives an opportunity to predict and justify solutions to application problems through a variety of strategies including proportional reasoning.
As the algebraic habits of mind evolve, students must be constantly taught to effectively communicate mathematical ideas using language efficient tools, appropriate units, and graphical, numerical, physical or algebraic mathematical models.
John bought a piece of land next to the land he owns. Now John has 25% more land than he did originally. John plans to give 20% of his new, larger amount of land to his daughter. Once John does this, how much land will John have in comparison to the amount he had originally?
This benchmark problem gives an opportunity to help the students to determine the percent increase or decrease for a given situation. This problem is also an example of a
Approach: To answer this, onemay be able to illustrate the pictorial approach using "1 unit = 100 ml” in each case as follows clearly illustrating the answer.
common misconception that leads to an incorrect solution. Most students believe that if there is percent increase followed by a percent decrease of the same amount (or vice versa), the answer returns back to the original amount.
To help them understand their misconception, one strategy is to start with 100 units of land as the original piece of land. A 25% increase would be the same as 1.25 (100) = 125 units of land. Now John plans to give 20% of his new piece of land which will leave him with 80% of the new piece of land. This is the same as 0.8 (125) = 100 units of land. One can also see this using the pictorial approach.
SPR Module: Along with QPR and APR, a good proportional reasoning curriculum must also develop a spatial sense through transformational geometry exercises. The proportional reasoning can be built through special exercises that will build students’ skills to generate similar figures using graph dilations including enlargements and reductions on a coordinate plane. They will also be trained to use proportional relationships in similar two-dimensional figures or similar 3D figures to determine missing measurements. This module will also provide an opportunity to use proportional reasoning to describe and verbalize how changes in dimensions affect linear, area, and volume measures. Consider the following problem that will allow students to describe how changing one measured attribute of the figure affects the volume and surface area.
Example: Given a rectangular prism with a length of 2, a width of 2, and a height of 1:
- 1. Without changing the length and width, change the height by a factor of n and create a table showing volume for increasing values of n.
- 2. Write in words the pattern you observe in the volumes.
- 3. Write an algebraic function for the pattern.
- 4. Without changing length and height, change the width by a factor of n and create a table showing volume for increasing values of n.
- 5. Write in words the pattern you observe in the volumes.
- 6. Write an algebraic function for the pattern.
- 7. Is the pattern in (iii) the same and why?
- 8. Without changing the length, change the height and the width by a factor of n and create a table showing volume for increasing values of n.
- 9. Write in words the pattern you observe in the volumes.
- 10. Write an algebraic function for the pattern.
- 11. Write a function to predict the volume if all three dimension change by a factor of n.
- 12. Repeat all the steps (i)-(xi) for calculating “surface area” instead of “volume.”
This exercise can help build spatial reasoning related to change in one dimension versus two dimensions and the effect of varying dimensions on volume and surface area. This activity also helps to explore patterns and discover the relationship between linear ratios, area ratios, and volume ratios.
At a summer institute focused on proportional reasoning, teachers made connections to solving real-world problems 6.RP.A.3. They were shown packages of 100-calorie snacks, given cereal and asked to show 100 calories of their favorite cereal. Figure 9.3 shows how one teacher illustrated the solution in multiple ways.
Teachers working together helped each other appreciate multiple strategies. One of the teachers commented, “I am so comfortable with mental math and using numbers. I find it hard to think in terms of manipulatives and pictures but seeing how other teachers solved it using these tools really helped me see how my students might approach it. I can truly see the value of hands-on manipulatives for my math students.” Other teachers shared, “Today using a ratio table, Karen showed me how to ‘pull apart’ a ratio so that I could manipulate it more easily.” Through the experience of relearning mathematics through multiple models, teachers felt more confident and more “strategically competent” using multiple models and posing rich proportional reasoning problems in class with their students.
The task of the teachers is therefore to help students connect their constructed knowledge to the powerful new ideas that they want to teach them. This, combined with the Common Core Standards, creates a great need to enhance teachers’ mathematics content and pedagogical knowledge with a special focus on modeling proportional reasoning. It is also essential to understand how these standards translate into classroom practices and assessment strategies. With a growing population of students identified as economically disadvantaged, LEP, and special needs in many school divisions across the nation, teachers need to be proficient in presenting mathematical ideas visually and through multiple representations.
Figure 9.3 100-calorie cereal portions using pictorial approach for the unitizing method. Source: Authors.
Think about it!
How do these visual representations (ratio tables, double number lines, bar models) help develop a deeper understanding of proportional reasoning?