6-7Statistics&ProbabilityMisconceptions

Proportional Reasoning

3 + 4 = 34

Look closely at errors in students’ work (formative assessment) to help you reflect

and make instructional decisions to suit all students’ needs.

7.RP.1 7.RP.2 7.RP.3

6.RP.1 6.RP.2 6.RP.3

Many students struggle with understanding the difference between proportional situations and additive ones. Students may not realize that although they may have added to find an equivalent numerator or denominator, they should not just add the same amount on both sides. Allowing students to manipulate physical objects and investigate with models deepens proportional understanding. It’s also important to encourage students to explain the meaning of numbers within the context of a problem. For example, what does it mean to have a 5:13 ratio? In the problem below the student added 13 + 13 to get 26 and then continued this additive process by adding 13 to 5. This student remembers the process of “do the same to the top and bottom,” but not multiplicative and ratio reasoning.

ADDITIVE APPROACH (MISCONCEPTION): PROPORTIONAL REASONING:

By allowing your students to struggle as they compare rates, to work on a variety of proportional reasoning strategies, and to slowly develop an understanding of the need for a unit ratio strategy, you can help them build a flexible knowledge of ratios.

As you continue to build on the additive vs. multiplication approach, give tasks that force a multiplicative approach. Although an additive approach is orderly, it may not be the most efficient and it does little to promote proportional reasoning. ADDITIVE APPROACH: The ratio table below is in response to the problem to find the number of pine trees on 75 acres. Students are likely to use the additive approach because the pattern is easily identifiable; they continue the pattern by adding 5 along the top and 75 along the bottom until the problem is solved. This type of problem does not force students to think about the multiplicative relationship of proportionality. Acres 5 10 15 20 Pine trees 75 150 225 MULTIPLICATIVE APPROACH: The examples below are three different responses to the question, “A person who weighs 160 pounds on Earth will weigh 416 pounds on the planet Jupiter. How much will a person weigh on Jupiter who weighs 120 pounds on Earth?”

The ratio-table approach focuses on multiplicative reasoning instead of rule oriented. This method forces students to rely on reasoning, not rote-algorithms to solve the problems.

Adapted from Van de Walle (2006, p. 163)

Proportional reasoning is difficult to define. It is not something that you either can or cannot do but is developed over time through reasoning. One way to describe proportional reasoning is to say it is the ability to think about and compare multiplicative relationships between quantities. These relationships are represented as ratios. The comparison of ratios helps students develop proportional reasoning.

ADDITIVE APPROACH: ANSWER: They both grew the same amount…3 inches. This answer is correct based on additive reasoning. PROPORTIONAL REASONING: ANSWER: The first flower grew 3/8 of its height or about 33%.

The second flower grew 3/12 of its height or 25%. The first flower grew at a faster rate.

The proportional reasoning approach compares the growth to the original height of each flower.

Adapted from Van de Walle (2006, p. 163)

Two weeks ago, two flowers were measured at 8 inches and 12 inches, respectively. Today they are 11 inches and 15 inches tall. Did the 8-inch or 12-inch flower grow more?

A vase holds yellow and red tulips. For every four yellow tulips, there are three white tulips. How many tulips might be in the vase? A common misconception is that there must be only 7 tulips in the vase instead of using 4:3 as a ratio to find equivalent proportions. Students need to recognize when they should find a unit rate and when they should use an equivalent approach. In this problem, students can mathematically find a unit rate, but based on the context of the problem, is it reasonable? MISCONCEPTION: WHAT TO DO:

Any ratio table provides data that can be graphed. A graph can be drawn and used to determine other equivalent ratios and a unit rate can be found by locating the point on the line that is directly above or to the right of the number 1 on the graph. Remember there are actually two unit rates for every ratio. Graphs provide another way to develop proportional reasoning, and they connect algebraic thinking to proportional thinking. All graphs of equivalent ratios are straight lines that pass through the origin and can be written in the form y = mx. The slope, m, is always one of the ratios. PROPORTIONAL REASONING:

Percent should be included with ratio and proportion conversations because a percent is one form of a ratio, a part-to-whole ratio. The better students are able to connect fractions, decimals, percent, ratios, and proportions, the more flexible and useful their reasoning will be. 6th GRADE: Students should use a variety of strategies to represent percentage problems. Students can combine or blend strategies together to help them make sense of percent problems. The traditional algorithm is not required in 6th grade. Students should focus more on relationships among equivalent ratios, and developing strategies to solve problems.

Image from Common Core Progressions, (2011, Pg. 7)

7th GRADE: In 7th grade, students extend their work to solving multistep problems. Problems involving percent increase or percent decrease, require careful attention to the whole. Consider the differences in the wholes for the problems below.

These problems are different because the 20% refers to different wholes. It’s important for students to pay close attention to the wording of problems so they can determine what the whole or 100% amount refers to.

Images from Common Core Progressions, (2011, Pg. 10)

Problem 1 After a 20% discount, the price of a Super Sick skateboard is $140. What was the price before the discount?

Problem 2 A Super Sick skateboard costs $140 now, but its price will go up by 20%. What will the new price be after the increase?