1 Katrina Madden
CCLM^2 Project
Summer 2012
DRAFT DOCUMENT. This material was developed as part of the Leadership for the Common Core in
Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee.
PART 1. STANDARD Grade: 6 Domain: RR-­‐Ratios and Proportional Relationships Cluster: Understand ratio concepts and use ratio reasoning to solve problems. Standard: 6.RP.2 PART 2. EXPLANATION AND EXAMPLES OF THE STANDARD I chose 6.RP.2 (Understand the concept of a unit rate a/b associated with a ratio a : b with b ≠ 0, and use rate language in the context of a ratio relationship. An example of this is a as follows “A recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup or sugar. We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger”). Something to note: Expectations for unit rates in 6th grade are limited to non-­‐complex fractions meaning that both the numerator and denominator of the original ratio will be whole numbers. I chose this standard because it really made me think. I had to read it several times before understanding what the standard was directly referring to. It is important for students to first understand what a ratio and rate are before digging deeper into a unit rate and the language associated. Before “unpacking” 6.RP.2 I did some online research, read the 6-­‐7 Ratios and Proportional Relationships Progressions document and referred to the book Developing Essential Understandings of Ratios, Proportions & Proportional Reasoning Grade 6-­‐8. I found many different definitions and insights on ratios, rates, and unit rates. I combined the readings to define them as follows: A RATIO-­‐ is a *multiplicative (and division) comparison of two or more quantities or measures that can be part-­‐to-­‐whole or part-­‐to-­‐part that sometimes have the same units and other times do not (can be identified and described by using “For every _____, there are _____”). A RATE-­‐ is a specific type of ratio that can have the same units and other times different units; CCSSM focuses on applying rates to situations with units that are the same, as well as units that are different (3 miles per 30 minutes or 5 miles per every 2miles) and is a set of many equivalent ratios. A UNIT RATE-­‐ expresses a ratio as a part-­‐to-­‐one, comparing the number of units (numerical part of the rate) of one quantity per one unit of another quantity (measurement conversions: 12 inches per one foot or 20 miles per hour). 2 Ratios have “associated rates” and ratio relationships ensure a constant rate. Using rate language refers to students using “per, for every, for each, etc.” Rate language is important for students to understand to be able to describe the relationship between the amounts of either quantity in terms of the other quantity. In 6th grade students should use reasoning and strategies to find unit rates or equivalent ratios, instead of an algorithm or rule, even though it is very traditional to quickly move students to setting up a proportion. The use of a proportion is introduced in grade 7. 6.RP.2 also states that unit rates are dependent on the situation encompassed. It is important to understand that when ratios are written in a context, the context does not dictate the order and a given situation may be represented by more than one ratio. Again the context does not determine the order of the quantities in the ratio; we choose the order depending on what it is we want to know. Below are two examples from the Illustrative math project that demonstrate this concept. These examples also get students to think about rate language. Example two indicates that there is almost always another ratio with its associated unit rate (the only exception is when one of the quantities is zero), encourages students to flexibly choose either unit rate depending on the question at hand. Example one: They were selling 8 mangos for $10 at the farmers market. Keisha said, “That means we can write the ratio 10 : 8, or $1.25 per mango.” Luis said, “I thought we had to write the ratio the other way, 8 : 10, or 0.8 mangos per dollar." Example two: The grocery store sells beans in bulk. The grocer's sign above the beans says, “5 pounds for $4.” At this store, you can buy any number of pounds of beans at this same rate, and all prices include tax. Alberto said, “The ratio of the number of dollars to the number of pounds is 4 : 5. That's $0.80 per pound.” Beth said, "The sign says the ratio of the number of pounds to the number of dollars is 5 : 4. That's 1.25 pounds per dollar." *Example of a ratio as a multiplicative and division comparison P : Q represents how many time larger P is than Q when P ˃ Q For example: (12: 18) is 1.5; 18 is 1.5 time larger than 12, so 12 ·∙ 1.5 = 18 and 18 ÷ 12 = 1.5. 3 **Examples below from Unpacked Content 6th Grade Mathematics using various contextual situations. Example one: There are 2 cookies for 3 students. How many cookies will each student receive (i.e. the unit rate)? As modeled each student (for every one student) would receive ⅔ of a cookie so the unit rate is ⅔:1. Example two: When riding his bike Jack can travel 20 miles in 4 hours. What are the unit rates in this situation (the distance traveled in one hour and the amount of time required to travel one mile)? 1 mile 1 hour Jack can travel 5 miles in one hour written as 5mi/1hr and it takes ⅕ of an hour to travel each mile which is written as ⅕hr/1mi. A representation between 20 miles and 4 hours is shown above. There are more examples below related to my text MathThematics. It is very interesting to think about the different ways to teach ratios and rates and this has given me a new way to think about my teaching. 4 Part 3. SCHOOL MATHEMATICS TEXTBOOK PROGRAM When reading 6.RP.2 I reflected back on my school year to think about how and when I taught ratios and rates, specifically unit rates. Ratios and rates are not specifically brought up in our 5th grade textbook, however students do a lot of work around division, making equivalent fractions, and looking at fractions as division problems which can lead into the some of the ideas of ratios, rates, and proportions. (Rates are often written in fraction notation in high school so it is important that ratio notation is distinct from fraction notation). th
When looking in my 6 grade textbook I was curious to see the definitions our book gave to ratios, rates, and unit rates. All definitions are closely related to those listed in the CCSSM but in our textbook the definition of a rate does not completely align with the CCSSM; which believe that a ratio/rate can be measure when units are the same and when units are different. A RATIO: is a special type of comparison of two numbers or measures. Ratios can be written in three different ways and order of numbers is important. A RATE: is a ratio that compares two quantities measured in different units; how one measure depends on another. A UNIT RATE: gives an amount per one unit (30 miles in one hour). In 6th grade our book gives a number of examples and asks students to find an equivalent rate with a denominator of one second. For example 5 seconds/15 feet = ? seconds/ 1 foot. Our book also has students think about rates/unit rates by using examples of tables as show below. A strategy that our book promotes in 6th grade is proportions, which is truly something students need not be introduced to until grade 7, after meanings and understandings are clear and mastered. Distance passed (feet) Time (seconds) 15 5 ? 10 ? 15 ? 20 ? 25 Our book touches on standard 6RP.2 but the overall understanding could definitely be built in more. I think students need to relate to ratios and rates in context and I think our book touches on some of the ideas but at a very low level. Our book does not get students to “flexibly” think about unit rates and choose which would best be appropriate. One way to better teach these ideas would be to implement tape diagrams. As listed in the progression document for Ratios and Proportional Relations ships on p. 4, “Tape diagrams are best used when the two quantities have the same units.” This page in progressions also touches on the topic that in tables, unit rates appear paired with one. 5 Moving onto our 7th grade textbook there area at most fifteen pages that directly discuss ratios, rates, and unit rates. Unit rates are defined as a comparison of rates for one unit of a given quantity. As in 6th grade, using a proportion (stating two ratios are equivalent) is the primary strategy, which in 7th grade quickly jumps into cross products. I think it would be clear to any educator that this transition of using cross products will become a rote strategy that kids “can do” but no understanding will be created. Our book uses the example of 2 ·∙ 9 = 18 and 3 ·∙ 6 = 18 so 2/6 = 3/9. From progressions p. 8, grade 7 students should be able to identify unit rates in representations of proportional relationships (foundation for the study of functions which continues in high school) along with building on ratio tables from grade 6. There is a continuum from grade level to grade level that builds year to year. We have to remember that students need to be exposed to different strategies and ways of thinking. We need to put problems (ratios and rates in this case) in context and use real-­‐world examples to better help students. Just because our book teaches a topic one way, doesn’t mean that is the only way. We as teachers need to think of what is in the best interest of the child, along with helping them see connections and build on previous knowledge. The new Math Practice Standards are a great place to start when trying to help change not only our thinking, but our students. 6.RP.2 focuses primarily on MP6 (precise use of language) and MP7 (looking for structure).