Standard: 7.RP.1

Stephannie Chiarelli
CCLM^2 Project
Summer 2012
This material was developed as part of the
Leadership for the Common Core in Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee.
Common Core State Standards for Mathematics
Standard: 7.RP.1 Part One: Grade, Domain, Cluster, Standard The grade this standard appears in is 7th Grade. The domain represented by this standard is Ratios and Proportions. (*refer to Curriculum Guide) The cluster the standard belongs to is Cluster 1. (*refer to Curriculum Guide) The exact standard, 7.RP.1 reads as following: “Compute unit rates associated with ratios of fraction, including ratios of lengths, areas and other quantities measured in like 𝟏
𝟏
𝟏/𝟐
𝟐
𝟒
𝟏/𝟒
or different units. For example, if a person walks mile in each hour, compute the unit rate as the complex fraction miles per hour, equivalently 2 miles per hour.” Part Two (a): Explanation and Examples of Key Terms within 7.RP.1 Ratios, fractions, rates, proportions, unit rates…these terms can be confusing! The Mathematical Practice Standards (*subsequently referred to as MP; *refer to Curriculum Guide) require that we ask students to attend to their explanations of their mathematical problem-‐solving strategies with precision (MP6). The following explanations and examples of key terms should assist you in helping your students to do just that. J Ratios: A ratio is an expression which compares separate quantities relative to each other, and usually involves exactly two quantities. In other words, one standard measurement is related to another standard measurement. Measurements may take the form of any attribute including time (e.g. seconds, hours, years); frequency (# of occurrences); distance (e.g., miles, kilometers); length (e.g., inches, feet, meters); amount, including measurements of weight, mass, area, density; etc. Attributes may be the same (for example, jogging 2 miles to every 3 miles walked; or different, jogging 2 miles per every 1 hour). Mathematically, ratios should be represented by separating each quantity with a colon, for example the ratio 2:3, which is read as the ratio "two to three" or “two out of three” or “two per three”. Once ratios are used in rates and proportions, they 𝟐 may be represented as , also read as “two to three” or “two out of three”, etc. 𝟑
Fractions: A fraction is an example of a specific type of ratio, in which the quantities are related in a part-‐to-‐whole relationship, rather than as a relationship between two separate quantities. Mathematically, they are represented by 𝟒
separating each part from its whole as in which is read as the fraction “four-‐fifths”. 𝟓
Complex Fractions: A complex fraction is a fraction where the numerator, denominator, or both contain a fraction, for example, 𝟏/𝟐
𝟏/𝟒
; 𝐚 complex fraction may also include mixed numbers or improper fractions. Rates: A rate is a ratio that relates a first and second attribute to the second attribute. For example, traveling a distance of 10 feet per every 6 seconds, a ratio of 10:6 is expressed as a rate of 10:6 per 1 second, 20:12 per 2 seconds, 30:18 per 3 seconds, etc. Rates may or may not have a proportional relationship. The context of the problem will indicate whether the relationship between the ratio and the rate is proportional or not. If a rate does not involve change over time or situation, as in the preceding example, it is proportional. If it does involve change, it is not proportional. Unit rates: A unit rate is a rate relating a quantity to 1 unit, for example, 50 miles per 1 hour. Complex fractions can be 𝟏
𝟏
𝟏/ 𝟐
𝟐
𝟏/𝟑
𝟏
𝟑
𝟏
𝟐
𝟏/𝟑
𝟐
𝟑
𝟐
𝟑
𝟑
expressed as ratios related to a unit rate. For example, teaspoon per cup can be expressed as per cup, or 𝟏/ 𝟐
/1. Ratios can be computed as a unit rate, for example, if a person walks mile in hour, he would walk mile per hour, or /𝟏. The 𝟏
𝟐
𝟑
𝟑
𝟏
𝟐
answer is computed as × = . A unit rate always has 1 as a denominator. Equivalency: Equivalent ratios have the same unit rate. In other words, they are proportional. Again, the context of the problem will indicate whether the relationship between the ratio and the rate is proportional or not. If the relationship is proportional, the numerator and denominator can both be multiplied or divided by the same unit. (See the example included in the explanation of rates.) Proportions: A proportion is an equation with a ratio on each side. It is a statement that two ratios are equivalent. An 𝟑
𝟔
𝟒
𝟖
example of a proportion would be 3:4 = 6:8 or = . Proportions may also be written as an expression containing complex fractions. !
So…similar to the example contained in the standard, if a person skates mile (fraction, measurement of !
!
the attribute of distance) in each hour (fraction, measurement of the attribute of time), compute the unit rate as the !
complex fraction !/!
!/!
(fractions contained in both the numerator and denominator) miles per 1 hour, or !/!
!/!
/1 (unit rate). Since the context of the problem gives no indication that the relationship is not proportional, the equivalent unit would be 2 miles per hour. We’re using visual aids (described below) to show the relationship between the faction and its unit rate !
!
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or simply computing × = = 2. Why is this even important? Two miles per hour is much more meaningful than the complex fraction. Part Two (b): Examples of Student Work within 7.RP.1 What will this standard look like in classroom practice? What are students expected to do? How should teachers and students be talking about these mathematical ideas? How can we model and look for and make use of structure (MP4,7) so that students will really understand “the math behind the math” BEFORE they are introduced to useful, but somewhat mindless step-‐by-‐step algorithms. The really nice thing about this standard is that students won’t be as tempted to ask, “Why do I need to know this?” The standard is all about real-‐world situations: taxes, tips, sports, cooking, shopping, building, scientific principles, other mathematical principles…and much, much more. The following examples will demonstrate how those real-‐world problems should look as you provide students opportunities to explore concepts related to the content standard using the mathematical process standards. A variety of visual tools will help your students understand the relationship between ratios and proportional rates in multi-‐
step problems. Examples of visuals include tables, double number lines, graphs, and tape diagrams. Table: (Consider also using a function table.) Consider the following problems (MP4,5,7,8): 𝟏
𝟏
𝟒
𝟐
“If a recipe for a dozen cookies calls for 𝐜𝐮𝐩 𝐨𝐟 brown sugar and cup of white sugar, how much brown sugar will you need for 5 dozen cookies?” !
*The table allows students to see that for 5 dozen cookies, you will need 1 cups of brown sugar in order to bake 5 !
dozen cookies. This answer demonstrates the idea of equivalency within rates. of cup of white sugar?” “What is the equivalent unit rate of 𝟏/𝟒
𝟏/𝟐
!
*The table shows that cup of brown sugar is needed per cup of white sugar !
Tables: Ratio of Amount of Brown Sugar to Amount of White Sugar Per Dozen as a Proportional Rate Dozens 1 dozen 2 dozen 3 dozen 4 dozen 1
1
3
!
Cups (1) Brown Sugar 4
2
4
!
1
2
Cups White Sugar !
!
3
2
(1) !
!
(2) 5 dozen 5
4
5
2
Dozens Cups of Brown Sugar 1
4
Cups of White Sugar 1
2
2 1
2
2
2
3 3
4
3
3
4 4
4
4
3
5 5
4
5
3
1 Double Number Line: Consider the following problems: 𝟏
𝟏
𝟖
𝟐
“If a farmer harvests of his crop every 2 days, how long will it take him to harvest his entire crop?” *The double number line allows students to see that in order to harvest the entire crop at a constant rate, it will take 12 days. “What is the equivalent unit rate of 𝟏/𝟖
𝟓/𝟐
per day?” *Since the number line does not show a direct relationship between the amount of crop harvested in a single day, students can use their reasoning skills to persevere at solving this problem (MP1,2), or they can make use of a standard algorithm, whereby the student should simply divide !
!
!
!
!
!
!"
!"
!/!
!/!
!
to arrive at the unit rate of of the crop harvested per day. This !"
will look like × = = of the crop per day. Double Number Line: Ratio of Amount of Crop Harvested to Number of Days as a Proportional Rate !
!
!
!
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←-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐→ Amount of Crop Harvested !
!
!
!
!
!
!
!
1 3 4 6 7 9 10 12 ←-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐l-‐-‐-‐-‐-‐-‐-‐-‐→ Number of Days Graph: Consider the following problems: 𝟏
𝟏
𝟐
𝟏𝟎
“If a champion swimmer consistently swims of a length of an Olympic-‐sized pool in minute, how long will it take him to swim 4 lengths of the pool?” !
*The line graph allows students to see that it would take a champion swimmer about of a minute to swim 4 !"
lengths of the pool. “What is the equivalent unit rate of 𝟏/𝟐
𝟏/𝟏𝟎
per minute? *Again, since the graph does not show the number of lengths one would swim in one minute, students could reason quantitatively (MP2) in order to persevere in solving this question (MP1). Otherwise, an algorithm could be used to !
!"
!"
!
!
!
compute the equivalent unit rate. This would look like × = = 5 lengths per minute. Minutes Line Graph: Ratio of Length of Pool Swam to Fraction/Decimal of a Minute as a Proportional Rate 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1/2 length 1 length 1 1/2 length 2 lengths 2 1/2 lengths 3 lengths 3 1/2 lengths 4 lengths Tape Diagram: Consider the following problems: 𝟏
𝟏
𝟐
𝟐
“If an avid reader reads of a 400 page book every 1 days, how many days will it take for that person to read three approximately 400-‐page books? *The tapes diagram clearly shows that it would take 6 days to read those three books. “What is the equivalent unit rate of 𝟐𝟎𝟎
𝟑/𝟐
pages per day?” *Since the tape diagram does not show a direct relationship between the number of pages read in a single day, again, students can use their reasoning skills to persevere at solving this problem (MP1,2), or they can make use of a standard algorithm, whereby the student could simply divide per day. This will look like !""
!
!
!""
!
!
× = 𝟐𝟎𝟎
𝟑/𝟐
to arrive at the unit rate of pages read , or ~ 133 pages per day. Tape Diagram: Ratio of Pages Read to Number of Days as a Proportional Rate 200 pages 400 pages 600 pages 800 pages 1000 pages 1200 pages !
!
!
3 days 6 days 6 days 1 days 4 days 4 days !
!
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Part Three (a): Textbook Development Standard: 7.RP.1 (Progressions: 6.RP.1, 2 and 3a, 3b, 3c, 3d→7.RP.1→8.EE.5) What’s in Glencoe? 6.RP.1 ,2, and 3a, 3b, 3c, and 3d The 6th grade standards include the domain Ratio and Proportional Relationships. Standards 6.RP.1 and 6.RP.2 are represented in Glencoe’s Course 1 for 6th graders. Chapters 10-‐1 and 10-‐2 devote ten pages to ratios and proportions. Students are introduced to the concepts of ratios, unit rates, and proportions. However, most of the problems simply require that students use the procedural, “Express each ratio as a fraction in its simplest form.” There is one example for finding unit rates that, again, uses a procedural: “Divide the numerator and denominator by 4 to get a denominator of 1.” There are some pictorial representations that appear to be very easy, and there is one set of function tables where students are asked to find a rule to represent the unit rate. Again, this task as presented would be very easy for most 6th graders and certainly not represent a very high level of cognitive demand. In terms of proportions, the procedural, “…cross products…multiply…and divide the numerator and denominator by the denominator…” is what is presented. There is a nice “Hands-‐on-‐Lab” where students work with tangrams to explore the relationship between ratios and area. This lesson would be useful for Standard 6.RP.3.d. 7.RP.1 7.RP.1, the standard for 7th graders around which this project is focused, is really not covered in Glencoe’s Course 2 for 7th graders. What is covered is mostly a review of Course 1. Chapter 7-‐1 covers ratios, Chapter 7-‐2 covers rates, and Chapter 7-‐3 covers proportions. Again, the material in heavy in procedural methods, through there is more in terms of the explanation of the concept of rates than there is in the 6th grade TE. I did note several function tables (very simple) and one graph depicting the relationship between quantity and time. There is a “Hands-‐on-‐Lab” where students would work with tile patterns to explore a constant rate of change. 8.EE.5 This standard is the 8th grade progression around proportionality. Rate of change is covered in Chapter 4-‐2, slope is covered in Chapter 4-‐3, and Proportions are covered in Chapter 4-‐4. In Chapter 4-‐3, students are given visuals including graphs and tables to show proportional relationships as the slope. There is evidence of using tables and graphs to represent the same rate. Part Three (b): Conclusions What’s Not in Glencoe? What to Do? th
Course 1/6 Grade Glencoe always does a good job of relating math to “real-‐world mathematics” (e.g., the “When am I ever going to use this?” section and plenty of word-‐problems), but the visuals that would provide students with an in-‐depth understanding of ratios and proportions are missing. In the Glencoe Course 1 TE, there are plenty of equations and word problems as well as a few simple tables and pictorial representations, but there are no tape diagrams, graphs, or double number lines to address the 6th grade standards in Ratios and Proportions. It seems that the vocabulary and concepts of 6.RP.1 and 6.RP.2 are introduced purely through algorithms, and 6.RP.3 is not represented well at all. Supplemental materials, strategies, and activities will need to be incorporated into 6th grade mathematics instruction in order for students to gain a strong foundation in ratios, rates, and proportions. Course 2/7th Grade Standard 7.RP.1 is explicit in asking students to represent complex fractions as an equivalent unit rates. There is nothing in the Course 2 TE that would satisfy the learning requirements for this standard. There are no complex fractions, and there are no tape diagrams or double number lines; only a few simple charts and one graph. I noted one multi-‐step problem. Again, supplemental materials, strategies, and activities will need to be incorporated into 7th grade mathematics instruction in order for students to truly understand the relationship between a ratio expressed as a complex fraction and its unit rate. Course 3/8th Grade With the information provided in Course 3, the standard 8.EE.5 can be adequately addressed. This is assuming students have had the proper preparation for the concepts of proportional relationships and unit rates through the related 6th and 7th grade progression of standards. A caveat is that I did not see a place where students would explicitly have an opportunity to compare two different proportional relationships using two different representations. The class instructor would have to pull that part of the standard from the TE in a cut and paste manner. Suggestions/Resources • Teachers at the middle school level, as well as teachers at the intermediate level, should take advantage of professional development opportunities at the school and district level around the RP strategies. • Please approach your Math Coach if you need assistance and support with RP strategies. • Math Coaches need to provide that PD which should include grade-‐level meetings during the school day, whole staff professional development, coaching, and modeling. • Milwaukee Public School teachers should refer to the “The Common Core State Standards for Mathematics Resources” found on the portal on the Milwaukee Mathematics Partnership resource page. • Navigating through Measurement in Grades 6–8 (with CD-‐ROM) (NCTM) “Real-‐world investigations…exploring proportionality, scaling, and similarity…investigate such rates as speed and density.” (MPS Supplemental Series) • Navigating through Number and Operations in Grades 6-‐8 (with CD-‐ROM) (NCTM) “Middle school students consolidate their understanding of integers and rational numbers, increasing their facility with fractions, decimals, and percents as well as proportionality…”(MPS Supplemental Series) • Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades 6-‐8; Joanne Lobato, Amy Ellis, Rose Mary Zbiek (Book)

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Standard: 7.RP.1

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