Andy Flater
CCLM^2 Project
Summer 2012
This material was developed for the Leadership for the Common Core in Mathematics (CCLM^2) project
at the University of Wisconsin-Milwaukee.
Part 1 Standard Grade: 7th Domain: Ratios and Proportional Relationships 7.RP.2b Cluster: Analyze proportional relationships and use them to solve real-­‐world and mathematical problems. Standard: Recognize and represent proportional relationships between quantities. b) Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Part 2 Explanation and Examples of Standards Explanation: The standard of Ratio and Proportional Relationships extends students’ work from the elementary grades. It extends students work in measurement and in multiplication and division. This standard is also a foundation for future work in Algebra and Geometry. Students will use this foundation when they discuss similar figures, slopes of lines and trigonometry ratios. It also addresses mathematics in the Sciences and in every day uses. Such as speed, acceleration, density, taxes, discount, and tips to name a few. The goal of this Standard is to recognize and represent proportional relationships between quantities. It is through learning multiple representations such as tables, graphs, equations, diagrams and verbal descriptions that students will be able to recognize if a relationship is proportional or not. The Mathematical Practice Standards this domain focuses on is #2 Reason abstractly and quantitatively and # 6 Attend to precision. Proportional relationships involve a collection of pairs of measurement in equivalent ratios. A collection of equivalent ratios can be graphed in a coordinate plane or seen by creating a table by determining the unit rate. The graph and table then represents a proportional relationship. Students in this standard will examine situations carefully to determine if they describe a proportional relationship. Students will then make connections to the graph and table to write equations of the relationship. Students must understand the structure of the problem which includes looking for and understanding the roles of “for every”, “for each” and “per”. These terms will solidify their understanding of a unit rate. The use of a ratio tables in Grade 6 will be crucial for students to develop in order to extend their understanding of unit rates. Examples: For every 5 cups grape juice, mix in 2 cups peach juice. In the table students should recognize the unit rate as the amount of increase in y cups as x cups increase
by 1 cup.
In the graph students should make the correlation that the unit rate as the vertical side length increase with
the horizontal side length of 1. This can be described as a “unit rate triangle” or “slope triangle”. These
terms will benefit students when they extend on these ideas in 8th grade in the Standards of Expression
and Equations and Functions.
For each 1 unit you move to the right, move up 2/5 of a unit.
When you go 2 units to the right, you go up
When you go 3 units to the right, you go up
units
units.
When you go 4 units to the right, you go up
units.
When you go x units to the right, you go up
units.
Once students are able to create tables and graphs to show proportional relationships they will then extend
them to developing equations from the tables and graphs.
Starting from (0; 0), to get to a point (x; y) on the graph, go x units to the right, so go up
units.
Therefore y =
This will lead to the formal equation of
Once students are able to use tables and graphs to formulate an equation, students will begin the process
of formulating an equation in a more tradition method to solve proportional relationships.
Looking back at the original question: For every 5 cups grape juice, mix in 2 cups peach juice. If we add to the question: If you have 50 cups of peach juice how many cups of grape juice will you need to create the same strength of mixture? In this case, students will view the 5 cups of grape juice mixed with 2 cups of peach juice as a unit rate
and the 50 cups of grape juice and the unknown as a rate. This is obtained from equivalent ratios so they
are proportional. 5 : 2 and 50 : t. These ratios can then be written as equivalent fractions because
equivalent ratios have the same unit rate.
Although students are able to write equations students need to be flexible in the strategies they choose for a specific situation. In this situation a student might revert to using a ratio table to solve rather than setting it up as a proportional relationship. Cups of grape 5 10 50 juice Cups of peach 2 4 20 juice Students at times have difficulty setting up the proportional relationship and may use ratio tables to solve the problem rather than in the more tradition method. Part 3 School Mathematics Textbook Program Textbook Development: I will be examining the Textbook development for the MathThematics Book 1, 2, and 3. Currently these books are being used for our 6th, 7th and 8th Grades respectfully. The MathThematics program use
thematic modules throughout the whole program. The mathematics is introduced by a particular theme.
Instead of having specific chapters on topics it is a spiral approach that develops concepts throughout the
entire program. These themes are intended to provide an opportunity for students to explore
mathematical concepts in context of “real world” application. The text are organized into modules and
further divided into explorations that begin with a “real world” connection and then explore mathematics
related to it.
The Domain Ratios and Proportional Relationships starts in the 6th grade. In 6th grade students should be
using their previous knowledge of multiplication and division to develop reasoning strategies to solve
ratio and rate problems. Students should be developing these skills through the use of tables of equivalent
ratios, plotting on coordinate plane, and in context “real world” problems. It appears that in Book 1 of
MathThematics students are exposed to comparing equivalent fractions and ideas related to this Standard.
However, I feel it is not explicit enough. It appears that the book heavily relies on a more traditional
method of setting up a proportion to solve most problems. Cross product is the strategy that is
emphasized the most. It seems if teachers are teaching as the book is suggesting that students will be
missing some experiences and strategies that might help in better understanding the concept and meet the
actual standards.
In 7th grade students are expanding on their knowledge of ratio and proportions form 6th grade. Students
are expected to solve multi step problems that relate to ratio and proportions. Graphing becomes an
integral part of the standard along with similar figure relationships. In Book 2 of MathThematics students
continue to use setting up a proportion to solve many ratio and relationship problems. Students are not
exposed to many of the strategies needed to address the Standard. Graphs and tables are under used in
this text.
Page 98 of Book 2
Although this question appears to address a portion of the standard with using a table, I feel it really
doesn’t address the Standards intent. The book goes directly to an equation rather than have the table
completed to create a graph and then develop the equation from identifying the constant of
proportionality. Students do not have the opportunity to determine if the two quantities show a
proportional relationship. It is just given to them through the equation. Students are also not given the
opportunity to graph this relationship to show the proportional relationship visually. Students may know
how to use an equation to complete a table or create a graph but may not have the understanding of
identifying the constant of proportionality.
In 8th grade students will use their knowledge on ratio and proportional reasoning when they start
formulating and reasoning about expressions and equations. Once again the book emphasizes the use of
setting up a proportion to solve ratio. It does not address the use of similar triangles to explain slope and
is weak on explaining how a unit rate can be interpreted as the slope.
Conclusion and Suggestions: It appears that the MathThematics Book1, 2 and 3 are heavily process orientated. It seems that it leads teachers towards going directly to a standard procedure. Although it has explorations and a few activities students do not appear to have enough time on the Standards. In fact, the books are weak when it comes to finding when the standards are being taught. Topics are revisited throughout the program it appears that not enough time is placed on Ratio and Proportional reasoning. The rates and proportions are in the text but the reasoning portion seems to be lacking. The texts do a nice job of reviewing topics with the spiral review through the progression of Book 1-­‐3. However, it does not always match up with the standard that should be taught at that grade level. It allows for some activities and exploration but lead teachers to teach one dominate method. In order to teach the standards the books do not have strategies that would help in doing so. Teachers will need to provide a lot of other opportunities besides what is in the books. An example of a problem that might benefit students is as follows. Plot the data from the table and connect the points with a line. Canoe Rental 2 Total Cost 5 A graph would need to be provided. 5 8 11 17 Is this a proportional relationship? Explain. 10 21