Mary Lock CCLM Project 2
July 15, 2011
CCSSM Interpretation Guide Part 1: Standard 7.RP.2 b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Part 2: Key Concepts and Terms: A change in one quantity with respect to another quantity is called the rate of change. Rates of change can be described using slope, where the change in “y” is divided by the change in “x”. You can find rates of change from an equation, a table of values, or a graph. A special type of linear equation that describes rate of change is called a direct variation. The graph of a direct variation always passes through the origin (0,0) and represents a proportional situation. In the equation, y = kx, k is called the constant of variation. It is also called the slope, or the rate of change. So, “constant of variation” and “slope” and “rate of change” are all synonymous. In the equation, y=kx, as x increases in value, y increases or decreases at a constant rate, k, OR y varies directly with x. Another way to say this is that y is directly proportional to x. The direct variation y = kx can also be written as k = y/x. Students can use simple algebra to see that these 2 equations are equivalent. In this form k = y/x you can see that the ratio of y to x is the same (constant) for any corresponding values of y and x. Y is also divided by x, so students can see that y is rise and x is run, and rise over run is the definition of slope. In other words, graphing proportional relationships represented in a table helps students recognize that the graph is a line through the origin (0,0) with a constant of proportionality equal to the slope of a line. When a relationship is proportional, all y over x ratios simplify to the slope of the line which is the constant of proportionality or the unit rate, with one as a denominator. Example: Consider the equation y = 60x. The slope is 60. The rate of change is 60 units for every one unit. The constant of variation is 60. Students should see that dividing the y by the x always get the same number, which is the unit rate or the slope of the line. Students should see the data displayed in a chart and test this idea with each set of numbers in the chart. For example, y = 5 when x = -­‐15, so the equation would be y = -­‐1/3x. For example, y = 24 when x = 4, so the equation would be y =6x. DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) Possible Confusing Aspects: The word “constant” here does not refer to a term without a variable. Unit rate is different than unit fraction, but has some similarity that could be explored, even though the 1 is in the numerator for unit rate and in the denominator for unit rate. A common error is to reverse the position of the variables when writing equations. Students may find it useful to use variables specifically related to the quantities rather than using x and y. For example, when the number of packs of gun is the x, the student may write g for gum instead of x. And when the cost in dollars is the y, the student may write d for dollars instead of y, so that the equation is d = 2g, instead of y = 2x to further connect the algebraic equation to the scenario. Students also could refer to the slope as 2/1 instead of just 2 so that they can connect the 2 as the rise or the change in y over 1, the run or the change in x. Students should be led to understand that a proportional relationship must go through the origin. In other words, there is no y intercept. Also, a listing of the x, y pairs will show that the equal sized “ up and overs” do not cross the y. Students should understand the difference between the two equations of y = kx and y = kx +b. Teacher Friendly Language: Students determine if two quantities are in a proportional relationship from a table. For example, the table below shows the relationship between time worked in hours and wages earned in dollars. One hour for 12 dollars is proportional to 2 hours of work for 24 dollars. However, the third line of the table shows 3 hours of work for 35 dollars. Therefore, there is not a constant of proportionality. The graph of all the x,y values in the table would not be linear and would not show a constant slope or rate of change due to the ordered pair (3, 35). The ordered pair (1, 12) indicates that one hour of work earns $12. This is the unit rate. The constant rate of proportionality is the unit rate. Time (h) Wage ($) x y 0 0 1 12 2 24 3 35 DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) The graph below shows the proportional reasoning behind the amount and cost of gas. http://www.cpm.org/pdfs/state_supplements/Proportional_Relationships_Slope.pdf Part 3: School Mathematics Textbook Program (a) Textbook Development Students in MTSD use Math Thematics Book 3 at Grade 7. In eighth grade, students switch to a more traditional text, Glencoe Pre-­‐Algebra. In Module 4, “Inventions”, of Math Thematics, students are expected to master the following: ü
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Find and interpret positive and negative slopes rise over run. Identify slopes of horizontal and vertical lines and the y-­‐intercept of a line. Write equations in slope-­‐intercept for from a graph. Use equations in slope-­‐intercept form and their graphs to model real world situations. DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) In eighth grade pre-­‐algebra, students are expected to master the following: Graph linear equations using intercepts Find the slope of a line Determine slopes and y-­‐intercepts of lines Graph linear equations using the slope and y-­‐intercept Write equations given the slope, and y-­‐intercept, a graph, a table, or two points. ü Solve a system of linear equations by graphing. ü
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(b) Conclusions On paper, the two sets of student expectations are very similar. However, in practice, most of my pre-­‐algebra students are nowhere near mastery of the 7th grade curricular objectives. In other words, the student expectations at 7th grade match the CCMS requirements at 7th grade, but the students seem to merely be exposed to the concepts and generally do not attain mastery of the concepts of Standard 7.RP.2.b. It is interesting to note that at 8th grade, we skip the lesson which most nearly matches this standard. In Chapter 8 of the Glencoe Pre-­‐algebra book entitled “Functions and Graphing”, we do not cover the lesson on rate of change that most closely matches this standard. Therefore, I will recommend that we include this very valuable lesson in the continuum of developing the concept of constant of proportionality. Furthermore, students at 7th grade need a richer, more conceptual background of the meaning of slope to master the more procedural expectations at grade eight and beyond. It is my hope that as teachers in Grade 7 and 8 become familiar with the demands of the CCSSM, we will develop lessons that integrate the Standards for Mathematical Practice and become more cognizant of new grade level expectations and hold students more accountable for their mastery. (c) Suggestions When unpacking the common core standards for grade eight in relation to the grade seven CCSSM standards, it is obvious we need to prepare lessons that are outside the realm of our current Glencoe pre-­‐algebra textbook: •
In standard 8.EE.6, students are asked to use similar trianges to explain why slope m is the same between any two distinct point on a non-­‐vertical line in the coordinate plane and derive equations based on the y-­‐intercept. At present, we are developing the y=mx+b equation, but we do not stress the concept of similar triangles in relationship to proportional points on the line. I now see how important this is the development of their conceptual understanding of this standard. DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) •
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In standard 8.F.2, students are required to compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) and determine which function has the greater rate of change. We currently do not have any lessons developed for this standard. In standard 8.F.3, students use equations, graphs, and tables to categorize functions as linear and non-­‐linear. Again, we do not stress this enough with our students. We need them to develop a deeper conceptual knowledge by seeing this idea in more varied situations, such as a verbal description, two ordered pairs, a table, a graph, or rate of change and another point on a line. Students need to more fluently move among all of the above variations on the theme of slope. Part 4: Check Point Formative Assessment that Reveals Student Thinking (a) Formative Assessment Task Ramon and his twin brother, Ricardo, run in a 50 meter race. Ramon gives Ricardo a 10 meter head start in the race, and they run at the same speed. Design a graph to show the relationship to time in seconds to distance in meters. Represent Ramon and Ricardo’s positions with equations. Explain the relationship of the two lines on the graph. Which line shows a proportional relationship? Use an x,y chart to prove your understanding. DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) (b) An example of proficient student work Design a graph to the relationship to time in seconds to distance in meters. Represent Ramon and Ricardo’s positions with equations. Ramon: y = 5x Ricardo: y = 5x + 10 Explain the relationship of the two lines on the graph. The lines are parallel because the speeds are equal and the slopes are equal. They are both lines which make them linear equations. Ricardo runs at a constant rate of 5 meters per second and has a head start of 10 meters. That is why Ricardo’s line starts at (0,10). Ramon does not have a head start and he is at 0 meters at 0 seconds, (0,0), the origin, at the beginning of the race. Which line shows a proportional relationship? Use an x,y chart to prove your understanding. Ramon’s line on the graph shows a relationship that is proportional because his equation is in y = kx format, where k represents a constant. I can tell that this equation shows a proportion because it crosses through (0,0). DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) Ramon constantly covers 5 meters for every 1 second he runs. For example, when he is 2 seconds into the race, he is at 10 meters. When he is 3 seconds into the race, he is at 15 meters. Here are x,y charts of the two equations: Ramon’s line Time (in seconds) Distance (in meters) 0 0 1 5 2 10 3 15 4 20 You can see that if I put each y (distance) over each x (time), which is the rise over the run, and then simplify each set, I always get the slope, which is 5. This 5 sits right next to the x, which is the spot where the slope sits. For example 20/4 is 5. So is 15/3, and so on. It is constantly this way with every x,y pair, so this relationship show on this line is proportional. This 5 is called the unit rate, which is also another name for slope. Ricardo’s line Time (in seconds) Distance (in meters) 0 10 1 15 2 20 3 25 4 30 Ricardo got a head start, so he does not start his race at 0 meters with the clock at 0 seconds. If I put rise over run for each x,y pair, I do not always get the unit rate. It is always different. For example, 15/1 is 15, but 20/2 is 10, and so on. This is not proportional. However, the slope is the same because they both travel at 5 meters per one second. Ricardo is the winner of the race because you can tell he got to 50 meters in 8 seconds and Ramon got to 50 meters in 10 seconds. DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) (c) Annotated student work Ramon: y = 5x Ricardo: y = 5x + 10 The lines are parallel because the speeds are equal and the slopes are equal. They are both lines which make them linear equations. Ricardo runs at a constant rate of 5 meters per second and has a head start of 10 meters. That is why Ricardo’s line starts at (0,10). Ramon does not have a head start and he is at 0 meters at 0 seconds, (0,0), the origin, at the beginning of the race. Ramon’s line on the graph shows a relationship that is proportional because his equation is in y = kx format, where k represents a constant. I can tell that this equation shows a proportion because it crosses through (0,0). Ramon constantly covers 5 meters for every 1 second he runs. For example, when he is 2 seconds into the race, he is at 10 meters. When he is 3 seconds into the race, he is at 15 meters. DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) Here are x,y charts of the two equations: Ramon’s line Time (in seconds) Distance (in meters) 0 0 1 5 2 10 3 15 4 20 You can see that if I put each y (distance) over each x (time), which is the rise over the run, and then simplify each set, I always get the slope, which is 5. This 5 sits right next to the x, which is the spot where the slope sits. For example 20/4 is 5. So is 15/3, and so on. It is constantly this way with every x,y pair, so this relationship show on this line is proportional. This 5 is called the unit rate, which is also another name for slope. Ricardo’s line Time (in seconds) Distance (in meters) 0 10 1 15 2 20 3 25 4 30 Ricardo got a head start, so he does not start his race at 0 meters with the clock at 0 seconds. If I put rise over run for each x,y pair, I do not always get the unit rate. It is always different. For example, 15/1 is 15, but 20/2 is 10, and so on. This is not proportional. However, the slope is the same because they both travel at 5 meters per one second. Ricardo is the winner of the race because you can tell he got to 50 meters in 8 seconds and Ramon got to 50 meters in 10 seconds. DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011)