Grade level of activity: Grade 9
Course used for: MPM 1D (Academic)
Ontario Curriculum expectations addressed:
- construct tables of values, graphs, and equations, using a variety of tools (e.g., graphing calculators, spreadsheets, graphing software,paper and pencil),to represent linear relations derived from descriptions of realistic situations (Sample problem: Construct a table of values, a graph, and an equation to represent a monthly cellphone plan that costs $25,plus $0.10 per minute of airtime.);
- describe a situation that would explain the events illustrated by a given graph of a relationship between two variables
- determine other representations of a linear relation, given one representation (e.g., given a numeric model, determine a graphical model and an algebraic model; given a graph, determine some points on the graph and determine an algebraic model);
- describe the effects on a linear graph and make the corresponding changes to the linear equation when the conditions of the situation they represent are varied (e.g., given a partial variation graph and an equation representing the cost of producing a yearbook, describe how the graph changes if the cost per book is altered, describe how the graph changes if the ﬁxed costs are altered, and make the corresponding changes to the equation).
- determine the equation of a line from information about the line (e.g. the slope and y-intercept; the slope and a point; two points)
- describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation (e.g., the cost to rent the community gym is $40 per evening, plus $2 per person for equipment rental; the vertical intercept,40,represents the $40 cost of renting the gym; the value of the rate of change,2,represents the $2 cost per person),and describe a situation that could be modelled by a given linear equation (e.g., the linear equation M = 50 + 6d could model the mass of a shipping package, including 50 g for the packaging material, plus 6 g per ﬂyer added to the package);
- determine graphically the point of intersection of two linear relations, and interpret the intersection point in the context of an application (Sample problem: A video rental company has two monthly plans. Plan A charges a ﬂat fee of $30 for unlimited rentals; Plan B charges $9,plus $3 per video. Use a graphical model to determine the conditions under which you should choose Plan A or Plan B.).
I designed the Activity “Does the Cat catch the mouse” to create a bridge between the “modeling with graphs” and “linear relations” units. I had my students work on this activity immediately after we finished the “modeling with graphs” unit and right before we started the “linear relations “unit. The main purpose of having the students do this activity was to connect the concepts that they just learned to the concepts they will learn. It was also designed as an activity to both evaluate students understanding of past concepts and introduce them to new concepts. I had students work in pairs for this activity; I paired students with similar abilities together.
The first four questions give the students enough information to represent the displacement of both the cat and the mouse on the same grid. What I found worked well was that students of all levels were able complete these questions. They just did them in different ways. Most of the students who did well in the modeling with graphs unit quickly took the information (rate of change and starting point) and created the equations for both cat and mouse and used them to graph both lines. Students who struggled with the previous unit were able to complete the first four questions by recognizing the pattern in both T – tables and simply extended these patterns to graph both lines. When it came time for them for them to determine the “speed” (rate of change) and equation of the lines, I had the opportunity to have discussions with them on how the patterns they used are related to what the speed (rate of change) is and how an equation can be arrived at that models the movements using the rate of change and starting points. I was happy that this activity gave me an opportunity correct any misunderstandings form previous unit.
Questions 5 and 6 gave students an opportunity apply understanding of how rates of change and starting points relate to the equation of a line. Question 5 gives the students the speed and starting points of the cat and mouse, generally students made the connection and easily created a table of values for both lines to help the graph the lines in order to answer the question. Question 6 required students to create linear relations that matched 3 different scenarios. What I was most pleased with was the amount of conversation this question generated between partners. Because there were infinite possibilities for the correct solutions, I found that students were discussing why some solutions make sense and others do not. Often times both partners came up with solutions that were different but both correct, it was fun watching them confirm with each other that they were both right! Question 6 also introduced them to some concepts (parallel lines, and negative rates of change) that they will work with in the next unit of study. I was happy with the way students were able give a “real life meaning” to what parallel lines represent (same speed) and negative rates of change (moving in opposite directions).