• real-life situations involving linear and non-linear relations:
  • the volume of fruit juice that can be made in comparison to the mass of fruit
  • the total amount earned in comparison to the number of hours worked
  • the volume of water in a pool over time as it is being emptied or filled
  • the number of times a piece of paper is folded in half compared to the number of layers of paper after each new fold
  • the monetary value of a vehicle over time
  • the height that a basketball rebounds after each bounce
• describing rates of change:
  • constant rate of change
  • zero rate of change
  • positive or negative rate of change
  • increasing rate of change
  • decreasing rate of change
• growing patterns:

• shrinking patterns:

Teachers can:

• pose situations to students and facilitate a conversation about what the shape of the graphical representation of each situation tells us about the situation;
• provide students with concrete materials and digital tools to build linear and non-linear growing and shrinking patterns, and support them in creating graphs to represent these patterns;
• support students in making connections between the shapes of the graphical representations and their rates of change (e.g., if the graph becomes steeper, what does that tell us about the rate at which the variables are changing?);
• encourage students to describe graphs and their rates of change using gestures, vocabulary, and other means that are accessible to them, along with mathematical terminology;
• share graphs that have connections to other strands in the course (e.g., appreciation, changing volume);
• facilitate a discussion about what strategies students might use to make predictions (e.g., interpolating and extrapolating) and when it is appropriate to use their graphs to make predictions.

• How can you tell from the shape of a graph when it:
  • is growing at a constant rate?
  • is growing at an increasing rate?
  • is growing at a decreasing rate?
  • has a rate of change of zero?
  • is shrinking at a constant rate?
• Explain how the shape of a graph can provide information about the situation it represents. How can the graph help you make predictions about the relationship?
• What are the differences between linear and non-linear relations? What are the differences in their graphs? In their rates of change?
• If you built a pattern to represent the graph below, what would it look like? Would the pattern be growing or shrinking as the independent variable increases? Would it go up by a constant amount or by a changing amount? How could you use the graph to make a prediction about the pattern?

• linear relations represented using:
  • concrete materials:

• tables of values:

• graphs:

• equations:
  number of tiles = 2 × (position number) + 3
  or
  T = 2p + 3
  or
  NumTiles = 2*PosNum + 3

Teachers can:

• ground the learning for this expectation in real-life examples and contexts that are relevant to students;
• provide students with opportunities to use concrete materials (e.g., colour tiles, interlocking cubes, cups, pine cones, beads), digital tools, and coding to represent linear relations;
• support students in developing ways to determine the rate of change and the initial value in each of the various representations;
• provide students with opportunities to make connections between representations by providing them with one or two representations and supporting them in generating the others;
• facilitate a discussion about which representations are most helpful for various purposes, including making and reflecting on both near and far predictions;
• continue to make connections between the rate of change and the initial value and what they represent in a given context;
• introduce the terms partial variation and direct variation as connected to initial values and proportionality;
• encourage students to identify relationships using functional thinking by making connections between the term value (dependent) and corresponding term number (independent) as well using recursive thinking by making connections from term value to term value.

• How can you determine the rate of change from each representation? Do you find it easier to determine the rate of change with some types of representations than with others?
• How can you determine the initial value from each representation? Do you find it easier to determine the initial value with some types of representations than with others?
• Given one of the four types of representations (concrete materials, table of values, graph, and equation), how do you create the others for the same relation?
• Given four representations of different types, how do you know they all represent the same relation?
• What does the rate of change mean in this context? What does the initial value mean in this context?

• compare graphically:
  • create a graph comparing distance in kilometres and time in hours:

• compare rates of change, initial values, and point of intersection
• compare algebraically:
  • compare rates of change and initial values by examining the equations
  • use the method of comparison to compare relation A: d = 8t + 2 to relation B: d = 12t by setting the equations equal to each other and solving for the variables to determine the point of intersection

Teachers can:

• sequence the learning to begin with two relations that allow students to accurately locate the point of intersection on a graph and move to relations that require an algebraic method of comparison to accurately determine the point of intersection;
• facilitate a discussion about the strengths and limitations of comparing relations graphically versus comparing them algebraically; 
• support students in interpreting the point of intersection in the context of the problem by highlighting that this is the point that satisfies both equations and the point where, for a specific independent value, both relations have the same dependent value;
• provide students with opportunities to use digital tools or coding to compare linear relations;
• support students in making connections between the algebraic method of comparison that they are learning now and the strategies they learned for solving equations with variables on both sides of the equal sign in Grade 8.

Note

The use of the algebraic method is intended to involve only the method of comparison, not to be extended to elimination, substitution, or other approaches to solving linear systems.

• What is similar and what is different about these two linear relations?
• What does the point of intersection (if any) mean in this situation?
• Compare the rates at which these relations are growing.
• Thinking about the context of a situation shown graphically, describe what is happening on the left and right sides of the point of intersection (if any) on the graph.
• List the strengths and weaknesses of both the graphical method and the algebraic method of comparing relations.
• Will two linear relations always have a point of intersection? In what cases will they not intersect?