Page 147 - Mathematics GRADE 9, DE-STREAMED (MTH1W)

Instructional Tips
Teachers can:
• create opportunities to reinforce math facts by asking students to list possible ordered pairs that make an equality true (e.g., for the relation x + y = 10, ask “What are some pairs of numbers that have a sum of 10?”);
• support students in moving from thinking about discrete points that satisfy equations and inequalities to thinking about continuous lines or regions and making connections to the concept of density of numbers;
• depending on student readiness, lead conversations that require students to reflect on the characteristics of different graphs, depending on the form of the algebraic equation. This will support them in building questions they can ask themselves to make conjectures about what a graph might look like; for example:
o Where would the graph of x + y = 10 intersect the x-axis? the y-axis? How do you know? o If two numbers, x and y, multiply to a positive number (i.e., xy = k, where k > 0), what has to be true about x and y? What would this look like in a graph? In which quadrants would
you find points on the graph? Explain your thinking.
• encourage students to make conjectures about what a graph will look like by first sketching the
graph and then using coding or digital tools to test their conjectures;
• highlight x = k and y = k as special cases of linear relations, and have students explore why they
are special cases;
• support students in noticing how y-values change, depending on linear and non-linear
relationships between x and y (e.g., ask: What do we know about y if xy = 10? if x + y = 10?);
• introduce students to the different ways of representing inequalities on a graph (e.g., if showing
the region x + y < 10, the line for x + y = 10 should be dotted as opposed to solid);
• support students in developing strategies for determining whether a given point satisfies an
equation or inequality represented on a graph, and how this point connects to various regions on
the graph;
• facilitate opportunities for students to explore, using coding and digital tools, multiple cases of
each of the equations and associated inequalities listed above in order to highlight their characteristics.
Teacher Prompts
• Think of two numbers that have a sum of 10. Do only whole numbers work? What integer values work? What about fractions or decimals?
• How does x = 10 compare to x > 10? What numbers satisfy x = 10? What numbers satisfy x > 10?
• What numbers satisfy x + y > 10? Where do these points lie on the grid in comparison to the line
x + y = 10?
• Think of two numbers that can be multiplied to get 180. Do only whole numbers work? What
integer values work? What about fractions or decimals? 146