• characteristics:
  • of graphs:
    • shape
    • direction or orientation
    • intercepts
    • rate of change (constant or not constant)
    • symmetry
  • of tables of values:
    • initial value
    • rate of change (constant or not constant)
    • symmetry
    • patterns or repetition
  • of equations:
    • degree
    • leading coefficient when terms are in descending order of powers
    • type of equation (e.g., y = x² (quadratic), y = 2^x (exponential), y = 2x (linear), $$y=\frac{1}{2 x}$$(reciprocal))
    • rate of change (constant or not constant)
    • initial value

Teachers can:

• encourage students to develop and expand their vocabulary for describing, explaining, and making connections among the characteristics of graphs, tables of values, and equations;
• facilitate discussions to highlight comparisons and connections between different representations of the same relation as well as between the same representation of different relations;
• support students in recognizing how and why calculating first differences in tables of values can be helpful for determining rates of change and identifying linear and non-linear relations;
• provide opportunities for students to use coding or digital tools to create graphs.

• How can you determine whether a relation is linear or non-linear by examining the graph? By examining the table of values? By examining the equation?
• Which of these graphs have a line of symmetry?
• What are the x- and y-intercepts of this graph, if any? Why does it make sense that the graph intersects the axes in these places?
• How does the graph of y = 2x compare to the graph of y = −2x?
• How are the graphs of y = 2x, y = x², and y = 2^x similar, and how are they different? How do these similarities and differences appear in their tables of values and in their equations?

• graphs of relations:

x = 10	y = 10
x + y = 10	x – y = 10
2x + 5y = 10	xy = 10

• associated inequalities:

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:
  • Where would the graph of x + y = 10 intersect the x-axis? the y-axis? How do you know?
  • 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.

• 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?
• What do you notice about where each of these graphs crosses the x-axis? The y-axis? Why does it make sense that they cross at these points?

• transformations:
  • translations:
    shift up, down, left, or right:

• reflections:
  over the x-axis or over the y-axis:

• rotations:
  clockwise or counterclockwise in increments of 90°:

Teachers can:

• make connections between learning from the elementary grades about transformations of points and shapes and the transformation of lines by providing students with graphs of lines (physical or digital) and asking them to perform the various transformations on them;
• support students in making connections between the changes in the equations and the changes in the graphs;
• provide opportunities for students to use coding or digital tools to test conjectures about how different transformations will affect the graph of a line and how changing the equation of the line will affect the graph;
• make connections to C1.3 when examining the effects of a translation up/down and a translation left/right; for example, point out that a shift down 4 (y = 2x − 4) has the same effect as a shift right 2 (y = 2(x − 2)), and ask “How are the equations similar and how are they different?”;
• support students in making connections between the y-intercepts and slopes of the lines and the equations of the lines;
• facilitate a discussion about the relationship between the equations of parallel lines when looking at translations and rotations of 180° or 360° and the relationship between the equations of perpendicular lines when looking at rotations of 90° or 270°;
• introduce the concept of combining transformations, depending on student readiness.

Note

The learning in this expectation connects learning from the elementary grades about transformations of points and shapes to the transformation of lines. This connection will then support learning about the transformation of more complex functions in future grades.

• How does translating, reflecting, and rotating a line connect to transforming shapes?
• What changes do you notice in this transformed graph? How is it the same? How is it different?
• If you start with the line y = 2x, what type of transformation would result in:
  • a parallel line with a y-intercept of 4?
  • a parallel line with an x-intercept of −3?
  • a line that is perpendicular to y = 2x?
  • a line that has a negative slope and the same y-intercept as y = 2x?
• Notice that a translation of 4 units down of y = 2x has the same effect as translating it 2 units right. How are the equations that represent the line after these transformations similar, and how are they different?
• What do you notice about the slope of a graph that is rotated 90° about the origin compared to the slope of the original graph?
• What do you notice about the slope of a graph that is rotated 180° about the origin compared to the slope of the original graph?

• determining the equation of lines from representations:
  • from a graph:

• the slope (rate of change) is $$\frac{4}{8}=\frac{1}{2}$$
• the line intersects the y-axis at 5, so the y-intercept (or initial value) is 5
• an equation for this relationship is $$y=\frac{1}{2} x+5$$
• from a table of values:

• the rate of change (slope) is $$\frac{2}{4}=\frac{1}{2}$$ since the dependent values increase by 2 as the independent values increase by 4
• by using the relationship and working backwards through the table, students can determine the initial value (or y-intercept) of 5
• an equation for this relationship is $$y=\frac{1}{2} x+5$$
• from a concrete representation:

• there are 2 more tiles every time the position number increases by 4, so the rate of change (slope) is $$\frac{2}{4}=\frac{1}{2}$$
• working backwards and using the relationship between the position number and the tiles, it can be determined that position 0 has 5 tiles
• an equation for this relationship is $$y=\frac{1}{2} x+5$$

Teachers can:

• pose problems that support students in moving from relations in the first quadrant to those that involve other quadrants;
• support students in making connections between the rate of change and the slope of the graph and between the initial value and the y-intercept by using concrete, numerical, and graphical representations;
• pose problems, including those involving relevant real-life contexts, that require students to predict values beyond the given information, and within the given information, by using graphs, tables, concrete representations, and equations;
• facilitate a discussion about the strengths and limitations of each representation (e.g., if the rate of change is fractional, it may not be possible to build every position of the concrete representation; if the values are very large, it may be hard to recognize them on a graph);
• encourage students to represent a relationship in a different way in order to work with a representation that they may be more comfortable with;
• support students in developing strategies for determining the rate of change from various representations and for various situations (e.g., fractional rates of change, negative rates of change, different scales on the x- and y-axes).

• How does the number of tiles added each time in a concrete pattern relate to the slope of the graph representing this relation?
• How can you determine the slope of a line from a graph? From a table of values?
• What information do you need to determine the equation of a line?
• Why can you use any two points on a line to determine the slope?
• What happens to this linear pattern if we move backwards from term 2 to term 1 to term 0? What might term −1 look like? How would you represent it on a graph?