standard_description: Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.
problem_activity: Activity
Note: if it is possible in your local environment, we recommend using the digital version of this activity.
This activity is intended to help students identify correspondences between parts of a table, graph, and an equation of a line through the origin in the first quadrant. Students are guided to notice:
Any pair of positive values
\((a, b)\)
determine a proportional relationship.
Given a point
\((a,b)\)
other than the origin on the graph of a line through the origin, the constant of proportionality is always
\(\frac{b}{a}\)
.
In an equation
\(y=\frac{b}{a} x\)
that represents the relationship, the constant of proportionality appears as the coefficient of
\(x\)
.
The constant of proportionality is the
\(y\)
-coordinate when
\(x\)
is 1, that is,
\((1,\frac{b}{a})\)
is a point on the graph.
In the print version, students plot one point and draw a ray that starts at the origin and passes through their point. They also create a table and an equation to represent the relationship. They respond to a series of questions about their representations. Then, they compare their representations of their relationship with the representations of other relationships created by members of their group.
In the digital version (recommended), students interact with a dynamic sketch while recording observations and responding to prompts. In the sketch, students are able to manipulate the graph of a proportional relationship, while changes to an associated table and equation automatically update.
Digital
Launch
For the print version: Arrange students in groups of 3. Assign each student in each group a letter: A, B, or C. Provide access to rulers.
For the digital version: Arrange students in groups of 2–3 and have them complete the task.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Speaking, Writing: MLR2 Collect and Display.
While pairs or groups are working, circulate and listen to student talk about the connections they see between the tables, characteristics of the graphs, and the equations. Write down common or important phrases you hear students say about the connections onto a visual display. Throughout the remainder of the lesson, continue to update collected student language and remind students to borrow language from the display as needed. This will help students read and use mathematical language during paired and whole-class discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
Explore the graph. Start by dragging the gray bar on the left across the screen until you can see both the table and the graph. Notice the values in the table and the coordinates of the labeled point. Grab the point and move it around.
What stays the same and what changes in the table? in the equation? on the graph?
Choose one row in the table and write it here. To what does this row correspond on the graph?
| \(x\) | \(y\) |
|-------|-------|
|       |       |
Grab and drag the point until you see the equation
\(y=\frac 32 x\)
.
Do not move the point. Choose three rows from the table, other than the origin. Record
\(x\)
and
\(y\)
, and compute
\(\frac yx\)
.
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|-------|-------|-----------------|
|       |       |                 |
|       |       |                 |
|       |       |                 |
What do you notice? What does this have to do with the equation of the line?
Do not move the point. Check the box to view the coordinates
\((1, ?)\)
. What are the coordinates of this point? What does this correspond to in the table? What does this correspond to in the equation?
Drag the point to a different location. Record the equation of the line, the coordinates of three points, and the value of
\(\frac yx\)
.
Equation of the line: _______________________________
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|-------|-------|-----------------|
|       |       |                 |
|       |       |                 |
|       |       |                 |
Based on your observations, summarize any connections you see between the table, characteristics of the graph, and the equation.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
For the print version: Arrange students in groups of 3. Assign each student in each group a letter: A, B, or C. Provide access to rulers.
For the digital version: Arrange students in groups of 2–3 and have them complete the task.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Speaking, Writing: MLR2 Collect and Display.
While pairs or groups are working, circulate and listen to student talk about the connections they see between the tables, characteristics of the graphs, and the equations. Write down common or important phrases you hear students say about the connections onto a visual display. Throughout the remainder of the lesson, continue to update collected student language and remind students to borrow language from the display as needed. This will help students read and use mathematical language during paired and whole-class discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
Your teacher will assign you
one
of these three points:
\(A=(10,4)\)
,
\(B=(4,5)\)
,
\(C=(8,5)\)
.
[IMG]
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|-------|-------|-----------------|
|   0   |       |       NA        |
|   1   |       |                 |
|   2   |       |                 |
|   3   |       |                 |
|   4   |       |                 |
|   5   |       |                 |
|   6   |       |                 |
|   7   |       |                 |
|   8   |       |                 |
|   9   |       |                 |
|  10   |       |                 |
On the graph, plot and label
only
your assigned point.
Use a ruler to line up your point with the origin,
\((0,0)\)
. Draw a line that starts at the origin, goes through your point, and continues to the edge of the graph.
Complete the table with the coordinates of points on your graph. Use a fraction to represent any value that is not a whole number.
Write an equation that represents the relationship between
\(x\)
and
\(y\)
defined by your point.
Compare your graph and table with the rest of your group. What is the same and what is different about:
your tables?
your equations?
your graphs?
What is the
\(y\)
-coordinate of your graph when the
\(x\)
-coordinate is 1? Plot and label this point on your graph. Where do you see this value in the table? Where do you see this value in your equation?
Describe any connections you see between the table, characteristics of the graph, and the equation.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The graph of an equation of the form
\(y = kx\)
, where
\(k\)
is a positive number, is a line through
\((0,0)\)
and the point
\((1,k)\)
.
Name at least one line through
\((0,0)\)
that cannot be represented by an equation like this.
If you could draw the graphs of
all
of the equations of this form in the same coordinate plane, what would it look like?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
When all students have completed the activity, ask them to share their responses with a partner or small group. After a minute, go around the room asking each group to share one thing from the last question, and display these for all to see. When each group has shared, ask if there were any important observations that were missed.
Ensure that all the important connections are highlighted:
A graph of a line through the origin and passing through the first quadrant represents a proportional relationship.
The value of
\(\frac ba\)
computed from any point
\((a,b)\)
on that line (other than the origin) is the constant of proportionality.
An equation of the relationship is given by
\(y = kx\)
where
\(k\)
is
\(\frac ba\)
for any point
\((a,b)\)
on the graph other than the origin.


aligns: no