Lesson 3: Algebra Walk—Identifying Linear and Nonlinear Tables and Graphs
Grade-Level Expectations Addressed:
A1B6 Represent and describe patterns with table, graphs, pictures, symbolic rules or words
A1D6 Identify functions as linear or nonlinear from a table or graph
Essential Question to Guide the Unit and Focus Teaching and Learning:
How do we determine whether a table or graph is linear or nonlinear?
How do we identify whether the rate of change is linear or nonlinear?
Specific Classroom Arrangement/Preparations:
Create a large 25 ´ 25 grid on a shower curtain with permanent marker or masking tape. Mark x- and y-axes at equal intervals. Be sure to leave enough space between the intervals for students to stand. Place the grid on the floor.
Materials:
• One set of nine blue index cards with one of the following on each card:
(0, x+1); (1, x+1); (2, x+1); (3, x+1); (4, x+1); (5, x+1); (6, x+1); (7, x+1); (8, x+1) (see Materials at the end of the lesson)
• One set of nine green index cards with one of the following on each card:
(0, 2x); (1, 2x); (2, 2x); (3, 2x); (4, 2x); (5, 2x); (6, 2x); (7, 2x); (8, 2x) (see Materials at the end of the lesson)
• One set of nine white index cards with one of the following on each card:
(0, x2); (1, x2); (2, x2); (3, x2); (4, x2); (5, x2); (6, x2); (7, x2); (8, x2) (see Materials at the end of the lesson)
• Three colored pencils—blue, green, black—for each student
• A hard writing surface for each student
• Human Graph and Table transparency with answer key (see Materials at the end of the lesson)
• One Human Graph and Table Recording Sheet for each student with answer key (see Materials at the end of the lesson)
• One Exit Assessment for each student with answer key (see Materials at the end of the lesson)
Step-by-Step Process:
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LEARNING ACTIVITIES |
QUESTIONS FOR STUDENTS |
TEACHER SUPPORT |
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“Today we will be making human graphs. We will be going outside for the first part of the lesson. Before we go out, let’s do some reviewing.” |
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Prior to the lesson, the teacher should have constructed a 25 ´ 25 coordinate grid on a shower curtain. |
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Distribute the blue, green, and white index cards (one card with one coordinate to one student) and one Human Graph and Table Recording Sheet to each student. (Each student who receives an index card will have a separate value of x.)
Remind students to take something hard as a writing surface. Assign one student to complete the Human Table transparency and one student to complete the Human Graph transparency. (They take turns entering their data on the transparency.)
Also distribute three pencils (in three different colors) to each student. |
“Who can tell me what a coordinate grid is?” (It is a 2-dimensional system in which a location is described by its distance from two perpendicular lines called axes.)
“How do we write the locations for points on a coordinate grid?” (We write them as ordered pairs. The first number x, tells the location on the x-axis and the second number y, tells the location on the y-axis.)
“How do we write the ordered pairs?” (x, y) |
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LEARNING ACTIVITIES |
QUESTIONS FOR STUDENTS |
TEACHER SUPPORT |
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“Some of you have a blue card with an ordered pair written on it that looks like this:
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“What is the x value?” (0)
“How would you find the y value?” (Take 0 and add 1 to it for a value of 1.)
“So what would the ordered pair be?” |
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Ask nine students with the blue cards to stand on the shower curtain on the x-axis value of their card. Other students should be watching where the students stand.
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LEARNING ACTIVITIES |
QUESTIONS FOR STUDENTS |
TEACHER SUPPORT |
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Allow time for the nine students to determine their y value.
“Now when I say go, walk to the value of your |
“How do you determine the y value?” (Add 1 to x.)
“How should you walk in relation to the |
Human Table 1 Coordinates (as per blue index cards)
Students will line up on the x-axis value on their card. The teacher should be monitoring to see who has difficulty with the x value. |
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LEARNING ACTIVITIES |
QUESTIONS FOR STUDENTS |
TEACHER SUPPORT |
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After all nine students have walked to their y value, have them explain how they know they are in the correct position on the coordinate grid. Other students should record in Human Table 1 the x and y values for each person standing on the shower-curtain grid and plot the points in blue pencil on the Human Graph in the Human Graph and Table Recording Sheet. |
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“To move from the (0, 1) to (1, 2) how many units up on the grid must you move?” (1)
“How many units over do you have to move?” (1)
“To move from (1, 2) to (2, 3), how many units up must you move?” (1)
“How many units over do you have to move?” (1)
“To move from (3, 4) to (4, 5), how many units up on the grid must you move?” (1)
“How many units over do you move?” (1) |
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LEARNING ACTIVITIES |
QUESTIONS FOR STUDENTS |
TEACHER SUPPORT |
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“What pattern do you recognize in your movements?” (You always move one unit up and one unit over to go from one point that we have plotted on the grid to the next.) |
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Since the moves up and over on the grid to get from one point to the next are always the same, we refer to this as a constant rate of change. |
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“What do you notice about where you are standing on the grid in relationship to the others?” (We are standing in a line.) |
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“The ordered pairs you’ve graphed form a straight line. We refer to your graph as a linear function because there is a constant rate of change from each x to the next x and a proportional constant change in the value from one y to the next.” |
“Can anyone describe a constant rate of change for x and y?” (As x increases by 1, y increases by 1.) |
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Make sure that all the students who were observing have the x and x + 1 values recorded in their tables before the next card group lines up on the x value. |
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LEARNING ACTIVITIES |
QUESTIONS FOR STUDENTS |
TEACHER SUPPORT |
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Have students move off the grid. Now, ask those that have the green cards to position themselves on their x value.
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Human Table 2 Coordinates (as per green index cards)
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“How do you determine the y value?” (Multiply x times 2 or double x.) |
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Allow time for students to determine their y value. |
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“Now when I say go, walk to the value of your y-axis.” |
“Since you do not want to change the value of your x-axis, how should you walk in relationship to the y-axis?” (You will walk parallel to the y-axis.) |
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LEARNING ACTIVITIES |
QUESTIONS FOR STUDENTS |
TEACHER SUPPORT |
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Ask the students who are observing to record in Human Table 2 the x and y values of each person standing on the grid and to plot the points with a green pencil on the Human Graph in the Human Graph and Table Recording Sheet.
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“How many units must you move up and over on the grid to get from the person standing at (0, 0) to the person standing at (1, 2)?” (Move up two units and over 1.)
“If you begin at (2, 4) and move two units up and one over on the grid, where will you be standing?” (3, 6)
“Is there a constant rate of change from one point on the grid to the next in terms of moves up and over?” (Yes.)
“What is the constant rate of change?” (Move up two units and over one.)
“What do you notice about where you are standing on the grid in relationship to the others?” (We are standing in a line.) |
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LEARNING ACTIVITIES |
QUESTIONS FOR STUDENTS |
TEACHER SUPPORT |
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“Why is this another example of a linear function?” (because the points graphed form a line, and there is a constant change from each x to the next x and a constant rate of change from each y to the next y)
“How would you describe the constant rate of change for x and the constant rate of change for y?” (As x increases by 1, y increases by 2.) |
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Have the group with the white index cards stand on the grid on their x value.
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Make sure that all the students who were observing have the x and 2x values recorded in their tables before the next group lines up on their x value.
Human Table 3 Coordinates (as per white index cards)
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LEARNING ACTIVITIES |
QUESTIONS FOR STUDENTS |
TEACHER SUPPORT |
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“How do you determine the y value?” (Multiply x times itself or square x.) |
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Allow time for students to determine their y value. |
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“Now when I say go, walk to the value of your y-axis. Since you do not want to change the value of your x-axis, you must walk parallel to the y-axis.” |
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Ask the students who are observing to record in Human Table 2 the x and y values of each person standing on the grid and to plot the points with a black pencil on the Human Graph in the Human Graph and Table Recording Sheet. |
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“To move from (0, 0) to (1,1) on the grid, how many units up and over must you move?” (one unit up and one unit over)
“To move from (1, 1) to (2, 4) on the grid, how many units up and over must you move?” (three units up and one unit over) |
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LEARNING ACTIVITIES |
QUESTIONS FOR STUDENTS |
TEACHER SUPPORT |
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“To move from (2, 4) to (3, 9) on the grid, how many units up and over must you move?” (five units up and one unit over)
“Do you always move the same number of units up and over to move from one point on the grid to the next?” (No.) |
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“We refer to this as a varying rate of change.” |
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“What do you notice about where you are standing on the grid in relationship to the others?” (We are not in a line.)
“If we call a graph that forms a line linear, what name do you think we would call this graph?” (nonlinear) |
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“Nonlinear graphs represent varying rather than constant rates of change. And since the change in this graph varies from one set of coordinates to the next, and the points do not make a line, it is a nonlinear graph.” |
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LEARNING ACTIVITIES |
QUESTIONS FOR STUDENTS |
TEACHER SUPPORT |
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Review the graphs (for the blue, green, and white index cards) on the Recording Sheets. Take some time to discuss the constant changes in the x and y values in the three tables. Also discuss that while there is a constant change in the y value on the nonlinear graph, the change in the x value is not constant and that to be a linear function, there must be a constant change in both. |
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Distribute the Exit Assessment to students to complete before leaving class. Answer key provided. |
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