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Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies and record and display their responses for all to see.
Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Mentally find values for and that make each equation true.
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
The purpose of this activity is for students to develop a quick method or formula to calculate the slope of a line based on the coordinates of two points. After students compute the slopes of three specific lines, they generalize the procedure by observing that they can calculate the vertical and horizontal side lengths of the slope triangle without a grid (eventually without even drawing the slope triangle) and that the slope is the quotient of these side lengths (MP8). It is not critical or even recommended to use the traditional formula with subscripts. It is more important that students know a technique or way of thinking about slope that works for them than it is that they memorize a particular way to express a formula.
In this partner activity, students take turns sharing their initial ideas and first drafts. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
Arrange students in groups of 2. Give them 5–6 minutes to calculate the slopes of the three lines in the first problem and write a draft for the second problem, followed by a whole-class discussion.
| vertical change | horizontal change | slope | |
|---|---|---|---|
| A | |||
| B | |||
| C |
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their procedure for finding the slope between any two points on a line from the second question. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. Have students take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.
Then close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.
If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.
After Stronger and Clearer Each Time, display the coordinates and for all to see and ask students to predict without calculating, whether the slope of the line that goes through these 2 points will be positive or negative.
Then display the graph below and plot both points. Draw the line that passes through the points and ask students again to predict whether the slope of the line is positive or negative. Invite students who predicted correctly before seeing the graph to share their strategies and record them for all to see.
Draw a slope triangle connecting the points and and use it to calculate the slope of the line (). Emphasize that since the line is going downhill from left to right, the slope must be negative. Discuss whether or not the procedure they described for calculating slope will work when the slope of the line is negative. (Yes.) Demonstrate how the slope can be calculated using just the coordinates. For example, the vertical change is and the horizontal change is , making the slope .
This activity gives students an opportunity to interpret the graph of a line in context, including the meaning of a negative slope and the meaning of the horizontal and vertical intercepts. Students also calculate the slope of a line with a negative slope from a graph.
Arrange students in groups of 2. Give students 3–4 minutes of quiet work time, followed by a partner then whole-class discussion.
Elena borrowed some money from her brother. She pays him back by giving him the same amount every week. The graph shows how much she owes after each week.
graph of a line on grid, origin O. horizontal axis, time in weeks, scale 0 to 7, by 1's. vertical axis, scale 0 to 20, by 5's. line crosses y axis at 0 comma 18 and crosses the x axis at 6 comma 0.
Answer and explain your reasoning for each question.
The goal of this discussion is to reinforce the ideas of what the slope and intercepts mean in context, with an emphasis on a context with a negative slope. Display the graph from the task for all to see.
Invite students to share their answers and reasoning for each question, indicating how the graph can be used to find the answers. For example, draw in a slope triangle for the first question, and emphasize how the slope is -3 because the number of dollars she owes decreases over time. Label the vertical intercept for the amount Elena borrowed and the horizontal intercept for the time it takes her to pay back the loan.
This activity gives students an opportunity to determine and request the information needed to recreate a design made out of different lines. Thanks to Henri Picciotto for permission to use these designs.
The goal is for students to recognize the information that determines the location of a line in the coordinate plane. Acknowledge students who use the following strategies, but keep the discussion focused on describing each line using slope and intercepts or coordinates of points on the line.
Describing one line as a vertical translation up or down of another line
Describing a line as parallel to another line and containing a particular point
Giving an equation to describe the line
Students are not expected to communicate by saying the equations of the lines, though there is nothing stopping them from doing so.
Tell students they will describe some lines to a partner to try and get them to recreate a design. Consider selecting a student as a partner to demonstrate the protocol to the class before distributing the designs and blank graphs.
Arrange students in groups of 2, in such a way that the partner drawing the design cannot peek at the design anywhere else in the room. Provide access to straightedges.
In each group, give a design card to one student and a blank coordinate plane to the other student. Once the first design has been successfully created, provide cards for the second design and instruct students to switch roles.
Your teacher will give you either a design or a blank graph. Do not show your card to your partner.
If your teacher gives you the design:
If your teacher gives you the blank graph:
When finished, place the drawing next to the card with the design so that you and your partner can both see them. How is the drawing the same as the design? How is it different? Discuss any miscommunication that might have caused the drawing to look different from the design.
Pause here so your teacher can review your work. When your teacher gives you a new set of cards, switch roles for the second problem.
Here are some questions for discussion:
"What details were important to pay attention to?" (the slope of the line, whether the slope was positive or negative, the vertical and horizontal intercepts)
"How did you use coordinates to help communicate where the line was?" (Coordinates could tell a location that the line had to go through.)
"How did you use slope to communicate how to draw the line?" (Slope told my partner how steep to draw the line and if it was going uphill or downhill.)
"Were there any cases where your partner did not give enough information to know where to draw the line? What more information did you need?" (Answers vary.)
Then display this blank coordinate plane, or one similar, for all to see.
Plot and label the point on the coordinate axes. Ask students to explain how to draw a line with a slope of -2 that goes through the point on the display. Demonstrate how to use the plotted point as one vertex of a slope triangle with a vertical length of 2 units and a horizontal length of 1 unit, and then draw in the third side of the triangle using a straightedge or ruler. Discuss:
“Can the point and slope used describe any other line on the coordinate plane?” (No)
“Can this same line be described in a different way?” (Yes)
“What are some other ways?” (a line with slope that goes through the point or any other point on the line, a line with slope with a vertical intercept of 9, the line or equivalent)
The goal of this discussion is for students to see how the procedure they wrote for finding the slope between any two points on a line can be written using a general formula.
Begin by inviting 1–2 students to share the second draft of their procedure. If necessary, emphasize the importance of subtracting the -coordinates for the two points in the same order as the -coordinates.
Then display this image for all to see:
Ask students to use the procedures they wrote for finding slope to find the slope of this line, using the variables , , , and . Have students share their thinking with a partner before continuing with the whole-class discussion. Record all correct responses next to the graph for all to see, including any equivalent expressions such as and .
One way to calculate the slope of a line is by drawing a slope triangle. For example, using this slope triangle, the slope of the line is , or . The slope is negative because the line is decreasing from left to right.
Another way to calculate the slope of this line uses just the points and . The slope is the vertical change divided by the horizontal change, or the change in the -values divided by the change in the -values. Between points and , the -value change is and the -value change is . This means the slope is , or , which is the same value as the slope calculated using a slope triangle.
Notice that in each of the calculations, the value from point was subtracted from the value from point . If it had been done the other way around, then the -value change would have been and the -value change would have been , which still gives a slope of .
If students calculate a slope and leave it in a form such as , consider asking:
What does the slope mean in context?
How is the slope different from ?