standard_description: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
problem_activity: Activity
This activity is intended to help students interpret features of graphs in terms of the proportional relationships they represent. This task highlights two fundamental ideas:
The steeper the graph, the greater the constant of proportionality.
If the graph represents distance of an object vs. time, the constant of proportionality is the speed of the object.
Students can reason abstractly (MP2) by picking an arbitrary time and comparing the corresponding distances, or picking an arbitrary distance and comparing the corresponding times. Ideally, both of these ways of reasoning are shared in a whole-class discussion of the task because they will be needed in future work.
While students work, monitor for these approaches:
At the same time, Asteroid x has traveled a greater distance (as in the graph with the vertical dashed line). We can’t tell exactly where 1 unit of time is on the graph, but wherever it is, we can tell that Asteroid x has covered a greater distance.
[IMG]
Asteroid x takes less time than Perseid 245 to cover the same distance. See the graph with the horizontal dashed line.
When distance is proportional to time and distance is graphed against time, the constant of proportionality represents the magnitude of the speed (unit of distance traveled per unit of time). It was shown in the previous activity that a steeper line has a greater constant of proportionality. Therefore, a line steeper than Perseid 245’s line represents a greater speed.
Digital
Launch
Keep students in the same groups of 2–3.
If using the digital activity, have students explore the applet to develop their reasoning around the following two concepts:
The steeper the graph, the greater the constant of proportionality.
If the graph represents distance vs. time, the constant of proportionality is the speed.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example: “First, I _____ because _____. Then, I….,” “I noticed _____ so I….,” and “I tried _____ and what happened was.…”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Meteoroid Perseid 245 and Asteroid x travel through the solar system.  Explore the applet to learn about the distance they had each traveled after a given time.
Is Asteroid x traveling faster or slower than Perseid 245? Explain how you know.
The graph shows the price of different lengths of two types of rope.
[IMG]
If you buy $1.00 of each kind of rope, which one will be longer? Explain how you know.
Student Response
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Print
Launch
Keep students in the same groups of 2–3.
If using the digital activity, have students explore the applet to develop their reasoning around the following two concepts:
The steeper the graph, the greater the constant of proportionality.
If the graph represents distance vs. time, the constant of proportionality is the speed.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example: “First, I _____ because _____. Then, I….,” “I noticed _____ so I….,” and “I tried _____ and what happened was.…”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Meteoroid Perseid 245 and Asteroid x travel through the solar system. The graph shows the distance each traveled after a given point in time.
[IMG]
Is Asteroid x traveling faster or slower than Perseid 245? Explain how you know.
The graph shows the price of different lengths of two types of rope.
[IMG]
If you buy $1.00 of each kind of rope, which one will be longer? Explain how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
It is important that students not assume “steeper always means faster,” but that they understand why it is in this case by reasoning abstractly and attending to the referents for points on the graphs. If the same relationships were graphed with distance on the horizontal axis and time on the vertical axis, a steeper line would indicate a slower speed. If the same relationships were graphed on separate axes, their scales could be different. Because the graphs share the same axes, it is implicit that comparisons between them occur relative to the same units.
Try to find students who took each approach, and invite each to share reasoning with the class. Important points to highlight are:
A steeper graph has a larger constant of proportionality and a larger constant of proportionality will have a steeper graph. (“When you look at two graphs of a proportional relationship, how can you tell which one has a greater constant of proportionality?”)
In a distance vs. time graph, a steeper graph indicates a greater speed. (“When you look at two distance vs. time graphs, how can you tell which represents an object traveling at a greater speed?”)
The two graphs in question 1 can be considered from two perspectives:
Same point in time, noting that one object has covered a greater distance. (“Which student focused on the two objects at the same moment in time? How did they know that Asteroid x was traveling faster?”)
Same distance, noting that one object took less time. (“The other student did not focus on the same moment in time. What did they focus on in her explanation?”)
Speaking: MLR8 Discussion Supports.
As students describe their explanations for which asteroid moved faster or slower, revoice student ideas to demonstrate mathematical language use. Press for details in students’ explanations by requesting that students challenge an idea, elaborate on an idea, or give an example. Show central concepts multi-modally by using different types of sensory inputs: acting out scenarios or inviting students to do so, using gestures, and talking about the context of moving objects. This will help students to produce and make sense of of the language needed to communicate their own ideas.
Design Principle(s): Support sense-making, Optimize output (for explanation)


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