The purpose of this Warm-up is to get students to think about a context that will be explored in the following activity and to reason about the speed, distance, and time that each animal is traveling in relation to one another. In the next activity, students will write equations for the bugs and graph these relationships.
By familiarizing students with a context and the mathematics that might be involved, this Warm-up prompts them to make sense of a problem before solving it. (MP1).
Student Lesson in Spanish
Launch
Arrange students in groups of 2. Display the image for all to see. Ask students to think of at least one thing that they notice and at least one thing that they wonder about. Give students 1 minute of quiet think time, and then 1 minute to discuss with their partner the things that they notice and wonder.
Activity
None
Student Task Statement
What do you notice? What do you wonder?
Four blank number lines. Each number line has a ladybug facing right and an ant facing left. Number line 1 is labeled ladybug start on the left end and 0 seconds. The ladybug is to the left of the number line and the ant is to the right of the number line. Number line 2 is labeled 2 seconds. The ladybug is at the 8th tick mark from the right. The ant is at the 16th tick mark from the left. Number line 3 is labeled 4 seconds. The ladybug is at the 16th tick mark from the right. The ant is at the 32nd tick mark from the left. Number line 4 is labeled 6 seconds. The ladybug is at the 24th tick mark. The ant is not shown in the number line.
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on the display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the idea of when the bugs pass one another and their speeds does not come up during the conversation, ask students to discuss this idea.
11.2
Activity
10 mins
Bugs Passing in the Night, Continued
Standards Alignment
Building On
Addressing
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
In this task, students find and graph a linear equation given only the graph of another equation, information about the slope, and the coordinates where the lines intersect. The purpose of this task is to check student understanding about the point of intersection in relationship to the context, while applying previously learned skills of equation writing and graphing.
Launch
Display the graph from the task statement. Tell students that this activity is about a different ant and ladybug than those in the Warm-up, and that we are going to think about their distances using a coordinate plane. Give 4–6 minutes for students to complete the problems, and follow that with a whole-class discussion.
Representation: Internalize Comprehension. Represent the same information through different modalities by using a number line and pictures similar to those in the Warm-up.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Activity
None
Student Task Statement
A different ant and ladybug are a certain distance apart, and they start walking toward each other. The graph shows the ladybug’s distance from its starting point over time and the labeled point indicates when the ant and the ladybug pass each other.
<p>Graph of a line, origin O, with grid. Horizontal axis, time in seconds, scale 0 to 5, by 1’s. Vertical axis, distance in centimeters, scale 0 to 24, by 2’s. The line passes through the origin and the point 2 point 5 comma 10.</p>
The ant is walking 2 centimeters per second.
Write an equation representing the relationship between the ant’s distance from the ladybug’s starting point and the amount of time that has passed.
If you haven’t already, draw the graph of your equation on the same coordinate plane.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to ensure that all students understand both how the labeled point in the task statement relates to the context and how to write and graph an equation from the given information.
Invite 2–3 groups with different approaches to share their solutions. Here are some questions for discussion:
“How does the ant’s rate of change show up in each method?” (It is the slope, which is the coefficient of in the equation and is related to the steepness of the line in the graph.)
“How can each method be used to find the starting distance between the 2 bugs?” (In one method, I counted back 2 cm per second to find the starting position. In the other method, I counted forward 2 cm per second to find the line, then drew the line and looked at where it crosses the vertical axis.)
If students struggled to graph the ant’s path, you may wish to conclude the discussion by asking students for different ways to add the graph of the ant’s distance onto the coordinate plane. For example, some students may say to use the equation figured out in the first problem to plot points and then draw a line through them. Other students may suggest starting from the known point, , and “working backward” to figure out that 1 second earlier at 1.5 seconds, the ant would have to be 12 centimeters away because .
11.3
Activity
15 mins
A Close Race
Standards Alignment
Building On
Addressing
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
In previous lessons, students encountered equations with a single variable that had infinitely many solutions. In this activity, students interpret a system with infinitely many solutions. A race is described using different representations (a table and a description in words). Students graph the relationships given by the descriptions and notice that the lines overlap so that both relationships are true for any pair of values along the graphed line.
This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.
Launch
Allow students 7–10 minutes of silent work time followed by a whole-class discussion.
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the table, without revealing the questions.
For the first read, read the problem aloud, and then ask, “What is this situation about?” (friends racing bikes). Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured (the distance and times for 2 points in Jada’s race).
After the third read, reveal the question: “Graph the relationship between distance and time for Jada’s bike race.” Ask, “What are some ways that we might get started on this?” Invite students to name some possible starting points, referring to quantities from the second read (plot the points).
Activity
None
Student Task Statement
Elena and Jada are racing 100 meters on their bikes. Both racers start at the same time and ride at constant speed. Here is a table that gives information about Jada’s bike race:
time from start (seconds)
distance from start (meters)
6
36
9
54
Graph the relationship between distance and time for Jada’s bike race. Make sure to label and scale the axes appropriately.
Elena travels the entire race at a steady 6 meters per second. On the same set of axes, graph the relationship between distance and time for Elena’s bike race.
Who won the race?
Student Response
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Building on Student Thinking
Activity Synthesis
The key point for discussion is to connect what students observed about the graph that they made to the concept of “infinitely many solutions” encountered in earlier lessons. Graphically, students see that there are situations where two lines align “on top of each other.” We can interpret each point on the line as representing a solution to both Elena’s and Jada’s equations. If is the time from start and is the distance from the start, the equation for Elena is . The equation for Jada is also . Every solution to Elena’s equation is also a solution to Jada’s equation, and every solution to Jada’s equation is also a solution to Elena's equation. In this way, there are infinitely many points that are solutions to both equations at the same time.
Ensure that students clearly understand that just because there are infinitely many points that are solutions, it does not mean that any pair of values will solve both Elena’s and Jada’s equations. In this example, the pair of values must still be related by the equation . So, pairs of values like , and are all solutions, but is not.
Lesson Synthesis
Display a set of axes for all to see. Ask each question one at a time, allowing students time to work through each problem. As students share their responses, add graphs of the lines described to the axes.
“A line goes through the point and has a slope of 1.5. What is an equation for this line?” ()
“A second line goes through the point and has a -intercept of . What is an equation for this line?” ()
“What does the point represent for these lines?” (The pair of values that is true in both situations.)
"A third line goes through this same point. How would that show up in a table representing the relationship for the third line?" (The number 2 would be in the column right next to the number 5 in the column.)
Student Lesson Summary
The solutions to an equation correspond to points on its graph. For example, if Car A is traveling 75 miles per hour and passes a rest area when , then the distance in miles it has traveled from the rest area after hours is
The point is on the graph of this equation because it makes the equation true (). This means that 2 hours after passing the rest area, the car has traveled 150 miles.
If you have 2 equations, you can ask whether there is an ordered pair that is a solution to both equations simultaneously. For example, if Car B is traveling toward the rest area, and its distance from the rest area is
We can ask if there is ever a time when the distance of Car A from the rest area is the same as the distance of Car B from the rest area. If the answer is yes, then the solution will correspond to a point that is on both lines.
Graph of 2 lines, origin O, with grid. Horizontal axis, time in hours, scale 0 to point 22, by point 0 2’s. Vertical axis, distance in miles, scale 0 to 14, by 2’s. One line passes through the origin and the point 0 point 1 comma 7 point 5. Another line crosses the y axis at 14 and passes through the point 0 point 1 comma 7 point 5
Looking at the coordinates of the intersection point, we see that Car A and Car B will both be 7.5 miles from the rest area after 0.1 hours (which is 6 minutes).
Now suppose another car, Car C, also passes the rest stop at time and travels in the same direction as Car A, also going 75 miles per hour. It's equation is also . Any solution to the equation for Car A is also a solution for Car C, and any solution to the equation for Car C is also a solution for Car A. The line for Car C is on top of the line for Car A. In this case, every point on the graphed line is a solution to both equations, so there are infinitely many solutions to the question, “When are Car A and Car C the same distance from the rest stop?” This means that Car A and Car C are side by side for their whole journey.
When we have two linear equations that are equivalent to each other, like and , we get 2 lines that are right on top of each other. Any solution to one equation is also a solution to the other, so these 2 lines intersect at infinitely many points.
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