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Arrange students in groups of 2. Give 3 minutes of quiet work time followed by time for a brief partner discussion for the last question.
Diego and Lin are drinking water.
Lin’s glass has 12 ounces and she drinks ounce per second.
Diego's glass has 20 ounces and he drinks ounce per second.
The purpose of the discussion is for students to practice describing graphs in words using correct mathematical terminology.
Select previously identified students to share their descriptions of the graphs. After each student shares, ask the class if the description is clear and to identify any vocabulary they heard that made the description precise or any vocabulary that is unclear. If any vocabulary needs to be reinforced for student understanding, this is a good time to discuss these words.
Students explore a system of equations with no solutions, in the familiar context of cup stacking. The context reinforces a discussion about what it means for a system of equations to have no solutions, both in terms of a graph and in terms of the equations (MP2). Over the next few lessons, the concept of one solution, no solutions, and infinitely many solutions will be abstracted to problems without context. In those situations, it may be useful to refer back to the context in this activity and others as a way to guide students toward abstraction.
In the digital version of the activity, students use an applet to graph the heights of two stacks of cups. The applet allows students to precisely graph the lines to see that they never intersect. Use the digital version if students have difficulty graphing precisely enough to show that the lines do not intersect or if appropriate tools for graphing are not readily available.
Give students 5–7 minutes of quiet work time followed by a whole-class discussion.
A stack of small cups has a height, , in centimeters, of .
A stack of large cups has a height, , in centimeters, of .
Graph the equation for each type of cup on the same set of axes. Make sure to label the axes and decide on an appropriate scale.
The key point for discussion is to connect what students observe about the graph to the idea (from earlier lessons) that equations can have no solution. Graphically, students see that the lines are parallel and are always separated by a distance that represents 3 cm. This means that a stack of n large cups will always be 3 cm taller than a stack of n small cups. Connecting this to the equations, this means that there is no value of n that is a solution to both and at the same time. By the end of the discussion, students should understand that these statements about this situation are equivalent:
A system of equations is a set of 2 or more equations, where the variables represent the same unknown values. For example, suppose that two different kinds of bamboo are planted at the same time. Plant A starts at 6 ft tall and grows at a constant rate of foot each day. Plant B starts at 3 ft tall and grows at a constant rate of foot each day. Because Plant B grows faster than Plant A, it will eventually be taller, but when?
We can write equations for Plant A and for Plant B, where represents the number of days after being planted, and represents height. We can write this system of equations.
Solving a system of equations means to find the values of and that make both equations true at the same time. One way we have seen to find the solution to a system of equations is to graph both lines and find the intersection point. The intersection point represents the pair of and values that makes both equations true.
Here is a graph for the bamboo example:
<p>Graph of two lines, origin O, with grid. Horizontal axis, time in days, scale 0 to 13, by 1’s. Vertical axis, height in feet, scale 0 to 12, by 1’s. A line, labeled Plant A, crosses the y axis at 6. A line, labeled Plant B, crosses the y axis at 3. The lines intersect at the point 12 comma 9.</p>
The solution to this system of equations is , which means that both bamboo plants will be 9 feet tall after 12 days.
We have seen systems of equations that have no solutions, one solution, and infinitely many solutions.
Provide students with access to straightedges. Keep students in groups of 2.
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem, without revealing the questions.
To further understand the situation, consider asking:
“When , where is Han? Where is Jada?” (Han is at the lake. Jada is 0.6 miles from the parking lot.)
“For times shortly after , is d decreasing or increasing for Han? Is d decreasing or increasing for Jada?” (Decreasing for Han and increasing for Jada.)
Give 2–3 minutes of quiet work time, and ask students to pause after they have completed the first problem to discuss their equation with a partner before starting to graph the equations. Give 5–7 minutes for students to complete the remaining problems with their partners, and follow that with a whole-class discussion.
There is a hiking trail near the town where Han and Jada live that starts at a parking lot and ends at a lake. Han and Jada both decide to hike from the parking lot to the lake and back, but they start their hikes at different times.
At the time that Han reaches the lake and starts to turn back, Jada is 0.6 miles away from the parking lot and hiking at a constant speed of 3.2 miles per hour toward the lake. Han’s distance, , from the parking lot can be expressed as , where represents the time in hours since he left the lake.
<p>Coordinate plane, origin O, with grid. Horizontal axis, time in hours, scale 0 to 1 point 75, by point 25’s. Vertical axis, distance from parking lot in miles, scale 0 to 5, by 1’s.</p>