Lesson Plan 1: Left Hand, Right Hand - Solving Systems of Equations
Teachers will need the following:
Students will need the following:
- Notebook or journal
- Graphing calculators
1. Introduce the lesson by asking students to think about linear situations the class has already studied. Ask several students to each share a particular situation. (Possible responses include: the time in seconds between seeing lightning and hearing the thunder as a function of one's distance from the lightning; the cost of renting a car as a function of the miles one drives.)
2. Ask students to work in groups and brainstorm two different linear situations that they could plot on the same graph so that the two situations could be compared.
3. Solicit suggestions from the groups. Clarify and make sure everybody understands all of the scenarios and why it would be reasonable to want to compare and/or contrast them. If some of the scenarios aren't feasible or reasonable, this should be part of the discussion as well. (Possible responses include: comparing costs of driving two different cars as a function of the distance you drive; comparing costs of renting a car from two different rental companies as a function of the miles you drive.)
Hand out the Right Hand/Left Hand activity sheets and ask students to turn to the full-page grid. Students should hold the paper in such a way that there are more squares running horizontally than vertically, and they should hold their pencil in their right hand. When told to begin, they should start writing M, D, M, D - one letter per square - across the page for three seconds. (Teachers can use any two-letter combination - in the video Jenny used the letters for the abbreviation of Maryland.) When time is up, students should put down their pencils.
Students should record the results of the first round in the table provided on the activity sheet (e.g., time: 3 seconds, squares: 7). They will repeat the process right handed four more times with the teacher timing them for different intervals (ranging from three to 10 seconds) each time. Students should record the resulting data in their table after each effort. (Note: The data should approximate a linear function, but there may be some variability caused by inconsistencies due to human error.)
3. Repeat the experiment with the left hand, recording the data each time.
4. Ask the students to draw a scatterplot for the two sets of data on the same graph, using the grid provided on the activity sheet.
5. Discuss whether or not the point (0, 0) should be part of the graph. Students should tell you that they do think the origin should be a part of the data because in zero seconds, zero letters can be written.
6. Have the students find an equation for the line of best fit for both sets of data. Most of the students will have two lines that intersect at the origin, but have different slopes.
7. Have students gather in groups and have each group discuss the similarities and differences of their graphs.
8. After the group discussions, ask students what an ambidextrous person's graph would look like. Have a student to come to the front of the class and demonstrate what he or she thinks the two lines would look like.
9. Ideally, the two lines should lie on top of each other because an ambidextrous person would fill the same number of squares with each hand over the same period of time.
10. At this point, the class has looked at individual student graphs and at a graph for an ambidextrous person. Students should realize that most of the graphs had one point of intersection - the origin - and they should understand that this intersection (0, 0) is the one point where the two sets of data are the same. They should also realize that an ambidextrous person's two lines would lie on top of one another and would thus share an infinite number of intersections (or solutions).
11. Hold a discussion in which the class investigates the graphs for several different scenarios. Using the three transparencies, show a graph on the overhead that has two intersecting lines, with two different slopes and two different y-intercepts. Ask students to think about the graph as it relates to solving systems of equations.
12. Show a graph of two parallel lines and ask students to draw conclusions. Students might comment, for instance, that this scenario will not work for the experiment they just performed. They should note that because these two lines never intersect, there is not a common point or solution.
13. Show a graph with two lines on top of each other. Ask students to draw some conclusions about the graph.
14. Ask students to think about all the possibilities when two linear equations are graphed on the same coordinate plane. At the end of the discussion, they should conclude that there are three possible scenarios:
15. Discuss the meaning of a "solution" for each of the three scenarios. At the end of the discussion, they should be able to understand that the two intersecting lines have one solution, the two parallel lines have no solution, and the two coincident lines have an infinite number of solutions.
- two intersecting lines
- two parallel lines
- two lines on top of each other (coincident lines).
16. Present the "Ace vs. Better Car Rentals" graph on the overhead. Ask the students to work in their groups to come up with a story, to discuss what is happening in this scenario, and to draw conclusions. The students should notice that the two rental companies charge different initial fees, making the y-intercepts different. They also charge different rates per mile, which give the lines different slopes. If a person drives 300 miles in one day, the cost is the same for both cars. If the person drives less than 300 miles, then Ace Car Rental is cheaper; more than 300 miles, Better Car Rental is cheaper.
17. Ask the groups to share their discussions.
Ask the students to discuss all the different concepts that arose in the lesson, and have each group write a summary of what they learned.