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Overview:
In this lesson, students compare the speed at which they write with their left hand to the speed at which they write with their right hand. The experiment serves as a vehicle to help students develop a conceptual understanding of the three different types of possible solutions to a system of two equations with two unknowns. This lesson plan is based on the activity Jenny Novak uses in the video for Part I of this workshop.
Time Allotment:
Two 50-minute periods
Subject Matter:
Linear equations
Systems of equations
Learning Objectives:
Students will be able to:
Standards:
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000:
NCTM Algebra Standard for Grades 6-8
http://standards.nctm.org/document/chapter6/alg.htm
NCTM Algebra Standard for Grades 9-12
http://standards.nctm.org/document/chapter7/alg.htm
Supplies:
Teachers will need the following:
Students will need the following:
Steps
Introductory Activity:
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.)
Learning Activities:
1. 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.
2. 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.
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.
Culminating Activity/Assessment:
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.
The questions below dealing with systems of linear equations have been selected from various state and national assessments. Although the lesson above may not fully equip students with the ability to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions.
7x + 3y = -8
-4x – y = 6What is the solution to the system of equations shown above?
A. (-2, -2)
B. (-2, 2) (correct answer)
C. (2, -2)
D. (2, 2)
The length of a rectangle is equal to triple the width. Which system of equations can be used to find the dimensions of the rectangle if the perimeter is 85 centimeters?
- l = w + 3
2(l + w) = 85- l = 3w
2l + 6w = 85- l = 3w
2(l + w) = 85 (correct answer)- l = w + 3
2l + 6w = 85
Marcos had 15 coins in nickels and quarters. He had 3 more quarters than nickels. He wrote a system of equations to represent this situation, letting x represent the number of nickels and y represent the number of quarters. Then he solved the system by graphing. How many coins of each type did Marcos have?
- (6, 9) (correct answer)
- (5, 10)
- (9, 6)
- (10, 5)
Industrial Electrical Use. A utility company offers electricity to industrial users at a rate of 8 cents per kilowatt-hour. The company also offers a fixed annual rate of $1,200,000 for unlimited use of electricity. Graph each of these two rates as a line on the grid in your answer booklet. Explain why a large industrial user of electricity would choose to pay the fixed annual rate. Use the information in your graph to support your answer.
Solution: If a user needs less than 15,000,000 kilowatt-hours of electricity in a year then the 8 cents per kilowatt-hour rate would be cheaper than the fixed rate. But, if they required more than 15,000,000 kilowatt-hours, then the fixed rate would cost less money. If the user requires exactly 15,000,000 kilowatt-hours, then the cost of the two plans would be the same.
Teacher Commentary: Systems of Linear Equations
This is the paper of an 11th grade student with a fairly low math skills level. After class discussion and examination of his paper, I observed that he was able to demonstrate understanding of the major concepts discussed in class today. He was able to make the connection between the slope of his line of best fit and the speed of his writing. He also explained that “when the time is zero, [that] the letters will be zero,” and was able to see that this should hold true for both the right hand and left hand data.
During class discussion, he was allowed the time to think through and adjust his answer as he was sharing with the class. His group’s conclusion paragraph shows that they were able to understand the three potential scenarios for a linear system: intersecting lines, parallel lines, or two lines sharing the same points. Overall, his work shows an understanding of the lesson objectives. I did notice a fairly common mistake on his paper. For question 3b, he was asked to find an equation for the line of best fit and he forgot to write y = for each equation. This will be something I will need to continue to emphasize.
After reading the students’ responses, I feel that this introductory lesson to linear systems went well. Students have been able to identify different scenarios for linear systems and have been able to make a real-world connection. As we continue to explore the systems unit, the students will have a rationale and purpose for learning how to solve a linear system. I plan to continue to discuss applications of linear systems for the second day of the unit. By having students work in groups and investigate and graph a system, they will be able to make important observations and comparisons of two plans. Subsequent lessons in the unit will lead students to learn the algebraic techniques for solving systems, while continuously stressing the connections to the real world.
Supplemental Curriculum Materials:
Picturing Pictures (PDF). From Baker’s Choice, Key Curriculum Press, 1150 65th Street, Emeryville, CA 94608, 1-800-995-MATH, www.keypress.com. Used with permission.
Profitable Pictures (PDF). From Interactive Mathematics Program Year 2, Key Curriculum Press, 1150 65th Street, Emeryville, CA 94608, 1-800-995-MATH, www.keypress.com. Used with permission.