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Cups and Chips – Solving Linear Equations Using Manipulatives
Overview:
In this lesson, students use manipulatives to represent visually the steps they take to obtain a solution to an algebraic equation. They develop an understanding of the connections between the solution involving manipulatives and the symbolic solution.
Time Allotment:
One 50-minute class period
Subject Matter:
Linear 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:
Introductory Activity:
1. As a warm up, present the following equations for students to solve:
2. Give students two minutes to complete the warm up problems individually.
3. Have students compare and discuss their solutions with a partner.
4. For each problem, consider student answers. For any problem with which students had difficulty, ask several students with different answers to present their solutions on the board or overhead, and help them clarify their understanding.
Learning Activities:
1. Distribute a bag of chips, a set of cups, and a large sheet of paper or dry-erase board to each group of students.
2. Explain that students will be using a cups and chips activity to solve the equation 2x + 6 = 12.
3. Present the following directions to students:
Then, ask students to show you the representation of 2x using the cups. They should all place two cups facing up on top of their paper or dry-erase board. Explain the following:
Have students use six yellow chips to represent +6. They should place these chips next to their two cups. Then, have them draw an equal sign to the right of the two cups and six yellow chips. Explain that they can represent +12 by placing 12 yellow chips on the other side of the equal sign.
4. Ask students what can be done to both sides of the equation to get rid of the six yellow chips (+6) on one side of the equation. Elicit from students that -6 should be added to each side (i.e., add six red chips to both sides); alternatively, +6 could be subtracted from each side (i.e., take away six yellow chips from each side).
5. On the overhead, add six red chips to the side with six yellow chips. Also add six red chips to the side with 12 yellow chips, and have students repeat these actions in their groups. Ask, “When you pair each red chip with a yellow chip, what happens?” Call on a student to explain that each pair is equal to 0.
6. Have students remove the pairs of red and yellow chips, leaving just two cups facing up and six yellow chips. Ask, “What equation do we have now?” Elicit from students that the cups represent 2x, the remaining yellow chips represent +6, and the equation now left is 2x = 6. Write this new equation on the overhead below the original equation.
7. Ask, “If two cups equal six chips, what does that tell us about one cup?” They should notice that there are three chips for each cup.
8. Demonstrate that the final equation is now x = 3, and write this equation on the overhead below the equation 2x = 6.
9. Give students the following problems to solve in their groups using cups and chips:
10.Circulate as students are solving these problems. Allow a few minutes for students to complete both problems.
11.Review the solutions to the problems with the class. For the second problem, be sure to discuss the final step, when students arrive at the equation 2x = 1. Ask, “Were you actually able to use the cups and chips to solve the problem? When you had 2x = 1, what operation did we have to do?” Elicit from students that both sides had to be divided by 2 (or that the chip needed to be split in half), to yield the answer x = ½.
12.Explain to students that you want them to try a problem with a negative coefficient. Give students the problem -2x + 3 = -5 to solve.
13.Ask, “What was the first step in solving this problem?” The students should notice that the first step is to subtract 3 from (or add -3 to) both sides of the equation, yielding -2x = -8.
14.Ask, “What is the next step to balance the equation and get x by itself?” Students may note that both sides need to be divided by -2, yielding x = 4. They may also state or demonstrate that they can turn over both the cups and the chips on both sides of the equation, which would represent multiplication by -1.
15.Ask, “How can we check this to make sure it is the correct answer?” Obtain from students that the value x = 4 can be substituted into the original equation to show that it works: -2(4) + 3 = -5.
Explain to students that now that they have solved the same equations using cups and chips and symbolic manipulation (or algebra), it’s time to try solving similar equations with symbolic manipulation (algebra) only. At 10 stations throughout the room, post various equations for the students to solve. Do not let them know that the solutions are given on the back of each piece of paper. Have students circulate in pairs through the stations, solving each equation and checking their answers. Give students 1-2 minutes at each station, as necessary. Below are some equations you might use (make sure some of the variables have negative and fractional coefficients):
16.Show students that they can turn over the papers to find the correct solutions. Give them a couple of minutes to verify their results, and then call the whole class together to review and clarify the solutions to any problems with which students had difficulty.
Culminating Activity/Assessment:
1.Once students have answered all questions, ask them to summarize the process of solving an equation. Solicit input from several students, and relate their descriptions to the cups and chips activity. Emphasize the need to add or subtract and then multiply or divide, and be sure to stress that the final step should always be to check the answer in the original equation.
2.Assign problems for homework.
The questions below dealing with solving linear equations have been selected from various state and national assessments. Although the lesson above may not fully equip students 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.
Clem’s balloon is 200 feet off the ground and rising at a rate of 5 feet per second. Mary’s balloon is 100 feet off the ground and rising at a rate of 9 feet per second. In how many seconds will the two balloons be at the same height? Show how you found your answer.
Solution: The height of Clem’s balloon can be represented as 200 + 5t, and the height of Mary’s balloon can be represented as 100 + 9t, where t is the number of seconds from now. The balloons will be at the same height when 200 + 5t = 100 + 9t, or when t = 25 seconds.
Solve the following equation for x.
3x – (2x – 3) = 2x – 9Solution:
3x – (2x – 3) = 2x – 9
3x – 2x + 3 = 2x – 9
x + 3 = 2x – 9
x = 12
Terry is going to the county fair. She has two choices for purchasing tickets, as shown in the table below.
Solution:
For Choice A, the equation is y = 6 + 0.5x; for Choice B, y = 2 + 0.75x.For the total costs to be equal, 6 + 0.5x = 2 + 0.75x, or x = 16; therefore, Terry would have to go on 16 rides.
For 14 rides, Choice A would cost 6 + 0.5(14) = $13. Choice B would cost 2 + 0.75(14) = $12.50. Terry should choose ticket B.
Solve for x:
2x – 3 = 7
A. -5
B. -2
C. 2
D. 5 (correct answer)
Teacher Commentary:
After reviewing the student’s work, I can see that this student has a strong understanding of the concepts discussed in class. She has shown a clear step-by-step procedure for solving the problems and has answered each question correctly. During the group activity, she was able to explain to her partners how to solve the problems. Her explanations were clear and thorough. The only thing I would like to see from this student is some indication that she is checking the solution each time. She has check marks on her paper that shows she has compared answers, but I would like to see her substitute her solution into the original equation to verify that her answer is correct.
I feel that today’s lesson was very successful. While some of the students were making common multiplication or sign errors in the beginning of class, I believe that the manipulatives helped the students to make sense of the algebraic operations needed to solve a two-step equation. I saw some of my weaker students show improvement as they practiced more problems. Because there was some discussion about typical algebraic mistakes, it may be helpful for a follow-up lesson to include an activity in which students analyze a series of problems worked out incorrectly. The students would be asked to look for any mistakes in the problem and then make necessary corrections to the problem. This may help my weaker students who may still encounter difficulty with determining the appropriate operation.