Insights Into Algebra 1: Teaching for Learning
Variables and Patterns of Change Lesson Plan 2: Cups and Chips
Cups and Chips – Solving Linear Equations Using Manipulatives
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.
One 50-minute class period
Students will be able to:
- Use manipulatives to solve linear equations of the form ax + b = c, where a, b, and c are constants.
- Use symbolic methods to solve linear equations.
- Verify and check the solutions to the equations by using substitution.
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000:
NCTM Algebra Standard for Grades 6-8
NCTM Algebra Standard for Grades 9-12
Teachers will need the following:
- A bag of 40 transparent chips (20 red, 20 yellow)
- 10 paper cups
- 10 equations for use at stations (the equation should appear on one side of a strip of paper, and the solution on the other side)
Students will need the following:
- A bag of 20 chips (red on one side, yellow on the other)
- 10 paper cups
- Individual dry-erase boards or large sheets of paper
Teachers Activities and Assignment
1. As a warm up, present the following equations for students to solve:
- x + 10 = 15
- y – 3 = -1
- 5 – m = -2
- w + 4 = -5
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.
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:
- If the variable is positive, place the cup(s) facing up.
- If the variable is negative, place the cup(s) facing down.
- The coefficient of the variable indicates the number of cups to use.
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:
- The chips represent the numbers.
- If a number is positive, the chip should be yellow side up.
- If a number is negative, the chip should be red side up.
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:
- 5m + 1 = -9
- 2x + 3 = 4
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):
- 3x + 2 = 14
- -3m – 1 = -10
- -7x + 5 = 12
- -w + 13 = 9
- ½d + 7 = 10
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.
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.
Related Standardized Test Questions
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.
- Taken from the Maine Educational Assessment, Mathematics, Grade 11 (2002:
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.
- Taken from the Massachusetts Comprehensive Assessment, Grade 10 (Spring 2002):
Solve the following equation for x.
3x – (2x – 3) = 2x – 9
3x – (2x – 3) = 2x – 9
3x – 2x + 3 = 2x – 9
x + 3 = 2x – 9
x = 12
- Taken from the Maryland High School Algebra Exam (2002):
Terry is going to the county fair. She has two choices for purchasing tickets, as shown in the table below.
- Write an equation for Terry’s total cost (y) for ticket Choice A. Then write an equation for Terry’s total cost (y) for ticket Choice B. Let x represent the number of rides she plans to go on.
- How many rides would Terry have to go on for the total cost of ticket A and ticket B to be equal? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.
- Terry plans to go on 14 rides. To spend the least amount of money, which ticket choice should Terry choose? Use mathematics to justify your answer.
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.
- Taken from the California High School Exit Examination (2002):
Solve for x:
2x – 3 = 7
D. 5 (correct answer)
Student Work: Linear Equations Assessment
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.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.