Lesson Plan 2: Cups and Chips - Solving Linear Equations Using Manipulatives
Teachers will need the following:
Students 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)
- 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
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.
- x + 10 = 15
- y - 3 = -1
- 5 - m = -2
- w + 4 = -5
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:
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:
- 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.
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.
- 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.
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.
- 3x + 2 = 14
- -3m - 1 = -10
- -7x + 5 = 12
- -w + 13 = 9
- ½d + 7 = 10
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.