Solving Linear Equations Using Manipulatives
There are many different online tools that can be used to solve linear equations. This one was chosen for this lesson because it is set up identical to the phsycial chips and cups manipulatives that will be used in this lesson. Two groups doing the exact same thing. One group using physical representation another group using online representation.
Teachers could also use:
2 Direct Instruction
At the time this lesson is presented, it is not a new concept, but it will be presented in a new way.
- 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)
- At least 10 computers or laptops
- Internet access
- Algebra Tiles Website http://my.hrw.com/math06_07/nsmedia/tools/Algebra_Tiles/Algebra_Tiles.html
Description of Lesson
Students are split into groups with four students in each group.
Each group using the physical manipulatives will be given:
- A bag of 20 chips (red on one side, yellow on the other)
- 10 paper cups
- large sheets of paper
Each student in the interactive manipulatives group will be given:
- An assigned computer with internet access
- Directions to access the Algebra Tiles website http://my.hrw.com/math06_07/nsmedia/tools/Algebra_Tiles/Algebra_Tiles.html
1. Explain that students will be using a manipulative to solve equations.
Half of the groups will be using cups and chips
Half of the groups will be using interactive online Algebra Tiles
2. Present the following directions to students:
The equation is 2x + 6 = 12.
- 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. 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.
Groups using the online tiles.
- Once the students open the interactive algebra tiles, the need to make sure that the top right option reads – Model/Solve an Equation
- Students will click on the positive (yellow) x tiles. The students will drag 2 positive (yellow) x tiles into the left box on the screen.
- The students will click on the positive (yellow) unit tiles and drag 6 of these tiles to the left box on the screen.
- The students will click on the positive (yellow) unit tiles and drag 12 of these tiles to the right box on the screen.
- The students will then click the box labeled – Set both sides equal. The equation 2x+6=12 will appear at the top of the screen
3. Ask students what can be done to both sides of the equation to get rid of the six yellow chips (+6 yellow unit tiles) 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, six red unit tiles to both sides); alternatively, +6 could be subtracted from each side (i.e., take away six yellow chips from each side; take away 6 yellow unit tiles from both sides).
4. 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.
On the interactive Algebra tiles have students click on the red negative unit tiles and drag six of them to the left box and six of them to the right box. Place the added tiles below the tiles for the original problem.
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.
5. Have students remove the pairs of red and yellow chips, leaving just two cups facing up and six yellow chips.
The students using the interactive tiles can drag a red tile on top of a yellow tile an it will disappear, signifying that the pair is equal to 0
Ask, "What equation do we have now?" Elicit from students that the cups (yellow x tiles) represent 2x, the remaining yellow chips (yellow unit tiles) represent +6, and the equation now left is 2x = 6. Write this new equation on the overhead below the original equation.
Show the students using the online manipulative that the screen now has 2x on the left side and 6 on the right side to represent the equation we have now.
6. 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. (3 unit tiles for every 1 x tile)
7. Demonstrate that the final equation is now x = 3, and write this equation on the overhead below the equation 2x = 6.
The students using the online tiles should drag one negative (red) x tile to the left side of the equation and place it over one of the yellow x tiles(It will disappear).
The students will then drag 3 (red) negative unit tiles to the right side of the screen an place them over 3 yellow tiles (They will disappear)
This represents dividing by 2
Students should now notice the new equation: x =3 (x on the left side, 3 on the right)
Students should now click the button labeled, check solution at the bottom left. If they have completed the steps correctly, the box at the bottom should display the words – “Well Done”
Students should click the box labeled Clear All to begin the next problem
3 Guided Practice
8. Give students the following problems to solve in their groups using cups and chips and the interactive algebra tiles:
5m + 1 = -9
2x + 3 = 4
9. Circulate as students are solving these problems. Allow a few minutes for students to complete both problems. Students using the cups and chips should be checking with the kids using the online manipulative to see if the are following the same process and getting the same answers.
10. 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.
Ask the students why Algebra tiles will not allow them to solve this problem.
11. Explain to students that you want them to try a problem with a negative coefficient. Give students the problem -2x + 3 = -5 to solve.
12. 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.
13. 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.
Students using the online manipulative should recognize that they have to change the color when multiplying.
14. 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.
Have the groups switch places from cups an chips to online interactive an visa versa. Redo the 3 problems again using the opposite kind of manipulative. Circulate the room to answer any questions.
4 Independent Practice
Explain to students that now that they have solved the same equations using cups and chips, interactive algebra tiles, 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
15. 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 an interactive algebra tiles 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.