Simplifying Equations of the Form ax+b=cx+d

Wagner, William F. Hyde Park Career Academy

Objective(s): (1) Students will be able to solve equations of the form ax+b=cx+d. (2) Students will be able to visualize the concept of "balancing" an equation. Apparatus Needed: Balance scale (easily obtainable if you have an understanding and cooperative science department) Assorted weights (I adapted a box of Cuisenaire rods) Yard stick (optional) Small stuffed animal or soft rubber toy (optional) Assorted "unknowns" (any objects that are of an unknown weight but can be weighed as a combination of rods) Recommended Strategy As a link between this lesson and previous lessons, I started with a review of the four basic types of equations, i.e. equations that can be solved by adding, subtracting, multiplying or dividing. I differentiate between the four types so that the students can remember them more easily by associating them with the four basic operations. Others have argued that there is only two or three basic types. I leave the final decision up to you. After this short review and discussion of methods for solving these types of equations, we show a more complicated equation of the ax+b=cx+d form and ask the students for suggestions on how to solve it based on present knowledge. After students have had an opportunity to give input on the solution, we demonstrate how equations can be displayed using a balance scale. Balance Scale Demonstration: Using the Cuisenaire rods, set up a situation you know is equal. For example, 4 light green rods with 4 white cubes weighs the same as 2 light green with 10 white cubes. Place these different combinations on opposite sides of the balance scale and watch as the scale balances (experimenting before is highly recommended to avoid an embarassing situation). Then ask the students what would happen if you started to remove pieces from different sides, (Desired response: the balance is upset). Next ask the students, "What should I do to put the balance back?" (Desired response: Whatever you did to the first side, now do to the other). Continue in this manner until the following situation is reached: 2 light green rods are balancing 6 white cubes. Now say, "If 2 green rods balance 6 white rods, what would happen if I took out half of the green rods?" (Desired response: Balance upset). Next say, "What should I do to restore the balance?" (Desired response: Take out half the white cubes). Upon doing this, the solution is now evident: 1 light green rod equals 3 white rods. Try this again with different combinations of rods (Again, experiment beforehand). As you go through each step, write on the board what the situation on the scale is, but in equation form, i.e. 4 light green with 4 white cubes balances 2 light green with 10 white cubes would become 4g+4=2g+10. After a few attempts, show that they are merely doing the same manipulations that they did on the basic 4 types with the following conditions: Adding or subtracting is done first with multiplication or division done last, each step should make the problem simpler than before and the ultimate end is to have all the unknowns on one side and all the constants on the other. Once this condition is met, the final step is to multiply or divide by some constant. This demonstration can be made more dynamic by allowing individual students to try their hand at the manipulations to produce equivalent situations. Other objects may be used as unknowns as students experiment to find other true statements. At the end of the lesson and as a taste of what is to come, I might use the yard stick and the stuffed animal to set up a situation to demonstrate inequalities. Place the yard stick so that half is on the desk (Use the stuffed animal as a weight by placing it on the end of the yardstick on the desk) and the other half off the desk. Then proceed to strike the free end of the yardstick sending the stuffed animal flying (experiment beforehand to be certain that the yardstick is strong enough and the stuffed animal light enough to make the flight). I then ask, just as the bell rings, "Why did the animal fly?". Without answering, class is dismissed and as they are leaving I tell them we will be discussing what they saw in our next class, which will be on inequalities.
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