Grade: Middle
Subject: Mathematics

#4116. Solving Equations (a balancing act)

Mathematics, level: Middle
Posted Thu Feb 21 17:14:11 PST 2008 by Kathrine Schmidt (Kathrine Schmidt).
Duarte High School, Duarte, USA
Materials Required: Index cards, Balance scale, weights, overhead/whiteboard
Activity Time: 50 minutes
Concepts Taught: Algebra 1 Standard 5

5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

Objectives- Students should have the ability and understanding of how to manipulate single variable equations to solve problems, including word problems.

Materials- multiple word problems yielding the same equation, balance scale with weights, weights wrapped in tissue paper to represent variables, whiteboard.

Preparation- Wrap 4 groups of 2 weights with tissue paper so that the students cannot see how many are inside each group. Write a "W" on the tissue paper around each group. These will be used as the variables in the demonstration of how to solve an equation.

1. Remind students what a variable is; that it is just a letter that is used to represent a number that we don't know yet. A variable is generally the value we are trying to find.
2. Divide students into groups of four and pass out one word problem to each group. Each word problem should be different, but give the same equation to solve. Try to make sure there is a mixture of student skill levels in the groups.
There are 1,000 students in a school in an area that is growing quickly. Each day it seems that almost 3 more students are entering the school. The school has a capacity of 1,800. In about how many days will the school be at capacity?
There is $1,000 in a school fund for a big dance. Tickets will be sold for $3 a piece. How many tickets will need to be sold to cover the anticipated cost of $1,800 for the band, the food, the decorations, and the publicity?
You want to go to Africa to see the wild animals. You can get $1,000 from your savings and your family, but you need $1,800. If you save $3 a week, how many weeks will it take you to save up for the trip?
The perimeter of the roof line of the warehouse above is 1,800 feet. If the front edge is 1,000 feet and the other three edges have the same length, what is the length of each of the other edges?
You need a total of 1,800 points to get an A in your favorite math class. Including tests and homework you have 1,000 points so far. If each homework assignment is worth 3 points, how many more do you need to do in order to get an A?
You have a new job and part of your salary is based on commission. You make $1,000 every month and an extra $3 per product you sell on top of that. If you need to make a total of $1,800 this month, how many products do you need to sell?
You are planning a road trip for you and your friends. You have collected a total of $1,000 from everyone, but you need $1,800 to do what you planned. If you invite 3 more people and the total cost doesn't change, how much would each of them have to pay to cover the rest?
You are working with a company that automatically gives raises after you work 1,800 hours. You have just completed your 1,000th hour, but your schedule got changed so you only work 3 hours a day. At this rate, how many days will you have to work before you get your raise?

3. Give the students time to see if they can come up with an equation to represent the problem they were given. If they have extra time they can try to solve it. While the students are working, check in with the groups and see how they are doing, asking questions and offering suggestions as needed.
4. After each group has had time to think and discuss, bring the class together and have one person from each group read their problem. Then talk about their results and how they came up with the answers they got.
5. Pick one of the groups to write the equation they got on the board and tell how they figured it out. Explain that even though each group got a different problem they should have all gotten the same equation as a result. If a group came up with something else, ask them how they came up with it and talk about what went wrong.
6. Remind (or teach) students how to solve problems with one variable.
a) Get out the balance scale, regular weights and wrapped weights you prepared (I will call them Ws through the rest of the lesson).
b) Write the equation, 4W + 3 = 11 on the board and put 4 Ws and three weights on one side, and 11 weights on the other.
c) Explain that the packages represent the variable, W, which is why you have 4 of them and you put the blocks and weights on the scale as you did because the equation says that they are equal.
d) In order to solve the equation you want to have only one W left because you want to know how much one variable is worth.
e) Because you need to keep both sides balanced, whatever you put on or take off from one side, you have to do the same thing to the other.
f) You need to start off by making it so you only have Ws on one side. No weights. In order to do that you have to take 3 weights away from the W side, but to keep them equal you have to take three away from the other side too. Underneath the equation you wrote on the board, write "4W + 3 -- 3 = 11 -- 3, 4W = 8"
g) Write "Addition Property of Equality" on the board.
h) Now to get one block instead of four, you need to divide the number of blocks by 4 or multiply by . Again you have to divide the other side by 4 as well to keep them equal. Students should see that you are left with 1 block equal to 2 weights.
i) Take away of the blocks and of the weights. Now under your equation write, "1/4 x 4W = x 8, W = 2"
j) Write "Multiplication Property of Equality" on the board.
7. Ask the students to look at the equation they came up with for their word problem and go through the steps to solve the problem with them.
8. Let them know that it is always a good idea to go back and substitute their answer to make sure it works. Do this with them so they can see how it is done.
Guided Practice- Give each group a set of problems to work through together. While the students are working, walk around and make sure the groups are on the right track and be available to answer any questions they might have.

Independent Practice- Give the students another set of problems to practice at home. Make sure there is a good mixture of thinking problems and real world problems.