Pct 2.1 Traditional Math: Word Problems, and Equations with Fractions and Variables on Both Sides
Seventh Grade
Working with fractions in equations causes seventh graders difficulty. Finding common denominators proves to be a great challenge, even for a problem requiring a whole number, like 3, to be expressed in terms of 9ths. To that end, it is a good idea to provide some review outside of this lesson to get students comfortable working with fractions. That said, because of the difficulties with and general dislike of fractions that students have, my tendency is to limit this lesson to accelerated classes for Math 7.
This lesson opens with word problems and then progresses to equations with fractions and variables and numbers on both sides of the equation. Although most textbooks these days combine all this in one lesson, I would recommend breaking this up into two days. It depends on how far the class gets. Certainly if this lesson falls on a “short day” (most schools have these), it should be split up.
Warm-Ups. Some focus on fractions and decimals is included in warm-ups as preparation for the material covered in this lesson.
1. A cake costs $1.20. It costs twice as much as a pie. Find the cost of 3 pies and 2 cakes. Answer: Cost of pie = $1.20/2 = $0.60. 3(0.60) + 2(1.20) = 1.80 +2.40 = $4.20
2. Solve. y/8 -1/2 = 7/8 Answer: y/8 =7/8 + 4/8; y/8 =11/8; y = 11
3. Solve. 1.1 + 1.2t = 5.9 Answer: 1.2t = 4.8; t = 4; or 11 + 12t = 59; t = 4
4. Solve. 3x -2 = x; Answer: 2x = 2; x = 1
5. Solve. 60 + 25u = 450 + 10u; Answer: 15u=390; u = 26
For Problem 2, students may need to be reminded that y/8 is the same thing as (1/8)y.
In Problem 3, students can solve it as is; that is, without eliminating decimals. This problem can be used as part of the lesson.
Word Problems. To start this lesson, we begin with some word problems, whose structure will be explored again in the unit on linear functions. The basic structure is the equation of a straight line; that is, y = mx + b.
We use problems that involve payment of a constant fee, like rental of bowling shoes at a bowling alley, combined with the cost of some number of games. Starting with an example of a bowling alley that charges $3 per game and $5 for rental of shoes, I ask what is the cost of 3 games. Students can use mini-whiteboards or their notebooks.
After hearing the correct answer ($14) I write the structure of the calculation on the board:
I do the same for 4 games and 5 games, after students supply the answers:
Finally, I’ll ask “How do we write the equation for bowling x games and the total cost is $23?”
I will circulate around the room to see what students are writing, and hint that they should look at the pattern for the three problems we just did. Having identified students who did it correctly, I’ll pick one to write the solution on the board.
Solving it, they get 6 games for the answer.
“We’ll now use the same structure to solve a more complex problem.” I put the problem on the board using a desk camera which most classrooms use now. (Lacking that, I would have the problem written in advance on poster paper.)
Cruise Company A charges a flat (one-time) fee of $75 plus $85 per day. Cruise Company B charges a flat fee of $30 plus $100 per day. Write and solve an equation to determine the number of days for which the charge for the cruises will be the same.
Students typically do not know where to start. I prime the pump: “What are we trying to find? What is the problem asking?”
Answer: The number of days for which total costs for either company will be equal.
“And if we don’t know the number of days, what do we use in place of a number?”
General response is “x”.
“Let’s write an expression for the cost for x days that Company A charges.” I’ll start this one for them. “What is the flat fee?” I write that down. “And it costs $85 per day for x days, so how do I indicate that part of the cost?”
Hearing answers that work, I will write down 75 + 85x.
“Now let’s do the same for Company B. Who can do it?” There’s enough information that they can copy the pattern: 30 + 100x
It may be necessary to remind them that the number of days is the same in both cases.
“Now how do we write this as an equation? What equals what?” I give them about a minute or so while I walk around the room, looking for correct answers. Finding one, I have that student write the equation on the board:
“The rest is easy,” I say. “Solve it.” Answer: 15x = 45; x = 3 days
I do a check of the answer: 75 + 85(3) =? 30 + 100(3); 75 + 255 =?30 + 300; 330=330; CHECK
(I also note that when we write an equation, we drop dollar signs and other unit symbols.)
Example 2: For this one, I have them work as independently as possible, writing the equation in their notebooks:
A rental car company offers two options: Option A: Flat fee of $25 and $0.45 per mile. Option B: Flat fee of $40 and $0.25 per mile. How many miles under each option must be driven for both options to cost the same?
The main prompt will likely be: “Look at how we did the last problem. What is similar?” Another might be: “What are we trying to find? How do we represent that?”
The answer:
There are a variety of such problems in most textbooks, which can be assigned as homework if this is the break point for the day’s lesson.