This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational. (Readers are encouraged to provide examples of mistakes that students will make for the particular topic being discussed. They will be incorporated into the ever-evolving text, so you can be a part of this next book!)
1. Introduction/Opening Monologue
Translating English into math is an important skill and is basic to solving all word problems. They have done this to a limited degree using arithmetic methods which I use in my opening monologue to the day’s lesson. I start off asking them to tell me how to write the numerical equation for various statements.
I quickly give a worked example to ward off students giving me the answer to the problem, though I can assure you that no matter how many warnings and instructions you give, that will occur: “The cost of ten yo-yos if each costs three dollars.” What I’m looking for is the numerical equation, not just the answer; in this case it’s 10 x 3 = 30.
Now I give the statements, telling students to write it down in their notebooks or on whiteboards:
“The number of students on three buses if each bus holds twenty two students” Answer: 3 x 22 = 66
“The amount of money Nina earned if she mowed the lawn for $15 and walked the dog for $4”. Answer: $15 + $4 = $19
“The number of students in each group if fifteen students are divided into five equal groups” Answer: 15 ÷ 5 = 3 or 15/5 = 3
The process of writing statements numerically is then extended to expressing statement in algebraic terms in which variables are used.
2. Translating into algebra
Warm-Ups. For this lesson I typically include warm-ups that provide a preview to what we will be doing, and serve as a segue to the day’s lesson. Examples include:
- Write the numerical expression for 5 more than 10. (Answer: 10+5 or 5 + 10)
- Write the numerical expressions for 5 less than 10. (Answer 10 – 5)
- Write as an expression: Seven times x (Answer: 7x)
Defining the Variable. Variables will represent unknown quantities. If I say “some number”, since we don’t know what that number is, it can be represented by a letter. Taking a problem from one of the warm-ups, I’ll point out that if I say “five more than ten” we write that as 5 + 10. I’ll ask: “Suppose I say ‘five more than some number’. How would I write that?” There is general consensus reached quickly that it is 5 + x, which can also be written as x + 5.
I give them numerical forms first and then extend that to variable. “Five times a number” (5x)
“A number divided by 3”. I’m looking for x/3, but if I get x ÷ 3, I accept it, and quickly point out that in algebra we write it in fractional form.
Now I start to use unknown quantities that name specific things, like “Two miles more than an athlete ran.” If there is a stunned silence, I’ll ask what the unknown quantity is, and they’ll pick up that the answer is 2 + x.
“Be careful on this next one,” I’ll warn. “I’m betting that at least people will get this wrong.” The problem: “5 less than some number.” Usually many people will say 5 – x. This particular error is the gift that keeps on giving. Some students will repeat the mistake through the year; so it is good to keep repeating such problems.
I advise at this point that if a statement is confusing, see how to write it using numbers, and I refer to one of the warm-up problems. For example “5 less than 10 is obviously 10-5.” The bottom line answer is five is being subtracted from a number.
It is helpful to mix in other terms for “less than”; for example “5 fewer than some number” and “some number decreased by 5” are equalivalent to 5 less than some number. At this point it is important to make the distinction between “less than ” and “is less than”. The phrase “5 is less than 10” is written 5 <10, and similarly “5 is less than some number” is written 5 < x.
After working with addition and subtraction, I turn to multiplication and division, and ramp the problems up so they are doing both. Starting with “Three times a number”, and “The cost of some number of games of bowling at $4 per game, for example, and ending with “Four less than two times a number.”
More Problems. After the students are fairly comfortable with the initial translations, I give more “story-oriented” problems, that increase with difficulty.
Here are some taken from Brown, et al (2000) that I like to use, and which I also include in homework worksheets. Students are to provide the expression in terms of the variable:
1, The Tigers had twice as many hits as the Yanks. If x = the number of hits by the Yanks, then ____ = number of hits by the Tigers. (Answer: 2x)
2. The length of a rectangle is four times the width. If x = the width, then ____ = the length. (Answer: 4x)
3. Mac is x years old. How old will he be next year? (x+1)
(This particular problem causes students difficulty because although they have been writing expressions in terms of a variable, some think that this problem is asking for a number. I remind them that they are to express Mac’s age in terms of x.)
4. Trish is t years old. How old was she 7 years ago? (Answer t-7)
5. Karen will be m years old next year. How old is she this year? (Answer: m-1)
6. Pete worked 4 hours more than Quinn. Quinn worked 2 hours more than Rob. If x= the number of hours Rob worked, then ___ = the number of hours Quinn worked and ___ = the number of hours Pete worked. (Answers: Quinn’s hours = x + 2; Pete’s hours: x + 6
Reference: Brown, Richard G., Smith, G.D., Dolciani, M.P.; (2000). Basic Algebra; McDougal Littell, Illinois.