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!)
After having practiced multiple one-step equations using addition or subtraction, the process is starting to be automatic. A problem like x + 5 = 15 can be solved by writing -5 on both sides of the equation, leading to x = 10. An alternative is to write that step in in-line form: x +5 - 5 = 15 – 5, which ultimately leads to a short-cut in which students leap to the final step of x = 15 -5.
This leap is something JUMP Math mentions, in which a number on the left side of the equation (“5” in this case) is moved to the other side, and changes its sign when it “crosses the border”—the border being the equal sign. This level of automaticity is how most people do it. It is not necessary to require students to do this, but if they are looking for ways to not spend so much time showing their work (which is a perennial complaint of many students), this might be an incentive.
Today’s lesson focuses on problems in the form ax = b and x/m = b. In general, students have an easy time with the first type, although it is not always obvious that they are dividing by a coefficient, and that a number divided by itself equals 1. A disappearing coefficient therefore means that it has been changed to “1” by virtue of dividing a number by itself. Students may not realize they are dividing because the divide sign (÷) is not used in algebraic equations or expressions. A fraction bar now represents division, which is something students will need to be reminded of throughout the year.
Furthermore, it is not at all obvious to students that dividing by the coefficient a is the same as multiplying by the reciprocal of the coefficient—that is, 1/a. This same relationship is at play with the second equation type: x/m = b, which is really in the same form as the ax =b, where a in this case equals 1/m. The understanding that a/b = a x 1/b will come in time for some students with more math courses; for other students it may not but will not represent a significant obstacle nor delay their progress in learning to solve equations.
Warm-Ups. In this lesson students need to recall that a number multiplied by its reciprocal equals 1, and that a variable or term with no coefficient means it has an unwritten coefficient of 1. E.g, x = 1x, ab = 1ab, and so on. Warm-Ups should therefore include problems that involve reciprocals and problems with unwritten coefficients as these examples illustrate. Students should be reminded to show their work, i.e., the steps for solving equations.
1. Solve for x. 2x – x + 3 = 14 Answer: x + 3 = 14; x=11
2. 4/5 x 5/4 = ? Answer: 1
3. 5 x 1/5 = ? Answer: 1
4. Solve for x. 15 = x + 20 Answer: x = -5
5. Combine like terms. -5x + 6k, - 19x +24x Answer: 6k
Students may balk at problem 4, claiming they haven’t been taught how to solve such problems. Ask them how to isolate x on the right hand side—what needs to be done? It might even be mentioned that it doesn’t matter whether the variable to be solved appears on the right or left hand side—the procedure is the same for either. For problem 3, students continually have trouble with multiplying a whole number by a fraction unless they write the whole number as a fraction with 1 as the denominator. In the case of Problem 3, this would look like 5/1 x 1/5. Also, they seem to be steadfastly stubborn in not seeing that the reciprocal of a whole number a is 1/a. It’s a good idea to keep such annoyances in front of them.
The division property of equality. This is a simple concept that students tend to grasp rather easily. I start off by recalling Problem 1 of the Warm-Up: 2x –x + 3 = 14.
They have done this problem successfully, but now I offer a slightly different version of the problem. Suppose we had 3x – x + 3 = 15. We would then obtain 2x = 12. How do we isolate the variable? To do this, I announce, we need the “division property of equality” which simply stated is:
When equal numbers are divided by the same number, the quotients are equal. (0 may not be used as divisor)
Thus, as an example, for the equation 50 = 50 each side can be divided by 2 to obtain 50/2 = 50/2, otherwise known as 25 = 25.
What about if we divided each side by 3? Is 50/3 = 50/3? We don’t need to convert to a mixed number to see that it is true.
A key example however is the following, since it is the operation they will be doing in this lesson when solving equations:
25 x 2 = 50
They can easily see that 25 x 2 divided by 2 equals 25, by doing the calculation in their heads. But it is important to show that it can also be expressed as 25/1 x 2/2. Multiplying the two fractions, one can see it equals (25 x2)/2.
I don’t dwell on this. Some students will get it, and others will be of the “if you say so” mode and want to know how to use it to solve 2x = 12.
I’ll ask if anyone knows how to solve it. Someone may shout out “6”, and upon asking how they did it, they will usually say “I know that 2 times 6 is 12.” Which is correct, and I will say so. I will also say that we can arrive at the same answer by dividing each side by 2, just like we did with 25 x 2 = 50. (There is the possibility that the person who shouts out “6” will explain that they divided each side by 2—it does happen on occasion!)
In any case, the bottom line is divide by 2, or more generally, divide by the coefficient of x. When we divide by the coefficient, we are “undoing” the multiplication.
I’ll have them do some examples at this point, reminding them that division is represented by a fraction bar, so when we divide both sides by a number, it looks like this: 2x = 16; 2x/2 = 16/2; x = 8. That is how I want the steps shown. I then have them do some examples in their notebooks or mini-whiteboards.
6x = 18
-5x = 20
3x = 15
4x = 26
For this last, they will get a mixed number, or they can leave it as a fraction which in lowest terms is 13/2.
Multiplication property of equality. Now I will write the problem 1/5(x) = 6
I’ll ask if anyone knows how to solve it. I sometimes hear “divide”, to which I will respond “Divide both sides by what?” They will say “1/5” and technically this is correct, because 1/5 ÷ 1/5 equals 1, and 6 ÷ 1/5 is 6 times the reciprocal of 1/5, or 6 x 5, which is 30.
I will say that this is correct and will point out that the last step was multiplying the right hand side by 5. What we did was multiply both sides by the same number—in this case “5”. I show this as follows:
5 x (1/5)(x) = 6 x 5. I will ask what 5 x 1/5 is, and I may have to resort to writing it as 5/1 x 1/5, and may also have to remind them of what we talked about earlier: that any number times its reciprocal equals 1.
What we are doing is multiplying both sides by the reciprocal of the coefficient of x, I will announce. And this is the multiplication property of equality, which I write on the board:
When equal numbers are multiplied by the same number, the products are equal.
For each a, b and c, if , then
I’ll try a few more:
1/6(n) = 6 Answer: n = 36
(-1/9)t = 8 Answer: t = -72
Now I try this, which generally results in blank stares: n/4 = 6 “What do we multiply both sides by?”
I may or may not hear a correct answer, but in any event, correct answer or not, I will say “Let me rephrase the question” and write: (1/4)n = 6. There are usually “Oh’s” of recognition, and they see it now is in the form they had just been working with. I tell them that n/4 is the same thing as (1/4)n. It is 1/4 multiplied by n/1, I will say to make it absolutely clear, so when the fractions are multiplied it becomes n/4. But we can also take it apart.
I then ask student to “take apart” u/9, t/7, -x/5. (The latter answer can be expressed as either –(1/5)x or (1/5)(-x). )
Two or three more examples follow with mixed format. Some in the form of, say x/7 = 8, and others in the form of (1/6)u = 5.
The homework should be a mix of the problems worked in class, including both notations of problems such as y/8 = 56, as well as 1/8(y)=56.