How to solve $a x = b$?

Question

I'm teaching algebra to lower ability grade 11 students. I've tried to give them fair grounding in algebraic manipulation. I'm trying to explain how to solve a linear equation like $2 x +1 =3$ (or $- 2.3 x + 1.2 =-3.7 $ but I'll treat the simpler one). The first step is clear "do the opposite of plus $1$" ie "subtract $1$ from both sides": $$ 2x+1=3 \\ 2x+1-1=3-1 \\ 2x+0=2 \\ 2x=2 $$ The middle steps are quite intuitive and my students seem happy skip them.

Next it comes to "do the opposite of multiply by $2$" ie "divide by 2" $$ \frac{2x}{2}=\frac22 \\ x=1. $$

How do I explain this step?

Ideas:

1. $\frac{x+x}2=x$
2. $\frac{2x}{2}=\frac22x=1x=x$
3. $\frac{2x}{2}=x\frac22=x1=x$
4. other explanations

I'll appraise these explanations.

1. seems intuitive but won't easily handle the complex example above.
2. The obvious choice, but why is it obvious and why is it true? Would I say that $\frac{a \times b}c = \frac{a}{c} \times b $? This seems confusing.
3. exchanges "$2$ lots of $x$" with "$x$ lots of $2$", which is valid but seems confusing.

Answer 1

I recommend that you carefully read the first part of the algebra book you use for your course. Usually there is a section on "Properties of Real Numbers", which effectively presents the axioms for the field structure of real numbers, and hence your course. Usually these are: (1) commutativity (add & multiply), (2) association (add & multiply), (3) distribution (multiply over add), (4) identities (add & multiply), (5) inverses (add & multiply).

Also there will be a formal definition of subtraction: $a - b = a + (-b)$, and division: $\frac{a}{b} = a \cdot \frac{1}{b}$. (Without saying this would be my top choice for axiomatic fundamentals, it is very standard.)

First note that properties of equivalent equations can be proven by basic properties of how equality is defined. E.g., For multiplication: If $a = b$, then $ac = bc$. Proof: $ac = ac$ by reflexivity; so $ac = bc$ by substitution ($a = b$ on right). Likewise for other operations.

Now, if desired, we can step through solving $2x = 2$ by the fundamental axioms. $\frac{2x}{2} = \frac{2}{2}$ by the equivalent-equation principle (prior paragraph, for division). This is equivalent to saying $2x \cdot \frac{1}{2} = 2 \cdot \frac{1}{2}$, by the definition of division. The left hand side is equivalent to $x \cdot 2 \cdot \frac{1}{2}$ (commutativity of multiplication), which is equal to $x \cdot 1$ (inverse property of multiplication), which is just $x$ (identity property of multiplication). The right hand side is immediately equal to $1$ (inverse property of multiplication). And so we have our $x = 1$.

Obviously, how much of these details you walk through with your students is up to you. Somewhat against popular opinion, I do actually find value and traction in my courses to walk through the axiomatic steps at least on one day, quizzing the class as a whole for which fundamental property was used at each step (as long as one or more students can give the answers in this presentation, I think it at least gives a model for everyone what real math looks like).

Ideally, of course, your students start generating an intuition that division to both sides of an equation cancels any multiplication there, as they did for the addition problem. The main problem I think we encounter is if they've previously been trained for addition problems to "move a term across the sides and change the sign", then that puts them on an incorrect path for any other operations.

Answer 2

@Pdmclean, My answer is a perspective :

Keeping it simple while comprehensive would work, but it needs time. Sometimes, trying really hard to explain a concept will end up confusing the students.

Keeping it simple, compact, and firm may let the students understand it and to them the concept is actually simple, logical, and easier to grasp.

Build through habits and repetition, I remember a quote from a mathematician, which i think that it is quite true, it is something like this :

"In mathematics, you don't understand things. You just get used to them."

This is quite makes sense, because mathematics is a tool that we have created, a logical way of using numbers and symbols.

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Example of explanation :

Solve $$2x + 1 = 4$$ The equation states that the value in left-hand side is equal to the value in right-hand side. If we change the value of either side, of course the equality will not apply anymore. So what we can do in order to get $$ x = .... (\text{something}) ? $$ To do this, manipulate each side the same way, adding or subtract by a number, divide or multiply by a number :

$$ 2x + 1 = 4 \implies 2x = 3 .....(\text{subtract by} -1)$$

Now what can we do to get the equation $x = \text{a number}$? divide both side by $2$.

$$ 2x = 3 \implies x = \frac{3}{2} .....(\text{divide by} 2) $$

*NOTICE that you can also do it by multiply by $\frac{1}{2}$

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After this, repeat with different examples, or same one with different manipulation.

Try to solve only one problem on the board, with the students watching. Then repeat this steps several times, to 5-6 problems. (like watching a video, but the difference is they can ask directly to you)

After this, students must try to solve problems themselves, give them time 10-15 minutes. If they cannot, do the previous step for this particular problem only. This way, after the students have tried working on it, they now know in which part they made mistakes, and therefore know how to fix.

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The key is that make the students get used to it, then the logic will eventually come to them.