Ask Marian
I have been told that my students should use more than one strategy for solving equations, but how else can you solve equations other than by isolating the variable?
There are lots of other ways to solve equations.
One way is often called guess and test, where students estimate a reasonable solution, test it, and then either increase or decrease the value based on what they learned from the test.
This actually makes sense. For example, if a student were given the equation 3n + 2 = 5n – 8, they might make a guess of n = 10; they would then figure out that 32 ≠ 42, so the 5n – 8 side is too high.
If they were to increase n, the difference between the right side and the left side would be even greater, so it would make sense to decrease n, for example, trying n = 8 or even lower.
Another strategy is to work backwards. For example, if the equation were 3n + 2 = 20, a student might think this: I multiplied by 3 and added 2 to get 20, so I can go backwards from 20 by subtracting 2 and then dividing by 3. This is a lot like isolating the variable but not quite.
Is it true that if I add 2 + 3 + 8 as 2 + 8 + 3, I am using the associative property?
Do you think students should keep track of how many multiplication facts they’ve learned and how many they still have to learn?
I am well aware that for some students, it could be motivating to see the number of facts they have learned going up and the number still to learn going down.
But the downside is that it might encourage students to continue to think of math as a checklist of small, isolated bits of knowledge rather than as an integrated whole.
I would prefer that they focus on the relationships between facts and how knowing one fact tells them about so many other facts. For example, if they know 5 × 6 = 30, they automatically know that 6 × 6 is 6 more, that 4 × 6 is 6 less, that 5 × 7 is 5 more, and so on.