Make Use of Structure

Task

Think about what these equations mean, and find their solutions. Write a sentence explaining how you know your solution is correct.

1. $x + 6 = 10$
2. $1000 - y = 400$
3. $100 = m + 99$
4. $0.99 = 1 - t$
5. $3a = 300$
6. $\frac12 p = 8$
7. $10 = 0.1w$
8. $1 = 50b$

IM Commentary

The purpose of this task is to help students reason about the meaning of equations and the solution of an equation, and to give them an opportunity to make connections with operations with fractions and decimals. The teacher might start off the problem by doing an example to help students get started. All of the sums, differences, products, and quotients students in this task should be doable by reasoning about the structure of the equation and mental math strategies, so students do not need to dive straight into an algorithm for solving the equations.

In a wrap-up discussion, the teacher can help students see that for each equation, we are finding an unknown value when the sum, difference, product, or quotient is known. This can help students see that they are using inverse operations, or "undoing" the given operation. They can then use the understanding of what it means to solve an equation in order to generalize the process of solving equations of the any equation of the form $x + p = q$ or $px = q$.

Solution

1. The number you add to 6 to get 10 is 4, so $x =4$.
2. The number you subtract from 1000 to get 400 is 600, so $y = 600$.
3. The number you add to 99 to get 100 is 1, so $m=1$. 
4. The number you subtract from 1.00 to get 0.99 is 0.01, so $t = 0.01$. (We might think of this as, what do we subtract from a dollar to get 99 cents?)
5. The number you multiply 3 by to get 300 is 100, so $a = 100$.
6. Half of a number is 8, and half of 16 is 8, so $ p = 16$.
7. One-tenth of a number is 10, and one-tenth of 100 is 10, so $w = 100$.
8. A number times 50 is 1, so it must be the reciprocal of 50, so $b = \frac{1}{50}$.