Reasoning about Solving Equations (Part 1)

Student Summary

In this lesson, we worked with two ways to show that two amounts are equal: a balanced hanger and an equation. We can use think about the weights on a balanced hanger to understand steps we can use to find an unknown amount in a matching equation.

This hanger diagram shows a total weight of 7 units on one side that is balanced with 3 equal, unknown weights and a 1-unit weight on the other. An equation that represents the relationship is 7=3x+17=3x+1.

Balanced hanger, left side, 7 squares, right side, 3 circles and 1 square.

We can remove a weight of 1 unit from each side and the hanger will stay balanced. This is the same as subtracting 1 from each side of the equation. 

Balanced hanger, and  to the side, an equation.
Balanced hanger, left side, 6 blue squares and one red squared being removed. Right side, 3 green squares, and one red square being removed. To the side, an equation 7 minus 1 = 3 x  + 1 minus 1, with each minus 1 written in red.

An equation for the new balanced hanger is 6=3x6=3x.

Balanced hanger, left side, 6 blue squares, right side, 3 green circles.  To the side, an equation says 6 = 3 x.

We can make 3 equal groups on each side and the hanger will stay balanced. This is the same as dividing each side of the equation by 3 (or multiplying each side by 13\frac13 ). In other words, the hanger will balance with 13\frac13 of the weight on each side.

Balanced hanger.
Balanced hanger, left side 6 blue squares, right side, three green circles.  A dotted line is drawn around each of 3 groups, each groups consists of two blue square s from the left side and one green circle from the right side. To the side, an equation says 1/3 * 6 = 1/3 * 3 x.

The two sides of the hanger balance with two 1-unit weights on one side and 1 weight of unknown size on the other side. So, the unknown weight is 2 units.

Balanced hanger, left side 2 squares, right side 1 circle.  To the side, an equation says 2 = x.

Here is a concise way to write the steps above:

7=3x+16=3xafter subtracting 1 from each side2=xafter multiplying each side by 13\begin{aligned} 7&=3x+1 & \\ 6&=3x & \text{after subtracting 1 from each side} \\ 2 &= x & \text{after multiplying each side by } \tfrac13 \\ \end{aligned}

Visual / Anchor Chart