## 7.1: Hanger Diagrams

In the two diagrams, all the triangles weigh the same and all the squares weigh the same.

For each diagram, come up with . . .

- One thing that
*must*be true - One thing that
*could*be true - One thing that
*cannot possibly*be true

Let’s see how a balanced hanger is like an equation and how moving its weights is like solving the equation.

In the two diagrams, all the triangles weigh the same and all the squares weigh the same.

For each diagram, come up with . . .

- One thing that
*must*be true - One thing that
*could*be true - One thing that
*cannot possibly*be true

On each balanced hanger, figures with the same letter have the same weight.

- $2 \boxed{\phantom{3}} + 3 = 5$
- $3 \boxed{\phantom{3}} + 2 = 3$
- $6 = 2 \boxed{\phantom{3}} + 3$
- $7 = 3 \boxed{\phantom{3}} + 1$

- Match each hanger to an equation. Complete the equation by writing $x$, $y$, $z$, or $w$ in the empty box.
- Find the solution to each equation. Use the hanger to explain what the solution means.

Here are some balanced hangers where each piece is labeled with its weight. For each diagram:

- Write an equation.
- Explain how to figure out the weight of a piece labeled with a letter by reasoning about the diagram.
- Explain how to figure out the weight of a piece labeled with a letter by reasoning about the equation.

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

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

Here is a concise way to write the steps above:

\(\begin {align} 7&=3x+1 & \\ 6&=3x & \text{after subtracting 1 from each side} \\ 2 &= x & \text{after multiplying each side by } \tfrac13 \\ \end{align}\)