# 4.1: Which Boxes

**At Grade**Created by: CK-12

You have lots of these boxes. If you wanted to make the pans balance, which boxes will you put in the empty pan? Can you solve this problem in more than one way?

### Which Boxes

In order to balance pans like the one above, use the problem solving steps to help you.

- First,
**describe**what you see and what information you are given. - Next, identify what
**your job**is and what you are trying to solve. In all of these problems, your job will be to figure out which boxes to put in the pan. You want to make sure that the pans will balance. - Third, make a
**plan**for how you will solve. - Fourth,
**solve**the problem. - Last,
**check**your answer by making sure that the combinations of boxes you found add up to the amount that you needed.

For the problem above, since the left side weighs 10 pounds, you want to make sure the right side will also weigh 10 pounds.

1.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you use a different set of boxes? Explain.

We can use the problem solving steps to help us solve this problem.

\begin{align*}& \mathbf{Describe:} && \text{Two pans. One pan is empty. Other pan has a 9-pound box.}\\ &&& \text{There are three other types of boxes.}\\ &&& G \ \text{is 4 pounds.}\\ &&& H \ \text{is 3 pounds.}\\ &&& L \ \text{is 2 pounds.}\\ &&& \text{There are lots of each type of box.}\\ & \mathbf{My \ Job:} && \text{Figure out which boxes will make the pans balance.}\\ & \mathbf{Plan:} && \text{Try to make 9 pounds with:}\\ &&& \qquad \text{Only} \ G \ \text{boxes}\\ &&& \qquad \text{Only} \ H \ \text{boxes}\\ &&& \qquad \text{Only} \ L \ \text{boxes}\\ &&& \qquad \text{Then try combinations:}\\ &&& \qquad \qquad \ G \text{s and} \ H \text{s}\\ &&& \qquad \qquad \ G \text{s and} \ L \text{s}\\ &&& \qquad \qquad \ H \text{s and} \ L \text{s}\\ &&& \qquad \qquad \ G \text{s}, \ H \text{s and} \ L \text{s}\\ & \mathbf{Solve:} && 3 \ H \ \text{boxes weigh 9 pounds}\\ &&& 1 \ H \ \text{and} \ 3 \ L \ \text{boxes weigh 9 pounds}\\ &&& 1 \ G \ \text{box and 1} \ H \ \text{box and 1} \ L \ \text{box weigh 9 pounds}\\ & \mathbf{Check:} && 3 + 3 +3 =9\\ &&& 3+2+2+2=9\\ &&& 4+3+2=9\end{align*}

2.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you use a different set of boxes? Explain.

We can use the problem solving steps to help us solve this problem.

\begin{align*}& \mathbf{Describe:} && \text{Two pans. One pan is empty. Other pan has a 5-pound box.}\\ &&& \text{There are three other types of boxes.}\\ &&& D \ \text{is 3 pounds.}\\ &&& E \ \text{is 2 pounds.}\\ &&& F \ \text{is 1 pounds.}\\ &&& \text{There are lots of each type of box.}\\ & \mathbf{My \ Job:} && \text{Figure out which boxes will make the pans balance.}\\ & \mathbf{Plan:} && \text{Try to make 5 pounds with:}\\ &&& \qquad \text{Only} \ D \ \text{boxes}\\ &&& \qquad \text{Only} \ E \ \text{boxes}\\ &&& \qquad \text{Only} \ F \ \text{boxes}\\ &&& \qquad \text{Then try combinations:}\\ &&& \qquad \qquad \ D \text{s and} \ E \text{s}\\ &&& \qquad \qquad \ D \text{s and} \ F \text{s}\\ &&& \qquad \qquad \ E \text{s and} \ F \text{s}\\ &&& \qquad \qquad \ D \text{s}, \ E \text{s and} \ F \text{s}\\ & \mathbf{Solve:} && 5 \ F \ \text{boxes weigh 5 pounds}\\ &&& 3 \ F \ \text{and} \ 1 \ E \ \text{boxes weigh 5 pounds}\\ &&& \text{2} \ E \ \text{boxes and} \ 1 \ F \ \text{box weigh 5 pounds}\\ &&& 1 \ E \ \text{box and 1} \ D \ \text{box weigh 5 pounds}\\ &&& 1 \ D \ \text{box and 2} \ F \ \text{boxes weigh 5 pounds}\\ & \mathbf{Check:} && 1 + 1 +1 + 1 + 1= 5\\ &&& 1+1+1+2=5\\ &&&2+2+1=5\\ &&& 2 + 3=5\\ &&& 3+1+1=5\end{align*}

#### Earlier Problem Revisited

We can use the problem solving steps to help us solve this problem.

\begin{align*}& \mathbf{Describe:} && \text{Two pans. One pan is empty. Other pan has a 10-pound box.}\\ &&& \text{There are three other types of boxes.}\\ &&& A \ \text{is 2 pounds.}\\ &&& B \ \text{is 3 pounds.}\\ &&& C \ \text{is 4 pounds.}\\ &&& \text{There are lots of each type of box.}\\ & \mathbf{My \ Job:} && \text{Figure out which boxes will make the pans balance.}\\ & \mathbf{Plan:} && \text{Try to make 10 pounds with:}\\ &&& \qquad \text{Only} \ A \ \text{boxes}\\ &&& \qquad \text{Only} \ B \ \text{boxes}\\ &&& \qquad \text{Only} \ C \ \text{boxes}\\ &&& \qquad \text{Then try combinations:}\\ &&& \qquad \qquad \ A \text{s and} \ B \text{s}\\ &&& \qquad \qquad \ A \text{s and} \ C \text{s}\\ &&& \qquad \qquad \ B \text{s and} \ C \text{s}\\ &&& \qquad \qquad \ A \text{s}, \ B \text{s and} \ C \text{s}\\ & \mathbf{Solve:} && 5 \ A \ \text{boxes weigh 10 pounds}\\ &&& 2 \ A \ \text{and} \ 2 \ B \ \text{boxes weigh 10 pounds}\\ &&& \text{One} \ A \ \text{box and} \ 2 \ C \ \text{boxes weigh 10 pounds}\\ &&& 2 \ B \ \text{boxes and one} \ C \ \text{box weigh 10 pounds}\\ & \mathbf{Check:} && 2 + 2 +2 + 2 + 2 = 10\\ &&& 2 + 2 + 3 + 3 = 10\\ &&& 2 + 4 + 4 = 10\\ &&& 3 + 3 + 4 = 10\end{align*}

### Vocabulary

To be ** equal** means to be the same. In this concept, we are trying to make weights equal. This means we are trying to make each side of the pan balance have the same weight.

### Examples

For each problem, put boxes in the empty pan to make the pans balance. Can you find more than one answer?

#### Example 1

\begin{align*}J + J + J + J + J + J = 12 \ pounds\end{align*}

- \begin{align*}K + K + K + K = 12 \ pounds\end{align*}
- \begin{align*}J + J + J + K + K = 12 \ pounds\end{align*}
- \begin{align*}J + L + L = 12 \ pounds\end{align*}
- \begin{align*}J + J + K + L = 12 \ pounds\end{align*}

#### Example 2

\begin{align*}R + R + R + R + R = 15 \ pounds\end{align*}

- \begin{align*}R + R + R + S = 15 \ pounds\end{align*}
- \begin{align*}R + S + S = 15 \ pounds\end{align*}
- \begin{align*}R + R + Q = 15 \ pounds\end{align*}
- \begin{align*}Q + S = 15 \ pounds\end{align*}

#### Example 3

\begin{align*}W + W + W + W = 16 \ pounds\end{align*}

- \begin{align*}Y + Y = 16 \ pounds\end{align*}
- \begin{align*}W + X + X = 16 \ pounds\end{align*}
- \begin{align*}W + W + Y = 16 \ pounds\end{align*}

### Review

For each problem, put boxes in the empty pan to make the pans balance. Can you find more than one answer?

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### Image Attributions

In this concept, students add whole numbers to determine which boxes should be placed in the empty pan of a two-pan balance in order to balance the pan. Students will use problem solving steps to help them.

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