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3.2: Balance the Pans

Difficulty Level: At Grade Created by: CK-12
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Balance the Pans – Balance-Equality

Teacher Notes

Given a two-pan balance with one pan empty and a weight in the other pan, students determine the boxes that can be placed in the empty pan to balance the pans. Since there is an unlimited number of boxes of each type, there is more than one set of boxes that make the pans balance. In other words, all problems have multiple solutions.

To identify and check the sets of boxes that balance the pans, students will add with more than two addends, all single 0 digit numbers 8 or less. Alternatively, students can count up to find sums. All sums are 18 or less.

Solutions

Balance the Pans 1

\begin{align*}C+ C = 6 \ pounds\end{align*}

\begin{align*}D + D + D + D + D + D = 6 \ pounds\end{align*}

\begin{align*}C + D + D + D = 6 \ pounds\end{align*}

Balance the Pans 2

\begin{align*}F + F + F = 9 \ pounds\end{align*}

\begin{align*}F + E + E + E = 9 \ pounds\end{align*}

Balance the Pans 3

\begin{align*}G + G + G + G + G = 5 \ pounds\end{align*}

\begin{align*}H + G + G + G = 5 \ pounds\end{align*}

\begin{align*}H + H + G = 5 \ pounds\end{align*}

Balance the Pans 4

\begin{align*}J + J + J + J + J = 10 \ pounds\end{align*}

\begin{align*}I + J + J + J = 10 \ pounds\end{align*}

\begin{align*}I + I + J = 10 \ pounds\end{align*}

Balance the Pans 5

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

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

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

Balance the Pans 6

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

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

\begin{align*}N + M + M = 16 \ pounds\end{align*}

(Teacher Page) – You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

\begin{align*}& \mathbf{Describe:} && \text{There are 2 pans.}\\ &&& \text{One pan holds a box. The box weighs 8 pounds.}\\ &&& \text{The other pan is empty.}\\ &&& \text{Box}\ A \ \text{weighs 4 pounds.}\\ &&& \text{Box}\ B \ \text{weighs 2 pounds.}\\ &&& \text{There are lots of boxes.}\\ & \mathbf{My \ Job:} && \text{Figure out which boxes to put in the pan.}\\ &&& \text{The boxes have to weigh 8 pounds in all.}\\ & \mathbf{Plan:} && \text{Try to make 8 pounds using}\ \text{A} \ \text{boxes only.}\\ &&& \text{Try using}\ B \ \text{boxes only.}\\ &&& \text{Try using both}\ A \ \text{and}\ B \ \text{boxes.}\\ &\mathbf{Solve:} && \text{These will make the pans balance:}\\ &&& 2 \ A \ \text{boxes weigh 8 pounds.}\\ &&& 4 \ B \ \text{boxes weigh 8 pounds.}\\ &&& \text{One}\ A \ \text{box and}\ 2 \ B \ \text{boxes weigh 8 pounds.}\\ & \mathbf{Check:} && 4 + 4 = 8\\ &&& 2 + 2 + 2 + 2 = 8\\ &&& 4 + 2 + 2 = 8\end{align*}

Balance the Pans 1

You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

Balance the Pans 2

You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

Balance the Pans 3

You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

Balance the Pans 4

You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

Balance the Pans 5

You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

Balance the Pans 6

You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

Extras for Experts: Balance the Pans – Balance-Equality

Solutions

Extra for Experts: Balance the Pans 1

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

\begin{align*}Q + P + P + P = 15 \ pounds\end{align*}

\begin{align*}Q + Q + P = 15 \ pounds\end{align*}

Extra for Experts: Balance the Pans 2

\begin{align*}R + R + R + R + S = 11 \ pounds\end{align*}

\begin{align*}R + S + S + S = 11 \ pounds\end{align*}

Extra for Experts: Balance the Pans 3

\begin{align*}Z + Z + Z + Z + Z + Z = 18 \ pounds\end{align*}

\begin{align*}Y + Y + Y + Z = 18 \ pounds\end{align*}

Extra for Experts: Balance the Pans 1

You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

Extra for Experts: Balance the Pans 2

You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

Extra for Experts: Balance the Pans 3

You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

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