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# 3.1: Introduction

Difficulty Level: At Grade Created by: CK-12

Which Boxes? – Balance-Equality

Teacher Notes

Given a two-pan balance with one pan empty and a weight in the other pan, students determine which boxes from a set of two or three boxes when placed in the empty pan will cause the pans to balance. Since there is an unlimited number of boxes of each type, there is more than one set of boxes that make the pans balance. So, all problems have multiple solutions.

Solutions

Which Boxes? 1

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

H+L+L+L=9 pounds\begin{align*}H + L + L + L = 9 \ pounds\end{align*}

G+H+L=9 pounds\begin{align*}G + H + L = 9 \ pounds\end{align*}

Which Boxes? 2

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

F+F+F+E=5 pounds\begin{align*}F + F + F + E = 5 \ pounds\end{align*}

E+E+F=5 pounds\begin{align*}E + E + F = 5 \ pounds\end{align*}

E+D=5 pounds\begin{align*}E + D = 5 \ pounds\end{align*}

D+F+F=5 pounds\begin{align*}D + F + F = 5 \ pounds\end{align*}

Which Boxes? 3

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

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

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

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

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

Which Boxes? 4

R+R+R+R+R=15 pounds\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 \ pound\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 use a different set of boxes? Explain.

\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*}

Which Boxes? 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.

Which Boxes? 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.

Which Boxes? 3

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you use a different set of boxes? Explain

Which Boxes? 4

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you use a different set of boxes? Explain

Extras for Experts: Which Boxes? – Balance-Equality

Solutions

Extra for Experts: Which Boxes? 1

\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*}

Extra for Experts: Which Boxes? 2

\begin{align*}P + P + P + P + P + P + P = 14 \ pounds\end{align*}

\begin{align*}N + P + P + P + P + P = 14 \ pounds\end{align*}

\begin{align*}N + N + P + P + P = 14 \ pounds\end{align*}

\begin{align*}N + N + N + P = 14 \ pounds\end{align*}

\begin{align*}M + M + P + P = 14 \ pounds\end{align*}

\begin{align*}M + M + N = 14 \ pounds\end{align*}

Extras for Experts: Which Boxes? 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.

Extras for Experts: Which Boxes? 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.

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Date Created:
Feb 23, 2012