UK Secondary (7-11)
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Weighing Equations (Investigation)
Lesson

Objectives

  • To understand that equations must always remain balanced.
  • To visualise how to balance equations.
  • To visualise how to solve one step equations.
  • To visualise how to do number problems.

Materials

  • 1 Clothes hanger
  • String
  • Scissors
  • Two identical small paper/plastic cups 
  • Tape
  • 20 Full Smarties candies
  • Paper
  • Marker
  • Long stick or pole (must be 60-90cm long)
  • Two Chairs

Procedure

To Make the Pan Balance
  1. First, take one of the small cups and loop the string around it, then double knot the string to secure it.
  2. Next, cut the string so it is just long enough to be tucked under the loop around the rim of the cup. Make sure there is enough string so that there is a little handle at the top of the cup.
  3. Double knot the string that was tucked under. 
  4. Repeat the first three steps on the second cup.
    Careful!

    Try to make the size of the handles at the top of both cups the same.

  5. Cut out $2$2 pieces of string that are both 25cm long.
  6. Tie one string to each side of the hanger. Make sure that the strings are tied in about the same place on either side.
  7. Tie each of the strings around the handle on the top of the cups.
  8. Place the two chairs back to back. They should be just far enough away from each other that either end of the stick can be placed on the tops of the chairs and balanced between them.
  9. Place the stick so that the ends of it are on the tops of the chair backs.
  10. Hang the clothes hanger with the cups attached from the pole. The final set up should something look like the picture shown.
  11. Make sure that the weight of the cups is evenly distributed and that the hanger is not tilting to one side before continuing.

 

To set up the smarties:

  1. Use your scissors to cut out $21$21 small pieces of paper about an 2cm square.
  2. Draw a negative sign, “-” on $10$10 of the pieces of paper.
  3. Draw a positive sign, “+” on $10$10 pieces of paper.
  4. Draw an ‘“$x$x” on $1$1 piece.
  5. Take $10$10 full Smarties packages and tape one piece of paper marked with a negative sign “-” to each.
  6. Take another $10$10 full Smarties packages and tape one piece of paper marked with a positive sign “+”  to each.

How it works:

  • One full Smartie package with a “+” sign on it will represent a +$1$1 value.
  • One full Smartie package with a “-” sign on it will represent a -$1$1 value.
  • The one piece of paper labeled with an “$x$x” on it represents the variable $x$x.
  • When there is 1 Smartie package with a “+” sign on it and 1 Smartie package with a “-” sign on it in the same cup they will cancel out (because $1-1=0$11=0).

 

Examples of how to use the pan balance

Example 1
  • Say you have the equation $x-1=1$x1=1 :

  • Place the piece of paper labeled with an "$x$x" into one cup along with $1$1 of the Smarties labeled with a “-.”

  • In the other cup, place $1$1 Smarties package with a “+” label on it.

  • Notice that the hanger tilts to one side and one cup is higher than the other.

  • Think about what value for “$x$x” will balance the equation and the hanger.

  • Here, I think that value will be $2$2 so I place $2$2 full Smartie packages with a “+” sign on them into the same cup that has the piece of paper labeled “$x$x.”

  • I notice there is 1 Smartie package with a “+” sign on it and 1 Smartie package with a “-” sign on it in the same cup as the “$x$x” now, so I remove them since they cancel each other out. This leaves only 2 Smarties packages in each cup.

  • The cups are now balanced so I know I was right and $x=2$x=2.

 
Example 2
  • Say you have the equation $2-x=1$2x=1:
  • Place the piece of paper labeled with an "$x$x" into one cup along with $2$2 of the Smarties labeled with a “+.”
  • In the other cup, place 1 Smarties package with a “+” label on it.
  • Notice that the hanger tilts to one side and one cup is higher than the other.
  • Think about what value for “$x$x” will balance the equation and the hanger.
  • Here, I think that value will be 1 so I place 1 full Smartie packages with a “-” sign on them into the same cup that has the piece of paper labeled “$x$x.”
  • I notice there is 1 Smartie package with a “+” sign on it and 1 Smartie package with a “-” sign on it in the same cup as the “$x$x” now, so I remove them since they cancel each other out. This leaves only 1 Smarties package in each cup.
  • The cups are now balanced so I know I was right and $x=1$x=1.

 

Questions

Work on your own or in small groups to answer the following questions using the pan balance and the Smarties as done in the examples.

  1. Determine if $x=4$x=4 is a solution to $x+1=5$x+1=5. Why or why not?
  2. Determine if $x=3$x=3 is a solution to $5-x=2$5x=2. Why or why not?
  3. Determine if $x=8$x=8 is a solution to $x-4=1$x4=1. Why or why not?
  4. Determine if $x=8$x=8 is a solution to $x-6=2$x6=2. Why or why not?
  5. Solve the equation using the pan balance: $x+2=7$x+2=7
  6. Solve the equation using the pan balance: $9-x=1$9x=1
  7. Solve the equation using the pan balance: $7-x=3$7x=3
  8. When $4$4 is subtracted from a number (call it $x$x), the result is $2$2, what is the number? Write an expression and use the pan balance to solve it for the value of $x$x.
  9. A number (call it $x$x) plus $3$3 equals $4$4. Write an expression and use the pan balance to solve it for the value of $x$x.
  10. Some number (call it $x$x) minus $6$6 is $3$3. Construct an equation and use the pan balance to find the value of $x$x.

Discussion Questions

  1. Why do you think the pan balance is a good representation of an equation with one variable?
  2. Are there multiple answers for the value of $x$x in these kinds of equations? Why or why not?

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