A Balancing Act

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Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
Specific Learning Outcomes

Solve problems using a combination of addition, subtraction, multiplication and division mental strategies.

Description of Mathematics

Number Framework Stage 6

Required Resource Materials

Counters and tens frames and paper to represent bags. Alternative materials are balances and weights, double-sided counters, Cuisenaire rods, or Animal strips.

Activity

Many students think the = sign means “get the answer”. This is, after all, what “equals” means on a calculator. Here the equals sign is extended from the current meaning of “find the answer” to include: “The left-hand side (of the equals sign) equals (balances) the right-hand side.” 

Because this is an important concept, a number of pieces of equipment can be used. Choose the equipment that works best for the students and is most readily available.

Using materials (Cuisenaire rods or animal strips)

Use the 1 (white), 2 (red), 3 (lime green), 4 (crimson), 5 (yellow), 6 (dark green), 7 (black), 8 (brown), 9 (blue), and 10 (orange) rods.
Choose one of the large Cuisenaire rods per pair of students.
As an example, they may take the 10 (orange rod) and put several smaller rods underneath it.
They then record the number sentence that matches what they have done.
If they have placed a 5 (yellow), a 3 (lime green), and a 2 (red), they would record 10 = 5 + 3 + 2 and read it as “Ten is the same as five plus three plus two.”
Have them explore other combinations that are equal to ten.
They then remove one of the small rods and show it to another pair.
The other pair records what they can see. If they removed the 2 rod from the previous example, they would write 10 = 5 + 3 + ?.
The pair would then look at the number sentence and identify the missing number needed to balance the equation.

Using imaging (with Cuisenaire rods or animal strips)

Problem: Behind this card, I am using a pair of rods. I will choose a 4 rod and a 6 rod. I am now picking up a 3 rod to go underneath the 4 and the 6 rod. What rod will I need to select next so that the total length of the two pairs of rods will be the same? How could we record that? (4 + 6 = 3 + ?) Read this as “four plus six is the same as three plus [seven].”

Using materials (with scale balances)

A scale balance can also be used to show equality.
If this is not available, use a coat hanger and string bags, or yoghurt pottles with handles tied to each end.
Marbles can be used in the pottles.

Examples:
4 + 3 = 5 + ?
5 + 1 = 4 + ?
6 + 1 = 5 + ?
5 + 2 = 4 + ? …

Using imaging (with scale balance)

Alternatively, an image of a scale balance can be used and a different combination of blocks placed on each side to build number sentences that balance.
Activity AL 7111 from the Assessment Resource Bank supports this idea – see the ARB web page

Using materials (double-sided counters)

Have the students work in pairs.
Both get eight double-sided counters.
They check to agree that they are both starting with the same number of counters.
Partner B takes some of their counters and hides them under a piece of card.
Partner A throws their counters on the floor and records what they see (e.g., “I have five red and three yellow counters”).
Partner B places the counters that are still in their hand down on the floor, all with the red side showing, and says: “I had four red counters in my hand. How many yellow counters must be hiding?”
They discuss how they could record this.
After discussion with the teacher, they record 5 + 3 = 4 + ?.
Partner B produces their counters from under the card and checks that what they have is represented by what they have recorded: 5 + 3 = 4 + 4.
They read: “five plus three is equal to four plus four”.
Repeat with other amounts.

Using imaging (with double-sided counters)

I have seven counters in my hand. I throw them down behind this card. Four are red, and three are yellow. Record the equation that shows this. Now I throw another seven counters, and this time five are red. How many must be yellow? How do we record this? (4 + 3 = 5 + l)

Examples: Word problems and recording for:
2 + 4 = ? + 3
5 + 5 = ? + 6
5 + ? = 4 + 3
? + 6 = 6 + 3
? + 6 = 8 + 2
1 + 7 = 4 + ? …

Using number properties

Work out 67 + ? = 66 + 44. Discuss why the answer is one less than 44 (because 67 is one more than 66 and balance has to be maintained). Record 43 in the square. 

Examples: Word problems and recording for:
42 + 38 = ? + 39
55 + 35 = ? + 34
105 + ? = 103 + 43
? + 65 = 67 + 33
? + 180 = 190 + 38
1 020 + ? = 1 010 + 67 …

Challenging examples:
442 – 38 = ? – 39
585 – 35 = ? – 34
105 – 56 = 103 – ?
? – 65 = 667 – 62
? – 280 = 790 – 290 …

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Level Three