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The Guides for Effective Instruction in Mathematics Revised

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C2.3 Identify and use equivalent relationships for whole numbers up to 50, in various contexts.

Activity 1: Exploration of Properties

Explore the Commutative Property

Example 1

Ask students to make a necklace by stringing 5 red beads and 2 blue beads, then suggest they write the corresponding math sentence.

5 + 2 = 7

A beaded necklace shows a sequence of 5 red beads and 2 blue beads.

Have students turn the necklace and write the corresponding math sentence.

2 + 5 = 7

A beaded necklace shows a sequence of 2 blue beads and 5 red beads.

Then suggest that students compare the two math sentences and answer the following questions:

  • What do you notice about the order of the beads?
  • What do you notice about the amount of beads?
  • Why is the quantity the same in both cases?
  • Change the amount of beads on the necklace and write two math sentences. Compare the math sentences. What do you notice?
  • If you follow the same approach using a large amount of beads on the necklace, such as 100, 300, or 500, what can you say about the equality situations you observe?

Note: Repeat the activity, asking students to move to look at the necklace in the other direction instead of turning it.

Example 2

Ask students to determine if the math sentences below are true or false without performing any calculations.

\(\displaylines{\begin{align}6 + 3 &= 3 + 6 \\ 15 + 14 &= 14 + 16 \\ 7 + 19 &= 19 + 7 \end{align}}\)

After introducing each math sentence, ask students the following questions:

  • How can you tell if this sentence is true or false?
  • What do you notice about the terms in this equality?

Explore the Role of the Number 0 in Addition

Present students with six toys and ask them to count them. Cover the toys with a cloth and tell them that you will add a mystery number of toys.

The figure has 3 airplanes and 3 tow truck in 2 rows. Row 1 has airplane, tow truck, tow truck and row 2 has airplane, tow truck, airplane.

Pretend to add toys, but do not add anything. Remove the cloth and ask the following questions:

  • How many toys are there now?
  • How many toys did I add?

Ask students to write a mathematical sentence that represents the situation (6 + 0 = 6).

Repeat the activity, adding 6 toys to 0 toys to represent (0 + 6 = 6).

Point out to students that since 6 + 0 = 6 and 0 + 6 = 6, we can conclude that 6 + 0 = 0 + 6 and make the connection with the commutativity property of addition.

Encourage students to propose a conjecture about the role of the number 0 in an addition, then ask questions such as:

  • Could you apply your conjecture to other numbers? Try it out.
  • Could you apply your conjecture to all numbers? Why?

Explore the Role of the Number 0 in Subtraction

Present 8 toys to students and ask them to count them. Cover the toys with a cloth and tell the students that you will remove a mystery number of toys.

Pretend to remove toys, but do not remove anything. Remove the cloth and ask the following questions:

  • How many toys are there now?
  • How many toys did I remove?

Ask students to write a mathematical sentence that represents the situation (8 - 0 = 8).

Encourage students to propose a conjecture about the role of the number 0 in a subtraction, then ask questions such as:

  • Could you apply your conjecture to other numbers? Try it out.
  • Could you apply your conjecture to all numbers? Why?

Note: Parentheses are used to ensure mathematical accuracy. However, with elementary students, it is best to stick to less abstract ways of highlighting groupings.

Example

Mathematical equation used to show different ways to regroup. Equation 1 has 2, plus, nine, plus, 5, equals, 2, plus, nine, plus, 5. Number 2, nine, nine, and 5 are circled.Equation 2 has 2, plus, nine, plus, 5, equals, 2, plus, nine, plus 5. Number 2, nine, nine, and 5 have a wave trace.

Activity 2: Using Strategies to Develop Algebraic Thinking

Add a Mystery Number

In pairs, ask each student to build a tower with six interlocking cubes and then ask them the following question: What do you notice when you look at the two towers? (The towers are the same height and the same number of cubes.)

Ask them to add any number of cubes to the towers, while making sure to keep the game tied.

Ask each team to explain how they maintained the tie. Then write their answers on a chart.

Example 1

Group Height of the towers at the beginning Added by student A Added by student B
1 6 3 3
2 6 4 4
3 6 1 1
4 6 5 5
5 6 2 2

Ask students to analyze the data and draw their attention to the fact that the same amount must be added to each of the towers for equality to hold. Point out that this is true for all the quantities added.

Ask them to check if it is possible to add a quantity to a single tower and maintain equality. Point out that the equality is maintained only if nothing is added (0 cubes).

Have one member of each team add a cube to their tower, then ask students the following questions:

  • What do you notice now as you look at the two towers? (The towers do not contain the same number of cubes; one more or one less depending on the situation)
  • What do you need to do to restore equality?

Note: Students should be able to see the inequality and explain it before they try to restore equality.

Have one member of each team add 2-5 cubes to their tower, as desired.

Ask the other student to make the two towers equal.

Ask each team to explain how they made the towers equal. Point out to students that each team had to add the same amount of cubes to the second tower as they added to the first tower to make them equal. Draw their attention to the fact that this applies to all the quantities added.

Decompose Numbers According to Position Values 

Introduce students to the mathematical sentences below and ask them if they are true or false:

\(\displaylines{\begin{align}47 &= 40 + 7 \\ 35 + 12 &= 35 + 10 + 2 \\ 17 + 21 &= 10 + 20 + 7 + 1 \end{align}}\)

To encourage students to justify their answers, ask them the following questions:

  • How can you tell if this mathematical statement is true (or false)?
  • Is it possible to test for equality using concrete materials, such as 10 frames, base 10 materials, and interlocking cubes?
  • How does this material help determine if the sentence is true or false?
  • Is it possible to verify the equality without concrete materials and without doing any calculations? How? (By comparing the tens and ones on each side of the = sign, it is possible to check the equality without using concrete materials)

Cancel Equal Terms or Expressions

Write the following mathematical sentence: \(3 + 14 + 32 = 32 + 14 + 3\).

Ask students the following questions:

  • Is this mathematical sentence true or false? How do you know? (Some students will use the commutative property.)
  • If I remove the numbers 32 on either side of the = sign, does the equality remain true?
  • If I remove the numbers 14 on each side of the = sign, does the equality remain true?

Example 2

Present students with the math sentences below and ask them to check whether they are true or false.

\(\displaylines{\begin{align}3 + 4 + 2 &= 3 + 3 + 2 \\ 5 + 6 + 8 &= 6 + 5 + 8 \\ 11 + 7 + 4 &= 5 + 6 + 7 + 4 \end{align}}\)

For each math sentence, ask students the following questions:

  • Is this sentence true or false?
  • If you cancel terms that are the same on either side of the = sign, is the equality still true? How do you know?

Compare Terms

Present the following mathematical sentences:

\(\displaylines{\begin{align}9 + 8 &= 10 + 7 \\34 + 17 &= 35 + 16 \\13 + 28 &= 16 + 25 \end{align}}\)

Ask students to check, using a double open number line, if these sentences are true.

For each sentence, ask students the following questions:

  • How can you say that this sentence is true?
  • What terms did you compare?
  • What do you notice about these terms?
  • Can you compare the terms in another way?

Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 101-108.

Activity 3: Representing Quantities to Understand Equivalence Relations

When numbers are decomposed, the sum of the parts is equivalent to the whole.

The various activities below lead students to understand that, even if the quantity is broken down in different ways, it remains the same. The concrete and semi-concrete representations can later be combined with a symbolic representation.

Example

\(\displaylines{\begin{align}50 &= 25 + 25 \\ 49 + 1 &= 1 + 49 \\ 20 + 20 &= 10 + 10 + 10 + 10 \end{align}}\)

Represent 40 by making different jumps on a number line or on a beaded line.

Mystery Number - Group students into pairs and invite them to represent a number in different ways (concrete or semi-concrete materials, symbols). Create a gallery where students circulate and see the different ways other teams have represented their number.

Activity 4: A Handful of Cubes

Summary

The purpose of this activity is to have students compare two sets of objects, that is, to determine which set has more objects, which set has fewer objects, or whether the quantity of objects in the two sets is equal. They will also need to re-establish equality between the quantities in the unequal sets.

Materials

  • Appendix MJ.1 (one wheel per team)
  • interlocking cubes of two different colours (30 to 40 per team)
  • sheets of paper (one per team)

Directions

Group students into pairs and model the activity with one team. Give each team a sheet of paper on which to record the score and 30 to 40 interlocking cubes of two different colours.

Here are the rules of the game:

  • Each student takes a handful of cubes of one colour.
  • A student spins the arrow on the wheel and reads the expression on which the arrow has stopped:
    • If the arrow has stopped in the "more than" section, the child with the most cubes wins a point.
    • If the arrow stopped in the "less than" section, the child with the least amount of cubes wins a point.
    • If the arrow stopped in the "is equal to" section, both students work together to equalize the quantities of cubes and each earns a point. The student must then determine the equivalence.
    • The game continues until a child accumulates 10 points.

During the math exchange, ask students to explain the strategy(ies) used to equalize quantities.

Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 122.

Activity 5: Feufino, the Dragon

Summary

In this activity, students represent, using mathematical sentences, all possible decompositions of the number 4.

Materials

Directions

After reading dragon tales to students, present them with the following imaginary situation:

A dragon named Feufino lives on a mountainside near Colouriville. In turn, all the dragons in this town have the four beautiful spikes on their tails painted in two different colours. When it's Feufino's turn, he can't decide how many points to paint in each colour.

Introduce students to the illustration of Feufino, the Sad Dragon (Appendix 1.5), saying:

"Feufino needs our help. To help him make a decision, we need to find all the ways he can get his four spikes painted."

Model a possible colour distribution using one of the sets of interlocking cubes; for example, one blue cube and three red cubes represent one of the colour distributions on the four points. Then ask students to find as many ways to distribute the colours as they can and write a mathematical sentence on a sheet of paper that corresponds to the colour distribution found (for the modelled example, it is 1 + 3 = 4 or 4 = 1 + 3).

Ask students to explain how they can know if they have all the possible distributions of two colours. Some might base their explanation on the following pattern:

\(4 = 3 + 1\)

\(4 = 2 + 2\)

\(4 = 1 + 3\)

(-1) (+1) (pattern in each column)

Change in A Relationship 

In a situation of equality, a change in one of the terms of the expression necessarily leads to a change in the other term.

Pool all possible decompositions of the number 4. Ask students questions such as:

  • Are all the distributions of two colours equal to 4? How can you tell?
  • Does the mathematical phrase (4 = 1 + 3 have the same colour distribution as the phrase 4 = 3 +1?

Give each student a copy of Appendix 1.6 and suggest that they colour the spikes on Feufino's tail according to their preferred colour distribution.

Note: This activity can be repeated using a different number of colours or spikes.

Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 143-144.