Expressions and equality

C 2.2

Determine whether given pairs of addition and subtraction expressions are equivalent or not.

Since young students often view equations in a format such as 4 + 5 = 9, they can develop misconceptions about the meaning of the equal sign. Specifically, students may think that the equal sign means “gives an answer of”, rather than “is the same quantity as”. Given 4 + 5 = __, students might answer “9”, not because they consider the equality on both sides of the equation, but because they believe the equal sign is a prompt for an answer. Students’ misconceptions about the equal sign are often not apparent to teachers until students are asked to complete an equation such as 3 + 5 = __ + 2. Students may believe that “8” is the missing numeral in the equation because they think that the equal sign is asking them to add 3 and 5. Students who have misconceptions about the equal sign also have difficulty recognizing that equations in unfamiliar formats are true (e.g., 7 = 3 + 4, 5 + 2 = 1 + 6).

It is important that young students understand the following: (a) the equal sign is a symbol that separates both sides of an equation, and (b) the equal sign states that the right side of the equation is the same quantity as the left side (e.g., in 5 = 3 + 2, 5 is the same quantity as 3 + 2). When students understand that the equal sign describes the relationship of equality between two quantities, they begin to focus their reasoning on the quantities represented by both sides of the equation.

In the following learning activity, students investigate equalities, using Cuisenaire rods. After finding combinations of rods that represent the same value, they translate these ideas of equality into symbolic equations. The activity provides students with an opportunity to develop an understanding that the equal sign means “the same quantity as” in equations.

• On a sheet of paper, students record the different combinations of Cuisenaire rods and their corresponding equations. (Students can find the value of each colour by referring to the chart that shows the numeric values for the different colours of Cuisenaire rods.) For example, the Cuisenaire rods illustrated on the previous page and their corresponding equations might be recorded in the following way:

  • Black = red + yellow 7 = 2 + 5

  • Black = purple + light green 7 = 4 + 3

  • Black = red + red + light green 7 = 2 + 2 + 3

• Provide students with an opportunity to explore different combinations of Cuisenaire rods that have the same length and to record corresponding equations.

• On a sheet of paper, students record the different combinations of Cuisenaire rods and their corresponding equations. For example, the Cuisenaire rods illustrated above and their corresponding equations might be recorded in the following way:

  • Black+red=yellow+purple 7+2=5+4

  • Black + red = light green + dark green 7 + 2 = 3 + 6

  • Black+red = red+red+red+light green 7+2 = 2+2+2+3

• As students find and record different combinations of rods and the corresponding equations, ask questions such as the following:

  • “What is the value of the two rods indicated by the spinner?”

  • “How can you find different combinations of rods that have the same value?”

  • “What does this equation mean?”

  • “How do you know that this part of the equation is equal to this other part?”

  • “How could you prove that this equation is true?”

• Provide students with an opportunity to repeat the activity of finding different combinations for two rods indicated by the spinner.

As you move around the room, take note of the various strategies being used. Routinely notice and discuss with students when they persevere. Give authentic feedback when students persevere (e.g., “I noticed that you rearranged the math manipulatives, bars, until you found the solution,” “I noticed you respectfully asked a peer for help and tried that suggestion,” “I noticed that you looked at the chart to decide which strategy you could try next.”).

Gather the students to discuss the activity. Pose the following questions to promote discussion:

• “What did you notice about the equations?

• “Once you found an equation, could you change it to come up with another one? How?”

• “How did this activity help you find equations?”

• “What are some of the equations you discovered?”

• “How could you use Cuisenaire rods to prove that 5 + 4 = 2 + 6 + 1?”

• “How could you use other manipulatives to represent this equation?”

• “For which two rods were you able to find many equivalent combinations of rods? Why?”

• Social Emotional Learning Consolidation: “Why do you think you liked this problem, especially?,” “Why do you think you like solving those kinds of problems/puzzles?,” “Will you share with me the strategy that helped you solve this problem?”.

For students who experience difficulties, simplify the activity by having them find combinations of rods that equal another rod. For example, students might find combinations that equal the brown rod (e.g., white and black, yellow and light green) and then record the corresponding equations (e.g., 8 = 1 + 7, 8 = 5 + 3).

Some students may not be ready to represent equations symbolically (e.g., 2 + 2 = 3 + 1). Have these students describe equalities orally and/or in written form (e.g., “7 and 2 is the same as 6 and 3”). Demonstrate how to record these ideas in symbolic notation by following a structure (i.e., __ + __ = __ + __ ).

For students who require a greater challenge, have them find different combinations that are equal to three Cuisenaire rods (e.g., find combinations of rods that equal the light green rod, the purple rod, and the yellow rod).

Assess students’ understanding of equality and equations by observing students during the learning activity.

• Are students able to find different equivalent combinations of Cuisenaire rods?

• Are students able to represent equations, using symbolic notation?

• How well do students explain the meaning of equation?

• How well do students explain the meaning of the equal sign?

Social Emotional Learning: SEL Self Assessments (French and English) / Seesaw Activities Library SEL Self Assessments for Primary