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Determine whether given pairs of addition and subtraction expressions are equivalent or not.
demonstrate an understanding of the concept of equality by partitioning whole numbers to 18 in a variety of ways, using concrete materials (e.g., starting with 9 tiles and adding 6 more tiles gives the same result as starting with 10 tiles and adding 5 more tiles);
represent, through investigation with concrete materials and pictures, two number expressions that are equal, using the equal sign (e.g., “I can break a train of 10 cubes into 4 cubes and 6 cubes. I can also break 10 cubes into 7 cubes and 3 cubes. This means 4 + 6 = 7 + 3.”).
Cuisenaire rods (1 set per student) Equal
“Combo Quest Spinner”, a paper clip, and a pencil (1 per student) Equation
Paper to record math thinking Combination
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.
Arrange students in physically distanced pairs. Provide each student with a set of Cuisenaire rods and a spinner (Combo quest spinner, a paper clip, and a pencil). Explain the activity:
Students spin the spinner and select the Cuisenaire rod indicated on the spinner.
Students find as many different combinations as possible of Cuisenaire rods that have the same length as the rod indicated by the spinner. For example, if the spinner shows “black”, students might find, among others, the following combinations of Cuisenaire rods:
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.
Students spin the spinner twice, select the Cuisenaire rods indicated by the spinner, and place the two rods end to end.
Students find different combinations of Cuisenaire rods that have the same length as the rods indicated by the spinner. For example, if the spinner indicates the black rod and the red rod, students might find, among others, the following combinations:
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
Relational Rod App (download from Catalogue)
Students will use the Mathies Relational Rods app to explore the Cuisenaire rods
Students will post their findings by taking screenshots of their work and equations from the Mathies app and post to Seesaw OR students may take screenshots of their work and use the recording and drawing tools in Seesaw to share their mathematical thinking
Rather than a spinner, invite students to choose any cuisenaire rods that they would like to explore.