The Equal Sign as “Same As” in a Missing-Addend Sentence
Mr. Patel’s second graders are working on number sentences. He writes 8 + 4 = __ + 5 on the board. Most students write 12. Aisha writes 7. When Mr. Patel asks her to explain, she says, “8 plus 4 is 12, and 7 plus 5 is 12, so both sides are the same.” Several students insist Aisha is wrong because “the answer comes after the equals sign.” Earlier in the year the class solved facts to 20 and used a balance scale once. The standard asks students to understand the meaning of the equal sign and find the unknown number in an equation. Mr. Patel has a balance scale, linking cubes, and a number line. He wants to build relational thinking without turning the balance scale into a rule-following procedure.
1. Why is 12 a predictable error, and what understanding do students need in order to see that 7 is correct?
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Second graders often learn the equals sign as a signal to compute: “8 + 4 =” means “give the answer.” In that operational view, the blank is the answer slot, so 12 feels correct. The students are not ignoring the + 5; they are treating the equals sign as a command rather than as a statement of equivalence. To see why 7 works, students need a relational view: the equals sign means “has the same value as,” and the entire expression on each side is a quantity. They must compare the value of 8 + 4 with the value of __ + 5, not simply put a number after the equals sign. Aisha’s explanation shows this: she computes both sides and checks that each is 12. The number 7 is correct because 8 + 4 = 12 and 7 + 5 = 12. The instructional goal is not to teach “subtract 5 from 12” as a trick but to build the idea that an equation is a balanced statement. Students should be able to decide whether a sentence is true or false and to justify with a representation, not just fill a blank.
2. Design a true/false number-sentence routine using the available materials that would develop relational thinking. Include specific examples and questions.
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Begin with sentences that are true or false and ask students to justify, not just compute. Examples: 8 + 4 = 12 (true); 8 + 4 = 7 + 5 (true); 8 + 4 = 12 + 5 (false); 15 = 9 + 6 (true); 6 + 6 = 10 + 2 (true); 5 + 3 = 3 + 5 (true). Write each on the board and ask: “Is this true or false? How do you know? What does the equals sign mean in this sentence?” For 8 + 4 = 7 + 5, some students will say false because “7 + 5 is not the answer to 8 + 4.” Ask them to build both sides with linking cubes: 8 and 4 on one side, 7 and 5 on the other. Then ask: “Do the two sides have the same total? What does that tell us about the equals sign?” Use the balance scale only after students have predicted; ask them to explain why it balances or tips, rather than reciting “both sides must be equal.” A useful follow-up is an open sentence such as 10 + 3 = __ + 6, where students can find more than one way to make the sentence true if the blank need not be a single number? Actually, for second grade, keep one unknown and ask for a value that makes it true.
3. How can Mr. Patel respond to the students who insist Aisha is wrong, while still validating their correct computation of 8 + 4?
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He can separate the correct part of their thinking from the assumption about the equals sign. For example: “You are right that 8 + 4 is 12. Aisha also says 7 + 5 is 12. So what would it mean if both sides are 12? Does the equals sign say ‘put the answer here’ or ‘these two sides are the same’?” This honors their computation while making the meaning of the symbol the object of discussion. He can write 8 + 4 = 12 and 8 + 4 = 7 + 5 side by side and ask: “What is the same about these two sentences? What is different?” The first has a single number on the right; the second has an expression. Students can test with cubes or the balance scale. If they still object that “the answer should be after the equals sign,” Mr. Patel can ask: “In 15 = 9 + 6, where is the answer? Does that sentence still make sense?” The goal is to broaden their definition of a number sentence, not to tell them their earlier learning was wrong. Aisha’s explanation should be positioned as a mathematical argument the class can test.