Misconceptions |
Examples for Preventing or Correcting |
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Some students believe repeated addition is the only definition of multiplication. |
Teach students that repeated addition is only 1 way to represent multiplication. Explain to students that if A and B are nonnegative numbers, then A x B is the total of A groups of B.^{1} |
Some students universally believe multiplication can be used when adding groups. |
Teach students that repeated addition of the same number of objects is 1 way of thinking of multiplication. Teach students that multiplication cannot be used when the number of objects in each group is not the same. Present examples and nonexamples (e.g., 5 + 5 + 5, 4 + 5 + 3). Demonstrate and have students differentiate between equal and unequal groups as well as identify when multiplication can and cannot be used.^{3} |
Some students believe that 4 x 3 and 3 x 4 have different answers. |
Illustrate the commutative property of multiplication, using array models to prove the total (product) is the same.^{2} |
Some students may believe that 30 = 5 × 6 is written incorrectly because the product (30) must follow the equal sign. |
Teach students the meaning of the equal sign and explain that the equal sign means “equals” or “is equal to” and that the expressions on each side of the equal sign have the same value.^{2} |
Some students do not connect the rows with the columns in a multiplication table. |
Draw attention to the row as each column is completed. Provide additional instruction on the commutative property of multiplication and the multiplication table’s design as needed. |
Some students believe that performing a strategy, such as doubling, changes the total number in the array. |
Teach students that the doubling strategy changes how the arrays look but not the total number. Use visualizations and manipulatives as needed. |
Students may assume that the commutative property also holds for division—for example, assuming that 15 ÷ 3 = 5, so 3 ÷ 15 = 5. |
Demonstrate an example, such as the following. Have 15 sheets of paper to share among 3 people. Ask students, "How many sheets of paper does each person get?” (5) Have 3 sheets of paper to share among 15 people. Ask students, "How many sheets of paper does each person get?" () For each demonstration, write the equation on the board. Draw attention to the quotients, which are different. |
Some students may confuse fact families with the set of a number and all its factors (12: 1, 2, 3, 4, 6, 12). |
Teach students that a family of facts consists of 3 numbers, 2 of which are the factors that when multiplied equal the product. |
Some students may believe a family of facts consists of any 2 factors of a product and the product. |
Emphasize that the equation constructed with these numbers must be true. For example, if students offer 5, 10, and 20, ask what the equation is (5 x 10 = 20) and whether it is true. |
Some students may need a more concrete model showing how multiplying by powers of 10 works. |
If so, use base-ten blocks to show 10, 100, and 1,000, as well as 20, 200, and 2,000. Show students how, in each place, the number in the second group is 2 times larger than the number in the first group: 2 is 2 times larger than 1, 20 is 2 times larger than 10, etc. The factor is always 2, and the number of 0s represents the other factor: 10, 100, or 1,000. |