## MSTAR Interventions

### Instructional Background

#### Preskills Knowledge and Skills for Students

• Understand that a multiplication problem A x B can be represented as the total in A groups of B.1
• Understand that the commutative property of multiplication states that A x B = B x A and apply the property when learning facts.
• Understand that given a whole number A, a whole number B that can be written as a whole number times A is called a multiple of A. (15 is a multiple of 5 because 15 = 3 x 5).1
• Identify the factors and product of a multiplication fact.
• Know what a product, a factor, and multiples are. Be able to find the product, factors, and multiples, given a number (or pair of numbers).

#### Mathematics Content Knowledge for Teachers

• One way to approach multiplication is through repeated addition. Also, equal partitioning is an approach for division. When using the repeated addition model, the second factor corresponds to the number that is repeated, and the first factor corresponds to the number of times it is repeated.2
• Arrays can be used to model multiplication and division problems.1
• Arrays are also valuable in proving the relationships between numbers in a fact family.3
• The quotient of A ÷ B, where A or B are not 0, can be viewed as looking for the unknown number of objects along 1 side of an array (the quotient) when the number of objects in the array (the dividend) and the number of objects along the other side of the array (the divisor) are known.
• The product of A x B, where A or B are not 0, can be viewed as looking for the unknown number of objects in an array (the product) when the numbers of objects along both sides of the array (the factors) are known.
• The commutative property of multiplication states that A x B = B x A. The statement A × B = B × A is true because the product A × B or B × A represents the total number of objects, but each expression represents the groups of objects and number of objects in each group in a different way. Therefore, 2 x 8 = 16 and 8 x 2 = 16.
• Multiplication and division are inverse operations, when 0 is not a factor, which means they can “undo” each other: 2 x 6 = 12, 12 ÷ 6 = 2.2
• Decomposing unknown facts into known facts is possible because of the distributive property: A x (B + C) = (A x B) + (A x C). Example: 7 x 3 = 7 x (1 + 2) = (7 x 1) + (7 x 2) = 7 + 14 = 21.
• Skip-counting by 10s, 5s, 2s, and 1s is the same as saying the multiples of 10s, 5s, 2s, and 1s.
• Products with like factors are multiples. For example, in 2 x 3 = 6, 3 x 3 = 9, and 4 x 3 = 12, the 3 is the like factor and the products (6, 9, and 12) are all multiples of 3.
• A product is the result of multiplying 2 or more numbers (factors).
• A and B are counting numbers and can be written in 1 of these three ways: A ÷ B,    or   . The result of A divided by B is called the quotient, the A is called the dividend, and B is the divisor.
• There are 2 ways to think about division: "How many groups?" or "How many in each group?"
• To answer the question “How many groups?”: If A and B are nonnegative numbers, and B is not 0, then A ÷ B means the number of groups formed when A objects are divided into groups with B objects in each group.1
• To answer the question ”How many in each group?”: If A and B are nonnegative numbers and B is not 0, then A ÷ B means the number of objects in each group when A objects are divided into groups with B objects in each group.1
• Students may use a variety of strategies to find unknown multiplication facts, using known facts. Any fact can be taken apart into different combinations. The most common strategies are listed below.
 2s Double the 1s facts 3s Take apart 3 into 1 + 2 4s Double the 2s 5s Skip-count 6s Double the 3s facts Triple the 2s facts Take apart 6 into 1 + 5 7s Take apart 7 into 2 + 5 8s Double the 4s facts Quadruple the 2s facts 9s Take apart 10 into 1 + 9 Take apart 9 into 4 + 5 12s Take apart 12 into 2 + 10

1. Beckman, S. (2011). Mathematics for elementary teachers with activity manual (3rd ed.). Boston, MA: Addison-Wesley.
2. Scott Foresman & Addison Wesley. (2009). enVision math Texas: Grade 3. Glenview: IL: Pearson Education.
3. Scott Foresman & Addison Wesley. (2009). enVision math Texas: Grade 5. Glenview: IL: Pearson Education.