N3.a. Add, subtract, multiply, and divide in multiple ways. Compare advantages and disadvantages of each method.

N3.b. Examine hypothetical or actual student calculation methods and decide if the methods are valid or not.

N3.c. Explain how to use properties of operations to make some calculations such as 24 x 25 easy to carry out mentally and write strings of equations, such as 24 x 25 = (6 x 4) x 25 = 6 x (4 x 25) = 6 x 100 = 600, to show how properties of operations support the “mental math.”

N4.a: Describe advantages and disadvantages of different concrete, pictorial, and abstract representations. Make connections amongst these representations and make connections between the representations and the numerical written methods, including the standard algorithm.

O2.a. Make sense of models for addition, subtraction, multiplication and division that progress from concrete through pictorial to abstract. Justify the choice of model for the context of a problem. Write and solve equations with the unknown in each location.

O2.b. State, recognize and apply:

• Associative property of addition

• Associative property of multiplication

• Closure of addition

• Closure of multiplication

• Commutative property of addition

• Commutative property of multiplication

• Distributive property

• Identity property of addition

• Identity property of multiplication

• Zero property of multiplication

• Additive inverse property

• Multiplicative inverse property

Understand that students may multiply in different ways using multiple representations.

Connecting the structure of multiplication to the properties of multiplication.

Connect multiplication with addition.

Connecting multiplication with different representations.

Making Sense Series - Multiplication Video (Graham Fletcher)

Developing Essential Understanding of Multiplication and Division for Teaching Mathematics in Grades 3-5 Book (Albert Otto, Janet Caldwell, Sarah Wallus Hancock, and Rose Mary Zbiek)

Beckmann (2017) Resources:

• Chapter 4

Billstein (2016) Resources:

• Sections 3-3, 3-5

Sowder (2017) Resources:

• Chapter 3

Van De Walle (2019) Resources:

• Chapter 12

N3.a: Add, subtract, multiply, and divide in multiple ways. Compare advantages and disadvantages of each method.

N3.b. Examine hypothetical or actual student calculation methods and decide if the methods are valid or not.

N3.c. Explain how to use properties of operations to make some calculations such as 24 x 25 easy to carry out mentally and write strings of equations, such as 24 x 25 = (6 x 4) x 25 = 6 x (4 x 25) = 6 x 100 = 600, to show how properties of operations support the “mental math.”

N4a: Describe advantages and disadvantages of different concrete, pictorial, and abstract representations. Make connections amongst these representations and make connections between the representations and the numerical written methods, including the standard algorithm.

O2.a. Make sense of models for addition, subtraction, multiplication and division that progress from concrete through pictorial to abstract. Justify the choice of model for the context of a problem. Write and solve equations with the unknown in each location.

Understand that students may divide in different ways using multiple representations.

Connecting the structure of division to the properties of multiplication.

Connect division with subtraction and multiplication.

Connecting division with different representations.

Making Sense Series - Division Video (Graham Fletcher)

Developing Essential Understanding of Multiplication and Division for Teaching Mathematics in Grades 3-5 Book (Albert Otto, Janet Caldwell, Sarah Wallus Hancock, and Rose Mary Zbiek)

O1.a:. Identify the operation(s) needed to solve specific word problems. Reinforce reasoning and meaning making for problem interpretation and reduce reliance on keywords. Use appropriate vocabulary.

O1.c: Use all of the different problem types for multiplication and division (Equal Groups, Arrays, Area, Multiplicative Comparison) to develop meaning for these operations.

O1.d: Explore and discuss the different ways remainders can be interpreted when solving division problems. In particular, identify the different meanings of the remainder in a partitive vs. a measurement context.

O2.a. Make sense of models for addition, subtraction, multiplication and division that progress from concrete through pictorial to abstract. Justify the choice of model for the context of a problem. Write and solve equations with the unknown in each location.

These categories help students develop a schema to separate important information and to structure their thinking (Van de Walle, Karp, Lovin, Bay-Williams, 2018, p. 117).

Helps students develop generalized strategies.

"Multiplication and division word problems: Improving students' understanding" Article in Arithmetic Teacher vol. 37, issue 2, pgs. 8-12.

"Children's solution strategies for equivalent set multiplication and division word problems." Article in Journal for Research in Mathematics Education vol. 20, issue 2, p. 147-158.

"When the answer is the question." Article in Teaching Children Mathematics vol. 23, issue 1, pgs. 30-37.