Understanding Multiplication and Division

Primary Standards:

MAFS.3.OA.1.1Interpret products of whole numbers, e.g. interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.  For example, describe a context in which a total number of objects can be expresses as 5 × 7. 

MAFS.3.OA.1.2Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. 

MAFS.3.OA.1.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 

Connecting Standards:

MAFS.3.OA.3.7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 • 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. 

Content Knowledge:

One of the most significant math goals for 3rd graders is developing a deep understanding of multiplication and division.  These operations are active, and understanding them is about recognizing the actions, or the situations, associated with each operation.  Rather than concentrate on finding solutions for problems, it is more important in this Unit that students focus with hands-on investigations through modeling of word problems, to notice situations that are associated with multiplication and division and recognize the actions of these operations. 

Visualizing problem situations helps students to better understand multiplication, for example group models, arrays, measurement situations and number lines.  Through opportunities to build or draw models for situations, students develop an understanding of the operation.  Using similar models to represent division problems helps students visualize division as the inverse of multiplication.  This will be explored more in the Connecting Division to Multiplication Unit.    Along with understanding situations of these operations, students must also understand the expressions/equations.  Having students tell what each number represents, or writing their own story problems, provides practice in interpreting the numbers and symbols in the expressions/equations.


GCG 1 – Learning Goal: As a mathematician, I can Use models to represent equal group multiplication MAFS.3.OA.1.1, MAFS.3.OA.1.3

  • Step 1: Use concrete and pictorial models to represent equal group situations (identify the number of groups and items in each group to find the total)
  • Step 2: Make connections of the relationship between repeated addition, counting groups, and multiplication equations
  • Step 3: Model and solve equal group multiplication problems

GCG 2 – Learning Goal: As a mathematician, I can Use models to represent array and measurement multiplication MAFS.3.OA.1.1, MAFS.3.OA.1.3

  • Step 1: Model and represent multiplication situations with arrays, equations, and expressions (identify the number of rows and items in each row to find the total)
  • Step 2: Model and represent multiplication situations with number lines, bar models, and equations/expressions (identify the number of groups and items in each group to find the total)
  • Step 3: Model and solve array and measurement multiplication problems

GCG 3 – Learning Goal: As a mathematician, I can Use models to represent equal group division MAFS.3.OA.1.2, MAFS.3.OA.1.3

  • Step 1: Use concrete and pictorial models to represent equal sharing (partitive) division situations
  • Step 2: Identify the unknown (as the number of groups or the number in each group) and connect models to division expressions/ equations
  • Step 3: Model and solve sharing (partitive) and grouping (measurement) division problems

GCG 4 – Learning Goal: As a mathematician, I can Use models to represent array and measurement division MAFS.3.OA.1.2, MAFS.3.OA.1.3

  • Step 1: Model and represent division situations with arrays, equations, and expressions (identify which part of the array represent the unknown – rows or number in each row)
  • Step 2: Model and represent grouping (measurement) division situations with linear models (number lines or bar models)
  • Step 3: Model and solve array and grouping (measurement) division problems

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