Understanding Area and the Distributive Property

Primary Standards:

MAFS.3.MD.3.5: Recognize area as an attribute of plane figures and understand concepts of area measurement.

a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.

b. A plane figure that can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

MAFS.3.MD.3.6: Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

MAFS.3.MD.3.7: Relate area to the operations of multiplication and addition.

a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of ab and ac. Use area models to represent the distributive property in mathematical reasoning.

d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

MAFS.3.OA.2.5: Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Content Knowledge:

In this Unit, students will explore another attribute of plane figures, area.  Area tells how much of a surface is covered, measured in square units.  The area of a figure is the number of same-sized square units that cover the figure without gaps or overlaps.  Students’ initial experiences with area are meant to help them visualize the concept as they explore finding areas by counting the number of square tiles that cover rectangular regions.

Problems with real-world contexts help students understand applications of area in our lives, such as how much carpet or tile is needed to cover a floor.  Exploring area through problems allows students to practice their measurement skills as well as develop a deeper understanding of when and how area is used. As students investigate the areas of various rectangular figures, they gather and observe data that lead them to insights about the connection between multiplication and area.  Without being told formulas, students are able to determine shortcuts for finding area without counting every square and are able to explain and prove this insight.

GCG 1 – Learning Goal: As a mathematician, I will be able to understand area and measure it by counting squares MAFS.3.MD.3.5, MAFS.3.MD.3.6

  • Step 1:  Identify and explain the concept of area as how much of a surface is covered
  • Step 2: Identify and explain that area is measured using square units without gaps or overlaps
  • Step 3: Determine the area of a rectangle, and other plane figures, by counting square units

GCG 2 – Learning Goal: As a mathematician, I will be able to connect arrays and multiplication to find the area of a rectangle MAFS.3.MD.3.7

  • Step 1: Relate the number of squares needed to cover a rectangle to arrays and multiplication facts/equations/expressions
  • Step 2: Determine the number of square units needed to cover a rectangle when not given enough squares (by using multiplication)
  • Step 3: Use side length measures to determine the area of a rectangle (by using multiplication)

GCG 3 – Learning Goal: As a mathematician, I will be able to model and apply the distributive property of multiplication to rectangles MAFS.3.OA.2.5

  • Step 1: Use concrete and rectangular models to discover that the process of breaking apart one factor does not change the product
  • Step 2: Use expression/equations (with parenthesis) to represent student models of distributive property
  • Step 3: Justify and apply the distributive property as an efficient way to multiply (use multiple ways to break up factors)

GCG 4 – Learning Goal: As a mathematician, I will be able to apply the distributive property to decompose/find the area of rectilinear figures MAFS.3.OA.2.5, MAFS.3.MD.3.7

  • Step 1: Decompose a concrete or gridded rectilinear figure into rectangles and use the area of each rectangle to find the area of the rectilinear figure (relate to equations and the distributive property)
  • Step 2: Recognize and explain how to decompose a rectilinear figure when given only side lengths (without a grid)
  • Step 3: Solve problems involving decomposition to determine the area of rectilinear figures
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