# Multiplication and Area

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2 Grade 3 Module 4 Multiplication and Area OVERVIEW In this 20-day module students explore area as an attribute of two-dimensional figures and relate it to their prior understandings of multiplication. In Grade 2, students partitioned a rectangle into rows and columns of samesized squares and found the total number by both counting and adding equal addends represented by the rows or columns. In Topic A, students begin to conceptualize area as the amount of two-dimensional surface that is contained within a plane figure. They come to understand that the space can be tiled with unit squares without gaps or overlaps. They make predictions and explore which rectangles cover the most area when the side lengths differ (but area is actually the same). Students may, for example, cut and fold rectangles to confirm predictions about whether a 1 by 12 rectangle covers more area than a 3 by 4 or a 2 by 6 rectangle. They reinforce their ideas by using inch and centimeter square manipulatives to tile the same rectangles and prove the areas are equal. Topic A provides students first experience with tiling, from which they learn to distinguish between length and area by placing a ruler with the same size units (inches or centimeters) next to a tiled array to discover that the number of tiles along a side corresponds to the length of the side. In Topic B, students progress from using square tile manipulatives to drawing their own area models. Anticipating the final structure of an array, they complete rows and columns in figures such as the example shown at the right. Students connect their extensive work with rectangular arrays and multiplication to eventually discover the area formula for a rectangle, which is formally introduced in Grade 4. In Topic C, students manipulate rectangular arrays to concretely demonstrate the arithmetic properties in anticipation of the following lessons. They do this by cutting rectangular grids and rearranging the parts into new wholes using the properties to validate that area stays the same, despite the new dimensions. They apply tiling and multiplication skills to determine all whole number possibilities for the side lengths of rectangles given their areas. Topic D creates an opportunity for students to solve problems involving area. Students decompose and/or compose composite regions like the one shown at right into non-overlapping rectangles, find the area of each region, and add or subtract to determine the total area of the original shape. This leads students to design a simple floor plan that conforms to given area specifications.

3 Terminology New or Recently Introduced Terms Area (the amount of two-dimensional space in a bounded region) Area model (a model for multiplication that relates rectangular arrays to area) Square unit (a unit of area specifically square centimeters, inches, feet, and meters) Tile (to cover a region without gaps or overlaps) Unit square (e.g., given a length unit, it is a 1 unit by 1 unit square) Whole number (an integer, a number without fractions) Familiar Terms and Symbols Array (a set of numbers or objects that follow a specific pattern, a matrix) Commutative Property (e.g., rotate a rectangular array 90 degrees to demonstrate that factors in a multiplication sentence can switch places) Distribute (e.g., 2 (3 + 4) = ) Geometric shape (a two-dimensional object with a specific outline or form) Length (the straight-line distance between two points) Multiplication (e.g., 5 3 =15) Rows and columns (e.g., in reference to rectangular arrays) Suggested Tools and Representations Area model Array Grid paper (inch and centimeter) Rulers (both centimeter and inch measurements) Unit squares in both inch and centimeter lengths (e.g., square tiles used for measuring area)

4 Lesson 1 Objective: Understand area as an attribute of plane figures. Shape A The area of Shape A is 6 square units. The units used to measure are squares, so they re square units! Shape B The area of Shape B is 10 square units. The units used to measure are squares, so they re square units! Lesson 2 Objective: Decompose and recompose shapes to compare areas. Both shapes are measured using the same unit, square inches, so they have the same area. Shape C = 4 square cm Shape D = 4 square inches These shapes both have 4 squares but different areas. Shape C is measured with square cm and Shape D is measure with square inches.

5 Lesson 3 Objective: Model tiling with centimeter and inch unit squares as a strategy to measure area. Lesson 4 Objective: Relate side lengths with the number of tiles on a side.

6 Lesson 5 Objective: Form rectangles by tiling with unit squares to make arrays. Lesson 6 Objective: Draw rows and columns to determine the area of a rectangle, given an incomplete array.

7 Lesson 7 Objective: Find the area of a rectangle through multiplication of the side lengths. Area can be found by multiplying the length and width of a rectangle sq units sq units 4 Lesson 8 Objective: Interpret area models to form rectangular arrays. Related facts can help determine an unknown length of a rectangle s side when you know the area and the length of one side.

8 Lesson 9 Objective: Analyze different rectangles and reason about their area. Lesson 10 Objective: Apply the distributive property as a strategy to find the total area of a large rectangle by adding two products There are 3 rectangles on the right: the large rectangle, the shaded rectangle, and the unshaded rectangle. Adding the areas of the shaded and unshaded rectangles will produce the area of the large rectangle.

9 Lesson 11 Objective: Demonstrate possible whole number side lengths of rectangles with areas of 24, 36, 48, or 72 square units using the associative property. Lesson 12 Objective: Solve word problems involving area. W = 32 4

10 Lesson 13 Objective: Find areas by decomposing into rectangles or completing composite figures to form rectangles. This figure shows a small rectangle cut out of a larger rectangle. We can find the area of the figure by subtracting the area of the smaller rectangle from the larger rectangle. This figure also shows a small rectangle cut out of a larger, rectangle. We can find the area of the figure by using the break apart strategy. Lesson 14 Objective: Find areas by decomposing into rectangles or completing composite figures to form rectangles.

11 Lesson 15 Objective: Apply knowledge of area to determine areas of rooms in a given floor plan. Lesson 16 Objective: Apply knowledge of area to determine areas of rooms in a given floor plan. Count the squares to figure out the length of each side. Break the object apart and find the area of each section sq units 2 Add together the area of each object to find the area for the whole shape sq units 2 7 Area = (2 x 5) + (2 x 7) = = 24 sq units

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