Grade: 3 Unit #4: Multiplication and Area Time: 20 Days

Unit OverviewThird grade students will recognize area as an attribute of two-dimensional regions. They will measure the area of a shape by finding the total number of same size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area, i.e. square cm, square in. Covering a region with “unit squares,” could include square tiles or shading on grid or graph paper. Based on students’ development, they should have ample experiences filling a region with square tiles before transitioning to pictorial representations on graph paper. Counting the square units to find the area could be done in metric, customary, or non-standard square units. Using different sized graph paper, students will explore the areas measured in square centimeters and square inches. Students can use geoboards, tiles, graph paper, or technology to find all the possible rectangles with a given area, e.g., find the rectangles that have an area of 12 square units. They will record all the possibilities using dot or graph paper, compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles. Students can then investigate the perimeter of the rectangles with a given area, such as 12. Further investigation will involve rectangles and squares with the same perimeter.

In this 20-day module students explore area as an attribute of two-dimensional figures and relate it to their prior understandings of multiplication. Students 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.  Students progress from using square tile manipulatives to drawing their own area models and manipulate rectangular arrays to concretely demonstrate the arithmetic properties. The module culminates with students designing a simple floor plan that conforms to given area specifications.

Connection to Prior LearningPrior work with arrays in 2nd and 3rd grade will help students understand how to model area and how to solve area problems using addition and multiplication.

Major Cluster StandardsGeometric measurement: understand concepts of area and relate area to multiplication and to addition. 3.MD.5 Recognize areas 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 which can be covered without gaps or overlaps by n unit squares is said to have an area of n squares.3.MD.6 Measure areas by counting unit squares (square cm, square m, square in., square ft., and improvised units). 3.MD.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.

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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 a x b and a x c. 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.

Major Cluster Standards UnpackedGeometric measurement: understand concepts of area and relate area to multiplication and to addition. 3.MD.5 Recognize areas as an attribute of plane figures and understand concepts of area measurement.

These standards call for students to explore the concept of covering a region with “unit squares,” which could include square tiles or shading on grid or graph paper. Based on students’ development, they should have ample experiences filling a region with square tiles before transitioning to pictorial representations on graph paper.

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3.MD.6 Measure areas by counting unit squares (square cm, square m, square in., square ft., and improvised units).

Students should be counting the square units to find the area could be done in metric, customary, or non-standard square units. Using different sized graph paper, students can explore the areas measured in square centimeters and square inches.

Again, student must have multiple opportunities to discover area concepts with tiles prior to moving to the pictorial representation. When moving to graph paper, care must be taken to connect the concrete to the pictures as this does not come natural to many students.

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3.MD.7 Relate area to the operations of multiplication and addition.Students can learn how to multiply length measurements to find the area of a rectangular region. But, in order that they make sense of these quantities, they must first learn to interpret measurement of rectangular regions as a multiplicative relationship of the number of square units in a row and the number of rows. This relies on the development of spatial structuring. To build from spatial structuring to understanding the number of area-units as the product of number of units in a row and number of rows, students might draw rectangular arrays of squares and learn to determine the number of squares in each row with increasingly sophisticated strategies, such as skip-counting the number in each row and eventually multiplying the number in each row by the number of rows. They learn to partition a rectangle into identical squares by anticipating the final structure and forming the array by drawing line segments to form rows and columns. They use skip counting and multiplication to determine the number of squares in the array.Many activities that involve seeing and making arrays of squares to form a rectangle might be needed to build robust conceptions of a rectangular area structured into squares.

Students should understand and explain why multiplying the side lengths of a rectangle yields the same measurement of area as counting the number of tiles (with the same unit length) that fill the rectangle’s interior For example, students might explain that one length tells how many unit squares in a row and the other length tells how many rows there are.

Students should tile rectangle then multiply the side lengths to show it is the same.

To find the area one could count the squares or multiply 3 x 4 = 12.

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Students should solve real world and mathematical problems

Example:Drew wants to tile the bathroom floor using 1 foot tiles. How many square foot tiles will he need?

Students might solve problems such as finding all the rectangular regions with whole-number side lengths that have an area of 16 area-units, doing this for larger rectangles (e.g., enclosing 24, 48, 72 area-units), making sketches rather than drawing each square. Students learn to justify their belief they have found all possible solutions. This work directly connects to multiplication and factors.

The commutative property can be developed through the models of 16 x 1 and 2 x 8 when compared to the models for 1 x 16 and 8 x 2.

This standard extends students’ work with the distributive property. For example, in the picture below the area of a 7 x 6 figure can be determined by finding the area of a 5 x 6 and 2 x 6 and adding the two sums.

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Using concrete objects or drawings students build competence with composition and decomposition of shapes, spatial structuring, and addition of area measurements, students learn to investigate arithmetic properties using area models. For example, they learn to rotate rectangular arrays physically and mentally, understanding that their areas are preserved under rotation, and thus, for example, 4 x 7 = 7 x 4, illustrating the commutative property of multiplication. Students also learn to understand and explain that the area of a rectangular region of, for example, 12 length-units by 5 length-units can be found either by multiplying 12 x 5, or by adding two products, e.g., 10 x 5 and 2 x 5, illustrating the distributive property.

The problem below illustrates the additive component of area. When an irregular figure is decomposed into two or more rectangles and then the area of each rectangle is added to find the total area:

Example:A storage shed is pictured below. What is the total area? How could the figure be decomposed to help find the area?

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With strong and distinct concepts of both perimeter and area established, students can work on problems to differentiate their measures. For example, they can find and sketch rectangles with the same perimeter and different areas or with the same area and different perimeters and justify their claims Differentiating perimeter from area is facilitated by having students draw congruent rectangles and measure, mark off, and label the unit lengths all around the perimeter on one rectangle, then do the same on the other rectangle but also draw the square units. This enables students to see the units involved in length and area and find patterns in finding the lengths and areas of non-square and square rectangles. Students can continue to describe and show the units involved in perimeter and area after they no longer need these.

Focus Standards for Mathematical PracticeMP.2 Reason abstractly and quantitatively. Students build toward abstraction starting with tiling a rectangle, then gradually moving to finishing incomplete grids and drawing grids of their own, then eventually working purely in the abstract, imaging the grid as needed. MP.3 Construct viable arguments and critique the reasoning of others. Students explore their conjectures about area by cutting to decompose rectangles and then recomposing them in different ways to determine if different rectangles have the same area. When solving area problems, students learn to justify their reasoning and determine whether they have found all possible solutions, when multiple solutions are possible. MP.6 Attend to precision. Students precisely label models and interpret them, recognizing that the unit impacts the amount of space a particular model represents, even though pictures may appear to show equal sized models. They understand why when side lengths are multiplied the result is given in square units. MP.7 Look for and make use of structure. Students relate previous knowledge of the commutative and distributive properties to area models. They build from spatial structuring to understanding the number of area-units as the product of number of units in a row and number of rows. MP.8 Look for and express regularity in repeated reasoning. Students use increasingly sophisticated strategies to determine area over the course of the module. As they analyze and compare strategies, they eventually realize that area can be found by multiplying the number in each row by the number of rows.

Understandings-Students will understand… How area and multiplication and addition are related. That area is measured using square units. How to find area of a rectangle. How to decompose irregular figures to find area.

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Essential Questions

How are multiplication and addition related to area?How is area measured?What is a square unit or unit squared?How can real world problems related to area be represented and solved?

Prerequisite Skills/Concepts: Students should already be able to…

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Advanced Skills/Concepts: Some students may be ready to…

Find figures with the same area, but different perimeter. Find figures with the same perimeter, but different areas. Solve more complex problems involving area.

Skills: Students will be able to …

Recognize areas as an attribute of plane figures and understand concepts of area measurement. (3.MD.5)

Measure areas by counting unit squares (square cm, square m, square in., square ft., and improvised units). (3.MD.6)

Explain area using what is known about multiplication and addition. (3.MD.7)

Solve word problems involving area of rectangular figures by modeling and multiplying side lengths. (3.MD.7)

Find the area of irregular figures by decomposing into rectangles and then adding the areas together. (3.MD.7)

Knowledge: Students will know…

A square unit is used to measure area.

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Transfer of Understanding-Students will apply…

Area measurement to real-world problem solving situations. Add To, Result Unknown problem solving situations to area problems. Example: If the area of a garden is measured in square feet, one

side of the garden is 8 feet and another side is 2 feet. What is the area of the garden? Solution: 8 + 8 = ? or 2 + 2 + 2 + 2 = 16

Academic Vocabulary

Area Area model (a model for multiplication that relates rectangular arrays to area) Square unit 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 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) = 2 × 3 + 2 × 4) Distributive Property Geometric shape Length (the straight-line distance between two points) Multiplication Factors Product Addition Addends Sum Rows and columns (e.g., in reference to rectangular arrays)

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Unit Resources

Pinpoint: Grade 3 Unit # 4

Connection to Subsequent LearningBased on work in third grade, fourth grade students learn to consider applications using rectangle area. Fourth graders use multiplication; spatially structuring arrays, and area, they determine the formula for the area of a rectangle A = l x w. Students will generalize that, given a unit of length, a rectangle whose sides have the length of 4 w (width) units and 3 l (length) units, can be partitioned into 4 w(width) rows of unit squares with 3 l(length) squares in each row. The product l x w gives the number of unit squares in the partition, thus the area measurement is l x w square units. These square units are derived from the length unit. Repeated reasoning about how to calculate rectangle area, will help students see area formulas as summaries of addition calculations. In Grade 4 and later grades, rectangle area problems become more complex. Students learn to apply these understandings and formulas to the solution of real-world and mathematical problems.

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