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1 3 rd Grade Mathematics Unit #3 : Relating Area to Multiplication and Division Pacing: 29 Days Unit Overview In this unit, students extend their understanding of multiplication and division (Unit 2) by applying it to real-world problems involving area. This unit is designed to be a short, yet thorough, dive into many of the measurement concepts of the grade. Students will reason abstractly and quantitatively by connecting concepts of area with multiplication and arrays (MP.2). The unit emphasizes modeling, as students are required to create area models and various polygons to solve area and perimeter problems (MP.4). Students also compare rectangles with the same area but different dimensions and look for patterns in the shapes of the rectangles (MP.7). Prerequisite Skills Vocabulary Mathematical Practices Find the sum of an equation with 3 or more addends. Multiplication facts 1-12 Partition (decompose) a rectangle into rows and columns of same-size squares. Apply the distributive property to find the product of two factors. Construct rectangular arrays. Write a multiplication equation that represents a given array. Perimeter Area Unit Square Square Unit Formula Dimensions Length Width Rectangular Composite Figure Decompose Partition Distributive property Rectilinear Shape Array Column Row Precision Gaps Overlaps Tiling Polygon Quadrilateral MP.1: Make sense of problems and persevere in solving them MP.2: Reason abstractly and quantitatively MP.3: Construct viable arguments and critique the reasoning of others MP.4: Model with mathematics MP.5: Use appropriate tools strategically MP.6: Attend to precision MP.7: Look for and make use of structure MP.8: Look for and express regularity in repeated reasoning 1 Page

2 Additional Standards (10%) Major Standards (70%) Common Core State Standards 3.MD.8: Solve Problems Involving Perimeters of Polygons 3.MD.5: Understand Concepts of Area Measurement and Square Units 3.MD.6: Measure Area by Counting Unit Squares 3.MD.7: Relate Area to Multiplication and Division 3.MD.7a: Find the Area of Rectangles by Tiling 3.MD.7b: Find the Area of Rectangles by Multiplying 3.MD.7c: Area and Distributive Property 3.MD.7d: Area Addition and Decomposition 3.OA.4: Unknowns in Multiplication and Division Equations According to the PARCC Model Content Framework, Standard 3.MD.7 should serve as opportunity for in-depth focus based on the content of this unit: Area is a major concept within measurement, and area models must function as a support for multiplicative reasoning in grade 3 and beyond. Measurement contexts for multiplication and division should serves as an example of opportunities for connecting mathematical content and mathematical practices: Students will analyze a number of situation types for multiplication and division, including arrays and measurement contexts. Extending their understanding of multiplication and division to these situations requires that they make sense of problems and persevere in solving them (MP.1), look for and make use of structure (MP.7), as they model these situations with mathematical forms (MP.4), and attend to precision (MP.6) as they distinguish different kinds of situations over time (MP.8). N/A N/A Progression of Skills 2 nd Grade 3 rd Grade 4 th Grade 2.G.2: Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. 2.MD.1: Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. 3.OA.3: Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. 3.MD.5: Recognize area as an attribute of plane figures and understand concepts of area measurement. 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. 3.MD.8: Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. 3.OA.4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8? = 48, 5 = _ 3, 6 6 =? N/A 4.MD.3: Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. N/A 4.MD.3: Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. 4.OA.2: Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. 2 Page

3 Big Ideas Perimeter is a measurement of the distance around a figure. Area is a measurement of the amount of flat space a figure takes up. What is a square unit? Why do we use square units to find and report the area of twodimensional shapes? Rectangles are composed of arrays (rows and columns). Therefore, we can use tiling and multiplication to find the area of a rectangle by multiplying the number of columns that compose its length by the number of rows that compose its height Given a perimeter, you can create shapes with various dimensions and therefore different areas. Given an area, you can create a rectangle with various dimensions and therefore various perimeter. Know/Understand Perimeter is additive: the sum of all side lengths equals the perimeter. Area is measured in square units. A square unit is a rectangle with equal side lengths. The size of the square unit affects the area. A figure that can be covered by n unit squares with no gaps or overlaps has an area of n square units. The dimensions of a rectangle are called its length and width. A composite figure is made up of two or more figures. Rectangles may have the same area, but different perimeters and vice versa. Students Will Be Able To Add side lengths to determine the perimeter of a figure. Solve for missing side lengths given the perimeter of a figure. Tile a rectangle and count square units to determine the area. Find the area and perimeter of a rectangle on a grid. Multiply side lengths of a figure to determine the area. Use the Distributive Property to find the area of rectangles with side lengths greater than 12. Find the area of a composite figure by decomposing into or composing rectangles The Distributive Property can be used as a tool to find the area of rectangles with greater side lengths. You can find the area of complex or composite shapes by decomposing them into familiar (rectangular) shapes 3 Page

4 SWBAT 1 Measure the perimeter of a figure in centimeters and inches using string and rulers 2 Use grid paper, rulers, and the perimeter formula to find the perimeter of a figure. Key Points/ Teaching Tips This should be a hands-on, kinesthetic, conceptual lesson about perimeter. In addition to using string and rulers/meter sticks, students may explore perimeter through geoboards and rubber bands, by walking around the perimeter of the room, by tracing the boundaries of shapes, or by searching the school for examples of perimeter (i.e. bulletin board boarders, door frames, etc.). Students will understand perimeter as the distance around the outside of a figure Students do not need to be adding side lengths until the next lesson. Students should measure the perimeter of objects to the nearest whole inch and centimeter in order to support standard 3.MD.2 and their understanding of the relationship between the size of a unit and the number of units needed. According to the unpacked standards guide, students should work through determining perimeter in the following progression: 1. Side lengths are marked off so that students can count unit lengths. 2. Side lengths are labeled with numerals. 3. Students mark off unit lengths with a ruler and label the length of each side. This lesson should give students an opportunity to practice all three. Unit Sequence Exit Ticket 1. Trace the perimeter of the shape below with a crayon: a. Explain in writing how you know you outlined the perimeter of the shape. 2. Use string and your rulers to find the perimeter of the shape above to the nearest inch and the nearest centimeter. Be sure to label your answers with the correct units. 3. Compare the perimeters of the objects in centimeters to the perimeters in inches. What do you notice? 1. Find the perimeter of the shape below: 2. Find the perimeter of Shape B below. Write an equation to support your answer. Instructional Resources 1 Chapter 13, Lesson 1 Module 7, Lesson 10 Homework Chapter 13, Lesson 2 Module 7 Lessons 12 & 13 1 Teachers should be aware that the curriculum covers geometry before perimeter; therefore, resources must be adapted accordingly. 4 Page

5 When using grid paper, students must attend to precision when counting around the corners (i.e. a common misconception is that students count square units instead of unit lengths). Students may benefit from being required to trace each square unit as they count. 3. Measure and label the side lengths of the shape below in centimeters. Then find the perimeter. 3 Given the perimeter of a figure, students will find a missing side length. Students should be able to name squares and rectangles and pay special attention to patterns in the side lengths of each. They should be able to use these patterns and what they know about the nature of squares and rectangles to reason about missing side lengths for these shapes. For other polygons, students should practice writing equations with letters or symbols standing for the unknown (missing side length). Sample PARCC EOY assessment question: 1. The perimeter of the triangle shown below is 48 inches. Find the unknown side length. 16 in. 2. The perimeter of the figure below is 26 feet. 10 ft.?? 10 ft. 20 in. 3 Chapter 13, Lesson 2 Module 7, Lesson 14 5 Page

7 6 Explore the area of rectangles by using tiles; attend to precision when tiling by ensuring there are no gaps or overlaps. According to the unpacked standards guide, students should have ample experiences filling a region with square tiles before transitioning to pictorial representations on graph paper. Students should understand area as the amount of flat space a shape takes up Students should be able to articulate why we use square units to measure area. Tiling a Tabletop provides a good foundation for an inquiry lesson on this topic. In addition to using notebook paper and index cards, students might also try describing the amount of flat space a shape takes up using circle counters so that they discover the importance of square units in being able to cover a figure. 1. Use square unit tiles to find the area of the shape below. Make sure to label your answer correctly. (Include picture of a rectangle that can be measured exactly using square unit tiles.) 2. Does the picture below accurately show the area of Rectangle A? Why or why not? 3. Why do we measure area in square units? Chapter 13 Foldable: 1 square unit LearnZillion: Find the Area of a Shape Tiling a Tabletop Error analysis: 7 Page

8 7 Use grid paper to find the area of a figure and to create figures with a given area. Students may benefit from exploring this objective with geoboards before working with grid paper. Students should understand that the size of the square unit affects the area of the figure. 1. Each is 1 square unit. What is the area of each of the following rectangles? Chapter 13 Lessons 3 & 4 Module 4, Lesson 3 Students may notice that multiple different shapes can have the same area. This will be revisited later in the unit. 2. Find the area of the figure below. Label your answer in square units. 3. Use the appropriate resources to draw a rectangle with an area of 8 square centimeters and a rectangle with an area of 8 square inches. What do you notice about these two rectangles? Explain. 1. Use a ruler to measure the side lengths of the rectangle in centimeters. Use your measurements to tile the rectangle with square centimeters. Find the area. 8 Use rulers or labeled side lengths to separate rectangles into square units. A common misconception is for students to fill a rectangle with square units without attending to the given side lengths. (For example, a student may make 4 columns even though the side length is labeled as 3 units.) This should be an explicit teaching point, and students should be required to attend to precision by checking their drawings to ensure that it matches the labeled side length. Chapter 13, Lesson 5 Module 4, Lesson 4 8 Page

9 Students should learn the word dimensions and recognize that this refers to the side lengths of a shape. Area = square centimeters 2. Use the given side lengths to tile the rectangle with square units. Find the area. 3 units 6 units 3. Why is perimeter measured in units and area measured in square units? 9 Relate area to arrays in order to apply skipcounting and multiplication strategies to determine the area of a rectangle. Students should draw on their foundation with arrays from Unit 2. They should be able to describe area as repeated groups of equal size of square units. 1. Darren has a total of 28 square-centimeter tiles. He arranges them into 7 equal rows. Draw Darren s rectangle and label the side lengths. a. Write an addition equation to find the total area. b. Write a multiplication equation to find the total area. 2. How is the area of a rectangle related to multiplication? Chapter 13, Lesson 5 Module 4, Lesson 5 9 Page

10 10 Complete rows or columns to complete arrays and form rectangles According to the unpacked standards guide, 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. This lesson is included to provide students with additional exposure to arrays. 1. Each represents a 1-cm square. Draw to find the number of rows and columns in each array. Then, fill in the blanks to make a true equation to represent the array s area Module 4 Lessons 6 & 7 x = square centimeters 2. Label the side lengths of Rectangle A on the grid below. Use a straight edge to draw a grid of equal size squares within Rectangle A. Find the total area of Rectangle A. 3. Mary skip-counts by 6s and Julio skip-counts by 7s to find the total number of square units in the array. Who is correct? Use pictures, numbers, and words to explain your answer. 11 Flex Day (Instruction Based on Data) Recommended Resources: Doubling, Halving, Tripling Find the Area Area Compare Module 7, Lesson 10 How Big is a Desk? 10 Page

11 12 Use equations and fact families to find the missing side length of a rectangle. Module 4, Lesson 8 Exit Ticket 1. Write a multiplication sentence to find the area of the rectangle below: Module 4, Lesson 8 Chapter 13, Lesson 6 2. Write a multiplication sentence and a division sentence to find the unknown side length for the rectangle below: 3. Ria draws a rectangle that has a side length of 4 inches and an area of 28 square inches. What is the other side length? Use words, pictures, and equations to support your answer. 11 Page

12 13 Apply the distributive property to find the area of rectangles. Pacing: 2 days Students should practice first with rectangles with grids, then with rectangles where only the side lengths are labeled. Inquiry-based question from Smarter Balanced: In the picture below, Rectangle A has a width of 3 ft., Rectangle B has a width of 2 ft., and both of them have a length of 4 ft. Rectangle C was formed by sticking Rectangle A and Rectangle B together along one of their lengths. This is shown with the dotted line. 1. Label the side lengths of the shaded and unshaded rectangles. Then find the total area of the large rectangle by adding the areas of the 2 smaller rectangles. Chapter 13, Lesson 7 Module 4 Lessons 9 & 10 Breaking Apart Arrays Find the area of the rectangle below by using the Distributive Property to decompose the longer side into a sum. 15 ft. 4 ft. a. What is the area of Rectangle A? b. What is the area of Rectangle B? c. What is the sum of the areas of Rectangles A and B? d. What is the width of Rectangle C? How do you know? e. What is the area of Rectangle C? f. Compare your answers in (c) and (e). Are they the same? Why or why not? 3. How are the operations of addition and multiplication used when finding area using the Distributive Property? 12 Page

13 15 Apply the distributive property to solve word problems about the area of rectangles. Students should have practice matching expressions with real world situations. See the example below from the NY State Assessment: The garden below was divided into regions one for carrots and one for peas. Sample PARCC assessment question: 1. There is a large mural made of colored tiles at the entrance to Rena s school. A part of the mural was damaged in a heavy storm. The part of the mural that was NOT damaged is 5 tiles long and 4 tiles high. Chapter 13, Lesson 7 Module 4 Lesson 12 Which expression represents the area, in square units, of the whole garden? a. (5 + 10) + (5 + 6) b. (5 x 10) + (5 x 6) c. (5 x 10) + (5 x 6) d. (5 + 10) x (5 + 6) Rena wants to know how many tiles need to be replaced. a. Choose an expression from the options box to write in each of the two blanks to make a true statement. Designing a Flower Bed Sample question from Arizona Common Core: Joe and John made a poster that was 4 by 5. Mary and Amir made a poster that was 4 by 3. They placed their posters on a wall side-by-side so that there was no space between them. How much area will the two posters cover? Draw a picture to help explain your answer. b. How many tiles need to be replaced in the mural? Explain how you found your answer. Sample Smarter Balanced: 2. Jasper used the expression 5 x (10 + 3) to find the area of a rectangular closet floor, in square feet. a. On a grid, draw a rectangle that could be the one Jasper measured. b. What is the area of the closet floor in square feet? 3. Jasper has 200 square feet of tile. He will use some of the tile to cover the closet floor. He will only use whole tiles. a. How many square feet of tile will Jasper have left after covering the closet floor with 13 Page

14 tile? b. Jasper wants to use some of the remaining tile to cover the floor of a kitchen. The kitchen is 12 feet long and 12 feet wide. Does Jasper have enough tiles to cover the kitchen floor? Show how you got your answer using drawings, mathematical expressions or equations, and words. 16 Use grid paper to find the area of composite figures. Provide an opportunity for students to do the heavy lifting by presenting them with a complex shape with given side lengths first ask them to calculate the perimeter, then give them think time to brainstorm how they could find the area (allow them to discuss in small groups). You may provide tiles for them to recreate the shape and test their theories. Students should begin their work with composite figures by using grid paper to: Decompose rectilinear figures into rectangles (2 or more) Compose rectangles in order to subtract a known area Module 4, Lesson 13 Concept Development Problems 1 & 2 (with grid paper), Problem Set #1, Homework #1-2 Students may benefit from coloring the rectangles they have created with different colors. They should be required to label the dimensions of these new shapes. 1. Find the total area of the figure below by decomposing it into 2 rectangles. 2. Find the total area of the figure above by composing a rectangle in order to subtract a known area. 3. For either #1 or #2, explain how you found the area. Include the following words in your answer: compose/decompose, add/subtract, area, and composite figure. Module 4, Lesson 13 Finding the Area of Complex Figures: Lessons 13 & Page

15 17 Find the area of composite figures in which all side lengths are labeled. Students should continue their work with composite figures by working without grid paper, but solving problems in which necessary side lengths are labeled. They must attend to precision when choosing which dimensions to use to find the area of the decomposed rectangles. 1. Find the area of the figure below three ways. Show two ways to decompose the figure and one way to compose a rectangle around it Module 4 Lessons 13 & 14 Chapter 13, Lesson 8 Module 4, Lesson 13 Problem Set #2 Module 4, Lesson 14 Problem Set #3-4 Chapter 13, Lesson 8 Example 2, Independent Practice #2-3, 5 Finding the Area of Complex Figures: Lesson Max found the area of the composite figure below: (10 x 4) + (8 x 3) = = 64 square inches Is Max correct? Use pictures, words, and numbers to explain your reasoning. If you believe he is incorrect, find the correct answer. 15 Page

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