Unit 3.3 Relating Area to Multiplication and Division

1 1 | Page 3rd 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 ( ). The unit emphasizes modeling, as students are required to create area models and various polygons to solve area and perimeter problems ( ). Students also compare rectangles with the same area but different dimensions and look for patterns in the shapes of the rectangles ( ). 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.

2 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 : Make sense of problems and persevere in solving them : Reason abstractly and quantitatively : Construct viable arguments and critique the reasoning of others : Model with mathematics : Use appropriate tools strategically : Attend to precision : Look for and make use of structure : Look for and express regularity in repeated reasoning 2 | Page Common Core State Standards Progression of Skills According to the PARCC Model Content Framework, Standard 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.

3 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 ( ), look for and make use of structure ( ), as they model these situations with mathematical forms ( ), and attend to precision ( ) as they distinguish different kinds of situations over time ( ). 2nd Grade 3rd Grade 4th Grade N/A : Recognize area as an attribute of plane figures and understand concepts of area measurement. N/A N/A : Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

4 : 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. : Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. : Relate area to the operations of Multiplication and addition. N/A : Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. : 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. : Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

5 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. : Determine whether a group of objects (up to 20) has an odd or even number of members, , by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. : 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 = ? : Multiply or divide to solve word problems involving multiplicative comparison, , by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. : Solve Problems Involving Perimeters of Polygons : Understand Concepts of area Measurement and Square Units : Measure area by Counting Unit Squares : Relate area to Multiplication and Division : Find the area of Rectangles by Tiling : Find the area of Rectangles by Multiplying : area and Distributive Property : area Addition and Decomposition : Unknowns in Multiplication and Division Equations Major Standards (70%) Additional Standards (10%) 3 | Page Big Ideas Students Perimeter is a measurement of the distance around a figure.

6 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 two-dimensional 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. 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 Be Able Perimeter is additive: the sum of all side lengths equals the perimeter. area is measured in square units.

7 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. 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 4 | Page Unit Sequence Key Points/ Teaching Tips Exit Ticket Instructional Resources1 1 Measure the perimeter of a figure in centimeters and inches using string and rulers This should be a hands-on, kinesthetic, conceptual lesson about perimeter.

8 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 ( 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 and their understanding of the relationship between the size of a unit and the number of units needed. 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.

9 3. Compare the perimeters of the objects in centimeters to the perimeters in inches. What do you notice? My Math Chapter 13, Lesson 1 EngageNY Module 7, Lesson 10 Homework (Appendix C) 2 Use grid paper, rulers, and the perimeter formula to find the perimeter of a figure. 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. 1. Find the perimeter of the shape below: 2. Find the perimeter of Shape B below. Write an equation to support your answer. My Math Chapter 13, Lesson 2 EngageNY Module 7 Lessons 12 & 13 (Appendix C) 1 Teachers should be aware that the EngageNY curriculum covers geometry before perimeter; therefore, resources must be adapted accordingly.

10 5 | Page When using grid paper, students must attend to precision when counting around the corners ( 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.