Unit 8 Perimeter and Area 1
Table of Contents Introduction...3 Unit Overview...4 Lessons...,...5 8-1 Perimeter...5 8-2 Perimeter with Missing Sides...7 8-3 Area...10 Mid Unit Quiz.13 8-5 Area...14 8-6 Area of a Parallelogram...17 8-7 Area of a Triangle...20 End of Unit Quiz 23 Math Assessment..24 Resources...25 Student Work...34 Reflection...37 References...39 2
Introduction The Area and Perimeter Unit is a curriculum unit adapted from the Everyday Mathematics Unit 8 Area and Perimeter. The first two lessons are on perimeter. The first lesson is a perimeter review and the second lesson is a new skill, finding the perimeter with missing dimensions. The next four lessons are focused on area. The first lesson is a review on how to find area by counting squares and hands-on activities. The following three lessons are how to determine area using a formula for rectangles, parallelograms, and triangles. Students will have background knowledge from previous chapters about these shapes. We used Everyday Mathematics as a guideline and added selfassessment, more manipulatives, modifications, and language objectives for ELL students to help the students learn. Each lesson includes a self-assessment because this unit was used as part of my teacher action research investigating the impacts of self-assessment on students math learning process and outcomes. The intention of the self-assessment was to help students be more reflective in their math work. As a teacher, I think that selfassessment and reflective learners are a very important part of education because it helps students understand why they are learning new concepts and take greater responsibility for their learning. I feel that it is important to encourage students to become lifelong learners and by integrating self-assessment students will be actively engaged in their own learning process. The school that I am teaching in is a school with a diverse population. The students come from very diverse socioeconomic levels throughout the city of Manchester, NH. The school is a Title 1 school. There is a large population of students in the school, approximate 750 students, Kindergarten to 5 th grade. The fourth grade students that I am teaching at grade level math students. 3
Lesson Overview Day 1: Perimeter Day 2: Perimeter with Missing Sides Day 3: Area of Squares and Rectangles Day 4: Mid Unit Quiz Day 5: Formula for the Area of a Rectangle Day 6: Formula for the Area of a Parallelogram Day 7: Formula for the Area of a Triangle Day 8: End of Unit Quiz Day 9: Math Assessment 4
8-1 Perimeter Essential Questions: 1. How do you find the perimeter of a rectangle and a square? 2. How do you use a ruler? 3. How do you find the perimeter of a rectangle and square using a number model with given numbers? 4. How do you measure feet, inches, centimeters, and millimeters and understand the differences between them? Learning Objectives: Students will measure and add distances and find the perimeter of a square or rectangle. Language Objective: Students will understand what finding the perimeter means and how to read the information that is given to them. The students will also able to verbally distinguish the difference between feet, inches, centimeters, and millimeters. Common Core State Standards: 4.MD.1: Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36) 4.MD.2: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 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. Assessments: Exit Slip 8-1- Finding the Perimeter of a Square with given dimensions Math Worksheet Measuring Perimeter Math Homework Everyday Math Study Link 8-1 Procedures: 1. Timed multiplication test (2-3 minutes). 2. Everyday Mental Math Teacher Guide p. 659 3. Everyday Math Message Give students centimeter graph and these directions: Draw a square with a perimeter of 16. Draw a rectangle with a perimeter of 16. 5
4. Elicit students prior knowledge. a. Ask the students if they know what it means to find the perimeter of a shape. b. Ask students if they have ever had to use perimeter. (Maybe outside of school) 5. Have a whole class discussion to create a definition of perimeter as well as distinguishing the number model for finding the perimeter of a shape. (Adding up all the sides). 6. Hand out rulers to each student and discuss with students how to use a ruler to measure the shapes. Remind students that their ruler has both centimeters and inches and they will be measuring in inches. a. Tell the students that they will be finding the perimeter of their whiteboards, desks, and math journal. b. Give the students some time to find the perimeter for each object. c. If possible have the students write their number models for each perimeter they found either one a piece of paper or their white board. d. Bring the students back together as a class and have students share how they found the perimeter (the steps that they took). 7. Check in with students thumbs up/thumbs down on their understanding of perimeter. 8. Pass out math worksheet. If students have thumbs down do a few problems as a class. If the students have thumbs up do one example and allow students to work independently or in partners on their worksheet. 9. Come back as a class and go over the perimeters that they came up with for each shape. Ask students to explain how they found the perimeter of each shape (squares, rectangles, and regular polygons) and show their number models for how they found the perimeter. 10. Hand out Exit Slip 8-1- Finding the perimeter of a square. Modifications & Accommodations: 1. Work with partners 2. Be pulled for group work during independent time 3. Use manipulative as a resource while independent time (Make sure to model and demonstrate the use of the manipulative, i.e. rulers) 4. Finding a spot in the room where the student feels most successful Materials (for teacher; for student; w/ resources): Overhead of math worksheet Rulers with inches and centimeters Math Worksheet Measure Perimeter Whiteboards Dry Erase Markers Pencils Homework Study Link 8-1 Perimeter Exit Slip 8-1 Finding the Perimeter of a square 6
8-2 Perimeter with Missing Sides Essential Questions: 1. How do you find the perimeter of a given shape? 2. How do you find the perimeter using a number model with given and missing numbers? Learning Objectives: Students will measure and add distances and find the perimeter of a polygon with missing numbers. Language Objective: Students will understand how to find the perimeter of a regular polygon and how to read the information that is given to them. The students will be able to verbally distinguish the difference between feet, inches, centimeters, and millimeters. Common Core State Standards: 4.MD.1: Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36),... 4.MD.2: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 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. Assessments: Exit Slip 8-2- Finding the perimeter of a polygon with missing dimensions Self-Assessment Math Worksheet- Examples of polygons with missing dimensions Math Homework Procedures: 1. Timed multiplication test (2-3 minutes). 2. Everyday Mental Math Teacher Guide p. 661 3. Everyday Math Message Find the Perimeter grid paper with missing dimensions. a. Students will be asked to find the perimeter of a polygon with missing dimensions. 7
b. They will have their own copy of a polygon with missing dimensions so the students can have a visual aid while they take some time to try the problem on their own. 4. Elicit students prior knowledge on how to find the perimeter and the definition of perimeter. a. Invite students to share their strategies of how they found the perimeter of a shape with missing dimensions. b. What might they have done that may have helped them with the math message? 5. Have multiple polygons on the board where there are missing dimensions. Make sure to pass out this practice sheet so the students can be working on it as the problems are being modeled. 6. Explain to the students that they will be solving what the missing numbers are to help them find the complete perimeter of the shape. 7. Model how to first find the missing numbers and then taking these numbers one step further to solve for the perimeter of the shape. a. When modeling, make sure to explain to the students that a helpful strategy is to first highlight all of the parallel lines. (Remind students to use different colored pencils for each set of parallel lines). b. After highlighting the parallel lines, show the students how the first set of parallel lines should add up to same measurement. (This will help to give the value of the missing dimension) c. Do this again with the second set of parallel lines. d. Once we have determined all of the missing numbers, have students add up all of the numbers to find the perimeter. Make sure they use the perimeter number model. 8. Ask student volunteers to go up to the board and solve the other examples. a. Have the students try to explain their process. 9. Check in with students thumbs up/thumbs down on their understanding of how to find the perimeter of polygons. 10. Pass out math worksheet. If students have thumbs down do a few problems as a class. If the students have thumbs up do one example and allow students to work independently or with a partner on the worksheet. 11. Come back as a class and correct the student worksheet to ensure that students have a stronger understanding of finding the perimeter of a shape and using the strategies given to find the perimeter of a shape with missing dimensions. 12. Have students explain how they solved each problem and how they found the perimeter on a polygon with a missing length. 13. Collect worksheet 14. Hand out self-assessments and Exit Slip 8-2 Finding the perimeter of a polygon with missing dimensions Modifications & Accommodations: 1. Work with partners 2. Be pulled for group work during independent time 8
3. Use manipulative as a resource while independent time (Make sure to model and demonstrate the use of the manipulative, i.e. rulers) 4. Finding a spot in the room where the student feels most successful Materials (for teacher; for student; w/ resources): Overhead of math worksheet Math Worksheet Lesson 8-2 Exit Slip 8-2 Self-Assessment 8-2 Colored Chalk Colored Pencils Homework Perimeter 9
8-3 Area of Squares and Rectangles Essential Questions: 1. How do you find the area of a rectangle or square by counting boxes inside the square or rectangle? 2. What is a square unit? Learning Objectives: Students will be able to find the area of a rectangle or a square by counting boxes inside the square or rectangle. Language Objective: Students will understand how to find the area of a rectangle or a square and how to read the information that is given to them. Common Core State Standards: 4.MD.1: Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36),... 4.MD.2: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 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. Assessments: Exit Slip 8-3 Finding the area of a rectangle Self-Assessment Math Journal p. 227 Math Worksheet Lesson 8-3 Area Math Homework Study Link 8-3 Procedures: 1. Timed multiplication test (2-3 minutes). 2. Everyday Mental Math Teacher Guide p. 670. Continue to review fractions, fraction of, and finding equivalent fractions. 3. Everyday Math Message p. 671 Ask students to think about a situation where they would need to find the area of a surface. 4. Elicit students prior knowledge. a. Have students think of examples outside of school where they would need to find area. 10
b. See if students understand why it is important to find the area of a surface. c. Explain and discuss the difference between area and perimeter. Area is the inside and perimeter is the outside. d. Explain and discuss what a square unit is and why it is used for area. 5. Share with the students the strategy of finding the area of a square. Then share with the students the strategy for finding the area of a rectangle by counting the boxes inside the shape. 6. Pass out and do the first problem on Lesson 8-3 Area worksheet. Have them find the area and perimeter of the square. 7. For problem 2, pass out 25 tile blocks to each student. Ask the students to make a square or rectangle with an area of 25. (There should be a variety of squares/rectangles that the students made ie. 5*5, 1*25). Ask the students for the dimensions of their shape. Once the students have shared these dimensions ask for the perimeter of each shape. a. While this discussion is going on create a chart on the chalkboard that will display the dimension of each shape, the area, and the perimeter. b. Ensure that the students are aware that different dimensions such as 5*5 and 1*25 have the same area but have different perimeters. 8. Repeat this exercise with 20 tile blocks. 9. Check in with students thumbs up/thumbs down on their understanding of how to find the area of squares and rectangles. 10. Have students open their Everyday Math Journal to page 227. On this page the students will be finding the area for a variety of different shapes. a. Before they begin, have the students look at problem 2. Explain to the students that sometimes there will be partial boxes and that they need to see how many partial boxes there are to make one full box. 11. Have students complete the journal page. Come back as a class and go over journal page have students explain how they found the area, especially if there are partial boxes. 12. Check in with thumbs up/down. If students still are not getting it, especially partial boxes do a few more examples on the board if time or continue in differentiation. 13. Hand out self-assessments and Exit Slip 8-3 Finding the area of a polygon. Modifications & Accommodations: 1. Work with partners 2. Be pulled for group work during independent time 3. Use manipulative as a resource while independent time (Make sure to model and demonstrate the use of the manipulative, i.e. rulers) 4. Finding a spot in the room where the student feels most successful Materials (for teacher; for student; w/ resources): Overhead of math journal page 227 Math Worksheet Lesson 8-3 Area 11
Math Journal p. 227 Tile blocks Study Link 8-3 Exploring Area Exit Slip 8-3 Finding the area of a polygon Self-Assessment 8-3 12
Mid Unit Quiz 13
8-5 Formula for the Area of a Rectangle Essential Questions: 1. How well can you count unit squares to find the area of a rectangle and square? 2. What is the formula to find the area of a rectangle and square? Learning Objectives: Students will be able to guide the development and use of a formula to find the area of a rectangle. Language Objectives: Students will be able to read the formula for finding a rectangle and verbally recite the formula using the words: area, length of base, and width of rectangle. Common Core State Standards: 4.MD.1: Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36),... 4.MD.2: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 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. Assessments: Exit Slip 8-5 Self-Assessment 8-5 Math Journal p. 232-233 Math Homework Study Link 8-5 Areas of Rectangles Procedures: 1. Timed multiplication test (2-3 minutes). 2. Everyday Mental Math Teacher Guide p. 682 3. Everyday Math Message p. 682 a. Have students complete the first problem on Everyday Math journal page 232. b. For this problem the students are being asked to find the area of 3 different rectangles. 14
4. Elicit students prior knowledge a. Have students turn and talk to a partner about the strategies they know for finding the area of a rectangle or square. b. See if any students know the formula that is used to find area and whether or not they used that for the math message. 5. Developing a Formula for the Area of a Rectangle a. Draw a rectangle on the board. Choose one of the sides and label it the base. The length of the base of a rectangle is called either the length or base for short. b. Explain that the shortest distance between the base and the side opposite the base is called either the width or height of the rectangle. Label it on the drawing. In a rectangle, the width is the length of a side adjacent to the base. c. On Everyday Math journal page 232, have students fill out the chart in number two. In this chart they will be recording the number of squares per row, the number of rows, the total number of squares, and the number model for each shape they had in problem 1. d. Have the students turn and talk. Have them look at the patterns in the table and generate a rule that could be used to find the area of any rectangle. Ask students to share their rules. e. Make sure to share if the length of the base and width of a rectangle are known, the area can be found by multiplying. i. Area of a rectangle = length of base * width of rectangle ii. This rule is called a formula. It can be abbreviated as A= l * w, where A stands for Area, L stands for length of the base and w stands for width of the rectangle. Another way of writing this formula is A = b * h, where b stands for length of base and h stands for height of the rectangle. f. Have students record a formula for area of a rectangle in problem number three. g. Tell students that the letters represent variables they can take any value or their values can vary. 6. Check in with students thumbs up/thumbs down on their understanding of how to find the area rectangles and squares. 7. Using a Formula for the Area of a Rectangle a. Have students count squares to find the areas of the rectangles in problem four on Math journal page 233 and record the results in the second column in the table. b. Have students share their results. Students should note that all rectangles except D contain half-squares, and that Rectangles G and I contain quarter squares. Make sure that students understand how to count partial squares. For example: i. Rectangle G 2 half-squares are the same as 1 square, and 4 quarter-squares are the same as 1 square ii. Rectangle H 5 half-squares are the same as 2 ½ squares 15
c. Next, have students record the length of the base and the height of each rectangle. Then have them use the formula to find the area of a rectangle by multiplying the base times the height. Do one or two together. d. Have students finish math journal page 233. 8. Come back as a class and go over the math journal page. Have students explain how they got their answers for finding the area. Make sure they understand both methods and that counting boxes can only be used when there are boxes in the rectangle or square. 9. Hand out self-assessments and Exit Slip 8-5 Modifications & Accommodations: 1. Work with partners 2. Be pulled for group work during independent time 3. Use manipulative as a resource while independent time (Make sure to model and demonstrate the use of the manipulative, i.e. rulers) 4. Finding a spot in the room where the student feels most successful Materials: Overhead of math journal Math Journal p. 232-233 Tile blocks (if needed) Exit Slip 8-5 Self-Assessment 8-5 Math Homework Study Link 8-5 Areas of Rectangles 16
8-6 Formula for the Area of a Parallelogram Essential Questions: 1. How do you find the area of a rectangle? 2. What is a parallelogram? 3. What is the formula to find the area of a parallelogram? Learning Objectives: Students will review properties of parallelograms and be able to use the formula for the area of a parallelogram to find the area. Language Objectives: Students will be able to read the formula for finding a parallelogram and verbally recite the formula using the words: area, base, and height. Common Core State Standards: 4.MD.1: Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36),... 4.MD.2: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 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. Assessments: Exit Slip 8-6 Self-Assessment 8-6 Math journal p. 237 Math homework Study Link 8-6 Areas of Parallelograms Procedures: 1. Timed multiplication test (2-3 minutes). 2. Everyday Mental Math Teacher Guide p. 688 3. Everyday Math Message p. 688 a. Have students construct a parallelogram. 4. Elicit students prior knowledge. a. Discuss how to find the area of a rectangle and remind students the properties of a parallelogram. 17
5. Developing a Formula for the Area of a Parallelogram a. Have students open their math journals to page 236. b. Point out that Parallelogram A on math journal page 236 is the same as Parallelogram A on Math Masters page 260 c. Guide students through the following activity i. Cut out parallelogram A from the master ii. Cut the parallelogram into two pieces along one of the vertical grid lines. Very important that they cut on one of the vertical lines. iii. Tape the pieces together to form a rectangle. iv. Tape this rectangle in the space next to the parallelogram in the journal 1. Discuss the relationship between the parallelogram and the rectangle formed from the parallelogram. a. Why must the parallelogram and the rectangle both have the same area? The rectangle was constructed from the parallelogram. Nothing was lost or added. v. Record the dimensions and area of the parallelogram and the rectangle. Length of base of parallelogram and length of base of rectangle = 6 cm; height of parallelogram and width (height) of rectangle = 2 cm; area of each figure = 12cm 2 vi. Check in with students thumbs up/thumbs down on their understanding of how to find the area rectangles and squares. vii. Have students repeat these steps with Parallelograms B, C, and D d. Bring students together to develop a formula of the area of a parallelogram. These are three possible lines of reasoning. i. The area of each parallelogram is the same as the area of the rectangle that was made from it. ii. The area of the rectangle is equal to the length of its base times its width (also called the height) iii. The length of the base of the parallelogram is equal to the length of the base of the rectangle. The height of that parallelogram is equal to the width (height) of that rectangle. Therefore, the area of the parallelogram is equal to the length of its bases times its height. Using variables: 1. A = b * h 2. Where b is the length of the base and h is the height iv. Have students record the formula at the bottom of journal page 237 6. Check in with students thumbs up/thumbs down on their understanding of how to find the area rectangles and squares. 7. Solving Area Problems a. Have students turn to math journal page 238. b. Work with the whole class on problem 6, math journal page 238. 18
c. Students can place an index card (or other square-corner object) on top of the shape, align the bottom edge of the card with the base, and then use the edge of the card to draw a line for the height. They will need a centimeter ruler to measure the length of the base and the height. If thumbs down do 7 and 8 together. If thumbs up have students complete 7 and 8 independently. 8. Come back as a class and go over journal. Have students explain how they found the area of a parallelogram. 9. Hand out self-assessments and exit slips. Modifications & Accommodations: 1. Work with partners 2. Be pulled for group work during independent time 3. Use manipulative as a resource while independent time (Make sure to model and demonstrate the use of the manipulative, i.e. rulers) 4. Finding a spot in the room where the student feels most successful Materials: Overhead of math journal p. 236-238 Math master 260 Math journal p.236-238 Exit Slip 8-6 Self-Assessment 8-6 Scissors Math Homework Study Link 8-6 Areas of Parallelograms 19
8-7 Formula for the Area of a Triangle Essential Questions: 1. How do you find the areas of rectangles and parallelograms? 2. What are the properties of a triangle? 3. What is the formula of the area of a triangle? Learning Objectives: Students will be able to use the formula for the area of a triangle. Language Objectives: Students will be able to read the formula for finding the area of a triangle and verbally recite the formula using the words: area, base, height, and divide by two (1/2 of base times height). Common Core State Standards: 4.MD.1: Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36),... 4.MD.2: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 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. Assessments: Exit Slip 8-7 Self-Assessment 8-7 Math journal p. 241-242 Math homework Study Link 8-7 Areas of Traingles Procedures: 1. Timed multiplication test (2-3 minutes). 2. Everyday Mental Math Teacher Guide p. 693 3. Everyday Math Message p. 693 4. Elicit students prior knowledge a. How do you find the areas of rectangles and parallelograms? b. Remind students of the properties of a triangle. 5. Developing a Formula for the Area of a Triangle 20
a. Draw a triangle on the board. Choose one of the sides the side on which the triangle sits for example call it the base. Label the base in your drawing. Explain that base is also used to mean the length of the base. b. The shortest distance from the vertex above the base to the base is called the height of the triangle. Draw a dashed line to show the height and label it. Include a right-angle symbol. c. Ask the class to turn to journal page 240 while you pass out Math Masters page 265. Point out that triangles A and B on the master are the same as triangle A on the journal page. Guide students through the following activity: i. Cut out triangles A and B from the master. Make sure students realize that the triangles have the same area and are congruent. ii. Tape the triangles together at the shaded corners to corm a parallelogram. iii. Tape the parallelogram in the space next to Triangle A in the journal. d. Discuss the relationship between the area of the triangle and the area of the parallelogram. Triangles A and B have the same area. Therefore, the area of either triangle is half the area of the parallelogram. e. Record the dimensions and areas of the triangle and the parallelogram. Base of triangle and parallelogram = 6cm; height of triangle and parallelogram = 4cm; area of parallelogram = 24 cm 2 ; area of triangle = ½ the area of parallelogram = 12 cm 2 f. Check in with students thumbs up/thumbs down on their understanding of how to find the area rectangles and squares. g. Have the students repeat these steps with Triangles C and D. Do them together if thumbs down. If thumbs up students can do independently. Do E, F, G and H if time. h. Then bring the class together to state a rule and write a formula for the area of a triangle. Since the base and the height of a triangle are the same as the base and the height of the corresponding parallelogram then: i. Area of the triangle = ½ the area of the parallelogram, or ii. Area of the triangle = ½ of (base * height) iii. Using Variables 1. A = ½ * b * h, or A = ½ of (b * h) iv. Have students record the formula at the bottom of journal page 241 6. Check in with students thumbs up/thumbs down on their understanding of how to find the area rectangles and squares. 7. Solving Area Problems a. Work with the class on problem 6 on math journal page 242. Students can place an index card (or other square-corner object) on top of the triangle, align the bottom edge of the card with base (making sure that 21
one edge of the card passes through point S) and then draw a line for the height. Students will need a centimeter ruler to measure the base and the height. b. If thumbs down do problem 7 and 8 together. If thumbs up allow students to do 7 and 8 independently. 8. Come back as a class and go over worksheet. Have students explain how they found the area of a triangle. 9. Hand out self-assessments and exit slips. Modifications & Accommodations: 1. Work with partners 2. Be pulled for group work during independent time 3. Use manipulative as a resource while independent time (Make sure to model and demonstrate the use of the manipulative, i.e. rulers) 4. Finding a spot in the room where the student feels most successful Materials: Overhead of math journal p. 241-242 Math Masters p. 265 (copy) Math journal p. 241-242 Scissors Exit Slip 8-7 Self-Assessment 8-7 Math homework Study link 8-7 Areas of Triangles 22
End of Unit Quiz 23
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Student Self-Assessment Sheet 8-2 Written Reflection Name: Date: How well did you understand the math objectives we have covered so far? Please explain your answer. Did you meet your math-learning goal you set for yourself last week? Why or Why not? What is a math-learning goal you would like to set for yourself for this week? (Same goal if you did not meet last week s goal.) Student Self-Assessment Sheet 8-3 Name: Date: How well do you think you can: 1. Find the area of a rectangle. Strongly Agree Agree Disagree Strongly Disagree 2. Understand the difference between area and perimeter. Strongly Agree Agree Disagree Strongly Disagree 30
Post-Quiz Reflection Unit 8 Quiz Part 1 1. How much did the self-assessment help you master the math objectives for part one of the unit? A lot Most of the time Sometimes Not at all 2. How would you score your effort on the quiz? 1 I tried my best 2 I did what I knew and passed it in 3 I didn t try much at all 4 I rushed through the quiz Student Self-Assessment Sheet 8-5 Name: Date: How well do you think you can: 1. Count unit squares or use a formula to find the area of a rectangle. Strongly Agree Agree Disagree Strongly Disagree 2. Use patterns in a table to develop a formula for the area of a rectangle. Strongly Agree Agree Disagree Strongly Disagree 31
Student Self-Assessment Sheet 8-6 Written Reflection Name: Date: How well did you understand the math objectives we have covered so far? Please explain your answer. Did you meet your math-learning goal you set for yourself last week? Why or Why not? What is a math-learning goal you would like to set for yourself for this week? (Same goal if you did not meet last week s goal.) Student Self-Assessment Sheet 8-7 Name: Date: How well do you think you can: 1. Find the areas of rectangles and parallelograms. Strongly Agree Agree Disagree Strongly Disagree 2. Develop a formula for calculating the area of a triangle. Strongly Agree Agree Disagree Strongly Disagree 3. Describe properties of and types of triangles. Strongly Agree Agree Disagree Strongly Disagree 32
Post-Quiz Reflection Unit 8 Quiz Part 2 1. How much did the self-assessment help you master the math objectives for part one of the unit? A lot Most of the time Sometimes Not at all 2. How would you score your effort on the quiz? 1 I tried my best 2 I did what I knew and passed it in 3 I didn t try much at all 4 I rushed through the quiz 33
Student Work Student 1 34
Student 2 35
Student 3 36
Reflection Overall, the Area and Perimeter unit went very well. I think that there are a few lessons that should have been two days. The students could have used more work with them, but we needed to finish the unit before April vacation. I think this unit following the fraction unit, made it challenging for the students because that had to completely switch gears and they struggled with fractions. The first lesson I felt went very well. Many students remembered what the perimeter is and were able to define it and find it. I was very pleased. The students struggled with the second lesson, finding perimeter with missing sides. This is one of the lessons that I wish we could have spent two days on. The students struggled with this lesson, even with using different colors for the parallel lines. Next time, I will plan on this lesson being two days, if needed. The third lesson went fairly well. The students remembered how to count the squares for the area. However, they struggled with how to count partial squares in the shape. This is another lesson I would like to see done in two days. I would do the basic counting squares the first day and then counting partial squares the second day, allowing the students more time to comprehend the concept. The quiz was mostly on fractions, and the students did not do very well. The quizzes were corrected but not graded. Next time, I would add perimeter and area to be the main focus on the quiz, and then have fractions for review. The fifth lesson went well. They understood the formula of the area. However, they needed more independent work on using the formula. Another thing that I noticed was that students were trying to measure the sides when a side was not given, for example one side of a square was labeled 7cm. The students did not understand that the shapes were not drawn to scale and could not measure to find the other sides. They 37
also did not remember that the lengths of the sides of a square are all the same. I think for this lesson, I would add in a review of properties of shapes. The sixth lesson went very well. The manipulatives used helped the students understand how to find the area of a parallelogram. I was very pleased at how this lesson went. Unfortunately, due to the time crunch we were unable to get to the seventh lesson. I think the students would have done well. The quiz went well. The students did well on the area and perimeter questions, except for the missing sides to find perimeter and counting squares to find the area. The other questions they did very well on. The test went very well. We did a very long review based on the test. This helped the students A LOT. I was very pleased to see this, as the class struggles when it comes time to take a test. The only thing I would change is how long the lessons are because the students would have benefited from more instructional and independent time as a whole for this unit. Overall, I was very pleased with how this unit went and cannot wait to implement this unit again. 38
References Bell, M., Bretzlauf, J., Dillard, A., Hartfield, R., Isaacs, A., McBride, J.,... Saecker, P. (2007). Everyday mathematics grade 4. Chicago, Illinois: Wright Group/McGraw-Hill. 39