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Perimeter, Area and Volume – Grade Four Ohio Standards Connection Measurement Benchmark D Identify appropriate tools and apply counting techniques for measuring side lengths, perimeter, and area of squares, rectangles, and simple irregular twodimensional shapes, volume of rectangular prisms, and time and temperature. Indicator 4 Develop and use strategies to find perimeter using string or links, area using tiles or a grid, and volume using cubes; e.g., count squares to find area of regular or irregular shapes on a grid, layer cubes in a box to find volume. Mathematical Processes Benchmarks J. Read, interpret, discuss, and write about mathematical ideas and concepts using both everyday and mathematical language. K. Use mathematical language to explain and justify mathematical ideas, strategies, and solutions. Lesson Summary: In this lesson, students develop and use strategies to find perimeter of regular or irregular shapes using string, paper clips, links and other objects. They find the area using tiles or counting squares of regular and irregular shapes on a grid. Students find the volume by layering cubes in a box. This lesson does not use formulas for calculating the perimeter, area and volume, but develops student’s understanding of the measurements by using other strategies. Estimated Duration: Three hours Commentary: A measurement is a number that compares the attribute of an object being measured to the same attribute of a unit of measure (Van de Walle, 1998). Through hands-on experiences related to real world contexts, students develop a deep conceptual understanding of perimeter, area and volume. To assist students’ development or the concepts, encourage them to think of “filling” with cubes, “covering” with squares and “matching” the lines (Van de Walle, 1998). Introduce the concepts separately. Make sure students have a firm grasp of one concept, before introducing another. Provide opportunities for students to compare what is being measured in a context, and then match the measure. Pre-Assessment: Pre-Assessment, Attachment A can be used in two different ways, individual or small group. Scoring Guidelines: Determine student’s knowledge of the three measurement areas (perimeter, area, and volume). If Pre-Assessment A is used individually, it can be scored and used as the base-line data to determine the instruction needed for perimeter, area, and volume. If Pre-Assessment A is completed in small groups and discussed as a class, the teacher can identify through observations and questioning student’s knowledge of the three measurement areas. Common misconceptions and problem areas that are noted informally should guide instruction. Post-Assessment: Distribute the Perimeter, Area and Volume, Post-Assessment B, to students. Allow students time to work, reminding them to explain their answers. 1 Perimeter, Area and Volume – Grade Four Use checklist, to assess students understanding of each measurement. See sample below. Name Determines Explains Determines Explains Determines Explains perimeter perimeter area area volume volume Organize skill groups (or re-teaching groups) based on the Post-Assessment. Work with individuals or groups for mastery. Scoring Guidelines: Skill checklist, Post-Assessment Checklist, Attachment C, allows you to identify individual student’s strengths and weaknesses. Identify students who have not mastered certain skills for reteaching/intervention activities. See Post-Assessment Answer Key, Attachment D, for answers and scoring guides. Instructional Procedures: Part One 1. Distribute a half-sheet of construction paper to each student and a small bag of paper clips. 2. Have students predict the number of paperclips needed to go around the edge of the construction paper and write the prediction on the paper. Allow time for students to share with their partner, how they decided on their prediction. Have volunteers share their strategy for predicting with the class. 3. Have students test their prediction by using the paper clips to measure around the rectangle. Walk around the room to observe students techniques. Make notes of the strategies used and identify areas of confusion. Are students measuring all four sides or just the two sides and doubling it. Are the paper clips lined up without spaces? What are students doing when they get to the corner? 4. Have students demonstrate how they used the paper clips to measure around the rectangle by placing the rectangle on the overhead and asking students to share their responses. Students can defend their answers. 5. Have students create different shapes using the same number of paper clips as the rectangle. Have students draw sketches of the shapes and label each side with the number of paper clips describing the measurement. Can you create a triangle with the same number of paper clips? How many different polygons can we make with this number of paper clips? 6. Allow students to share their shapes and measurements in small groups. 7. Ask students what the number of paper clips represents. (The perimeter, or the distance around the shape) Have students write the word perimeter in their mathematics journal or notebook and describe perimeter in their own word. Explain to students that they can change their description as they learn more about perimeter. Encourage students to sketch a picture to help them remember. 2 Perimeter, Area and Volume – Grade Four Instructional Tip: Post mathematics vocabulary words as you introduce them. These words stay up in the classroom for the unit or school year. You can refer to them during review times by asking students to find the word on the “Mathematics Wall”. 8. Have students fold the rectangle in half. Have half of the class fold horizontally (↔) way and the other half fold vertically (↕). Then, cut the rectangle in half along the fold. 9. Ask them to predict the number of paper clips first, talk to their partner their prediction and then use the paper clips to find the perimeter. 10. Circulate around the room to identify students’ procedures for finding the perimeter, and asking questions such as: What do you do when the paper clips are not exactly at the end of the shape? What do you do when the paper clip goes past the end of the shape? 11. Select students to share the perimeters of the two shapes. Compare the perimeters of the two shapes. 12. Give students perimeters of polygons and have students create the polygons with the paper clips. Make a triangle with a perimeter of 13 paper clips. Make a quadrilateral with 11 paper clips. Make a square with 12 paper clips. Make a concave quadrilateral with 16 paper clips. Make a square with 14 paper clips. (Not possible) Discuss why this is not possible with students. Students should realize that the number of paperclips used has to be a multiple of four. 13. Distribute Shapes on Grid Paper, Attachment E. Ask students to determine the perimeter of the shapes on the grid. Ask what unit of measurement could be used. What part of the grid is used to measure perimeter? It is important that students understand to use the line segments on the grid as units of measurement. 14. Have students draw shapes on the grid paper with given perimeter measurements. Model the shape for the students as needed. Observe students and assist as needed. Sample shapes include: Draw an “L”-shape with a perimeter of 20 units. Draw a rectangle with a perimeter of 24 units. Draw a “T”-shape with a perimeter of 30 units. 15. Provide different measuring tools for practice. These may include: How many new unsharpened pencils, would I need for the perimeter of the chalkboard or bulletin board? Why should I use unsharpened pencils? (the unit we are measuring in needs to be the same length) Demonstrate by showing different size pencils would give a different measurement. How many of the chalkboard erasers are needed to find the perimeter of the desk? How many crayons (new) would I need for the perimeter of my Math book? 16. For homework, assign students to find the perimeter of the polygon, Attachment F, Perimeter Homework. Give each student a paper clip to use for the first part. Explain they will measure 3 Perimeter, Area and Volume – Grade Four first with the paper clip. Then they will use another object, like pennies, dry cereal, toothpicks, etc) Stress the importance of the measuring tool being all the same size. Part 2 17. Share homework assignment on perimeter. Have students compare their measurements. Discuss how different units of measurement resulted in different measurements. Pose the question to discuss using an example from a student’s paper. How can the perimeter be 12 paper clips but 30 dimes? Why are the numbers different? (After discussion, students should be able to notice that the smaller the unit we are measuring with, the higher the measurement. When you measure in larger units the number for perimeter will be lower.) 18. Distribute one-inch color tiles to each student or use one-inch grid paper squares. 19. Give students a half sheet of paper and ask students to predict the number of the square tiles needed to cover the paper without measuring. Walk around the room to observe students techniques. Are students measuring all of the sides or just the two sides and multiplying. Are the tiles lined up without spaces? Do the squares go off the paper? Or not all the way to the edge? 20. Have students share the number of squares it took to cover the paper and compare it to their prediction. Talk about differences in responses and have students explain their techniques for finding the number of squares. 21. Ask questions to introduce the concept of area. Can you think of any time in real life that you would need to know how many squares cover a space? (tiling or carpeting a floor, building a patio or deck, etc.) What measure are we finding when we find the number of squares needed to cover something?” (area) How did we find the area? (We covered it with square tiles and counted.) 22. Have students record the word area in their mathematics journals and describe it in their own words. Add the word, area, to the “Mathematics Wall”. 23. Have students fold and cut the rectangle in half, some vertically and the rest horizontally. Have students predict the number of squares that would cover the shape, discuss the prediction with a partner and use the squares to determine the number needed to cover the shape. 24. Have students discuss the number of squares needed. Have them compare it to the first shape. Students should see they needed half of the number of squares. 25. Have students compare area and perimeter. Have them sketch a shape in their journal. Have them use a red crayon to highlight the perimeter and a blue crayon to shade the area. (Perimeter is the distance around something, but area is the number of squares that cover it) 26. Ask students to make a rectangle with an area of 6. Walk around and observe students’ displays. 27. Select students to show different models with an area of 6 squares on the overhead. 28. Have students make rectangles with the area of 10. Ask, “If the area is 10, how many color tiles are you going to use?” (10) 29. Select students to share their area models on the overhead projector. 4 Perimeter, Area, and Volume– Grade Four 30. Collect color tiles. Distribute a sheet of grid paper and sheet of construction paper to each pair of students. Assign each pair an area measurement (12, 14, 15, 16, 18, 20, 24, 28, 30, 32, 35, 36, 42). 31. Ask students about measuring area using grid paper. Ask students, What part of the grid paper is used to measure the area of a shape on grid paper? How would you describe the measurement? (squares) Explain that they use the grid sheets to create shapes with their assigned area number. 32. Model making shapes with areas of four square units on the overhead projector. Model a few additional area measurements for students, as needed. 33. Observe students as they shade in appropriate areas. Encourage multiple shapes of the same area. Ask students to explain and show how they determined the shape. Students should explain they number of squares and count them out. Have students cut out the shapes and glue onto construction paper. Instructional Tip: Students may create irregular shapes; however, as long as they create a polygon of some kind; it should be accepted and shared for the creative thought. 34. Have pairs share their work with other pairs. They should explain the different shapes created and how they determined the area. 35. Distribute to students a sheet of one-inch grid and a piece of string (about two feet per student). Explain to the students that their assignment for homework is to: a. Trace their shoe print on the grid sheet b. Use the string to find the perimeter. Demonstrate how they place the string along the traced line. Cut the string that represents the perimeter of the shoe. c. Find the area by counting the number of squares. Explain that students use some estimation since their shoe print is irregular and curved. Demonstrate if necessary. Part Three 36. Have pairs of students share their understanding of perimeter and area. Assign Partner A to talk for one minute about perimeter, then Partner B to talk for one minute about area. Instructional Tip: Allowing students to talk about what they know with a partner, allows for active participation for all students. One minute seems a long time since a definition takes only seconds to recite. This forces students to expand their thinking and describe activities that relate to the concept they are learning. Model this strategy of discourse before having students try it. 37. Review the Shoe Print homework assignment. Have students compare and order the string lengths for perimeter and the square units for area. Display student work. 38. Display your own grid print of your shoe and the string for perimeter. 39. Pose the question: How can I use the measurements I have found to find a box that would hold my shoe? Display a variety of gift boxes and shoe boxes. (Include very small boxes and large ones, too.) Allow students to discuss what they know about the shoe. (the length of the string around the shoe, the number of squares to cover the bottom of the shoe) 5 Perimeter, Area, and Volume– Grade Four 40. Demonstrate by placing the string for perimeter and the cut-out shoe print in a variety of different boxes. Some boxes, such as the gift boxes may have enough area on the base to contain the shoe print and the string. Is knowing the area of the shoe print enough? Why? Can we determine if the shoe fits inside the box using just the string? (length) Why? What do we need to know about the shoe to determine if it will fit in the box? 41. Refer students to the boxes. Ask students how we might find out the amount of space inside the boxes. Suggest the string and squares. Ask students if these units would be easy or difficult to use. Model an attempt of measuring with the squares. Begin to place squares in the box and think aloud that there would be too many squares to count. Ask students what a unit to measure space needs. (height) 42. Explain that one way volume can be determined is by counting the number of cubes needed to fill the box. Ask, How are the cubes different from the square tiles or squares from the grid? (the squares have length and width, but the cubes have length, width and height) How are the cubes different from the string? (The string only measures length.) 43. Have students make a tower made with 12 cubes. Model building a tower for students, if necessary. Observe students as they work. 44. Have students share their towers. 45. Place the word volume on the Mathematics word-wall and have students include volume in their mathematics journals. Explain that volume is the measure of space. Space has length, width and height. Therefore, we need a unit that has length, width and height like a cube. 46. Display a tower of cubes as shown. Ask, How many cubes make this figure? (four) What is the volume of this tower? (four cubes) How much space does this tower take up? (four cubes of space) 47. Have students build towers with given volume measurements. Observe and assist students by connecting the number of cubes to measuring volume. 48. Have students share their towers. Make comparisons of the towers. Students should realize, no matter the shape or dimensions of the towers, they take up the same space, or have the same volume. 49. Distribute a small cardboard box (raisin box, chalk stick box, jewelry box) and a supply of cubes to each pair of students. Have students to find the volume of the box. Ask the students for a process to find the volume. Reponses should include filling the box with cubes. 50. Have students predict the number of cubes first. Record their predictions and ask them to explain how they decided upon their prediction. 51. Allow time for the students to fill their boxes. As you walk around the home, guide a discussion using questions similar to these: 6 Perimeter, Area, and Volume– Grade Four Why should you not just dump the cubes in the box? (Because there would be space between them) How is finding the volume different from the area or perimeter? (Volume is the measure of space, how much a three-dimensional shape will hold. Area measures the number of squares that cover something. Perimeter measures around the outside of a figure.) What should you do when the cubes do not fit perfectly? (estimate) If we used bigger cubes, would the number of cubes needed be greater or less than the current number of cubes? (The bigger the cube, the fewer number of cubes are needed to fill it) 52. Have students share their measurements with the class. 53. Have students write what they know about volume in journals. Encourage students to sketch a picture to help them remember volume, and what unit is used to measure volume. Differentiated Instructional Support: Instruction is differentiated according to learner needs, to help all learners either meet the intent of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified indicator(s). Provide additional hands-on experiences with measurements. Use larger units to cover and measure desktops, areas on the floor, windows, chalkboard, etc. Give students strings of different lengths (or paper clips, links, etc) and have them find objects in the classroom that have that perimeter. Have students create a floor plan of their dream bedroom on grid paper. Students can determine the area and perimeter of the room, their bed, dresser, and all other items on the floor plan. They can then determine the amount of wallpaper border, carpet to order, and amount of paint. Have students create a box with the given volume. How many different size boxes can be created with the same volume? Provide cubes for students to build the tower, then use paper to create the sides. Extensions: Have students create a bulletin board display, entitled Mathematics in the Real World. Students make illustrations showing real-world uses for perimeter, area, and volume. Ask students to find ads in the newspaper that involve perimeter, area, and volume. For example, an ad for carpet or tile (area), ad for fencing or border (perimeter), and an ad for refrigerator or microwaves that shows the capacity in cubic feet (volume) Assign students to make a flipbook with one page for each of the three measurements (perimeter, area, and volume). Each page should include the description of what each measures, strategies to find each, and examples of each. Home Connection: Consider the ideas for homework embedded in the Instructional Procedures. 7 Perimeter, Area, and Volume– Grade Four Materials and Resources: The inclusion of a specific resource in any lesson formulated by the Ohio Department of Education should not be interpreted as an endorsement of that particular resource, or any of its contents, by the Ohio Department of Education. The Ohio Department of Education does not endorse any particular resource. The Web addresses listed are for a given site’s main page, therefore, it may be necessary to search within that site to find the specific information required for a given lesson. Please note that information published on the Internet changes over time, therefore the links provided may no longer contain the specific information related to a given lesson. Teachers are advised to preview all sites before using them with students. For the teacher: sentence strips for word wall, overhead, unsharpened pencils, new crayons, grid paper, a variety of different size shoe boxes, empty cereal box, blocks or cubes 1 For the students: set of paper clips, sheet of construction paper, scissors, color tiles, 2 journals, jewelry boxes-1 per pair, set of cubes, 12” by 18” sheets of construction paper Vocabulary: perimeter area volume Technology Connections: Use software programs that have students find area, perimeter, and volume without using formulas Attachments: Attachment A, Pre-Assessment with Answer Key Attachment B, Post Assessment Attachment C, Post Assessment Checklist Attachment D, Post Assessment Answer Key Attachment E, Perimeter of an Irregular Shape Attachment F, Perimeter Homework 8 Perimeter, Area, and Volume– Grade Four Attachment A Pre-Assessment Name_______________________________ Date__________________________ 1. The perimeter of the rectangle is _________ links. How do you know? _______________________________________________________ ______________________________________________________________________________ __________________________________________________________________ 2. What is the area of the shape below? Measure in square units. __________ How do you know? ___________________________________________ ___________________________________________ ___________________________________________ ___________________________________________ ___________________________________________ 3. Each cube below equals 1 cubic unit. What is the volume of the three-dimensional figure in cubic units? ____________ How do you know? ________________________________________________________ ________________________________________________________________________ 9 Perimeter, Area, and Volume– Grade Four Attachment A (continued) Pre-Assessment Directions: Circle the best answer for each. 4. You are putting a new tile floor in the bathroom. You need to find the number of square tiles to buy to cover the floor. The number of tiles represents the … A. perimeter B. area C. volume D. area and volume 5. The principal has decided to put a fence around the playground at school. He will measure around the playground to find the … A. perimeter B. perimeter and area C. area D. volume 6. The number of cubes to fill the box represents the ___________of the box. A. perimeter B. area C. area and volume D. volume 7. From the illustration, you can tell… A. The perimeter is about 18 circles C. The volume is 18 about circles B. The area is 18 about circles D. The perimeter and area are same 8. The area of the polygon below is how many squares units? A. 4 square units B. 6 square units C. 8 square units D. 16 square units 9. How many cubes represent the volume of the rod below? ______ 10 Perimeter, Area, and Volume– Grade Four Attachment A (continued) Pre-Assessment Answer Key 1. 12 links - I counted all the links going around the rectangle or students may have written that they added the fives and the two ones to get 12. 2. 15 square units - I counted the squares by drawing lines over the black rectangle or added 3 five times or 5 three times. 3. 12 cubic units - I counted the cubes or I counted the ones in the front and doubled it for the ones behind it. 4. B - area 5. A - perimeter 6. D - volume 7. A - The perimeter is 18 circles 8. C - 8 square units 9. 4 cubes 11 Perimeter, Area, and Volume– Grade Four Attachment B Post-Assessment Name_______________________________ Date__________________________ Directions: Read and respond to the following questions. 1. Latoya has made a quilt for her bed. She has decided to put lace trim around it, and needs to know how much lace to buy. She does not have measuring tape or a ruler, but she does know that her dad’s shoe is 1 foot long. Describe how Latoya should use the shoe to figure out the amount of lace she needs for the trim around her quilt. 2. Which is the closest estimate to the perimeter of the square, measured in ○? A. 7 circles B. 14 circles C. 28 circles D. 24 circles 3. Which is the closest estimate to the perimeter of the triangle, measured in links? A. 2 links B. 4 links C. 6 links D. 8 links 4. If the principal of your school decides to put a fence around the playground, which measurement will the principal find? ______ A. area B. perimeter C. volume D. perimeter and volume 12 Perimeter, Area, and Volume– Grade Four Attachment B (continued) Post-Assessment 5. The cafeteria at school will be getting a new tile floor. Which measurement will they need to know how many tiles are needed? ______ A. area B. perimeter C. volume D. perimeter and volume Find the area of the shapes below. 6. 7. Area = _________square units Area = _________square units 8. Which shape has the greatest area? How do you know? ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ 13 Perimeter, Area, and Volume– Grade Four Attachment B (continued) Post-Assessment 9. Shade in the grid to make four different shapes with the area of 12. 10. What is the volume of the figure shown? Explain how you know. _______________________________ _______________________________ _______________________________ _______________________________ 11. Estimate the volume of the box in cubes? Use the cube to help you estimate. Explain how you found the estimated volume. ______________________________________________________________________________ ______________________________________________________________________________ 14 Perimeter, Area, and Volume– Grade Four Attachment C Post Assessment Checklist Name Determines perimeter Explains perimeter Determines area Explains area Determines volume Explains volume 15 Perimeter, Area, and Volume– Grade Four Attachment D Post-Assessment Answer Key 1. Use the rubric to describe the level of performance. 2 points – Describes the process of placing the shoe around the border of the quilt, and counting the number of shoes it takes. The number of shoes is equal to the number of feet of lace she needs. 1 point- Describes using the shoe to measure the quilt, but is not specific about the border or the amount of lace to buy as it relates to the number of shoes. 0 points- Writes no response or gives wrong information 2. C. 28 circles 3. C. 6 links 4. B. perimeter 5. A. area 6. 7. Area= 13 square units Area= 16 square units 8. Use the rubric to describe the performance. 2 point Identifies #2 as having the largest area, and explains that it has the greatest number of square units with 21. 1 point Identifies #2, but gives no explanation or explanation is unclear. 0 points Gives incorrect shape. 9. Use the rubric to describe the performance. 2 points Shaded 4 shapes with 12 shaded squares. 1 point Shaded only 2 or 3 correct shapes with 12 shaded squares. 0 points Shaded only 1 or none at all correctly. 10. Volume = 40 cubes The volume is the number of cubes and I counted them. 11. Estimates may include 16 or 20 cubes I counted the number of squares going across, up and behind. 16 Perimeter, Area, and Volume– Grade Four Attachment E Shapes on Grid Paper 17 Perimeter, Area, and Volume– Grade Four Attachment F Perimeter Homework Name_______________________________ Date__________________________ Directions: Find the perimeter of the polygon below. How many paper clips make the perimeter of the polygon. ________ Explain how you know. ___________________________________________________________________________ 18 Perimeter, Area, and Volume– Grade Four Attachment F Perimeter Homework Name_______________________________ Date__________________________ Directions: Find the perimeter of the polygon below, using two different measurement objects. First, measure using the paper clip provided. Then find another object in your home to use to measure. You could use pennies, toothpicks, dry cereal, or any other items you have at home. Respond to the question below. The perimeter = ____________paper clips. I found the perimeter by Attach or trace the measuring tool you used from your home to this sheet. The perimeter = ____________. Why are the 2 perimeters you found different? __________________________________ ________________________________________________________________________ THINK: If someone uses pennies to find the perimeter, and another person uses toothpicks, how will their measurements for perimeter be different? Explain _______________________________________________________________________ 19

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