Linear, Square and Cubic Units Grade Five


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1 Ohio Standards Connection Measurement Benchmark F Analyze and explain what happens to area and perimeter or surface area and volume when the dimensions of an object are changed. Indicator 4 Demonstrate understanding of the differences among linear units, square units and cubic units. Mathematical Processes Benchmarks J. Communicate mathematical thinking to others and analyze the mathematical thinking and strategies of others. K. Recognize and use mathematical language and symbols when reading, writing, and conversing with others. Lesson Summary: In this lesson, students demonstrate understanding of the differences among linear, square and cubic units. They investigate packages with measurements such as a box of aluminum foil, package of ribbon and bag of mulch. They determine the dimensional attributes of each measurement unit and the measurable attributes of a box. By matching the attributes of the units to the measurable attributes of the box, students understand and label measurements appropriately. Estimated Duration: One hour 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). Students must understand the attributes of the units and the attribute being measured. For example, attributes of a square unit include length and width or it is described as twodimensional. When measuring the length and width attributes of an object, a square unit is used because the attributes of the unit and object being measured are the same. Encourage students to think of filling with cubes, covering with squares and matching the lines (Van de Walle, 1998). PreAssessment: Have each student individually complete the following instructions on a sheet of blank paper: 1. Draw a line that is two inches long and label the measurement of the line. 2. Draw a shape with an area of two square inches and label the area and the dimensions. 3. Sketch an object that is two cubic inches and label the object. (If sketching is difficult for the student, have them name an object that is two cubic inches.) Collect the papers to informally assess student understanding of linear, square and cubic measurements. Scoring Guidelines: Informally review the students preassessment papers. Use the following guidelines: a. Draws a line measuring two inches and labels as linear measurement b. Draws a rectangle of two square inches and labels measurement using square unit. 1
2 c. Sketches cubic object (two oneinch cubes placed next to each other or one on top of another) and labels measurement using cubic unit. Identify the level of student understanding of the concept of linear, area and volume and the labels used for each concept. Level of Understanding Adequate Understanding Partial Understanding Limited Understanding Performance Descriptor Provides pictures which are appropriately drawn and labeled. Measurements are accurate, dimensions are given and labeled with appropriate unit. A student who experiences difficulty drawing may use a ruler and/or build models with squareinch tiles and cubicinch cubes to complete the task. Provides pictures for one or two of the measurements appropriately. Uses a rough sketch, not actual measurements. Continue to assess using manipulatives as explained in scorepoint two to determine understanding of the units. Provides little evidence of understanding units through drawing pictures or building models with tiles and cubes. Continue to assess by asking for description of the measurable attributes. For example: Which model is a square? a cube? Post Assessment: Have each student independently complete PostAssessment Units, Squares or Cubes, Attachment A. Students pair and share their conclusions for each situation. Informally observe students. Collect the papers to evaluate student understanding of linear, square and cubic measurements. Scoring Guidelines: Use PostAssessment Answer Key Units, Squares or Cubes, Attachment B, to evaluate student s understanding of identifying the appropriate unit of measurement and an adequate explanation of the choice. Instructional Procedures: Instructional Tip: Gather objects such as a box of aluminum foil, bucket, package of giftwrap paper, bag used to sell mulch, plastic storage bin, spool of ribbon, package of a paper tablecloth, package of shoe laces, etc. Make sure the measurement of the object with unit label is on the object. 1. Show students an example of a linear unit such as a oneinch piece of ribbon, a square tile and a cube. Have small groups discuss and describe attributes of these units that can be 2
3 measured. Select students to share the attributes and record in a table on the board or overhead projector. Direct students to record table in their mathematics journal. Oneinch ribbon Square tile Cube Length, weight Length or width of side, diagonal, weight Length, width, height, weight, amount of material 2. Display the collection of objects and ask students questions to discuss measurable attributes. a. Identify the measurements on the packages of the items. Record the measurements in a table on the board. b. Facilitate a discussion about the measurement attributes of the items, the unit on the items and the meaning of those units. For example: What does 200 ft. on a package of ribbon mean? (how long the ribbon is or the length of the ribbon) What does 75 sq. ft. on a box of aluminum foil mean? (If students can not answer this question, ask if this means the aluminum foil is 75 feet long. Ask what square means. Expect them to relate this to a square that has length and width.) What does 3 cu. ft. on a bag of mulch mean? (Students may relate cubic and cube. Ask them to describe a cube.) Which of the items were measured using the same attributes? How do you know? (They have the same unit label.) Have the class group the items according to the unit of measurement. Instructional Tip: When buying a yard of fabric, the fabric is sold by the linear yard. The width of the fabric might range from 35 inches to 54 inches. Although one square yard is purchased, the fabric may measure more or less than a square yard. 3. Gather students into small groups. Provide each group with a small empty box. Ask students to brainstorm what they could measure on these boxes. Write all the suggestions on the board. Expect answers such as sides, length, inside, perimeter, area, volume, weight, etc. 4. Have students discuss what can be done with the box or any box. Expect answers such as fill it, wrap it, measure around the box. Record the responses on chart paper or the board. 5. Ask students to discuss and record measurable attributes of what can be done with the box. Ask questions such as: What is measured when measuring around the box? (the length) What unit would I use? (inches or centimeters) What tool would I use? (ruler, students may suggest wrapping ribbon around the box then measuring the ribbon) How would I determine the measurement using the ruler? (Match the ruler and the length of the box.) Explain that the ruler is matched to the length and the length is compared to the ruler to determine the measurement. 3
4 What can be measured on the box to cover it with wrapping paper? (the length and width of the box) What unit would I use? (square units) Describe the action when wrapping a box. (covering) Which unit has the same attributes as the paper? (Square units) If filling the box, what determines how much the box will hold? (the length, width and height on the inside of the box) What unit would I use? (cubic units) 6. Encourage students to focus on a unit that has the same attribute as what is being measured. Refer them to the questions and responses in Step 2. Have students make a chart of the unit and its attributes for their journals or notebooks. Discuss linear as onedimensional, area as twodimensional and volume as threedimensional. 7. Present the following scenarios using the overhead projector or chalkboard: a. Have students answer the questions independently, and then share with a partner. b. Have each pair of students discuss the answer to each question with another pair of students. c. Bring the class together to discuss the answers and any misconceptions. Encourage students to add any reference to the chart that will assist them in knowing which unit is appropriate for a measure. Tiffany is wallpapering her kitchen. She will also put a border in the kitchen. What does Tiffany need to know when purchasing the border? (Perimeter, length, no other special unit name) What does she need to know to determine the amount of wall paper to buy? (length and width of walls, square units) The school board is planning to build a new playground at the middle school. A basketball court is included in the plans. What does the school board need to know about the basketball court? (Area of the court to fit in the plan, square) The landscaper will put flowers around the house when building the house is complete. What does the landscaper need to know before buying the flowers? (Perimeter of the ground around the house, length) The landscaper wants to purchase mulch for the flower bed. What does the landscaper need to know to determine the amount of mulch to buy? (Length and width of bed, thickness of mulch, cubic units) Tom needs to fill a hole in his yard with concrete. What does he need to know about the hold before ordering the concrete? (Volume, cubic) Would it be easier to fill the hole with squares or cubes? Aaron is carpeting his room. In order to purchase the carpeting, what does Aaron need to know? (Area of the floor to be carpeted, square) 8. Have students write summaries in the journals telling what they have learned about units of measurement. Tell them to describe the measurable attributes of the units and provide examples of objects which have identical attributes of the unit. For example, a cube has length, width and height or it is threedimensional. The space in a box has length, width and height. To measure the space, fill the box with cubes. 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). Allow students to use grid paper, tiles and cubes when finding perimeter, area and volume. 4
5 Extension: Invite professionals who measure in their careers to speak to the class. Speakers can discuss how important measurement skills are in their field and describe what could happen if measurements were not accurate. Home Connection: Ask students to make a list of situations or examples they find at home that require measuring in square units or cubic units in their journals. Have students share the examples with the class, having them verify the examples. 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: For the student: Packages of items with measurement labels (E.g., package of foil or wrapping paper, ribbon or string, buckets or plastic mulch bag), overhead projector, rulers, grid paper, square tiles, centimeter cubes, chart paper Boxes, paper, pencil, rulers, cubes, square tiles and grid paper (if needed) and chart paper for each group Vocabulary: area attribute cubic perimeter square volume width Research Connections: Marzano, Robert J., Jane E. Pollock and Debra Pickering. Classroom Instruction that Works: ResearchBased Strategies for Increasing Student Achievement, Alexandria, VA: Association for Supervision and Curriculum Development, Van de Walle, John A., Elementary and Middle School Mathematics: Teaching Developmentally, Addison Wesley Longman, Inc
6 General Tips: Have student create a chart the unit of measurement and attributes of the unit. Attachments: Attachment A, Units, Squares or Cubes Attachment B, Units, Squares or Cubes Answer Key 6
7 Attachment A Units, Squares or Cubes Name Date Directions: Read each situation and determine the attribute that is measured. Write the appropriate unit of measurement and explain your answer. 1. Tom is building a doghouse and kennel for his dogs. He wants to put a fence around the doghouse and kennel. What does Tom need to know for the fencing he needs? 2. Sarah is painting the walls in her room. What does Sarah need to know when buying the paint? 3. Sean wants to give his sister a sandbox. He needs to place the sandbox in the yard and buy sand to fill the sandbox. a. What does Sean need to know about the place for the sandbox? b. What does Sean need to know to fill the sandbox? 4. Anna needs to have water put in her swimming pool. What does Anna need to know about the amount of water she needs for the swimming pool? 5. Dennis is putting new carpet in his house. What does Dennis need to know when he buys the carpet? 6. Harry is putting a garden behind his house and needs to purchase topsoil to cover the garden to a specified depth. What does Harry need to know before he buys the topsoil? 7
8 Explanations may vary. Linear, Square and Cubic Units Grade Five Attachment B Units, Squares or Cubes Answer Key 1. Tom is building a doghouse for his dogs. He wants to put a fence around the doghouse. What does Tom need to know for the fencing he needs? The fence goes around the doghouse and kennel, Tom needs to find the length, which is measured in linear units. 2. Sarah is painting the walls in her room. What does Sarah need to know when buying the paint? To paint the walls of her room, Sarah needs to know the area of the walls. Area is measured in square units. 3. Sean wants to give his sister a sandbox. He needs to place the sandbox in the yard and buy sand to fill the sandbox. a. What does Sean need to know about the place for the sandbox? Sean needs to know the area of the place for the sandbox. Area is measured in square units. b. What does Sean need to know to fill the sandbox? Sean needs to know the volume of the sandbox to determine how much sand the sandbox will hold. Volume is measured in cubic units. 4. Anna needs to have water put in her swimming pool. What does Anna need to know about the amount of water she needs for the swimming pool? Anna is filling the swimming pool with water, which is volume. Volume is measured in cubic units. 5. Dennis is putting new carpet in his house. What does Dennis need to know when he buys the carpet? Carpeting is covering the floor, so Dennis needs to know the area of the floor. Area is measured in square units. 6. Harry is planting a garden behind his house and needs to purchase topsoil to cover the garden to a specified depth. What does Harry need to know before he buys the topsoil? Harry needs to know the volume of the topsoil that will cover the garden to the specified depth. 8
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