String, Tiles and Cubes: A Hands-On Approach to Understanding Perimeter, Area, and Volume Teaching Notes Teacher-led discussion: 1. Pre-Assessment: Show students the equipment that you have to measure with: string, tiles, and cubes. An introductory question might be: What could you measure with each of these tools? This question will yield important information, allowing the teacher to do a quick and informal pre-assessment about students understanding of perimeter, area, and volume. It is not necessary to formally use the terms perimeter, area, and volume at this point - it is essential that the students eventually generate those terms based on the experiences involved in the lesson. If not sooner, students should be ready to use those terms in the Summary section of Activity #2. Depending on student responses to the introductory questions, the teacher will need to decide how many of the example problems listed below that they want to do. Three examples are included 1 each for perimeter, area, and volume. More or less examples could be used depending on the knowledge that a particular class has about these ideas. For an advanced group, it would be interesting to have no teacher leadership for Activity #1. 2. Examples: These examples will show students how to proceed with Activity #1. Example A - Cereal Box: (I would cut down the cereal box to 3-4 inches tall so that students and you don t have to fill an entire box with cubes it is also easier to manipulate a box that doesn t have it s top flaps getting in the way!) If we want to measure the amount of cereal inside the box, what tool would we use: string, square tiles, or cubes? Hopefully, discussion would ensue about the appropriateness of the three tools. Could we fill the box with string? Certainly, but what would that mean? How does the length of a string match with the amount of cereal? Could we fill the box with tiles? Yes, but is that the best tool to use? Unit cubes are the best tool to measure the amount of cereal that would fill the box. Note: Measurement of 1-dimensional and 2-dimensional attributes is hard because all of our measurement tools are 3-dimensional objects. It is easy for students to think about filling a box with tiles or string even though we know (in the abstract), that it makes no sense. 1-dimension and 2-dimensional measurements should not fill a box, but with the physical limitations of the tools, they do! If students get distracted by this idea, conversation about the best tool available should get them back on track. Strings, Tiles and Cubes Teacher Materials, Page 1 of 11
Example B - Amount of carpet in the classroom: We want to measure the amount of carpet it would take to carpet the floor of our classroom. What would be the best tool to use: string, square tiles, or cubes? Classroom discussion might include the following ideas: Wouldn t larger tiles make it easier? Couldn t we just measure the sides and multiply to find the area? Your response to this question would be to complement the strategy and then talk about what units you re creating when you do that string, square tiles, or cubes? Couldn t we just spread out the cubes? Your response to this needs to bring the students back to the primary attribute that they are measuring we only need the surface of the cube and the square tiles are all we need. Example C - Desk Perimeter: We want to add a band of gold trim to the edge of your desk. What tool would we use to measure the amount of trim we need: string, square tiles, or cubes? Classroom discussion should arrive at the conclusion that string measured around the edge of the desk is the best solution. Unlike counting tiles or cubes, though, after the length of string is determined, students will need to use a meter stick to find a numerical length. Students might come up with the idea of measuring two sides and then doubling the answer you could talk about what shapes that would work for and when you wouldn t be able to take that shortcut. Student Activities: Please refer to the attached student activity pages for the notes below. Student Activity #1 Estimating/Measuring perimeter, area, and volume using string, tiles, and cubes. The main purpose of this activity is for students to have the kinesthetic experience of actually measuring these attributes. As they physically deal with string, square tiles, and cubes, their understanding and familiarity of 1-dimensional, 2-dimensional, and 3- dimensional attributes will increase. It is extremely important in this activity to let the students generate the ideas of perimeter, area, and volume and to make the connections between those formal ideas and the measurements that they are making. Students will have an official opportunity to use the words perimeter, area, and volume at the conclusion of Activity #2. Before students get materials, pass out the worksheet and have them read it first! You ll want to take a few minutes to make sure that they understand what they are going to do before you have them pick up manipulatives. Strings, Tiles and Cubes Teacher Materials, Page 2 of 11
TC-1: Depending on your class experience in working with small groups, you may need to discuss these roles and make sure that everyone understands what the roles are before they start the activity. I would pass the worksheet out first and let students read it before materials are picked up! TC-2: Tools. The bigger the square tiles and cubes are, the easier it will be to manage the manipulatives. Cubic inch cubes are significantly easier to handle and count than cubic centimeters cubes! The same is true for the square tiles. We certainly get a more accurate estimate with smaller cubes and tiles, but that s not the main focus of this lesson. I would encourage you to use larger materials to help with the management of the manipulatives. As you discuss the results from Activity #1, there will be opportunity to talk about the size of the tiles and cubes and how that impacts accuracy. TC-3: Items to be measured. The diagrams are all included and need to be photocopied (1 set per group). I would choose cereal or food boxes that are appropriately sized for how many cubes you have for students to use. If you don t have a large set of cubes, then choose smaller boxes (macaroni and cheese boxes would work) and cut off the top half or 2/3! Again, I would think about the size of my measurement tools when assembling these items. I ve included a circular lid just to remind the students that we can deal with circles also. TC-4: Vocabulary the word attribute may not be one that is terribly familiar to your students. It is defined as a specific characteristic or......... Student should be able to understand the word from the context of the list, but it is worth a few seconds to make sure. Have the resource manager(s) from each team gather supplies and let them begin. You ll need to circulate around the room conversations that you will have with teams might include accuracy, patterns, more efficient ways to find answers, etc. TC-5: The answers for the first column of the chart are: string, tiles, tiles, string, string, cubes, and cubes. TC-6: Students might need some help with the second column. The goal is to have them verbalize their measurement methods. Answers might include: We wrapped the string around the edge of the shape, being careful at the corners. When we found the total length of string, we measured it against the meter stick and found our total in cm. TC-7: Answer will vary based on the manipulatives that you use. Groups should be able to compare their answers for diagrams A, B, C, and D. Strings, Tiles and Cubes Teacher Materials, Page 3 of 11
While teams are taking measurements and filling out the data collection page, you should be circulating around the room and listening to the math talk from students. You could keep a word box on the board or overhead to compile a list of the vocabulary words that are generated. Some teams will need more guiding questions as they work while other teams may get frustrated with having to measure with the tools provided. Some students will be eager to find shortcuts and use formulas as soon as possible. It is important to have the students take the physical measurements. The experience of using the manipulatives is essential to their internalizing the 1- dimensional, 2-dimensional, and 3-dimensional tools. As teams are finishing the data collection sheet, they could write their answers for the diagrams on the board. In the follow-up discussion, accuracy should be a major topic. Follow-up Discussion Have students share answers about the easiest and hardest attributes to measure. Talk about accuracy. Questions might include: How could we improve the accuracy of our answers? Why are our answers to Diagrams A, B, C, and D all slightly different? Have students try to verbalize the differences between what they measured with string, tiles, and cubes. They are going to formalize this in writing in Activity #2, but it might be appropriate to let students have some initial verbalization of the similarities between what they measured with string, what they measured with square tiles, and what they measured with cubes. Student Activity #2 Matching quantities to be measured with appropriate tools and summary. The main purpose of this activity is to cement the idea of perimeter, area, and volume being measured with string, square tiles, or cubes. Certainly, the minimal measuring tools are getting old, but the value of this lesson is in the simplicity of the connections. At the end of the activity, students should be thinking, When I find the distance around something, I need to use something like string, and it s called the perimeter. When I want to find the amount of space inside a flat object, I use square tiles, and it s called area. When I want to find the amount of space inside a box or 3-dimensional object, I use cubes, and it s called volume. TC-8: Part 1 This activity should be completed individually at first and then students could team up to share and discuss answers. The correct answers are: 1. string 2. square tiles 3. cubes 4. string 5. cubes 6. string 7. cubes 8. square tiles 9. square tiles 10. string Strings, Tiles and Cubes Teacher Materials, Page 4 of 11
TC-9: Part 2 The summary questions give students the opportunity to synthesize information from Activity 1 and the first part of Activity 2. The questions are scaffolded to help students develop their conclusions and the word boxes are included to help students make the leap from our initial tools of string, tiles, and cubes to the more useful units of measure. The Summary questions could be done individually or in teams, but it is important for them to share answers and edit their work based on conversations. After students finish part 2, it is very important to discuss their answers. They could make posters for 1 of the three ideas, they could report out to other teams, or you could lead a full class discussion. At the end of that discussion, have the students go back to Part 1 and choose 1-2 appropriate units from the word boxes for each of the examples in Part 1. Student Activity #3 This is a more open-ended, creative activity that has students find items that could be measured with the given tool/units. The final question has them summarize everything they know about perimeter, area, and volume. TC-10: If a student clearly understands perimeter, area, volume and the units that each is measured with, they should be able to create their own situations and problems that could be measured with a given unit. This is certainly an open-ended activity with infinitely many answers, but it is important for students to brainstorm and create their own situations. They have many models from the previous 2 activities, but hopefully they ll think beyond those examples and come up with some clever and ingenious items! TC-11: This summary question is the culmination of the entire sequence of activities. After student have written their own ideas about the three topics - hopefully sounding something like: When I find the distance around something, I need to use something like string, and it s called the perimeter. When I want to find the amount of space inside a flat object, I use square tiles, and it s called area. When I want to find the amount of space inside a box or 3-dimensional object, I use cubes, and it s called volume. I would provide an opportunity to share those final ideas. This could be done in combination with the summary questions from Activity 2. Posters or presentations are a great way for students to formalize learning. Students could use some of the vocabulary words or math talk that you ve recorded on the board if they are struggling with this summary. This would also be an appropriate time to introduce the formal definitions of perimeter, area, and volume. I anticipate that the student will have created definitions that are very close to that. Strings, Tiles and Cubes Teacher Materials, Page 5 of 11
String, Tiles, and Cubes Student Activity #1 Name Directions: For this activity, your team will be measuring attributes of different objects and shapes. You will record your findings in an organized chart. Team roles: Each team of 3-4 students needs a: resource managers/materials coordinator (1-2 people) (TC-1) measurement specialists (1-2 people) time manager. (TC-2) (TC-3) Measurement tools needed for each team: 60 inches (approximately) of string or fine gauge wire and a meter stick a container of square tiles a container of cubes Items to be measured for each team: Diagrams A, B, C, and D. 2 cereal or food boxes 1 circular lids from yogurt or other food containers Data Collection Sheet 1 Attributes to be measured: (TC-4)Diagram A measure the distance around the outside edge of the shape. Diagram B measure the amount of space inside the shape. Diagram C measure the amount of space inside the shape. Diagram D measure the distance around the outside edge of the shape. Circular Lid measure the distance around the outside edge. Box #1 measure the amount of space inside the box. Box #2 measure the amount of space inside the box. Questions (to be answered after you finish collecting your data): Which attribute was the easiest to measure? Why? Which attribute was the hardest to measure? Why? Strings, Tiles and Cubes Teacher Materials, Page 6 of 11
String, Tiles, and Cubes Student Activity #1 Data Collection Sheet (TC-5) Team Names (TC-6) (TC-7) Item being measured Diagram A distance around outside edge. Did you use string, tiles, or cubes? Describe how you found your answer. Answer Diagram B amount of space inside the shape. Diagram C amount of space inside the shape. Diagram D distance around outside edge. Circular Lid distance around outside edge. Box #1 amount of space inside the box. Box #2 amount of space inside the box. Strings, Tiles and Cubes Teacher Materials, Page 7 of 11
String, Tiles, and Cubes Student Activity #2 Name Part 1 For the questions below, choose the best tool available to find the requested measurement. Your only choices are string, square tiles, and cubes. 1. The school is going to build a new fence around the edge of the new ball fields. What is the best measurement tool? (TC-8) 2. The school is going to plant grass and needs to figure out how much grass is needed to cover the new ball fields. What is the best measurement tool? 3. The school needs to bring in dirt before they plant the grass and they want it to be 6 inches deep. They need to know how much dirt will be needed. What is the best measurement tool? 4. Mary wants to find the distance around the edge of her math book. What is the best measurement tool? 5. Jonathon needs to find the amount of water that fills his dog s water dish. What is the best measurement tool? 6. Your mom and dad are buying new trim for around the edge of their windows. They need to find the total distance around the edge. What is the best measurement tool? 7. Rosa has a box and needs to know how much space is inside the box so that she can fill it with books. What is the best measurement tool? 8. Tameka gets to pick out new carpeting for her bedroom! But first she needs to find out how much carpet she needs. What is the best measurement tool? 9. Hassan s parents are going to put a new bamboo floor in their kitchen and need to find out how much flooring they need. What is the best measurement tool? 10. George is going to build a path around the edge of their backyard. He needs to find out how long the path is. What is the best measurement tool? Strings, Tiles and Cubes Teacher Materials, Page 8 of 11
Part 2 Summary 11. String Summary: (T-9) a. For which questions in part 1 did you choose string? b. What are the similarities between those situations that made you choose string? c. What is the proper vocabulary word for that measurement? d. From the word boxes below, choose the units of measure that match this idea. 12. Square Tiles Summary: a. For which questions in part 1 did you choose square tiles? b. What are the similarities between those situations that made you choose square tiles? c. What is the proper vocabulary word for that measurement? d. From the word boxes below, choose the units of measure that match this idea. 13. Cubes Summary: a. For which questions in part 1 did you choose cubes? b. What are the similarities between those situations that made you choose cubes? c. What is the proper vocabulary word for that measurement? d. From the word boxes below, choose the units of measure that match this idea. inches cubic miles square yards U.S. Units square cubic miles yards feet cubic inches yards square feet square inches miles cubic feet Centimeters Kilometers Square centimeters Metric Units Square kilometers Cubic meters Cubic kilometers Cubic centimeters meters Square meters Strings, Tiles and Cubes Teacher Materials, Page 9 of 11
String, Tiles, and Cubes Student Activity #3 (T-10) Name For each measurement unit listed, find 1-3 different items that could be correctly measured with that unit. Be creative! 1. Miles - _ Centimeters - 2. Square kilometers - _ Square inches - 3. Cubic feet - Cubic meters - Strings, Tiles and Cubes Teacher Materials, Page 10 of 11
4. The three types of measurements that we have worked with are Perimeter, Area, and Volume. In your own words, write a few sentences about everything you know about these three ideas. Perimeter: (TC-11) Area: Volume: Strings, Tiles and Cubes Teacher Materials, Page 11 of 11