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# Math Discovery: Area and Perimeter

Module by: Julie Janzen. E-mail the author

Summary: Learning about area and perimeter can be a difficult to conceptualize for student. The following series of three lessons creates a way for students to explore these important mathematical concepts in multiple ways to increase understanding. The lessons also include the California state standards covered as well as the accommodations meant to “reach out” or “extend” to all types of learners.

Day 1

Lesson Topic: Area and Perimeter of Figures

Measurement and Geometry

1.0 Students choose and use appropriate units and measurement tools to quantify the properties of objects:

1.1 Choose the appropriate tools and units (metric and U.S.) and estimate and measure the length, liquid volume, and weight/mass of given objects.

Mathematical Reasoning

1.0 Students make decisions about how to approach problems:

1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.

2.0 Students use strategies, skills, and concepts in finding solutions:

2.1 Use estimation to verify the reasonableness of calculated results.2.2 Apply strategies and results from simpler problems to more complex problems.2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.2.4 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.

3.0 Students move beyond a particular problem by generalizing to other situations:

3.3 Develop generalizations of the results obtained and apply them in other circumstances.

Learning Objectives:

• To explore what square units are by creating square units of different sizes (square centimeter, square inch, square foot, square meter)

Assessment Strategy:

• Participation in activity - observation
• Group discussion
• Review random selection of journals

Vocabulary to Develop: Vocabulary will be developed during the introduction, discussion and during and after the activity

• Area
• Perimeter
• Square units (in terms of square centimeters on graph paper)

Materials Needed:

• Sheets of paper, a few very large ones (butcher paper)
• Rulers
• Scissors
• Journals
• Pencils

Accommodations:

• Reaching out (to ELL students, students with special needs and other learning styles):
• Reduce the number of square objects to make
• Motor skill difficulties: Have student direct someone else how to draw and cut out objects
• ELL: Make only square centimeters and meters if not familiar with inches and feet.
• Extend (GATE Students):
• What other square measurements could be made if you had the materials? (i.e. miles, kilometers)
• What could these measurements be used for in the real world?

Teaching Strategies:

• Process Skills Emphasized: reasoning, communication, connections and representation.
• Intro: (approx five minutes)
• “Today we are going to learn about square units. Does anyone know what a square unit is? Do you have a guess?”
• Talk about how a square unit has the same measurement on each side. For instance, “A square inch measures an inch on each side”.
• Procedure: (approx 30 minutes)
• “I have paper of all different sizes up here. I would like each of you to make a square unit for each of the following:”
• List on board and say, “square centimeter, square inch, square foot and square meter.”
• “Do the first three on your own and the square meter as a table group. If you finish early you can make more square centimeters and square inches.”
• Reflection and Sharing: (Approx 15 minutes)Have a class discussion.
• Does anyone know what perimeter is? Let’s try to figure out the perimeter of each of your square units? What if you put two together? How does the number change?
• Does anyone have an idea what area is? What if you made a bunch of square inches? How could this help you measure the area of something?
• Beyond:
• Discussion/Extension Questions: Have you ever heard of using square feet before? Where? [Discuss buildings/homes/rooms]

Assess/Closure:

• Journal Writing Activity - “What are some important things we learned today? What can we use them for? What do you want to learn more about?”
• Write down anyone who did not participate fully in the activity. Think about why. Review selection of journals.

Day 2

Lesson Topic: Area and Perimeter of Figures

Measurement and Geometry

1.0 Students choose and use appropriate units and measurement tools to quantify the properties of objects:

1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them.

Mathematical Reasoning

1.0 Students make decisions about how to approach problems:

1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.

2.0 Students use strategies, skills, and concepts in finding solutions:

2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.2.4 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.

Learning Objectives:

• To explore the basic concepts of area and perimeter as related to square and rectangular figures
• Explore use of physical models (students themselves) to help think about math concepts

Assessment Strategy:

• Participation in activity observation
• Group discussion
• Review written explanation of what was learned

Vocabulary to Develop: will develop before, during and following lesson in context what is known and what is involved in activity.

• Area
• Perimeter
• Length
• Width
• Square unit vs. unit

Materials Needed:

• Student journals (or paper) and pencils for each student

Accommodations:

• Reaching out (to ELL students, students with special needs and other learning styles):
• Suggest picture explanation instead of words
• During activity allow students to talk and seek help from neighboring students
• For write-up assure that there is no wrong answer, it is just to see what you picked up from activity
• Pay attention and make sure group is including everyone in discussion
• Extend (GATE Students):
• What other formations could the students have stood in?
• What differences would this have made?
• Illustrate your ideas and write about what you think that may mean.

Teaching Strategies:

• Process Skills Emphasized: problem solving, reasoning, communication, and representation.
• Intro: (approx 5 minutes)
• Draw a couple of square units on the board.
• “What did we learn yesterday about square units?”
• “What did we say we could use them for?”
• “How could a make a rectangle out of square units?” Draw.
• “What is the perimeter? Length? Width?”
• Procedure: (approx 30 minutes)
• “Now with those ideas in mind, let’s all go outside together and explore the concepts. Bring your journal and a pencil with you.”
• “I am going to have you all line up in rows that create a rectangular shape. Pretend that someone is looking down on all of you and can see the shape.”
• Line up children in rows. If the number of children cannot be used to make an exact rectangle have the extras help with counting/drawing and make sure they are the first to make up the next rectangle or square. Have them draw a model of what they represent using dots. Suggest asking someone next to them for help if needed.
• “When we talked about perimeter we came up with the idea that it means the number of units on the outside. Let’s pretend that each of you is one square unit, like what we learned to create yesterday. First we will count only the people on the outside to find the perimeter. Call out the counting number when you are pointed to. We are going to count each corner twice. Think about why we are doing that and we will discuss later.”
• Any extra students can walk around and point to students or you can fill this role. Have the students on the outside count aloud. Have them each write down the final number.
• “When we talked about area, we decided that it means the number of units that make up the whole shape. So now we will count all of you by rows.”
• Have the students record the answers.
• Now rearrange the students into different rectangular shapes and squares. Make sure all students get to be part of the shapes and that they are drawing a model and recording the number.
• Reflection and Sharing: (Approx 15 minutes)
• Now have students go inside and study their pictures. Have them write up a few facts about what they learned from the outside activity. Ask them to include length and width.
• After they have written for 10 or so minutes, have them share what they came up with in their table groups. Have them come up with one item to share as a group
• Group discussion: each group shares their one item, discuss until end of class
• Beyond:
• Possible discussion/extension questions, depending on where discussion goes: What is the difference between a unit and a square unit? Why did we count the corners twice? What does perimeter have to do with length and width? What do all of these things have to do with area?

Assess/Closure:

• Have students leave journals open to today’s activity for your later review and assessment before end of day. Revise next day’s lessons to incorporate unclear concepts if necessary.
• For homework have students take home their journals and reflect on what they thought of for the concepts area and perimeter and what the other groups came up with. Have them write up a description (on a separate piece of paper) of what they think about their own and others’ ideas and why or why not they may or may not help determine the area and perimeter of shapes. Also have them write any questions they may still have.

Day 3

Lesson Topic: Area and Perimeter

Measurement and Geometry

1.0 Students choose and use appropriate units and measurement tools to quantify the properties of objects:

1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them.

Mathematical Reasoning

1.0 Students make decisions about how to approach problems:

1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.

2.0 Students use strategies, skills, and concepts in finding solutions:

2.2 Apply strategies and results from simpler problems to more complex problems.2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.2.4 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.

3.0 Students move beyond a particular problem by generalizing to other situations:

3.3 Develop generalizations of the results obtained and apply them in other circumstances.

Learning Objectives:

• To explore the basic concepts of area and perimeter as related to square and rectangular figures
• To use models drawn on graph paper to explore concepts

Assessment Strategy:

• Participation in activity – observation (roam around room to check understanding)
• Group discussion
• Turn in drawings and review journal work (check for understanding)

Vocabulary to Develop: Vocabulary will be developed from a review of yesterday’s class and after the activity in the discussion.

• Area
• Square Units (in terms of centimeters on graph paper)
• Perimeter
• Length
• Width

Materials Needed:

• Graph paper – a few sheets for each student
• Rulers
• Pencils
• Journals

Accommodations:

• Reaching out (to ELL students, students with special needs and other learning styles):
• Use hands on manipulatives to arrange instead of graph paper
• Clarify that it does not matter how many shapes are completed in the time frame…just experiment
• Extend (GATE Students):
• Encourage them to explore other shapes or combinations of squares and rectangles.
• Think of possible formulas that may help solve these problems faster than counting

Teaching Strategies:

Process Skills Emphasized: reasoning, communication, connections and representation.

• Intro: (approx five minutes)
• Review questions from the homework of the day before first in desk groups and then as a class. Review activity and things learned. Have them turn in the homework to assess where the students were at the end of the previous class.
• Procedure: (approx 25 minutes)
• “Now we know what a square unit is from two days ago and have learned how we can change square and rectangular objects. Today we are going to do something similar only with graph paper and drawing.”
• Hand out graph paper to table groups.
• “First I am going to give you about ten to fifteen minutes to work on your own. With the graph paper, I would like you to draw square and rectangular figures of different shapes and sizes. Then I want you to find the perimeter and then the area of each. Can anyone remind us how we learned to do that?”
• Reflection and Sharing: (Approx 5-10 minutes)
• “Now I would like you to work in your table groups and show each other what you drew. Talk about what you drew, why you drew it, and what you discovered. Did you notice any patterns?”
• Beyond: (Approx 5-10 minutes)
• Bring Discussion to a larger group and ask each group for at least two things that they discovered. Have them think about what they might use the information they have learned in the last few days for.
• To write in journals: Ask what their favorite part of the last few lessons on area and perimeter were? What helped their understanding the most?

Assess/Closure:

• Turn in journals for review of last three lessons (not for correction, if work is done credit is given). For own purposes grade as adequate, partial, or limited understanding. Assess student knowledge and modify/add lessons to clarify concepts that may still be unclear.

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