## Fair and Square: Using Concrete-Pictorial-Abstract Activities to Maximize Area

- Lesson

Students discover the
relationships between area and perimeter as they prep for playing Square Off, a
wonderful Calculation Nation^{®}
game. This lesson helps students understand the math of area and perimeter, which
will help to maximize their scores when playing the game. Creating human-sized
rectangles and working with geoboards provide concrete experiences before
moving on to pictorial and abstract work with area and perimeter of rectangles.

Successful learning takes place when a child has a true understanding of “the why” before learning “the how.” Using a concrete-pictorial-abstract approach can maximize learning, which is why this instructional technique is used throughout this lesson.

In this lesson, the concrete portion will occur with two activities, “Human Rectangles” and “Geoboard Connections.” Small groups will construct rectangles of a given perimeter with themselves as the vertices in “Human Rectangles.” Then in “Geoboard Connections,” each student will construct rectangles of a given perimeter but with different areas on geoboards.

Students move to the pictorial or representational stage when they draw and label the rectangles on the grid paper.

The abstract part occurs when students use numbers and symbols or formulas to determine area and perimeter. Students use their prior concrete and pictorial experiences to visualize the models, which helps them to understand the abstract concepts of area and perimeter.

This lesson involves three parts, which may be completed on three separate days or consecutively during the same day, if time allows.

To prepare for this lesson, make your own grid on the playground, or use masking tape to make a grid in your classroom on the floor, or simply use the floor in your classroom if it is covered with square tiles.

Prepare enough string loops so that there is one for each group of four students in your class. Each string should be 16 units long. Units can be one foot or any other unit that will work in your classroom. If you have 9-inch square floor tiles, then use 9 inches as the length of your unit. The exact length of the unit is not important, as long as the unit matches the length of the side of the squares in your grid.

Mark the units on your string in one of the following ways: a permanent marker; tie knots; tie another short piece of string at each unit mark.

Tie the beginning and end of the string together to form a loop.

Also, make enough copies of the four worksheets: group data sheet (1 for each group), individual (1 for each student), class (1 for each student or an overhead copy to display), and 1 copy of perimeter war cards for each group (print cards and answers back-to-back on card stock).

Individual Data Activity Sheet

Perimeter War Cards Activity Sheet

**Square Off**

To build exigency for learning the concepts in this lesson,
you may want to allow students to play a few games of Square Off on Calculation Nation^{®}. Playing the game successfully requires an understanding of the relationship
between area and perimeter. Students may have a difficult time winning the game
if they don’t understand the concepts involved. After students play one or two
games, announce, “Let’s learn some strategies that will help you win.”

Calculation Nation: Square Off

**Human Rectangles**

On the first day, divide class into groups of four students each. Give each group a string of 16 units.

All four students need to hold the string at a unit mark so that the loop forms a rectangle.

Before they begin, tell them that they will be asked the following questions after the activity.

- How do you know it’s a rectangle? [Opposite sides have the same the same length. We are standing along the grid lines, so we can see that we have four right angles.]
- What part of the rectangle are you holding? [A vertex.]
- What are the dimensions? [Answers will vary.]
- What can you do to figure out the area? [Count each square on the grid or floor tiles OR count how many spaces there are between the knots across and down the string, and then, multiply.]
- What is the perimeter of your rectangle? [16 units]
- What is the area? [Answers will vary.]

Have students record their data on their group data activity sheet.

At some point, students may attempt to make a rectangle with dimensions that use fractions. Tell them that although there are many rectangles in real life with those measurements, for today’s lesson, they should only consider dimensions that are whole numbers.

Have the group try to make a rectangle with different dimensions. Before they begin, have them discuss what they will do. It would be fun to see if they could do it with their eyes closed. (If the string is marked with knots, they could feel the units.)

Have students look at their data sheet. As you circulate, discuss the following questions to members in each group:

- Do you see any patterns? [As the length increases by 1, the width decreases by 1; the sum of the length and width is always the same; the sum of the length and width is half of the perimeter.]
- How does the area change as you change the dimensions? [They may not see the relationship yet, but eventually we want them to see that the narrower the rectangle, the less the area is. The more like a square, the greater the area. Once students realize this, you may want to reinforce the concept that a square is a rectangle by definition.]

Invite the groups to record their data on a class data activity sheet. Ask the following questions: Do you think you have made all the possible rectangles using this perimeter? [Answers may vary.] How do you know? [Answers may vary. Sample answer: We have used all the possible whole numbers from 1-15 as the length and width.]

**Geoboard Connections**

On the second day or second part of the lesson, give each pair of students a geoboard and rubber bands.

Refer to the previous activity, and ask the following questions:

- How did you keep the perimeter the same and change the area? [The string stayed the same size, so that was the perimeter, but we could make different size rectangles with it so the area changed.]
- When you use a geoboard, will you be able to keep the same perimeter and make rectangles of different areas? [Rubber bands stretch and the string we used yesterday did not, so the only way to keep the perimeter the same is to count the units.]

You may want to review counting spaces instead of pegs to find length on a geoboard.

Explain that today’s activity will allow each student to experiment with constructing rectangles on their geoboards with a given perimeter. They will start with a perimeter of 12 units. Have students find the area of each rectangle they make and record its dimensions on the data sheet.

Ask questions such as:

- Which rectangle has the greatest area? [A 3x3 square.]
- Which has the least area? [A 1x5 square.]
- Can you make a generalization stating how to maximize area for a given perimeter? [The rectangle should be a square.]
- Can you think of a situation when you would want to try to get the greatest area for a set perimeter? [You have a set amount of fencing and you want to fence off the greatest area for your dog to run around in.]

Next, have students use the grid paper to create rectangles with a perimeter of 18 units. See if they can create the rectangle with the greatest area first, and then the least area. Have them record their data on the individual data activity sheet.

**Perimeter War**

During the third part of the lesson, tell students to look at their data from the previous activities. Ask if they notice a relationship between the dimensions of their rectangles and the perimeter. If necessary, give the following hints:

- What is the sum of each rectangle’s length and width?
- How does the sum of the length and width relate to the perimeter?
- How can you find dimensions of a given perimeter without drawing them or using a geoboard? [Look at the perimeter and find half. Then look for two numbers whose sum is half of the perimeter.]

Have them discuss their answers with a partner.

Have the students try all the possible rectangles for a perimeter of 20 units. Then ask, “How do you know that you’ve found all of them?” [Rectangles that measure 1 × 9, 2 × 8, 3 × 7, 4 × 6, and 5 × 5 are the only ones possible. Their dimensions have a sum of 10 units, which is half of the perimeter.]

Later in the lesson, students will play Square Off, a game
on Calculation Nation^{®}.
Maximum scoring within Square Off occurs when a player makes a rectangle with
the greatest area that captures the most spaceships. Points are also earned by making
a rectangle that has the greatest area for a given perimeter. (By now, the
class should understand that the closer the rectangle is to a square, the
greater the area for a given perimeter. Students should realize that the title
of the game is a hint to getting the best score.)

Prior to playing Square Off, students will play Perimeter War. Begin this part of the lesson by putting the following numbers on the board.

24 16 10 14 12

Ask, “Which of these numbers could be the perimeter of a square (with integer sides)? How do you know? Why don’t the others work?” [Only 24, 16, and 12 could form a square with integer sides. They are divisible by 4, or said another way, when you divide by 2, the result is an even number.] Note that 14 and 10 could also be the perimeters of a square, but it would not have integer sides. Give students enough time to explore these perimeters. For those who need to revert to concrete or pictorial representations, allow them to use geoboards or grid paper.

Then ask, “What would be the closest to a square you can get for the perimeters of 10 and 14?” [A 2 × 3 rectangle is possible for a perimeter of 10 units; a 3 × 4 rectangle is possible for a perimeter of 14 units.]

Tell the students that they will be playing Perimeter War to help them get quicker at finding dimensions of rectangles for a given perimeter. Model the game first. Hold up a card and ask them to raise their hand when they have the most efficient dimensions, that is, the dimensions closest to making a square. For example, if the card is 36, the most efficient dimensions would be 9 × 9; if the card is 22, the most efficient dimensions would be 5 × 6. Show students that the answer is on the back of the card. When playing the game, the student who answers first correctly gets to keep the card. The student with the most cards at the end of the game is the winner.

Students can play in small groups. (Try to match students of like ability as playing partners.) Each group will need a dealer who puts the card in play and knows the answer, so she can award the card to the player who answers correctly first. As a modification for low-achieving students, play the game by allowing them to win cards by dividing the perimeter in half. Once they get good at that, they can play the game with the full rules.

Square Off also has another element that you may want to practice with your students. Students have to choose among four different perimeters when they make their rectangles.

Write the following four numbers on the board. Then ask, “For these perimeters, which would allow you to make a rectangle with the greatest area?” [18.]

14 16 10 18

You can then ask, “Is it true that the larger the perimeter, the larger the area? Can you give an example when this is not true? Work with your partner.” [If you are efficient in choosing the best dimensions for a rectangle, this will be true: the greater the perimeter, the larger an area that can be formed.]

**Playing Square Off**

As a culmination of the lesson, allow students to play Square Off. They can choose to play against the computer by choosing the “Challenge Yourself” option, or they can play against others by choosing “Challenge Others.” Players within Calculation Nation have anonymous usernames, but students within your class can share their usernames with one another so they know who they are challenging.

Afterwards, lead a discussion about the game. You can use the Questions for Students to lead this discussion.

- Group Data Activity Sheet
- Individual Data Activity Sheet
- Class Data Activity Sheet
- Perimeter War Cards Activity Sheet
- Computers with access to the internet
- Sturdy string
- Tiled floor or masking tape
- Geoboards
- Rubber bands
- Grid paper
- Cardstock

**Assessment Options **

- The best assessment option for this lesson is to circulate while students are working on the various activities. Ask questions and listen to student discussions as they are working. Their communication will be your best indication of whether they understand the relationship of area and perimeter.
- Record scores for five games of Square Off before and after the activities. You may also want to record how long it took to complete five games before and after the lessons. It may take students more time to play the game after the lessons, because they now understand what they have to do. Hence, they may spend more time trying to maximize their score. Ask students to respond to the following prompt in their journals: “What changes in strategy allowed your scores to increase when playing Square Off?”
- Use teacher observation of students playing Perimeter War. Check to see that students have gained understanding of the strategy used to find the most efficient dimensions.

**Extensions **

- What happens if you have a
given area? Can you have different perimeters? Use a geoboard or grid paper to
make as many rectangles with a given area, and determine the perimeter for each.
Can you make a generalization about the relationship between the area and
perimeters? [For a rectangle with an area of
*n*square units, the perimeter can range from 2(*n*+ 1) units (when the rectangle has dimensions 1 ×*n*) to*n*units (when the rectangle is a square).] - What would happen to the area and perimeter of a rectangle if you double the length and the width? What if you double the length only? Use a geoboard or grid paper for your investigation. Can you make a generalization about what happens? [If both dimensions are doubled, the perimeter doubles, but the area quadruples. If only the length doubles, then the area will double, also; the perimeter will increase, but the amount in increases depends on the exact dimensions of the rectangle.]
- Earlier, some students may have tried to use fractions for their dimensions. As an extension, ask the following question: If you were not limited to using whole‑number dimensions, how many different rectangles could be made with a given perimeter? [There is no limit to the number of rectangles that could be made.]

**Questions for
Students **

1. What happens to the area when you change the dimensions of a rectangle with a given perimeter? Which type of rectangle maximizes the area?

[The area changes if you have different lengths and widths for the same perimeter. The closer the rectangle gets to a square, the greater the area is. The rectangle will look more like a square as the length and width numbers get closer together. For example- a 5 × 6 rectangle will look more like a square, than a 4 × 7 rectangle.]

2. What did you do to be able to speedily find dimensions of a rectangle given a perimeter?

[Take the perimeter and split it in half, because opposite sides of a rectangle are equal. So if the length and width are added, the result would be half the perimeter. Once half the perimeter is known, then two numbers with that sum need to be found. For example, if the perimeter of 10, then half the perimeter is 5. Two possible rectangles are 4 × 1 or 3 × 2, both of which have a perimeter of 10 units.]

3. After going back to the game Square Off, what else besides choosing the best perimeter and most efficient rectangle will help increase your score?

[Capturing the most ships.]

4. Is there ever a time when it would be better to choose a lower perimeter or make a less efficient rectangle when playing Square Off?

[Yes, sometimes you need to do that to capture more ships.]

**Teacher Reflection **

- Was students’ level of enthusiasm/involvement high or low? Did it remain the same for the abstract lesson? Explain why or why not.
- Think of a concept that has been difficult for you to teach or your students to master. How might you incorporate the C-P-A approach to that lesson to help your students be more successful?
- Did you challenge the achievers? How?
- How did the C-P-A approach help your weaker math students?
- Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?
- What worked with classroom behavior management? What didn't work? How would you change what didn’t work?
- Could you have used technology to improve this lesson? How?

### Learning Objectives

Students will:

- Be able to explain the relationship between area and perimeter.
- Formulate a rule for determining the best dimensions for maximizing area with a fixed perimeter.

### NCTM Standards and Expectations

- Use geometric models to represent and explain numerical and algebraic relationships.

- Build and draw geometric objects.

- Create and describe mental images of objects, patterns, and paths.

- Recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.

### Common Core State Standards – Mathematics

Grade 3, Measurement & Data

- CCSS.Math.Content.3.MD.C.6

Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

Grade 3, Measurement & Data

- CCSS.Math.Content.3.MD.D.8

Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

Grade 4, Measurement & Data

- CCSS.Math.Content.4.MD.A.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.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP4

Model with mathematics.

- CCSS.Math.Practice.MP7

Look for and make use of structure.