Level C Lesson 25 Perimeter of Polygons

In lesson 25 the objective is, the student will solve problems involving perimeter of polygons.

The skills students should have in order to help them in this lesson include multiplication facts zero through nine.

We will have three essential questions that will be guiding our lesson. Number 1, how can the perimeter of a square be found using a formula? Number 2, how can the perimeter of a rectangle be found using a formula? And number 3, how can a missing side of a polygon be determined if the perimeter of the polygon is known?

The SOLVE problem for this lesson is, Margo is helping her dad build a pen in the backyard for her new puppy. She got the puppy for her birthday. The pen is in the shape of a rectangle. The length of the pen is eight feet, and the width of the pen is six feet. What is the perimeter of the pen?

We will start by Studying the Problem. First we want to identify where the question is located within the problem and we will underline the question. What is the perimeter of the pen? Now that we have identified the question we want to put this question in our own words in the form of a statement. This problem is asking me to find the pen’s perimeter.

During this lesson we will learn how to determine the perimeter of polygons. We will use this knowledge to complete this SOLVE problem at the end of the lesson.

Throughout this lesson students will be working together in cooperative pairs. All students should know their role as either Partner A or Partner B before beginning this lesson.

We will start this lesson by Investigating Perimeter of Polygons, creating concrete and pictorial representations. Each pair of students will need a set of toothpicks and colored pencils to participate in this activity.

First let’s trace your finger around the edge of your desktop. This is the perimeter of the desk. What do you think perimeter means? It is the distance around a figure. When might it be useful to know how to measure the perimeter of any surface? It may be useful when putting up a border on the bulletin board, or when putting up a fence in a yard. Using toothpicks, let’s create a square! A square has four sides, so we will use four toothpicks. What do you notice about the square? All of the sides are equal, or the same size. How many sides does the square have? The square has four sides. So how many toothpicks were used to create the square? We had to use four toothpicks. If each of the toothpicks is considered one unit of measurement, what is the perimeter or distance around the edge of the square? The perimeter is four units. Let’s transfer the toothpick square to the grid paper and shade. Our toothpick square was one unit on each side. So we will shade in one of our gridded squares so that our square is one unit on each side. What is the perimeter of the square? Remember that the perimeter is the distance around the edge of the square. Each side is one unit. So the perimeter of the square is four units. We will label our square on the grid paper, that the perimeter equals four units.

Let’s look at our toothpick square again, and add another square by adding three more toothpicks. We have now turned our square into a rectangle. How many units are on the top of the rectangle? Let’s count, one, two. There are two units on the top of the rectangle. How many units are on the bottom of the rectangle? Let’s count, one, two. There are two units on the bottom of the rectangle. How many units are on the left side of the rectangle? Again, let’s count there is one unit on the left side of the rectangle. And how many units are on the right side of the rectangle? Let’s count, there is one unit on the right side of the rectangle. So what is the perimeter of the rectangle? One, two, three, four, five, six, the perimeter of the rectangle is six units. Let’s transfer the toothpick rectangle to the grid paper and shade. How many squares are in the toothpick rectangle? There are two squares. So we will shade in two squares on our grid paper. How many units of perimeter are shown for the shaded rectangle? There are six units for the perimeter of the shaded rectangle. We will label the shaded rectangle that the perimeter equals six units.

Now let’s Investigate Perimeter of Polygons. What is the shape we see here? This shape is a square. If side one is the top of the square, let’s identify the number of units for this side. There are one, two, two units. Let’s record that there are two units for side one in the chart that is provided. Side one is two units. If side two is the right side of the square, identify the number of units for this side; one two. There are two units for side two. Let’s record this in the chart that is provided. Side two is two units. If side three is the bottom of the square, identify the number of units for this side; one, two. There are two units for side three. Let’s record this information into the chart that is provided. Side three equals two units. And if side four is the left side of the square, identify the number of units for this side; one, two. There are two units for side four. Let’s record side four into the chart that is provided. Side four equals two units. Now let’s look at the column in our chart that’s labeled perimeter. We will use p to represent the perimeter. We need to add the units on the sides together to find out what the perimeter of this square is. The perimeter equals two plus two plus two plus two, which is a total of eight units. The perimeter of the square is eight units.

Let’s look at another example together. How is this problem different from the first problem? There are no units showing inside of the square, but the dimensions are given. Let’s identify the units for this square. Each of the dimensions is labeled in feet. The units for this square are feet. Now describe the sides of the square. Each side of the square is four feet. All of the sides are equal. Next, let’s identify the measure of each side to find the perimeter of the square. Side one is four feet. Let’s place four feet into the chart for side one. Side two is four feet. Let’s place the measurement for side two into the chart. Side three is four feet. Let’s place the measurement for side three into the chart. And side four is four feet. Let’s place the measurement for side four into the chart. In the column labeled perimeter we are going to use p as an abbreviation for perimeter and add the units on the sides together. The perimeter equals four plus four plus four plus four. The perimeter equals sixteen feet.

Let’s look at another problem together. In this problem, we don’t have a picture, but we have dimensions. How can we find the perimeter? We can add all of the dimensions given to find the perimeter of this square. Each side has a dimension of six meters. The perimeter equals six plus six plus six plus six for a total of twenty four meters.

Let’s look at another example. This time our shape is a triangle. How is the problem different from the other problems? The perimeter is given, but we are missing a side. We need to start by identifying the number of meters for side one. The chart and the picture tell us that side one is three meters. Next, let’s identify the number of meters for side two. The chart and the picture tell us that the number of meters for side two is five. So what is the sum of the known sides? We can add the known sides together to find the sum. Three meters plus five meters equals eight meters. How can we find the missing side? We can add the two sides that we know together, then subtract from the perimeter, which has also been given to us. The perimeter is twelve, and the two sides that we know have a sum of eight. We can subtract eight from twelve. Twelve minus eight equals four meters. So side three equals four meters. We will label our chart four meters for side three. Let’s check our answer by adding all three sides together to see if we get the same perimeter as what is given. The perimeter equals three plus five plus four which equals twelve meters. So we know that our answer is correct.

In this next example we want to use length and width to help us to find the perimeter. What does length mean? The length is how long an object is. Looking at the picture, what is the length of the rectangle? The length is four units. Let’s record four units for the length in the chart. What does width mean? The width is how wide an object is. What is the width of the rectangle? Looking at the picture the width of the rectangle is one unit. Let’s record one unit for the width in the chart. How did you find the perimeter of the rectangles in the last activity? We added each side together to find the total, or sum of all sides. What is the total length of the top and the bottom of the rectangle? The top and the bottom of the rectangle represent the length which is four units. So the total length of the top and the bottom of the rectangle is eight units. How did you determine the total length? We can multiply the top or the bottom length by two, four times two equals eight, or we could double one length, four plus four equals eight. What is the total width of the two sides of the rectangle? The width of each side is one unit. So the total width of the two sides of the rectangle is two units. How did you determine the total width? You could multiply one side length by two, one times two equals two, or double one width, one plus one equals two. Can you think of a way to find the perimeter of the rectangle using what you know about the length and the width? We can multiply the length by two, multiply the width by two and add the products together. What letters might represent the perimeter, length and width? So far we have used p to represent the perimeter. We will use p for perimeter, l for length, and w for width. Can you think of a way to describe finding the perimeter using the letters in a formula? P which represents the perimeter equals two times the length plus two times the width. P equals two l plus two w. Let’s record the formula in the chart underneath perimeter. P equals two l plus two w. We want to use this formula and substitute in the values of the length and the width to find the perimeter of this rectangle. The length is four and the width is one. So our formula now reads perimeter equals two times four plus two times one. Let’s solve. Two times four equals eight plus two times one equals two. So the perimeter equals eight plus two. When we add eight plus two together we get ten. The perimeter of the rectangle is ten units.

Let’s take a look at the next problem together. This time our shade is a square. How did you find the perimeter of the rectangles in the previous activity? We added each side together to find the total, or the sum. So how can we find the perimeter of the square using only the length of one side? We can multiply one side by four, because each of the four sides measures three units. The length of our square is three units and the width of our square is three units. What letters might represent the perimeter and side? We can use p for perimeter, and s for side. Let’s establish a way to describe finding the perimeter using one side length, beginning with p equals. P equals four s. S represents one side and all four sides of our square have the same length. We can record the formula in the box underneath perimeter in our chart. P equals four s. Since each side is three units we can substitute the number three for s in the formula. P equals four times three. Four times three equals twelve. So the perimeter of this square is twelve units.

Let’s take a look at the next square. Even though we only know one side of the figure, what do we know about a square? We know that the square has four equal sides and four equal angles. So if the length of the square is five inches, what is the width of the square? The width is also five inches. Let’s record the length and the width in the chart. The length is five inches and the width is five inches. What formula will we use to find the perimeter of the square? We will use p equals four s. Remember that p represents the perimeter and s represents the length of one side. We can record this formula in the box underneath perimeter p equals four s. We will substitute the value of one side of the square into the formula to find the perimeter. P equals four times five. We can record this step in the box for perimeter as well. Four times five equals twenty. The perimeter is twenty inches. Could we use the formulas for the perimeter of a square and a perimeter of a rectangle for finding the perimeter of a triangle or trapezoid? No, because a triangle has three sides, and they are not always equal. A trapezoid has sides that are different lengths and widths.

Now we will create a foldable for Perimeter and Area helping us to organize the information we have learned in this lesson for future reference. We will start with one sheet of paper, placed horizontally in front of you. We are going to fold the top of the paper down to near the bottom of the page, leaving about half of an inch at the bottom. We will crease the fold at the top. Next we going to turn the paper vertically and fold up once, then fold the paper again. This will create four sections. We want to turn the paper back horizontally and we are going to lift the top flap and cut along each crease to create four flaps. We will cut along the first crease and then fold this flap down. Cut along the second crease and fold this flap down. And cut along the third crease and fold both flaps down. Once we have pushed the four flaps down we are going to fold the half of an inch of paper at the bottom up over the flaps. On the outside of the first flap at the top we are going to write Perimeter of Polygons and include the definition of the perimeter. Inside on the first flap at the top we are going to draw a square with the side measurement of four inches. And we will record how to find the perimeter of a square by counting each side. On the page showing beneath the flap we are going to draw a rectangle with side measurements of five, three and five meters, leaving the fourth side blank. You will work with your teacher to record how to find the missing side by adding the sides and subtracting from the perimeter of sixteen meters for the rectangle. On the second flap you will write Perimeter of Polygons Using Formulas. When you lift this flap you will work with your teacher to include information about Perimeter of Polygons Using Formulas under this flap. We will complete this foldable in a future lesson.

We are now going to go back to the SOLVE problem from the beginning of the lesson. Margo is helping her dad build a pen in the backyard for her new puppy. She got the puppy for her birthday. The pen is in the shape of a rectangle. The length of the pen is eight feet, and the width of the pen is six feet. What is the perimeter of the pen?

At the beginning of the lesson we Studied the Problem. We identified where the question was located within the problem and underlined the question. What is the perimeter of the pen? Then we took this question and put in our own words in the form of a statement. This problem is asking me to find the pen’s perimeter.

In Step O, we will Organize the Facts. First we will identify the facts. Margo is helping her dad build a pen in the backyard for her new puppy, fact. She got the puppy for her birthday, fact. The pen is in the shape of a rectangle, fact. The length of the pen is eight feet, fact, and the width of the pen is six feet, fact. What is the perimeter of the pen? Now that we have identified the facts, let’s eliminate the unnecessary facts. These are the facts that will not help us to find the pen’s perimeter. Margo is helping her dad build a pen in the backyard for her new puppy. This will not help us to find the perimeter of the pen. So we will eliminate this fact. She got the puppy for her birthday. Knowing when she got the puppy will not help us to find the perimeter of the pen. So we will eliminate this fact as well. The pen is in the shape of a rectangle. It is important that we know the shape of the pen in order to find the perimeter. So we will keep this fact. The length of the pen is eight feet. Knowing the length of the pen will help us to find the perimeter. So we will keep this fact as well. And the width of the pen is six feet. We also need to know the width of the pen to find the perimeter. So we will keep this fact. Now that we have eliminated unnecessary facts, let’s list the necessary facts. The pen is a rectangle; the length is eight feet; and the width is six feet.

We are now ready to move on to Step L, Line Up a Plan. First we need to choose an operation or operations to help us to solve the problem. We know that the length and the width of the pen are, and that it is in the shape of a rectangle. Thinking back to the formula for finding the perimeter of a rectangle, we know that we will need to use addition and multiplication to solve this problem. Let’s write in words what your plan of action will be. First we want to multiply the length by two and multiply the width by two. Normally we do not use numbers in Step L. But because the number two is a part of the formula that does not change, we need to use those numbers here. Then we add these values together.

Let’s move on to Step V, and Verify Your Plan with Action. First let’s estimate your answer. We know that the length of the pen is eight feet and the width is six feet. We can estimate our answer at about thirty feet. Next we will carry out your plan. We said that the plan was to multiply the length by two. The length is eight, so we will multiply two times eight. And multiply the width by two. The width is six, so we will multiply two times six. Two times eight plus two times six equals sixteen plus twelve. Since we have found the product of two times eight and two times six we are now ready to add these values together. Sixteen plus twelve equals twenty eight. The perimeter of the pen is twenty eight feet

Let’s Examine Your Results. Does your answer make sense? Here compare your answer to the question. Yes, because I am looking for the perimeter of the pen. Is your answer reasonable? Here compare your answer to the estimate. Yes, because it is close to my estimate of about thirty feet. And is your answer accurate? Here you want to check your work. Yes, the answer is accurate. We are now ready to write your answer in a complete sentence. The perimeter of the pen is twenty eight feet.

Now let’s go back and discuss the essential questions from this lesson.

Our first question was, how can the perimeter of a square be found using a formula? We can use the formula p equals four s, and substitute the value of one side for the s. We multiply four times the value of one side to find the perimeter.

Our second question was, how can the perimeter of a rectangle be found using a formula? We can use the formula p equals two l plus two w, and substitute the value of the length for l and the value of the width for w. We will multiply the length by two and the width by two and then add the two values to find the perimeter.

And our third question was, how can the missing side of a polygon be determined if the perimeter of the polygon is known? We can add the values of the given sides and subtract the sum from the total perimeter.