Level C Lesson 30
Shapes with Equal Areas
In lesson 30, the objective is, the student will be able to split polygons into parts with equal area and name the areas as a fraction of the original polygon.
The skills students should have in order to help them in this lesson include being able to split shapes into unit fractions and understanding that area is the amount of space a two-dimensional figure covers.
We will have three essential questions that will be guiding our lesson. Number 1, when a larger shape is split into smaller shapes that are equal in size, explain how to find the area of one of the smaller shapes. Number 2, if a triangle is one-sixth the area of a larger polygon, how many equal triangles form that polygon? And number 3, if a polygon is split into ten equal triangles, what is the area of each triangle when compared to the area of the polygon?
The SOLVE problem for this lesson is, in Sharice’s gym class the students are drawing shapes in the parking lot to play a game. Sharice drew a pentagon, which she then has to split into five equal sections. One student is going to stand in each of the sections. What part of the total area of the pentagon is each student standing in, and what shape is each section?
We will begin by Studying the Problem. First we want to identify where the question is located within the problem and we will underline the question. What part of the total area of the pentagon is each student standing in, and what shape is each section? 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 shape and area of each part, when compared to the entire pentagon.
During this lesson we will learn how to determine the fractional area of equal parts of a shape. 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.
Each pair of students should have the page with the shapes provided on it as well as two pairs of scissors and glue. Student pairs will need to cut out the shapes on the Copy Master. Pairs will need to cut the paper in half and work together on this task. Student pairs will be sharing all of the shapes with their partner, once cut out each pair of students should have a set of the shapes seen here.
Let’s take a look at the first problem together. How many triangles can fit inside the rhombus? Let’s see, one, two. There are two triangles that fit inside the rhombus. You will glue the triangles inside the rhombus in your book.
How many triangles can fit inside the trapezoid? Let’s see, one, two three. Three triangles fit inside the trapezoid. Again you will glue the triangles inside the trapezoid in your book.
How many triangles can fit inside the hexagon? Let see, one, two, three, four, five six. Six triangles can fit inside the hexagon. Again you will glue the triangles inside the hexagon in your book.
And how many trapezoids can fit inside the hexagon? Let’s see, one two. Two trapezoids can fit inside the hexagon. Again you will glue the trapezoids inside the hexagon in your book.
In problems one through four, what did the pasted shapes do to the original shape? They split the original shape into smaller pieces. Are the smaller shapes within each shape the same? Yes. We can place the left over triangles and trapezoids on them to see that they are exactly the same size.
Let’s look back at problem one. We want to label the two triangles inside the rhombus as the unit fraction of one-half. Since the triangles are exactly the same and there are two of them, they each form one-half of the rhombus.
Now let’s look at problem two. Label the three triangles inside the trapezoid as the unit fraction of one-third. Since the triangles are exactly the same and there are three of them, they each form one-third of the trapezoid.
Let’s look at problem three. Label the six triangles inside the hexagon as the unit fraction of one-sixth. Since the triangles are exactly the same and there are six of them, they each form one-sixth of the hexagon.
And let’s look at problem four. Label the two trapezoids inside the hexagon as the unit fraction of one-half. Since the trapezoids are exactly the same and there are two of them, they each form one-half of the hexagon.
What does area mean? Area is the amount of space a two-dimensional figure covers. Because area is the amount of space that a figure covers, we can describe the smaller triangles in the rhombus of problem one as each taking up one-half of the area of the rhombus. If the rhombus is a pizza and you eat one triangle, would you be eating half of it? Yes, because you would be eating one piece out of two of the exact same size pieces. This means that the triangle is exactly one-half of the area of the rhombus. It does not matter what the area of the rhombus is, the triangle covers half of the area, and so each triangle’s area is half of the area of the rhombus. The unit fraction one-half that you wrote on each piece also represents the smaller shaped area when compared to the larger shape.
In problem two, the trapezoid was split into three triangles. How much of the trapezoid’s original area does one triangle cover? One-third. The unit fraction one-third that you wrote on each piece also represents the smaller shapes area when compared to the larger shape. This means that the triangle is one-third of the area of the trapezoid.
In problem three, the hexagon was split into six triangles. How much of the hexagon’s original area does one triangle cover? One-sixth. The unit fraction one-sixth that you wrote on each piece also represents the smaller shaped area when compared to the larger shape. This means that the triangle is one-sixth of the area of the hexagon.
In problem four, the hexagon was split into two trapezoids. How much of the hexagon’s original area does one trapezoid cover? One-half. The unit fraction one-half that you wrote on each piece also represents the smaller shaped area when compared to the larger shape. This means that the trapezoid is one-half of the area of the hexagon.
Now let’s look at the graphic organizer of polygons. We will be splitting the shapes into smaller shapes of equal areas. In the first column the shape is drawn. The second column tells how many equal parts to split the shape into. In the third column, you will be writing in the name of the smaller shapes. In the last column, you will fill in the blank about the area. We will fill in the first row of the graphic organizer together. How can we split the triangle into two equal parts? We can split the triangle into two equal parts by placing a vertical line from the top angle to the bottom of the triangle. What shapes were created when we drew the line? Three-sided figures, or triangles. We will place the word triangles in the third column. Are these two shapes equal in size? Yes, because the original triangle was split into two equal pieces. The smaller triangle is what area of the larger triangle? The smaller triangle is one-half the area of the larger triangle. So in the last column we will complete the statement that the smaller shape is one-half the area of the larger shape.
Now let’s take a look at an example Splitting Shapes into Shapes with Equal Areas. Aaron has a wooden piece of plywood. It is in the shape of a rectangle. He is painting the rectangle three different colors – red, blue and green. Each color will take up the same amount of space on the rectangle. Color the rectangle below to show how Aaron painted it. What is the shape of the figure? The figure is a rectangle. How many equal parts are we dividing the rectangle into? And why? We are going to divide the rectangle into three equal parts, because we have three different colors that Aaron is going to paint it. How do we divide the rectangle into three equal sections? We can draw vertical lines so that there are three sections of the same size. Now we will color the three sections, one red section, one blue section, and one green section. How should we label each section? Each section is one-third of the original rectangle, so we can label each section one-third. Let’s do that now. The red section is one-third the area of the rectangle’s area.
Now we are going to go back to the SOLVE problem from the beginning of the lesson. In Sharice’s gym class, the students are drawing shapes in the parking lot to play a game. Sharice drew a pentagon, which she then has to split into five equal sections. One student is going to stand in each of the sections. What part of the total area of the pentagon is each student standing in, and what shape is each section?
At the beginning of the lesson we Studied the Problem. We underlined the question. What part of the total area of the pentagon is each student standing in, and what shape is each section? And then we took this question and put it in our own words in the form of a statement. This problem is asking me to find the shape and area of each part, when compared to the entire pentagon.
In Step O, we will Organize the Facts. First we will identify the facts. In Sharice’s gym class, fact, the students are drawing shapes in the parking lot to play a game, fact. Sharice drew a pentagon, which she then has to split into five equal sections, fact. One student is going to stand in each of the sections, fact. What part of the total area of the pentagon is each student standing in, and what shape is each section? Now that we have identified the facts we want to eliminate the unnecessary facts. These are the facts that will not help us to find the shape and area of each part when compared to the entire pentagon. In Sharice’s gym class, knowing that Sharie is in gym class will not help us to find the shape and area of each part when compared to the entire pentagon. So we will eliminate this fact. The students are drawing shapes in the parking lot to play a game. This also does not help us to find the shape and area of each part. So we will eliminate this fact as well. Sharice drew a pentagon, which she then has to split into five equal sections. Knowing what shape she drew and how many sections she is going to split it into is important to finding the shape and area of each part. So we will keep this fact. One student is going to stand in each of the sections. Knowing what’s going to happen with each of the sections when she is done drawing the shapes is not going to help us to find the shape and area of each part. So we will eliminate this fact as well. Now that we have eliminated the unnecessary facts, we will list the necessary facts. The shape is a Pentagon, and it is split into five equal sections.
In Step L, we will Line Up a Plan. First we want to choose an operation or operations to help us to solve the problem. To solve this problem we will need to draw a picture, so we will not have an operation or operations to help us. Now let’s write in words what your plan of action will be. We can draw a pentagon and split it into five equal sections.
In Step V, we Verify Your Plan with Action. First let’s estimate your answer. We can estimate that the shape is a triangle and each triangle will be one-fifth of the entire pentagon. Now let’s carry out your plan. First we need to draw a pentagon. And now we need to split it into five equal sections. Each of the sections is a triangle. Because there are five sections, each triangle is one-fifth of the pentagon.
In Step E, we Examine Your Results. Does your answer make sense? Here you want to compare your answer to the question. Yes, because I am looking for the shape and area of the sections. Is your answer reasonable? Here compare your answer to the estimate. Yes, because the pieces are triangles, and the area of each triangle is one-fifth. And is your answer accurate? Here you want to check your work. Yes. The answer is accurate. We can now write your answer in a complete sentence. The shapes will be triangles, and each has an area that is one-fifth the area of the pentagon.
Now let’s go back and discuss the essential question from this lesson.
Our first question was, when a larger shape is split into smaller shapes that are equal in size, explain how to find the area of one of the smaller shapes. The numerator is one, and the denominator is the number of equal shapes the original shape was split into.
Our second question was, if a triangle is one-sixth the area of a larger polygon, how many equal triangles form the polygon? Since the denominator of the area is six, there must be six equal triangles.
And our third question was, if a polygon is split into ten equal triangles, what is the area of each triangle when compared to the area of the polygon? The area of each triangle is one-tenth of the area of the original polygon.