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Level C Lesson 29 Quadrilateral Attributes In lesson 29 the objective is, the student will be able to identify different types of quadrilaterals based on attributes, and draw examples of quadrilaterals that do and do not belong to subcategories. The skills students should have in order to help them in this lesson include knowledge of polygons and quadrilaterals. We will have three essential questions that will be guiding our lesson. Number 1, explain the difference between a rectangle and a square. Number 2, how do you know if a quadrilateral is a parallelogram? And number 3, sketch an example of a quadrilateral that is not a rectangle, square, parallelogram, rhombus or trapezoid. The SOLVE problem for this lesson is, The Flannery family is putting an in-ground pool in their backyard. The pool company told them that they could use any shape quadrilateral they wanted for the pool. Mrs. Flannery told her daughter, Chloe, that the pool was going to have two right angles and one pair of parallel sides. What quadrilateral will the shape of the pool represent? We are going to 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 quadrilateral will the shape of the pool represent? 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 quadrilateral that represents the shape of the pool. During this lesson we will learn how to classify quadrilaterals based on attributes. 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. First you are going to do an activity with your class called getting into groups. You will begin the lesson by sorting students into groups based on a certain characteristic. Go ahead and do that now. How are your sorted? By shirt color, hair color, pant color, type of shoes, or are you sorted in another way, talk about this with your class. Now let’s do it again! You want to be sorted in a different way. How are you sorted this time? Talk about it again as a class. Now that you have had a discussion as a class about the characteristics, that were used to sort you into groups. We are going to take a look at several shapes. We need to begin by cutting out the shapes on the Copy Master that is provided by your teacher. We don’t want to cut directly on the lines, because we don’t want to cut off the congruency marks on the shapes. Once all of your shapes are cut out, Partner A, set your shapes aside, and Partner B, keep your shapes out. You should have a group of shapes that is the same as what you see here. We can use characteristics of the shapes to sort and organize them. Describe some characteristics that we can use to sort the shapes. We could sort the shapes by right angles or no right angles, congruent sides, or parallel sides. Take a look at the group of shapes. We see that several of the shapes have right angles, parallel sides and congruent sides. I’m sure that you could come up with other characteristics that we could use to classify our shapes. Each of the shapes that we cut out is called a Quadrilateral. Create a definition for quadrilateral based on your observation of the shapes. A quadrilateral is a closed figure that has four sides. Explain why each of the shapes are quadrilaterals. Each shape has four sides and is a polygon. We have many different types of quadrilaterals. We will learn how to classify them using attributes. Attributes are what help you distinguish the shapes from each other. One of the attributes we will be looking at is congruent sides. Take a look at Figure A. Figure A has four congruent sides. What does it mean if sides are congruent? It means that the sides are exactly the same length. Identify any markings on Figure A. We see that there are four lines, one on each side of Figure A. What do you think those lines mean? In a shape, the sides that have the same number of lines through them are congruent. Remember that congruent means that the sides are exactly the same length. Take a look at Figure D. How many sides are congruent? All four sides are congruent. We see that each of the sides has one line through it. This line means that the sides are congruent. Now let’s take a look at Figure I. How many sides are congruent? There are two different sets or pairs of sides. The sides with one line are congruent to each other. The sides with two lines are congruent with each other. Another attribute we will look at is if quadrilaterals have right angles. Identify a quadrilateral that has a right angle. Figure A, C, I, and J all have right angles. How do you know that a quadrilateral has a right angle? it makes an “L” shape and has a square inside of it. Take the corner of a piece of paper, which is a right angle, and see if it fits perfectly into the corner of Figure A. We can see that the corner of our paper fits perfectly into the corner of Figure A. This is one way to check and see if an angle is a right angle. What other ways could we check an angle to see if it is a right angle? The corner should form a perfect capital letter “ L.” Put a corner of a sticky note or an index card in it to test the angle. Now let’s draw a right angle for Question 6. To draw a right angle, we start by drawing the two sides. To check to see that our angle is drawn correctly, we can take the corner of a piece of paper and see if it fits perfectly into the corner of our drawing. Last we draw a little square in the corner to represent that this is a right angle. So far we have talked about congruent sides and right angles. The last attribute we will look at will look at is parallel sides. What is the meaning of parallel sides? They are sides that could go on forever but will never touch or intersect. For example, the two sides of railroad or roller coaster tracks always have to be the same distance apart. Now that we have talked about the three attributes, congruent sides, right angles and parallel sides, we are going to take a look at a graphic organizer. The graphic organizer on Student page two hundred seventy eight shows how to sort quadrilaterals based on the attributes we’ve talked about in this lesson. Let’s start by determining which of the shapes are quadrilaterals. Looking at the shapes we can see that they are all quadrilaterals, because they are four-sided, closed figures. Let’s take a look at the next box on the organizer. First we will look at Parallelograms. A Parallelogram is a quadrilateral with two sets of parallel and congruent sides. We know that all of the shapes that we have are quadrilaterals. We need to look for the attributes of parallel and congruent sides. Explain how you can identify congruent sides. We need to look for the lines on the sides to show that they are congruent. How many sets, or pairs, of parallel sides must a quadrilateral have in order to be a parallelogram? The definition tells us, it must have two sets of parallel sides. Now separate the shapes that have two sets of congruent sides. Looking at the shapes we can see that shape A, C, D, E, G, and I have two sets of congruent sides. Let’s take these shapes away from the others to study them more carefully. Are the pairs of congruent sides also parallel? Explain. Yes, I can put my pencils over the sides, and they could go forever without touching, similar to railroad tracks. Are all of these parallelograms still quadrilaterals? Yes, because they have four sides. Are all parallelograms quadrilaterals? Yes, all parallelograms are quadrilaterals, because they are closed figures with four sides. Are all quadrilaterals parallelograms? No. A trapezoid is a quadrilateral but it does not satisfy the definition of a parallelogram. A trapezoid does not have two sets of parallel and congruent sides. Now let’s talk about rectangles. A rectangle is a parallelogram with two sets of congruent sides and four right angles. Explain which shapes we should look at to identify the rectangles. We need to consider only the parallelograms because the definition tells us that rectangles are parallelograms. Which shapes do you think are rectangles? Shape I, C, and A are all rectangles. They are parallelograms with four right angles. How are the rectangles still parallelograms? They have two sets of parallel and congruent sides. We will take these three shapes and place them under Rectangles, as all three are parallelograms with two sets of congruent sides and four right angles. Now let’s talk about which shapes would be classified as a Rhombus. A Rhombus is a parallelogram with four congruent sides. Explain which shapes we should look at to identify the rhombuses. We need to consider only the parallelograms because the definition tell us that they are parallelograms. Which shapes do you think are rhombuses? Shape A, and D are both rhombuses. They are parallelograms with four congruent sides. Let’s take shapes A and D and place them under Rhombus. Explain how these rhombuses are still parallelograms. They both have two sets of parallel and congruent sides. Now let’s talk about the shapes that are Square. The definition of a square is a parallelogram with four congruent sides and four right angles. Explain which shapes we should look at to identify the squares. We need to consider only the parallelograms because the definition tells us that squares are parallelograms. Which shapes do you think are squares? Shape A is a square. Figure A is a parallelogram with four congruent sides and four right angles. Let’s take shape A and place it under the definition for a square. Explain how this square is still a parallelogram. We still have two sets of parallel and congruent sides. Explain how this square is still a rectangle. It is a parallelogram with two sets of congruent sides and four right angles. And explain how this square is still a rhombus. It is a parallelogram with four congruent sides. Now let’s talk about those shapes that are Trapezoids. The definition of a trapezoid is a quadrilateral with one set of parallel sides. Explain which shapes we should look at to identify the trapezoids. We need to consider only the shapes that were NOT parallelograms because a trapezoid has only one set of parallel sides, where a parallelogram has two sets of parallel sides. We can remove all of the parallelograms from our group of shapes when we are trying to see which of the remaining shapes are trapezoids. Looking at the shapes that are not parallelograms, which ones do you think are trapezoids? Shapes B, H and J are trapezoids. They each have one set of parallel sides. Let’s take these shapes and place them under Trapezoid. Are trapezoids still quadrilaterals? Yes, because they have four sides. Are the trapezoids rectangles? No, because they do not have four right angles and two sets of congruent sides. Are there any shapes left in the quadrilateral area? Yes, there are two shapes left, shapes F and K. What do you think this means? These figures are quadrilaterals because they have four sides, but they do not fit in any of the other categories. So let’s summarize what we’ve done with these shapes. The shapes all started at the top as quadrilaterals. As the arrows move down on the graphic organizer, so do the shapes. Some shapes moved down to parallelograms and didn’t stop there. They also moved down to rectangle, rhombus or square. Some of the shapes have many names. The best name for a shape is the category that it ended up in, although it can be called anything that was connected with an arrow above it. We are now going to go back to the SOLVE problem from the beginning of the lesson. The Flannery family is putting an in-ground pool in their backyard. The pool company told them that they could use any shape quadrilateral they wanted for the pool. Mrs. Flannery told her daughter, Chloe, that the pool was going to have two right angles and one pair of parallel sides. What quadrilateral will the shape of the pool represent? At the beginning of the lesson we Studied the Problem. We underlined the question, what quadrilateral will the shape of the pool represent? And put this question in our own words in the form of a statement. This problem is asking me to find the quadrilateral that represents the shape of the pool. We will now Organize the Facts. We will start by identifying the facts. The Flannery family is putting an in-ground pool in their backyard, fact. The pool company told them that they could use any shape quadrilateral they wanted for the pool, fact. Mrs. Flannery told her daughter, Chloe, that the pool was going to have two right angles, fact, and one pair of parallel sides, fact. What quadrilateral will the shape of the pool represent? 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 what quadrilateral the shape of the pool will represent. The Flannery family is putting an in-ground pool in their backyard. Knowing that they’re putting the pool in their backyard is not going to help us to know the shape of the pool, so we will eliminate this fact. The pool company told them that they could use any shape quadrilateral they wanted for the pool. This fact does not help us to find what quadrilateral the shape of the pool will be. So we will eliminate this fact as well. Mrs. Flannery told her daughter, Chloe, that the pool was going to have two right angles. Knowing that the pool will have two right angles will help us to find out what type of quadrilateral the shape of the pool represents. So we will keep this fact. And one pair of parallel sides, knowing that the pool has one pair of parallel sides is also important to finding out the shape of the pool. Now that we have eliminated the unnecessary facts, we will list the necessary facts. The pool is a quadrilateral, it has two right angles, it has one set of parallel sides. In Step L, we will Line Up a Plan. First we need to choose an operation or operations to help us to solve the problem. In this problem we will not have an operation as we are looking to classify the shape of the pool. We will write the word none for our operation. Now let’s write in words what your plan of action will be. We can look at the chart of quadrilaterals to see what quadrilateral has the listed attributes. We are now ready to Verify Your Plan with Action. First we estimate your answer. Since we are using the chart of quadrilaterals to see which quadrilateral has the listed attributes will not have an estimate for our answer. We will move right into carrying out your plan. The attributes on Step O were that the pool has two right angles and it has one set of parallel sides. Looking at the chart of quadrilaterals the only quadrilateral that has the attributes of one set of parallel sides is a trapezoid. A trapezoid has one set of parallel sides. So the pool will be in the shape of a trapezoid. In Step E, we will Examine Your Results. Does your answer make sense? Here compare your answer to the question. Yes, because I am looking for the shape of the pool. Is your answer reasonable? Here compare your answer to the estimate. Since we did not have an estimate for this problem, this question is not applicable. And is your answer accurate? Here you want to check your work. Yes our answer is accurate. We are now ready to write your answer in a complete sentence. The pool will be in the shape of a trapezoid. Now let’s go back and discuss the essential questions from this lesson. Our first question was, explain the difference between a rectangle and a square. The rectangle has two separate pairs of congruent sides, and a square has all four sides congruent. Our second question was, how do you know if a quadrilateral is a parallelogram? A quadrilateral is a parallelogram if it has two pairs of congruent parallel sides. And our third question was, sketch an example of a quadrilateral that is not a rectangle, square, parallelogram, rhombus or trapezoid. Answers will vary, but none of the answers should have parallel sides. Here are some examples: