This is an illustrative math problem that has the students looking at the area of 4 triangles to decide which one has the greatest area. Before starting on the problem, have the students make a prediction about their solution and why they feel that way. Then, have students should work independently on this problem. They can use the formula, count squares, or compose them into rectangles to justify their answers.
Students will find out that they all have the same area. I want them to explain how they know. So for each triangle, I should see some work.
Tools: Do now triangle problem.
Before going through the problems with the students, I want them to look at the shape and answer the following questions:
Students should be able to see that the shape can be broken into 2 rectangles. In the power point, I’ve used animation to show this. The original shape will show first so we can have the discussion. Then have the students explain how they could make it into two rectangles. Then click the power point again and the irregular figure will appear again with the dotted line showing one way of breaking it into two triangles. Ask the students what the formula for a rectangle is? Then apply it to the two rectangles shown. In some shapes, we may need to find a missing side length. It would probably be a good idea to have a discussion about how to do this as it may be something they have to do later on.
I’m going to show them two methods to solving the same problem.
Method 1 is to break the shape into two shapes and add the areas together
Method 2 is to compose the shape into one shape we know and subtract the areas.
Either method is useful, but depending on the shape, it may be easier to use one over the other.
The students will be practicing some problems on their own. The problems are in their notes. Remind students to ask themselves “what shapes can I make this in to so I can use a formula I already know”. They should draw the dotted lines to help them see the shapes and to find the missing side lengths. (SMP 2)
Method 1: Break into two rectangles and add the areas. I would have them find the missing side length for practice
Method 2: Make one large rectangle and subtract to find the area
Only method 1 will work. Students should see a triangle and a trapezoid. They should find the areas and add them together.
Method 1 works best. Break the shape into a triangle and a rectangle. They will need to find the base of the triangle by using the corresponding side. Also, the 8 cm length is on the outside of the shape, but it should be used as the height of the triangle. If students have trouble with this, tell them to visualize it inside the triangle. They should see that it would represent the height because it meets the base at a right angle.
Students are looking for an entry point to each of these problems supports SMP 1. Then connecting what they know to a formula is using SMP4.
The students will be working in their cooperative groups to complete a roundtable on irregular figures. Students should be using formulas of figures they already know to solve these problems. You should see them making the dotted lines to show the different figures. Students should check the work of their tablemates before starting a new problem. If the problem is incorrect, they should peer tutor and coach to help their tablemate out. (SMP 3)
The students may have trouble with problem 3 because they will need to create 3 different rectangles. We’ve been practicing with creating 2 rectangles. This may be a stretch for them, but I would encourage them to break it into shapes they know.
The students will be writing about how they know how to find the area of irregular shapes. Writing in math helps create pathways to the brain to help them remember. This is in their notes, but the writing portion can be done on a separate sheet of paper and collected as evidence of student learning.
Tools: Closure question.