Sample secondary mathematics lesson involving area
This resource presents a complete description of one high school math lesson for your consideration. Once you have read it, you might want to analyzing it using the questions presented in Reflecting on a sample mathematics lesson.
Context: The students are at the end of the unit and have two days before their unit packets are due and they have to take their exam. The students have a good handle on the areas of rectangles and triangles and the methods that can be used to find them. The students still need convincing about how the areas of similar triangles are related.
Goals: Wrap up the discussion on the areas of similar triangles. Help the students see that if the scale factor between the sides of similar triangles is x, then the difference between their areas is x2 . I will answer any questions that the students have from the unit to help them become more comfortable with the material.
Materials: Dry-erase board, dry-erase markers and the Triangle Area Review handout.
1:35 – Have warm-up on the board. Find the area of the obtuse triangle with a base of 20 and a height of 10. The students can do the problem alone or in groups for the next ten minutes. Problem ended up taking closer to fifteen minutes. The goal of this warm-up was to open the students' minds to the possibility that the area of any triangle can be found when the base and the height are given. I drew the triangle this way because I knew that if it were tilted, the students would have a more difficult time with the problem.
1:37 – The students are working. Melissa asks if this is a right triangle. “No, this is not a right triangle.” After quite a bit of talking from the class, I clarify that it is possible to find the area of the triangle and that Heron's Formula does not need to be used. The students still seem skeptical. I had drawn the triangle with the height on the inside, but was unsure of whether that would give too much away. Prior to this lesson I debated whether I should have the height on the inside or outside of the triangle and whether it made much of a difference. I am still unsure, but I think that having the height marker on the inside makes it a little more obvious that 1/2*b*h can be used to find the area.
1:48 – Most of the students appear to be done, so I give them one more minute.
1:49 – Ask the students for their methods and solutions. I did not expect this problem to be so difficult for them. A student tells me that they think the answer is 100 cm2 , so I ask how they came to that conclusion. I expected most of the students to use their methods from the floor plan exercise. The student who offered their answer did, indeed, draw a box around the triangle. “Did anyone find an answer without drawing a box?” No answer. “Everyone drew a box?” The students respond with “yup.” I expected that some students would use the method 1/2*b*h, but would not be able to explain why they knew that this method could be used. I repeatedly asked why the students knew that this particular triangle was half of a rectangle. With the triangle drawn in this way, it is not always easy to see the concept of a triangle being half of a rectangle.
1:52 – Jacob comes up to the board to draw the rectangle around the triangle with the length of the rectangle the same as the base of the triangle. The height of the triangle breaks the rectangle into two rectangles and Josh successfully shows the rest of the class that the triangle is in fact half of the rectangle. Some students are still confused and ask to try another problem.
1:54 – Draw on the board a triangle, which appears to be isosceles. After some questioning, I explain to the students that you cannot assume that this triangle is isosceles. The triangle has a height of 15 and a base of 10. The problem is not very different from the previous problem. After a minute some of the students ask if they just need to do the same thing as the first warm-up problem. Helen continues to claim that 1/2*b*h can only work for right triangles despite the fact that we had shown in the first problem that the formula worked. The majority of the students seem to be engaged in the lesson, and one student asked about a triangle with a crooked base. I showed him that the triangle could simply be rotated until the base seemed to be horizontal.
1:59 – Nate asked about obtuse triangles that did not look like the first warm-up problem. I drew the following triangle and asked if the students could find the area. I had thought about bringing this up, but I decided that I would only do a problem like this one if a student brought it up. This problem made me push everything back a little, but I was really interested to see what the students would come up with. “Do you guys think you can find the area?” Most of the students said "Yes" and some even said that it would be easy. Once they started working, they realize how unusual the problem is.
The students had the height of 4 and the base of 12, which creates an impossible problem unless I give the students the missing dotted portion that is horizontal. I expected the students to draw a rectangle around the triangle to figure out the area. Ashley drew a perpendicular from the left vertex of the base. Where that line intersected the longest side, she drew a perpendicular line to the left. She quickly realized that this would not work. I had hoped that once the students realized that 1/2*b*h could be used on an obtuse triangle, such as this one, they would see that the formula could be used on any triangle. I am still not sure that all of the students fully accepted this conclusion.
2:06 – Helen asked me why I divided by 2. Amy tried to explain the concept to Helen. Helen was still confused so I went through the math step by step. I went back through the parts where Helen was getting stuck until she was ready to move on.
2:09 – I asked Samantha to recap for the class the conjecture that she had made the day before about similar triangles. Samantha explained that if the scale factor between the sides of similar triangles is x then the scale factor between their areas is x2. The class was still confused about the relationship between similar triangles so I asked them to test Samantha's conjecture with another problem. I let the students decide the scale factor to prove that the conjecture was not working because of the numbers that I had chosen for the problem. The height of the smaller triangle was 4 and the base was 10. With a scale factor of 25, the students found that the height became 100 and the base became 250. Kristen then found the area of the larger triangle to be 12,500. Samantha's conjecture worked for this problem, but I was not convinced that the students knew why.
2:13 – There was silence when I asked the students why Samantha's conjecture worked, so I placed the students into groups to have a discussion about the conjecture. This was not part of my original plan, but I wanted to know that the students understood before we moved on to the review. The students worked together for eight minutes.
2:21 – Amy, Felipe and Kevin came up to the board to present their explanation. These three got confused with their explanation and, by the end of their presentation, everyone was still confused. I told the class that I was going to recap their reasoning and see if I might be able to explain it in a different way. I took them through step by step and left the sides as variables to see if that would make it clearer. I showed that the area of the first triangle is 1/2*b*h and that since the scale factor is 25, that the base of the larger triangle would then be 25*b and the new height would be 25*h. The class had a very hard time with this because they wanted to call the base and height of the larger triangle b and h like they had with the smaller triangle. Therefore, the new area would be 1/2*(25*b)*(25*h) = 1/2*(25*25)*(b*h) = 1/2*b*h*252 . I asked if everyone understood. Durell told everyone to say that they understood so that they could move on. I told them that I did not want them to just agree because I wanted them to get it. I wanted to make sure that everyone knew that I cared about how well they did in the class. Everyone understood so I moved on.
2:27 – I asked the students to pull out a sheet of paper so that they could turn in a summary about what they had learned that day. I recapped that we had learned that the area of a triangle could be found when we are given the base and the height and that we had learned about the areas of similar triangles. The students were to write a couple of sentences about each. Only about three-fourths of the students began to write. I handed out the review as they worked.
2:30 – By about this time, all of the students were working on their summaries.
2:33 – With only two minutes before the class ended, I reminded them that their summaries were due at the end of the hour. I made the announcement that their reviews should be done by the next day so that I could answer all of their questions before the quiz on Thursday. I had them add their Triangle Area Review sheets to the Table of Contents in their Unit Packets. The class was moving around a lot so I reminded the class a few times that their reviews should be done by the following day, but I am not sure how many were truly listening at that point.
2:35 – End of the hour.
Assessment: At the end of the hour, the students are asked to write a paragraph explaining what they know now about triangles. They were to write a couple of sentences on the areas of triangles and a couple of sentences about how the areas of similar triangles are related. With this summary, I hoped to gain some insight on what the students are comfortable with and what I may need to go over at the beginning of class the next day. I did get a rough idea, but the students were not very complete with their explanations of what they understood and what they were confused about. In the end, I decided to let them ask their questions from the review.
Homework: Triangle Area Review. Tell the students to finish the review so that I can answer all of their questions the next day. I will not tell them that they will have about ten minutes the next day to work with their neighbors to finish their reviews.
You can analyze this lesson by using the questions in Reflecting on a sample mathematics lesson.