Amanda Orr

MATH TALK: THINKING THROUGH LESSON PROTOCOL

Small Group Problem: Find the Perimeter and Area of the figure:

SET-UP

Selecting and setting up a mathematical task:

What are your mathematical goals for the lesson (i.e., what is it that you want students to know and understand about mathematics as a result of this lesson)?

Students will be able to find the perimeter and area of a figure with missing lengths. Students will be able to find the missing lengths. Students will be able to work in small groups and explain their thought process and discuss the problem with peers.

In what ways does the task build on students’ previous knowledge?

What definitions, concepts, or ideas do students need to know in order to begin to work on the task?

Students need to know the definitions of perimeter and area. They need to know the general concept of being able to find the perimeter and area of a figure. It would also be beneficial if they knew how to find the missing lengths of the figure.

What are all the ways the task can be solved?

Which of these methods do you think your students will use?

What misconceptions might students have?

What errors might students make?

The first mistake that students might make is starting to find the perimeter without finding the missing lengths first. Students may also confuse the definitions of perimeter and area.

The following explains the ways that the task may be solved, and the problems students may have in the process:

Missing Lengths = 2 units, 2 units

Because this figure is drawn to scale on a poster board, students can use post-it notes to find the missing lengths. They can line up the post-it notes on the line and count them up to get the missing length.

Students might put the wrong number of post-it notes on the figure.

Students can draw in the missing portion. They will realize that because the opposite side is 5, the missing part will have to get the whole side equal to 5. Thus, they will do 5-3, which is equal to 2. For the second missing length, students will realize that when the missing part is drawn in, the whole side is equal to the top, which is 8. They have 6 at the bottom, and will realize that they need 2 more to get to 8. Thus, they will do 8-6 = 2, to get the missing length of 2.

Students may make an error subtracting.
Students could compare the wrong sides.

Perimeter = 26 units

Students must find the missing lengths first (see above)
Students will add up the length of each side.

Students may make mistakes adding.

Because the figure is drawn to scale, students can put post-it notes on the sides of the figure. They can count how many post-it notes there are.

Students could put the post-it notes on the inside of the figure instead. If they count the post-it notes, it will give them less than the actual perimeter. If they put the post-it notes on the inside, they will have to count the corner post-its twice, which would likely cause confusion.

Area = 36 square units

Because the figure is drawn to scale, students can insert post-it notes on the inside of the figure. They have to line them up so that they match the lengths on the outside of the figure.

Students may not fill up the entire figure with post-it notes. They might just do one set of post-its next to the lengths. This will cause them to get the incorrect answer.

Students can break the figure into two rectangles. They can use the length x width formula to find the area of each rectangle. The larger rectangle will be 30, while the smaller will be equal to 6. They will add the two areas together to get the area of the entire figure. The students could use the same method, but break it into two longer rectangles. One rectangle will have the area of 24, while the other will have an area of 12. This also gives the area of 36 square units.

When breaking up into two rectangles, students could end up using the wrong side lengths.
Students may forget the formula for area.

The students can break up the figure into two rectangles (two different ways to do this, see above). They can use post-it notes to fill in the figure, lining up the post-its correctly with the right length. They can count the post-it notes in the larger figure, and then the smaller figure. They can add their totals together to get the area of the whole.

Students may not fill up the entire rectangle with post-it notes, which will cause them to get the wrong answer.
Students might make a mistake counting the post-it notes.

What are your expectations for students as they work on and complete this task?

What resources or tools will students have to use in their work?

How will the students work – independently, in small groups, or in pairs – to explore this task?

How will students record and report their work?

I expect the students to work with their peers to try to solve the problem. I expect them to realize that there are missing lengths. I expect them to be able to solve the problem by working together.

The problem will be drawn on a poster board. Students will be able to draw and show their work by writing on the poster board with marker. Students will also be given a worksheet with the problem on it if they would rather work it out that way. Students will record their answers on their worksheets. Students will also be explaining their process, while the teacher listens. Students will essentially be reporting their work verbally to the teacher.

Students can also use post-it notes to assist them in solving the problem.

This task will be done with a small group of students only. The group will likely contain 3-5 students. There will be no more than 5 students.

How will you introduce students to the activity so as not to reduce the demands of the task?

I will explain that students will work together in their small groups to solve a problem. I will tell them that they have to explain their thought processes while they work. I will mention that I will not give them too many hints, as they have to work with their peers to solve the problem. Then, I will show them the poster with the figure drawn on it. I will ask, “What is the perimeter of the figure?” I will not mention that there are missing lengths, as they have to realize that there are lengths missing.

What will you hear that lets you know students understand the task?

The teacher should hear the students working together because this is a group task. Students may say things such as:

“There are missing lengths. We have to find what the missing parts are.”
“To find the perimeter, we have to add all the outside sides together.”
“To find the area we have to count all the blocks on the inside”
“To find the area, we can break the figure in half, and find the area of each. Then, we can add it together.”

EXPLORE

Supporting students’ exploration of the task

As students are working independently or in small groups:

What questions will you ask to focus their thinking?

“How do you find the perimeter of a figure?”
“How do you find the area of a figure?”
“Do we have all the information we need to solve the problem?”

What will you see or hear that lets you know how students are thinking about the mathematical ideas?

I will be able to physically see their work, as they are able to write on the poster board. I will be able to see if they use post-it notes correctly to find the perimeter or area. I will be able to see the math they have written on the poster.
I will be able to hear them discussing the problem and how to solve it with their peers.
Students will realize that there are sides of the figure that are missing. Students will discuss how to find those measures.
The teacher should hear mathematical terms such as perimeter area, adding, multiplying, and other mathematical terms related to solving the problem.

What questions will you ask to assess students’ understanding of key mathematical ideas, problem solving strategies, or the representations?

“What did you get for the perimeter of the figure?”
“What did you get for the area of the figure?
“How did you find the perimeter?”
“How did you find the area?”
“What steps did you take? Why did you take those steps?”

What questions will you ask to advance students’ understanding of the mathematical ideas?

I will provide additional problems that are similar to the ones they just solved so they have extra practice with the task.
I will ask where we might see a problem like this in real life (example: gardening, building a fence, etc.)

What questions will you ask to encourage students to share their thinking with others or to assess their understanding of their peer’s ideas?

"Do you agree or disagree with the other members in your group?”
“Can you explain what he/she did?”
“Why did he/she do this step?”

How will you ensure that students remain engaged in the task?

Allow them to use post-it notes
Allow them to write on the poster board with markers
Discuss where a similar problem could be found in the real world.

What will you do if a student does not know how to begin to solve the task?

Ask, “Do we know the length of every side?” “Do you see anything missing?”
Ask them if they remember how to find the perimeter and area. If they do not remember, briefly review with a simple figure such as a square with sides that are measured with 2 units. Quickly model how to find the perimeter and area.

What will you do if a student finishes the task almost immediately and becomes bored or disruptive?

For students who finish early, hand them the pre-prepared worksheet with additional practice problems on it, including a challenge problem that involves writing an explanation of how to solve the problem

What will you do if students focus on nonmathematical aspects of the activity (e.g., spend most of their time making beautiful poster of their work)?

If students are not paying attention, focus them back onto the task, prompting them onto their work. Ask them questions about the problem. This will be easier to monitor since the task will involve small groups.

SHARE, DISCUSS, AND ANALYZE

Sharing and discussing the task

How will you orchestrate the class discussion so that you accomplish your mathematical goals?

Discussion will be in a small group. Students will mainly lead the discussion by talking about the problem with peers. The teacher will regulate by making sure one person speaks at a time while the others listen carefully. The teacher will ask questions to the students along the way.

Specifically:

Which solution paths do you want to have shared during the class discussion?

In what order will the solutions be presented?

Why?

In what ways will the order in which solutions are presented help develop students’ understanding of the mathematical ideas that are the focus of your lesson?

First, have students explain how they found the missing lengths of the figure. It is important that they explain this first because they need the missing length in order to find the perimeter and area of the whole figure.

Missing Lengths = 2 units, 2 units

Because this figure is drawn to scale, students can use post it notes to find the missing lengths. They can line up the post-it notes on the line and count them up to get the missing length. This should be first because the students can get a visual representation,
Students can draw in the missing portion. They will realize that because the other side is 5, the missing part will have to get the whole side equal to 5. Thus, they will do 5-3, which is equal to 2. For the second missing length, students will realize that when the missing part is drawn in, the whole side is equal to the top, which is 8. They have 6 at the bottom, and will realize that they need 2 more to get to 8. Thus, they will do 8-6 = 2, to get the missing length of 2. It is important to do this second because it is more of a mathematical equation that represents the visual representation. It shows what the visual portion stands for and means,

Finding the perimeter should come next. Tools from the perimeter may be useful in finding the area.

Perimeter = 26 units

The visual portion will go first so students can see and understand what the formula represents. Because the figure is drawn to scale, students can put post-it notes on the sides of the figure. They can count how many post-it notes there are.

Students could put the post-it notes on the inside of the figure instead. If they count the post-it notes, it will give them less than the actual perimeter. If they put the post-it notes on the inside, they will have to count the corner post-its twice, which would likely cause confusion.

Students will add up the length of each side. This comes second because it supplements what the visual representation stands for. It generalizes the visual.

Area = 36 square units

Because the figure is drawn to scale, students can insert post-it notes on the inside of the figure. They have to line them up so that they match the lengths on the outside of the figure. This is first so students can see a visual representation. It will give them a better understanding of what the formula actually means.
Students can break the figure into two rectangles. They can use the length x width formula to find the area of each rectangle. The larger rectangle will be 30, while the smaller will be equal to 6. They will add the two areas together to get the area of the entire figure. The students could use the same method, but break it into two longer rectangles. One rectangle will have the area of 24, while the other will have an area of 12. This also gives the area of 36 square units. This comes second so students realize what the visual means.
The students can break up the figure into two rectangles (two different ways to do this, see above). They can use post-it notes to fill in the figure, lining up the post-its correctly with the right length. They can count the post-it notes in the larger figure, and then the smaller figure. They can add their totals together to get the area of the whole. This is last because this may be a way that some students may not have initially thought of. Additionally, it applies the skills from the other explanations.

What specific questions will you ask so that students:

make sense of the mathematical ideas that you want them to learn?

expand on, debate, and question the solutions being shared?

make connections between the different strategies that are presented?

look for patterns?

begin to form generalizations?

“How do you find the perimeter of a figure?”
“How do you find the area of a figure?”
“Do we have all the information we need to solve the problem?”
“What did you get for the perimeter of the figure?”
“What did you get for the area of the figure?
“How did you find the perimeter?”
“How did you find the area?”
“What steps did you take? Why did you take those steps?”
"Do you agree or disagree with the other members in your group?”
“Can you explain what he/she did?”
“Why did he/she do this step?”
“Where might we see a similar problem in real life? What are some real life examples of where we might need to find the perimeter and area of an irregular figure?”
“What are similarities and differences between finding the perimeter and finding the area?”
“Can you use your method to find the area of any irregular figure?”

What will you see or hear that lets you know that students in the class understand the mathematical ideas that you intended for them to learn?

Describe questions you would ask on an assessment to determine if students understand the mathematical ideas in this lesson.

The teacher should see that the students have solved the problem correctly. Students will have shown their work on either the poster or their individual worksheet. The teacher should also hear collaboration and discussion between students on how to solve the problem. Students should use mathematical terms in correct contexts while speaking.

On an assessment, students will have to solve a similar problem. They will have to find the area and perimeter of an irregular figure that has missing lengths. The teacher could ask students to write a quick explanation to show how they solved the problem.

What will/would you do tomorrow that will/would build on this lesson?

What future mathematical topics might this lesson build on?

Students will do extra practice on finding the perimeter and area of figures. The activity will build on any activity that involves finding the perimeter or area. Eventually, students may use the skills applied in finding perimeter and area for tasks such as finding surface area.