Students will leverage the relationship between triangles and rectangles to uncover the formula for calculating the area of a triangle.

In this task, students will explore the relationship between rectangles and triangles. They will use their prior knowledge of the area of a rectangle to uncover the formula for calculating the area of a triangle. Some of the big ideas that may emerge in this unit include:  

• The area of a rectangle can be determined by multiplying the length of its base by the length of its height; 
• The area of a rectangle is used to determine the area formulas for other polygons; 
• Using “base” and “height” rather than “length” and “width” can support student understanding when determining the area formulas for triangles and all quadrilaterals; 
• A parallelogram is any quadrilateral with two sets of parallel sides; 
• A rectangle is a parallelogram; 
• Any triangle can be doubled to create a parallelogram; and,
• Congruent shapes have the same size and shape, but not necessarily the same orientation.

Show students this video:

The video shows a square backyard with dirt and sod (pre-cut pieces of grass to use for a lawn) being “rolled out” to cover the backyard area. However, this might not be initially obvious to students.

Then, ask students:

What do you notice?
What do you wonder?

Give students 60 seconds (or more) to do a rapid write on a piece of paper.

Then, ask students to share with their neighbours for another 60 seconds.

Finally, allow students to share with the entire group.

Some of the noticing and wondering that came up in a class recently included:

• I notice a square.
• I notice small green squares being placed.
• I wonder if this is a yard.
• I wonder if the green is grass.
• I wonder how many pieces of grass are placed in the yard. 

At this point, you can answer any notices and wonders that you can cross off the list right away. 

For example: 

• This is a backyard.
• The backyard is being covered in sod. 
• Sod is pre-cut sections of grass that is used to cover a yard quickly instead of planting grass seed and having to wait for it to grow.

Next, share the following visual prompt briefly and ask the following question:

Each piece of sod (grass) placed on the ground in the backyard measures 1 square metre. Estimate how many square metres will cover the entire area of this backyard?

We can now ask students to make an estimate (not a guess) as we want them to be as strategic as they can possibly be. This will force them to use reasoning to try and come up with a reasonable measurement.

As students share their estimates, consider drawing a number line to represent the range of estimates and lead a discussion about how confident the group is that the actual number of pieces of sod will land within the range.

Monitor student thinking by circulating around the room and listening to the mathematical discourse. Encourage students to use precise mathematical language and listen for student understanding of how to determine the area of a rectangle. Are students using the dimensions of the square to determine the total number of square units? 

Allow students to share their estimates with neighbours first, then with the class. Write down their estimates on the chalkboard/whiteboard/chart paper so students feel their voices are being heard and so they feel they have a stake in solving this problem.

You might offer students a visual that shows the lawn partitioned into 1 square-metre sections of sod and then ask them to:

Update your estimate.

Monitor how students update their estimates. Do their approaches change? Record their updated estimates on the number line. Be sure to note the change in the range of estimates. Is the range tighter now?

Share the following image revealing that the yard is 12 metres by 12 metres long, which gives 144 square-metres of area to cover with sod.

Since you have already taken some time to set the context for this problem and student curiosity is already sparked, we have them in a perfect spot to help push their thinking further and fuel sense making.

Share the following silent animation that shows the same 12 metre by 12 metre backyard being split diagonally to have a triangular section of sodded grass area and a triangular section of concrete patio.

You can verbally share the following prompt with students:

The homeowners have decided to transform a part of their back yard into a patio. They are going to be covering the patio area in concrete. How many square metres of concrete will they need? Justify your answer.

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Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.

Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.

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Show students the following reveal video that partitions the yard diagonally into two triangles that each cover half of the yard. One of the triangles will flip to “cover” the other triangle to make it clear that each triangle represents half of the square (rectangular) yard.

You can consider sharing the screenshot of the final frame of the video as well.

Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.

Consolidation Prompt #1:

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Consolidation Prompt #2:

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Consolidation Prompt #3:

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We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.

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