Students will leverage the relationship between rectangles and other parallelograms to emerge the formula for calculating area of a parallelogram.

In this task, students will explore the relationship between rectangles, other parallelograms. They will use their prior knowledge of the area of a rectangle and triangle to uncover the formula for calculating the area of any parallelogram through the context of comparing the area of different rugs. 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;
• Congruent shapes have the same size and shape, but not necessarily the same orientation; 
• Any parallelogram can be decomposed and recomposed to build a rectangle with the same area.

Show students this video:

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 squares on the floor.
• I wonder if they are tiles.
• I wonder if this is a room.
• I notice three different things covering the tiles.
• I wonder if they are rugs.
• I wonder how big they are.
• I wonder if they are the same size. 

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

For example: 

• This is an animation of a tiled room in my house.
• I am interested in buying a rug for this room.
• I was trying to see how these three rugs would look in the space.

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 video and re-iterate the prompt below:

I want to buy a new rug for this room. I am going to choose the rug that covers the most area. Which rug should I buy? Defend your answer.

Provide students with a handout of the three rugs.

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You might choose to also share a visual of the rugs with the dimensions labelled.

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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.

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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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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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