Students will compare the areas of squares off the sides of rectangles to see if a relationship exists with the areas and the type of triangle.

In this task students will explore how the relationship between the area of squares off the sides of a triangle can be used to determine if a triangle is right, obtuse, or acute. [The converse of the Pythagorean Relationship].

Show students the following picture and have them estimate.

Show students the following video:

Consider pausing this video at various points to have students estimate the number of squares in the larger squares.

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.

Replaying the video and/or leaving a screenshot from the video up can be helpful here.

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

Finally, allow students to share with the entire group. Be sure to write down these noticings and wonderings on the blackboard/whiteboard, chart paper, or some other means to ensure students know that their voice is acknowledged and appreciated.

Some of the noticing and wondering that may come up includes:

• I see three triangles with squares
• Some squares are full and some aren’t.
• I wonder why the figure’s squares fit and the others don’t.
• Why are we doing this?
• I notice three different types of triangles.

After we have heard students and demonstrated that we value their voice, we can land on the first question we will challenge them with:

How does a triangle’s type affect the area of the squares off the sides of that triangle?

Allow students to make conjectures with their elbow partners in a think-pair share format.

Engage your students in a class discussion about classifying triangles by their angles (obtuse, acute, right) and/or by their lengths (scalene, equilateral, isosceles).

Ask students to brainstorm how they can reach conclusions to any conjectures they have.

Group your students (randomly is best) in partners or groups of 3.

Distribute each group a copy of BLM 2.1 TRIANGLES.

A downloadable PDF blackline master of this investigation can be downloaded and used with your students in your classroom here: [ OPEN | DOWNLOAD ]

Students are to cut out each triangle.

For each triangle:

• Measure the side lengths;
• Measure the angles;
• Classify the triangle;
• Draw the square off each side length (similar to the video) and determine its area.

Students may wish to record their measurements on BLM 2.2 RECORDING SHEET to keep organized. Other students may wish to record and share their thinking on whiteboards or other non permanent surfaces.

A downloadable PDF blackline master of this investigation can be downloaded and used with your students in your classroom here: [ OPEN | DOWNLOAD ]

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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.

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