# More on Triangles: Lesson One

## Objectives

Big Idea – We can describe, measure, and compare spatial relationships.

• to build on students' knowledge of triangle and develop students' reasoning with properties of different classes of triangles
• to develop students' language related to different triangles
• to engage students in the process of defining triangles/geometric shapes which is useful in their subsequent work in proposing definitions of other mathematical concepts (e.g., quadrilaterals)
• to highlight inclusive relations between triangles (e.g., recognizing equilateral triangles as special types of isosceles triangles)

Overarching Aims:

Informal descriptions of properties that coordinate their visual impressions with geometric language.
• Properties of movement:
- Students may notice that some points can be dragged directly and move freely, that some cannot be grabbed but may move by means of another point.
You may hear students say words like ‘paralyzed’ triangle for isosceles and equilateral triangles after noticing the restrictive movements of these triangles upon dragging.
• Properties of symmetry:
- Students may notice symmetry in the isosceles and equilateral triangles and non-symmetry in the scalene triangles

Informal recognition of these properties may be helpful in recognizing formal geometric properties like symmetrical movement, equal angles, and equal sides.

## Background

You'll be using Sketch 1 of this file which has three different types of triangles - scalene, isosceles, equilateral.

Each triangle type has a different colour (pink for scalene, red for equilateral, blue for isosceles), which allows you and the students to refer to the triangles without having to introduce their formal geometric names.

Sketch 1 is intended to help the students attend to the similarities and differences between the three triangles.  While initially very similar looking, these triangles behave very differently under dragging.

## Activity – Whole Group

• "What kinds of shapes do you see on the screen?"
• "Are there any differences?" (typical responses: colour, thicker line segments, two are closer together)
• "Is there anything the same?" (typical responses: size, all triangles, all coloured, all have three sides, all pointed the same way (orientation), all have angles).

2.  After the initial discussion about the static triangle images, have one student explore dragging the pink triangle. Have the other students act as detectives, noticing what changes when the triangle is dragged and what stays the same.

• "Can you make the triangle really long and skinny?"
• "Can you make it really small/big?"
• "Can you make it upside down?"
• "What other things can you try?"
• "If we had to give it a name, what name could we give it?"

3.  Repeat the process for both the red and the blue triangles.

• "Why can't we make the red triangle long and skinny?"
• "Is there anything that stays the same?"

4.  Now have a student explore trying to overlap the blue triangle onto first the red triangle and then the pink triangle. (Have several students explore this - they need to be convinced that the blue triangle can't overlap the pink triangle).

• "Why could we put the blue triangle on the red triangle?"
• "Why can't we put it on the pink triangle?"
• "Is it possible to overlap a triangle that has two equal side lengths onto one that doesn't?"

## Assessment – Individual or Partner

1. Have the students draw a design on paper using only triangles.
• Ask them to label the types of triangles with the names they chose.
• Provide correct geometric names for students who ask.

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