More on Triangles: Lesson Two

Objectives

Big Idea

• 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 triangle.

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 Sketches 2, 3, and 4 of this file to explore which coloured triangles could fit into the given triangle outlines.

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

Key questions to pose for all three sketches are:

• Why can't you use the red/pink/blue/green triangle in the given outline?
• Why do you think the '...' triangle would work on this outline?

Activity – Whole Group

Working with Sketch 2

Project Sketch 2 on the screen and ask the students to explore which coloured triangles could fit into the given triangle outlines.

For this sketch, the eyes and nose outlines are equilateral triangles and the mouth is an isosceles triangle.

The intent of this sketch is to help students see that any triangle except the green (right) triangle could fit in the eyes and nose interchangeably, whereas in the mouth only pink (scalene) and blue (isosceles) triangles could fit.

This would help in drawing children’s attention to the differences/similarities in the eyes, nose and the mouth outlines.

Working with Sketch 3

Project Sketch 3 on the screen and ask the students to explore which  coloured triangles could fit into the given triangle outlines.

For this sketch, all the outlines look like equilateral triangles. So all the triangles can be used for filling the outlines except the green (right) triangles.

This design is intended to draw children’s attention to the fact that any kind of triangle can be used interchangeably in these particular (equilateral) outlines. This would implicitly initiate an understanding of inclusion relation of equilateral triangles in isosceles and scalene triangles.

Working with Sketch 4

Project Sketch 4 on the screen and ask the students to explore which coloured triangles could fit into the given triangle outlines.

For the sketch, all the outlines are made of right triangles of different sizes. Note that the equilateral triangles cannot be used and isosceles triangles will fit only into the two left-most outlines.

The anticipation is that students will first use the green (right) and pink (scalene) triangles based on visual similarity of green triangle with the outlines and their experience of easy morphing of scalene triangle during previous sketches respectively. This would lead to a ‘stuck’ moment.  The teacher may need to prompt the children to reconsider their choices for filling the outlines and help them to see the inclusion relationships.

In our teaching experiment, the children fit the green (right) triangles into the two right-most outlines and the pink (scalene) triangles in the two-left most outlines. This prompted the need to reconsider the fitting of pink (scalene) triangle from left-most outlines to two middle outlines.

Assessment – Individual or Partner

Have the students draw a 'family of three or four' for each type of triangle (isosceles, equilateral, scalene, and right triangle).

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