Sample Lesson Plans Grade 1-­‐‑3 More on Triangles Developing conceptions of equilateral, isosceles and scalene triangles using Triangle Shapemakers (2-­‐‑3 Lessons) Main activity Students interact with pre-­‐‑constructed triangle shapemaker sketches of different classes of triangles on the projector screen to explore the concept of equilateral, isosceles and scalene triangles. Grade level Grade 1, Grade 2, Grade 3, Grade 1/2 split, Grade 2/3 split class Overall • To build on students’ knowledge about triangle and objective/goals develop students’ reasoning with properties of different classes of triangles • To develop students’ language related to different triangles, which involves inclusive descriptions of classes of triangles (i.e., recognising equilateral triangles as special types of isosceles triangles). Computer tasks Summary of tasks Paper-­‐‑ and –pencil tasks Geometer’s sketchpad, Computer connected to a projector, Materials Interactive whiteboard Observation guides Students’ use of math language Students’ use of gesture Students’ use of diagrams and drawings Key Questions How do you know? What helped you? Why did you say that? Summary: Students’ See Appendix B thinking 1 © Sinclair, Kaur & Ng, 2014 Sample Lesson Plans Grade 1-­‐‑3 More on Triangles Detailed Plan of Lesson 1 Introduction to Sketch 1: The triangle Shape Makers sketches are constructed with different types of triangles (scalene, isosceles, equilateral triangles, right triangle). Each triangle type has a different colour (pink for scalene, red for equilateral, blue for isosceles and green for right), which can enable the teacher and the children to refer to the triangles without having to introduce their formal geometric names. The sketch 1 (figure 1) is intended to help the children to attend the similarities and differences between the three triangles with the help of dragging facility. While initially very similar-­‐‑looking, these triangles behave very differently under dragging (see details in Appendix A) Figure 1: Triangles Shape Maker sketch Exploring the sketch and simultaneous class discussion: (20-­‐‑30 minutes) • Teacher begins by projecting sketch 1 on the interactive whiteboard. • Teacher asks the students what kinds of shapes they see on the screen. After the students recognise the shapes being triangles, teacher invites them to describe the ways in which the three triangles are the same and ways in which the three triangles appear different. • After the initial discussion about static triangles in the sketch, the teacher invites the three students to be the drivers to drag one of the three different triangles each. The teacher asks other students to be the detectives trying to describe what kinds of triangles can be made, what can change and what stays the same in each one of these triangles upon dragging. • The teacher invites the first student to drag the pink triangle to different sizes and orientations and simultaneously asks other students to describe the behaviour of pink triangle. Repeat the same process for red and blue triangles respectively. 2 © Sinclair, Kaur & Ng, 2014 Sample Lesson Plans Grade 1-­‐‑3 More on Triangles • For comparing the two triangles simultaneously, teacher fixes the scalene triangle to one particular position and asks the students to explore if they can overlap the red or blue triangle over the given pink triangle without touching/distorting the given pink triangle. During this exploration, students may notice some differences between pink and red or blue triangle (e.g. Symmetric vs. non-­‐‑symmetric). •
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The teacher focuses the intervention on the geometrical properties of the triangles and does not encourage the use of the measuring tools (such as length and angle). With the teacher-­‐‑led discussion when student notice the invariances of angles and sides in the triangles, the teacher can introduce the vocabulary words like equilateral for the red triangle, isosceles for the blue triangle and scalene for the pink triangle. Some exploratory questions that teacher can ask: • Can you make this pink triangle really long and skinny? • Can you make it really small? • What other things you can try? • Can you make the red triangle long and skinny like your friend did for the pink triangle? • What is the matter with the red triangle that makes it difficult to make it long and skinny? • Even when the red triangle is getting bigger and smaller, is there anything that staying the same (simultaneously drag the vertex of the red triangle slowly)? • What can you say about the sides of the red triangle upon dragging? Similar questions can be asked for the blue triangle during the exploration and simultaneous discussion. Paper-­‐‑and-­‐‑Pencil Task: Drawing a design by using only triangles (10 minutes) The teacher ask children to draw a design by using only triangles in their math journals and label the type of triangles in their design. 3 © Sinclair, Kaur & Ng, 2014 Sample Lesson Plans Grade 1-­‐‑3 More on Triangles Detailed Plan of Lesson 2 Exploring Shape Makers with Outlines For the sketches 2, 3 and 4 shown in figure 2(a, b, c) respectively, the teacher asks the students to explore which coloured triangles could fit into the given triangle outlines. (a) – Sketch 2 (b) – Sketch 3 (c) – Sketch 4 Figure 2 (a, b, c): Shape Makers with Outlines Working with the Happy Face sketch (figure 2a): (10 minutes) The teacher projects the sketch 2 on the screen and asks the students to fill in the outlines using the given triangles. In this sketch, for the happy face, the eyes and nose outlines are made of equilateral triangles and the mouth is made of 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 the sketch in figure 2b: (10 minutes) The teacher projects the sketch 3 on the screen and asks the students to fill in the outlines using the given triangles. For this sketch shown in figure 2(b), all the outlines look like equilateral triangles. So all the triangles can be used for filling the outlines except the right triangles. This design is intended to draw children’s attention to the fact that any kind of triangles 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. It is worth noting that there will be no usage of green (triangles) during the students’ working with the sketches shown in figure 2(a) and 2(b). Working with the sketch in figure 2c (15-­‐‑20 minutes) 4 © Sinclair, Kaur & Ng, 2014 Sample Lesson Plans Grade 1-­‐‑3 More on Triangles The teacher projects the sketch 4 on the screen and asks the students to explore which coloured triangles could fit into the given triangle outlines. For the sketch shown in figure 2(c), all the outlines are made of right triangles of different sizes. Note that the equilateral triangles cannot be used in this sketch 2(c), 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 encourages the children to provide explanations for actions and claims and to try to convince their peers with their arguments. •
The teacher prompts the children to reconsider their choices for filling the outlines (when required) and helps them to see the inclusion relationships. In our teaching experiment, the children fitted 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. Key questions that teacher can ask for sketches 2, 3 and 4 • 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? Paper-­‐‑and-­‐‑Pencil Task: Drawing families of different triangles (10 minutes) The teacher asks children to draw a “mom”, “dad” and two other “family members” for each type of triangle (isosceles, equilateral, isosceles, and right triangle). 5 © Sinclair, Kaur & Ng, 2014 Sample Lesson Plans Grade 1-­‐‑3 More on Triangles Detailed Plan of Lesson 3 Working with Shape Makers for Comparison in pairs (15-­‐‑20 minutes) Next the teacher invites children to work with the sketches 5 and 6, shown in 3(a) and 3(b) respectively. The objective is to use these sketches as a way of focusing attention on the inclusive relations, after the children had worked with the outline challenges previously. Sketch 3 (a) is focused on exploring whether a scalene triangle can fit into the given equilateral triangle (top) and whether an equilateral triangle can fit into a given scalene triangle (bottom). The sketch in figure 3(b) is focused on exploring whether an equilateral triangle can fit into the given isosceles triangle (top) and whether an isosceles triangle can fit into a given equilateral triangle (bottom). These two sketches differ from the previous sketches, as their intent is to make the explicit comparison between the pairs of different types of triangles. Sketch 5 -­‐‑ Comparison 1 Sketch 6 -­‐‑ Comparison 2 Figure 3 (a, b): Comparison sketches Key questions that teacher can ask for sketches 5 and 6 • Make a guess before trying; can we fit a scalene triangle into an equilateral triangle? • How come scalene can make equilateral triangle? • Why can’t you fit the equilateral triangle into the given scalene triangle? • Why can we turn isosceles into equilateral, but we can’t turn equilateral into isosceles? 6 © Sinclair, Kaur & Ng, 2014 Sample Lesson Plans Grade 1-­‐‑3 More on Triangles Appendix A Behaviour of dynamic scalene, isosceles and equilateral triangles In the sketch shown in figure (1) only the middle (red) triangle is constructed to be equilateral; the bottom right (blue) one is constructed to be isosceles while the top left (pink) one is scalene. Although no vertex was labelled in the sketches, this is done so in Figure 1 in order to explain the dragging behaviour of different triangles. In the scalene (pink) triangle, dragging any one vertex (A, B or C) does not move the other two vertices, whereas in equilateral (red) triangle, dragging vertex E or F (which determine the size of the triangle) rotates and enlarges the entire triangle around vertex F or E respectively; dragging vertex D simply translates the triangle from one place to another. In the isosceles (blue) triangle, dragging vertex I does not move the other two vertices, dragging vertex H or G moves the entire triangle except the vertex G or H respectively. Vertex I can only be dragged along a circular trajectory with GH as the radius of the circle. 7 © Sinclair, Kaur & Ng, 2014 Sample Lesson Plans Grade 1-­‐‑3 More on Triangles Appendix B Students’ Thinking Summary: Initially students might pay attention to comparing the visual appearance and spatial/graphical properties like colour, size (skinny, long, small, big) and orientation (up, down). Then students can justify their comparison through informal properties check. These informal properties may include the sketchpad-­‐‑based dynamic properties like movement of a point (characteristic of its trajectory, speed), movement dependencies between points (points that can be dragged directly and moving freely, point that cannot be grabbed but may move by the mean of another point). For example, while comparing scalene and isosceles triangles, students may notice the different movement behaviour of triangles under dragging or they can recognize some geometric properties like symmetry in blue (isosceles) and non-­‐‑symmetry in pink (scalene) triangles. Also, they can use the words like ‘paralysed’ triangles for isosceles and equilateral triangles after seeing the restrictive movements of these triangles upon dragging. This initial informal properties check can be further helpful in recognizing formal geometric properties like symmetrical movement, equal angles, and equal sides. 8 © Sinclair, Kaur & Ng, 2014