#LessonStarter is a Twitter hashtag, used particularly by Matt Skoss, to collect together ideas that might start (or take over!) a lesson. A #LessonStarter is usually a provocative image, but could also be an intriguing mathematical prompt. For me, lesson starters are often spontaneous. Today, a few serendipitous moments meant that I had a lesson starter just before walking into class.
This morning I woke up to a tweet from Bryn Humberstone.
— Bryn Humberstone (@brynhumberstone) August 14, 2017
I love using topical moments — like today’s date — as a lesson starter. While I understand the fatigue that days like Pi Day (14 March) can cause, I think it can be a great opportunity to briefly introduce some maths that might not be in the lesson plan. (Sara Van Der Werf has a recent and positive blog post on math(s) holidays that you should read.) Plus, I like my students to witness the joy I have in mathsy moments — even if they think I’m weird!
To top it off, my plan for today’s #math1070 class was to review last week’s task: choose one of the four images below and tell me what you Notice and Wonder about it. You might be able to see how my lesson starter was going to unfold!
I started with today’s date: 15/8/17. I said it was an interesting mathematical date. Could anyone work out why? I then showed this image. We had a quick refresher of the Pythagorean theorem. This image already throws up some interesting questions: is it possible to always colour the squares on the hypotenuse with a complete square of one colour and a double thickness ‘half-border’ of the other?
Next, I showed the image from brilliant.org. I was careful to draw the squares on the sides of the 3-4-5 triangle, and connect Pythagoras’ theorem to the area of the squares that can be drawn on each side of the triangle. Then I asked them what they noticed and wondered. What do you notice and wonder? We had a little discussion based on this prompt, including generalising the side lengths.
Finally, I showed the image from their weekly task. Those who chose this image as their prompt generated a lot of noticings and wonderings; they are at the bottom of this post. For now, I directed their attention to the number of isosceles trapeziums making up the larger trapeziums on each side. The counts are 9, 16 and 25. What do you wonder now?
We explored whether this was a 3-4-5 triangle (it is) using the long base of the smaller trapezium as one unit. The overwhelming question in the room was why Pythagoras’ Theorem works for non-square shapes. I acknowledged that it works for similar shapes, and left it there. (By now, the lesson starter was threatening to take over the lesson!)
I finished with a quote from Chapter 7 of Tracy Zager’s book, from Peter Hilton: “Computation involves going from a question to an answer. Mathematics involves going from an answer to a question.” I hope that these images and initial discussion prompted many more questions that they could go on to explore in their own time.
Incidentally, it was another serendipitous find this morning, shared on Twitter by John Golden, that sharply reminded me of this quote. John shared this blog post by Pat Ciula. It uses the same trapezium image to launch a complete different exploration. I love this post; do check it out for yourself.
This wasn’t really what I had planned for today’s class, but when these moments come along, you need to grab them!
‘Noticing and Wondering’ from my students
These are from last week’s task, submitted in advance of today’s class. I’ve made some minor edits.
I notice (that) …
- Different colours are used: red and pink.
- A triangle has been made in the middle.
- The triangle formed is a right-angled triangle.
- All three shapes are the same.
- All three shapes are made up of smaller versions of the overall shape.
- The shapes are created by wooden blocks all put together.
- Each individual piece is a trapezium.
- There are three larger trapeziums made out of individual tessellating trapeziums.
- That the four-sided polygons go in an anti-clockwise movement ranging from smallest to largest.
- The blocks are arranged in different ways in each trapezium.
- The three trapezoids are similar trapezoids.
- There are nine sides forming the outer perimeter of the shape.
- In the centre of the three larger trapezium shapes, the points of the bases of these shapes make a right-angled triangle.
- If you were to move the largest trapezium shape so it was opposite where it is now, the shape in the middle made by the sides of the trapeziums would still be a triangle.
- Each tile is the same shape – trapezoids. In each tile, three sides (top and the two edges) are equal, and the base (the longest side) is twice the length as the other three sides.
- If the length of the base of the tile is one unit, then the length of the middle triangle’s three sides are 3 units, 4 units and 5 units. The layers of the tiles within the trapezoids are also 3, 4, and 5.
- Small trapezium is made out of 9 smaller trapeziums. Bottom trapezium is made out of 16 smaller trapeziums. Larger trapezium is made out of 25 smaller trapeziums.
- The number of tiles on each side of the triangle follows Pythagoras Theorem: 32 + 42 = 52, which is 9 + 16 = 25.
- The image includes an aspect of scale.
- The small pink polygons shapes make a negative internal outline of a right-angled scalene triangle.
- That the fewer pink polygons on each section related directly to the length of the sides of the internal outline of the triangle, that is, fewer pink polygons, shorter sides.
- I noticed that the depth and width of the outer shapes was linked, that is, left side shape is 3 pieces deep and 6 ‘lengths’ wide and consists of 9 pieces, bottom shape is 4 pieces deep and 8 ‘lengths’ wide and consists of 16 pieces and finally upper right shape is 5 pieces deep and 10 ‘lengths’ wide and consists of 25 pieces.
I wonder …
- Is the angle exactly 90 degrees?
- How many trapeziums there are?
- How long did it take to make?
- Why are there two different colours?
- How many blocks would it take to fill the right-angled triangle?
- Are sides ‘a’ and ‘b’ equal to ‘c’ in length?
- If the bottom section would fill the triangle perfectly?
- Are the small trapeziums arranged in a particular way for a reason? Or is it random to fit the desired shape?
- Do the larger shapes go down in size proportionally and is there a specific number to make up each shape?
- Would the area of the trapeziums be bigger or smaller than the area of the triangle in the middle?
- What the two remaining internal angles would be (excluding the right-angle).
- Why using trapezoids to represent Pythagoras Theorem also works? Is it because the ratio of the area of the trapezoid to the square is fixed?
- I wonder if the sizes of the trapezium differed, what the effect would be on the right-angled triangle on the middle? Is it possible to make an isosceles or equilateral triangle in the centre?
- How many triangles can each Isosceles Trapezoid be split into?
- How many other shapes are present in the Isosceles Trapezoids? For example – I have already noticed that triangles can be found — what other shapes are there?
- What is the reasoning for this image? Why was it created?
- How I could start a conversation with the students regarding this shape and what they may come back with?