Sticks and Shadows

Paul Lockhart’s excellent book Measurement (a constructive follow-up to his infamous diatribe about the start of maths education, A Mathematician’s Lament), includes this gem:


I adapted for mixed-ability Yr9 class. Some students were thinking deeply about the shape of the curve, others were practicing measuring angles and lengths and plotting coordinates.

From formal to concrete
Multiple ways to measure the shadow…


Geogebra app proved the most popular, since the least brainpower required to use it. Maybe only give this to the students once they have struggled with a physical metre-ruler or the unit circle first?


Lovely conversations about how a shadow can have a negative length, and why we would want to think of the metre-ruler sticking straight into the ground and still casting a shadow.

Some students found four squares representing 30 degrees a challenge. Which graph is correct?

Spot the difference.
Student explanation

Self-Differentiated Graphing


Continuing my thoughts on using technology with graphing (see previous blog post). Students choose which family of graphs to draw. Drawing multiple lines on one coordinate grid encourages thinking about similarities and differences.

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One student draws quadratic curves (excellent BIDMAS  and substitution practice) while her neighbour draws straight lines through the origin (excellent for understanding what a graph is and how to use coordinates accurately). Both feel that they are doing the same task, bought into it through choice of difficulty, and able to help each other.

I like this type of mixed ability 


Structured Exploration, Proactive Self-checking

When I began to use Desmos and Geogebra in maths lessons, the tasks were lacking in structure. “Click on this link and have a play” I would say optimistically, hoping the students would be self-motivated enough to wonder and ask questions themselves. Varied success. Here are two first attempts at providing specific purposes for technology in the Maths classroom.

Structured Exploration

Simpy put: “Click on this link, have a play, and then do this subsequent exercise”. Allows teacher assessment, students given some (but not total) direction. Examples:

Proactive Self-checking

Here are three levels of marking:

  1. Teacher holds the answers, and marks herself. Teacher has good knowledge of student’s understanding, but the delay between question and feedback can be long, and this is time-consuming for teacher.
  2. Student is given the answers somehow (back of textbook, on a separate slide…) and passively checks their work against it. Immediate feedback, but often lack of cognitive depth in feedback
  3. Student works out the answer in a different way. This level encourages building of links between separate areas of maths, and allows the student to continually assess their progress. Let’s call it Proactive Self-Checking.

Here are some examples of proactive self-checking (do you think it is important enough to need capitalisation?):

When practising plotting points from an equation, students can compare their hand-drawn graph to a desmos graph.

Desmos doesn’t have that weird kink in the line – what have I done wrong?
The line on Desmos is going up but mine is horizontal – something is wrong!

This is a bit obvious, no real links made. How about encouraging the use of area models to check expanding brackets?

I might use a more efficient method to calculate, but by using this method to check I am constantly reminded of the link

How about starting to understand simultaneous equations,to check that I solved any equation? If I am using bar modelling or the balance method to solve my equation, then being able to link the answer to a picture is deep stuff!

2x + 5 = 3 has a solution at x = -1

How about using pictures to make sure you can simplify expressions?

I don’t know what I have done wrong, but at least now I know I need to ask a peer for help!

Your next steps:

Think about providing structured exploration for angle rules?

Design proactive self-checking for your next lesson?

Use this collection of exercises as an intro to Geogebra with your class?

Search the excellent Geogebra database or Teacher Desmos Site for ideas?

Try this challenge?


Desmos to spot mistakes

Desmos is an excellent graphing calculator. Nick had a really great conversation with himself. The graphing calculator disagreed with the line that he had plotted. Which was correct? He oscillated between faith in himself and faith in his ipad, before spotting that 2 x 2 was not 2. Technology used as a self-marking tool can be powerful. To check at the end of the journey, rather than as an overly helpful crutch from the beginning of the journey.

Who is right??