A starter
Give individuals a few moments to look at the graph on their own.
At this point you may chose to hide the image
Allow 5 minutes for students to discuss what they have seen with their partner.
Follow this with whole class discussion, giving students time to get a feel for the properties that are likely to be of use in later work. Example questions and prompts with possible responses:
• What do you see? [diagonal line of 2 by 2 squares]
• What are the coordinates of the centre of A? [(1,1)]
• How about the centre of the square one to the left of A? [(-1,-1)]
• Two to the left of A? [(-3,-3)]
• Three to the left? [(-5,-5)]
• Where is the top right vertex of square A? [2,2]
• And the square two to the right of A? [6,6]
• What do you notice? [x and y coordinates same}
• How about the bottom left vertices? [A: (0,0); (2,2);(4,4)…]
• How would you describe pattern to someone who could not see it? [e.g. a 2x2 square has one vertex at the origin and an opposite vertex at (2,2). An infinite line of congruent squares is formed by placing them so the diagonally opposite corners of adjacent squares touch each other and are all on the line y=x]
Main activity
One or a mixture of the following:
Activity 1
Start with enough copies of each graph so that each student has one graph.
Use just the odd number graphs if you are going to do both activities.
Students work in pairs.
They must not show the graph they are given to their partner until the end of the task. They describe their graph to their partner, who will try to reproduce it from the information given. The aim is to focus attention on the detail and relevant information.
- Students spend 5-10 minutes working on their own getting to know their graph, identifying details and making any notes ready to share with their partner.
- Pairs take it in turns to describe and draw graphs.
You could:
• Allow the describer to watch what their partner is doing and amend the information they give using what happens as feedback.
• Ask the person trying to draw the graph to keep it hidden from their partner until all the information is given. They can ask questions if required.
When they think they have finished with each graph. Pairs should spend time discussing the strengths and limitations of the descriptions based on what was produced.
Activity 2
Using odd number graphs from resources below.
Pairs choose one of their graphs and try to define it using the minimum amount of information. They share the information with the rest of the class so that those with the same graph can recognise the description and put their hands up if they think they have the graph being described.
Answers:
For example:
- The centre of one square is (2,2) the centre of the adjacent square is (4,4) the square has side 2 and a pair of sides are horizontal [four pieces of information].
- The top right hand corners of two squares of side 3 are( 1,4) and (2,7) [three pieces of information]
Keep a note of good examples to discuss at the end.
And Finally
- Show tilted square image [See resource list].
- Hide the image after a minute or so.
- Ask students to reflect on what they saw and important features that would help you reproduce the image. The aim is to focus students’ attention so that they develop a real sense of the structure they are dealing with.
- Then share ideas in pairs and then with the whole class (think-pair-share). Once you start whole class discussion, allow time for different ways of seeing to emerge by giving room for different answers to the same question.
- Students may start to talk about squares as ‘diamonds’ or neglect to mention the background grid. This would need challenging.
- If they have not said so, ask what colour the lines/grid/ squares are.
- How and where were the axes labelled?
- Where were the arrows, what direction and what colour?
- Does the pattern pass through (4,5)? [yes, it passes through all integer points]
- Which direction is the arrow pointing? [ e.g. this point is one up from (4,4), points where the ordinates are equal the arrows in the first quadrant go down to the right so the arrow at (4,4) will be going up to the left, or, first quadrant, odd sum of the ordinates, up to left]
- If the pattern continues forever, what can you say that you know about the way it continues? [it will pass through all the points in the plain]
- Does the pattern pass through (-45, 50)? [yes]
- Which direction is the arrow pointing then? [e.g. down to the left]
- Can you give any general rules that will help you determine the direction of the arrow for any point? [e.g. squares of even diagonal unit lengths go clockwise, if the sum of the ordinates is odd the arrow is pointing counter-clockwise…]
You might say:
“Did anyone see that differently” or
“Did anyone get to the same answer in a different way?
Try recreating the image as the students describe it, saying when you are not sure what to do. For example:
You may need to show the graph again if memories are failing.
Questions to follow up with and for looking deeper:
Resources
Grid 1
Grid 2
Grid 3
Grid 4
Grid 5
Grid 6
Grid 7
Grid 8
Grid 9
Grid 10
Tilted Square
More coming soon
