Opposite vertices
Can you recreate squares and rhombuses if you are only given a side or a diagonal?
Problem
Opposite Vertices printable sheet - squares
Opposite Vertices printable sheet - rhombuses
Charlie has been exploring squares with vertices drawn on the points of a square dotty grid.
Unfortunately, he rubbed out some of his work and only left behind one side of each square.
Can you recreate the squares he drew?
Is there more than one possibility?
Could any line joining two points be the side of a square whose vertices lie on grid points?
How can you be sure?
Alison has been drawing squares and their diagonals. Here are some of the diagonals she drew:
Can you recreate the squares she drew from her diagonals?
Is there more than one possibility?
Can you find a method to draw a square when you are just given the diagonal?
Could any line joining two points be the diagonal of a square whose vertices lie on grid points?
Can you find a way to help Alison decide whether a given line could be the diagonal of such a square?
Charlie and Alison played around with rhombuses next.
Charlie said "Whenever I join two points to make a line, I can use my line as a side of several different rhombuses".
Do you agree with him?
When you are given a line, is there a quick way to work out how many rhombuses can be drawn using that line as one of the sides?
Alison said "When I draw a rhombus, it shares its diagonal with infinitely many other rhombuses."
Do you agree with her?
Not all lines can be the diagonal of a rhombus. Is there a quick way to decide which lines could be the diagonal of a rhombus?
Getting Started
The interactivity in Square Coordinates may help you to visualise tilted squares.
You can challenge yourself by playing Square It against the computer.
Student Solutions
Here is how Emily from Beeston Rylands Junior School started on the problem.
Damini from Newstead Wood School For Girls discovered that not all diagonal lines that join two grid points make a square:
You find the centre of your diagonal line and go from there away from the line in a straight line (at right angles) until you find the right point. Then join the point to one end of the diagonal line and you have a side! How easy was that. Not all diagonal lines that join two grid points make a square. If you take a piece of square dotty paper and join up any two dots it will be hard to find the square, as there might not even be a square.
I agree with the fact that a straight line can be the side of many different rhombuses but to me I see no real way of determining how many rhombuses go to one line.
I also agree that the diagonal of one rhombus can have the same diagonal line as infinitely many other rhombuses as if you take one line you can keep editing the size of the rhombus without actually changing the length of the first diagonal.
Alexander from Wilson's School suggested a strategy for finding the number of possible rhombuses that could be made when he is given a side:
To find how many different rhombuses can be found with a single line that is between two dots, you need to choose one end of the line. You then look at "one side" of that point, and see how many different ways the line can be "twisted". You then multiply it by four because the same number of "twisted" lines can be found on the "other side" of that point, and on the "two sides" of the other end of the line. The answer to this operation determines how many rhombuses can be made from one line on a dotted piece of paper.
(Editor's comment: I think that Alexander has double counted here; each of the "twisted lines" found at one end will make rhombuses with each of the "twisted lines" found at the other end of the line.)
Niharika from Leicester High School for Girls explained that if the vector $\pmatrix{a\cr b}$ represents the diagonal line, the square will have vertices at grid points if a+b (and therefore a-b) is even.
She also suggested a strategy for finding the number of rhombuses that could be drawn if one of the sides is given. Her solution is here.
Niharika also contributed a solution to Vector Journeys, which uses vector notation so solve a related problem.
We also received good responses from the Maths Galaxy Explorers from North Walsham Junior School.
Teachers' Resources
Why do this problem?
You may be interested in our collection Dotty Grids - an Opportunity for Exploration, which offers a variety of starting points that can lead to geometric insights.
Possible approach
These printable worksheets may be useful: Opposite Vertices - squares
Opposite Vertices - Rhombuses
10mm Dotty Grid
This problem requires students to draw tilted squares reliably. The interactivity in Square Coordinates might be helpful to demonstrate to students what a tilted square looks like. Students could play Square It until they can reliably spot tilted squares on a dotty grid.
- the side of at least one square,
- the diagonal of at least one square,
- the side of at least one rhombus,
- the diagonal of at least one rhombus.
Alternatively, start by showing the image of Charlie's rubbed out squares, and give students some time to recreate the squares on dotty paper. Once they have finished, ask them to compare in pairs - have they always drawn the same square? Are there any other possibilities?
Bring the class together and share the techniques students were using to complete the squares.
"Draw any line you like on your dotty paper, and then try to complete a square using your line as one of the sides."
...
"Has anyone found a line that they can't use as a side of a square?" (If someone has found one, display it on the board and ask the class to help.)
"Do you think we can always draw a square using any line?"
Give students some time to discuss this and come up with justifications for their answer.
Next show the image of Alison's diagonals and give students time to recreate the squares.
Again, ask them to share approaches.
"Draw any line you like on your dotty paper, and then try to complete a square using your line as the diagonal."
This time, it can't always be done. If students are struggling to work out when it can and can't be done, suggest that they draw some squares and see what the diagonals have in common.
Then move on to rhombuses.
"Draw any line you like on your dotty paper, and then try to complete a rhombus using your line as one of the sides. How many different rhombuses can you draw using your line?"
Give students time to do this for a number of different lines of their own choosing, and then bring the class together to share their findings.
"When we worked on squares, there were some lines that we couldn't use as a diagonal. Will the same be true for rhombuses?"
Once again, give students time to explore.
Finally, bring the class together to share their ideas and justify their findings.
One technique for testing ideas at the end is to set a specific challenge, for example, to draw some lines and ask them to determine how many rhombuses could be drawn using each line as a diagonal.
Key questions
Possible support
The interactivity in Square Coordinates helps students to visualise tilted squares.
Possible extension
Vector Journeys challenges students to explore similar relationships using vector algebra.