2024 marks the 10th anniversary of CAMI (!) and to honor all we have learned and all the ways we have grown as a group, we are going into the vaults for a few CAMI meeting, to reopen and revisit some of our early explorations together. This month’s meeting was a new take on a problem we explored in June 2016 at Making and Testing Conjectures: The Diagonal Problem.
We started with a Which One Doesn’t Belong?
Directions: Which of these things is doing their own thing? Find something that makes one of the numbers unique. If there is time, you can find what is unique about more than one number. The question Which one doesn’t belong means “how is this number different from the other 3?”
3 is the only one with 1 digit.
27 is the only one without the number 3 as a digit. It is the only one that is even if read backwards. It is the only one that is a perfect cube (3x3x3).
123 has digits in numerical order. It is the only one with 3 digits. It is the only age that Carol has not yet been.
31 is the only one not divisible by 3. The sum of the digits is not divisible by 3.
Next, we looked at two images and shared what we noticed and what we were wondering about.
WHAT WE NOTICED
- On the 5×5 square, the diagonal splits each square it passes through
- On the 10×7, each section of the diagonal passes through 2 or 3 squares – the pattern is 2, 2, 3, 2, 3, 2, 2
- If you bisect the rectangle in the other direction, you get symmetry in the amount of blue on
- There are several lines of symmetry we could draw
- In the 5×5, the diagonal creates an isosceles triangle
- The shaded squares show which squares the diagonal passes through
- In the 10×7, none of the shaded squares are bisected by the diagonal.
- In the 5×5, the diagonal forms right angles
- In the 10×7, the diagonal does not hit a single corner of any square it goes through (except at the ends)
- No adjacent blue squares in the square are shaded
- The square is blue if the line passes through
WHAT WE WONDERED
- In the 10×7 the diagonal doesn’t seem to hit the corner of any square – could we come up with some different dimensions (maybe multiples) that would hit the corners?
- Is there a pattern in the number of squares in each row?
- If we made this rectangle larger, but in proportion, would the pattern (2,2,3,2,3,2,2) keep? (for example, 20×14)
- Why does the 10×7 produce rows of 3 shaded squares?
- If we expanded the rectangle, would it extend the pattern of shaded squares or would it be different?
- If I just had the dimensions, could I predict the pattern of shaded squares?
To guide our explorations, the group decided to focus on 2 questions:
What is the relationship between the dimensions of each rectangle and the number of shaded squares?
Can you predict the number of shaded squares based on the dimensions of a rectangle?
TOOLS FOR EXPLORING DIAGONALS
- Geogebra Applet – https://www.geogebra.org/m/ppBPT4HW
This applet allows you make rectangles with different dimensions and shows the diagonals and the number of shaded squares. It also allows you to hide the shaded squares (which might be helpful for testing any conjectures you come up with along the way).
Take some time on your own to play with the applet. HAPPY EXPLORING!
SOME OF THE IDEAS UNCOVERED DURING OUR GROUP EXPLORATIONS
