T-Totals - mr barton maths

Getting Started

1

On a 9×9 grid, place the T-shape at different positions. Record the T-number and the T-total each time. What do you notice?

2

Can you find a formula connecting the T-total to the T-number?

Hint: if the T-number is n, express each cell in the T-shape in terms of n and the grid width.

3

Test your formula by predicting the T-total for a T-number you haven’t tried yet. Were you right?

Changing the Grid Size

4

Switch to a 7×7 grid and repeat your investigation. Does your formula still work? What changes?

5

Try the 8×8 and 10×10 grids. Can you find a general formula that works for any grid size g?

Hint: the T-total = 5n − 7g for an upright T on a g×g grid. Can you prove why?

Rotating the T-shape

6

Rotate the T-shape 90° clockwise. Investigate the relationship between T-number and T-total for this new orientation. How does the formula change?

7

Now try 180° (upside-down T) and 270°. Find formulas for each rotation.

8

Can you explain algebraically why each rotation gives a different formula? Express each cell’s offset from the T-number in terms of the grid width g.

Changing the Starting Number and Step

9

Change the starting number to 0. How does this affect your formulas? What about starting at 5?

10

Change the step to 2 (so the grid counts 1, 3, 5, 7…). How does your formula change? Try step = 3.

11

Can you find a master formula that includes the grid size g, starting number s, and step d?

Reversing the Grid

12

Switch the direction to ← ↑ so numbers count down. How does the T-total change for the same position? Can you explain why?

Different Shapes

13

Switch to the L-shape. Investigate the relationship between the anchor number and the L-total. Find a formula.

14

Try the + shape (plus/cross). This one is symmetric — does the formula simplify compared to T and L?

15

For the + shape, the centre number always equals the total divided by 5. Can you prove this algebraically?

Deeper Thinking

16

If the T-total is 200 on a 9×9 grid, what is the T-number? Can you work backwards from any T-total to find the T-number?

17

Two T-shapes on the same grid give totals of 37 and 167. Without looking at the grid, can you work out how far apart they are?

18

Design your own shape (perhaps an E, S, or Z). Define its cells relative to an anchor point and find its total formula. Share your shape with a partner.