Grid Staircases

A: The Middle Row

1. What does the middle row of this staircase add up to?If you move the staircase around, so that it has different top numbers, how does the sum of the middle row change?

2. Can you predict the sum of the middle row from the top number? How do you do this?

3. Why is the sum of the middle two numbers always an odd number?

4. Can you work out the top number from the sum of the middle row? How? 

5. Try these:

• The sum of the middle row is 31, what is the top number?

• The sum is 69.

• The sum is 171.

• The sum is 139. There is something a bit strange about this one. Can you spot it?

• The sum is 353. Even stranger!

• The sum is 8,765.

6. So now we have extended our square so that it really only exists in our heads. Maths is like that!

B: The Bottom Row

1. Try the same for the bottom row of the staircase. Move it round to find a relation between the number at the top and the sum of the bottom row.

2. Can you predict the sum of the bottom row from the number at the top?

3. Can find the number at the top from the sum of the bottom row? 

4. Try these:

   • 84, 240, 522, 861, 48 807.

C: The Whole Staircase

1. Find out what top number gives a staircase where all the numbers, including the top one, add up to 326,460

D: The Gaps between the Stairs

1. What number would give a staircase where all the numbers add up to 105? Remember the 100 square is in your head now, so you are not restricted to the numbers on the paper version.

2. Explain carefully how you work this out.

3. Now try for a sum of 500.

4. How about 1000?

5. What about 24?

6. Can you find a staircase that has a sum of zero?

Further Questions and Challenges

E: More Stairs

1. What would the relationships be in a 4 level staircase?

2. How would you find the sum of the fourth row from the top number?

3. How would you find the top number from the sum of the fourth row?

4. How would you find the sum of the whole staircase from the top number?

5. How would you find the top number from the sum of the whole staircase?

F: More Questions

1. Try with a fifth row.

2. Can you see any patterns?

3. What would the formulas be for a staircase with n rows?

4. What would happen if you changed the width of the grid so there were only 9 numbers in each row? What if there were 8 in each row, or 11?

5. What if the numbers were not increasing by 1 each time, but by 2? Or by 0.5?

6. What if the grid went from the bottom up instead of top down, or from left to right or right to left?

7. Can you come up with your own questions to investigate next?

Extension

While it is not required at Year 7, some basic understanding of algebra may help you to simply and solve the problems above.  You may wish to take a look at topics such as "substitution", "collecting like terms", "forming expressions" and "solving equations".  The relevant skills can be found on  DrFrostMaths, CorbettMaths, MyiMaths and Eedi.  Watch any video and/or go through any online lesson as you see fit.