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# Times Tables Shifts

##### Age 7 to 11Challenge Level
Woodlea Junior School pupils sent in their solutions:
Ayla, Eleanor and Georgia wrote a solution each that said that they solved it by finding the difference between the numbers to find the times table.  Then they found the difference between the normal times table and the shifted one, and that was how much it was shifted up/down by.

Hannah wrote:
To find out the table I found out the difference between the numbers e.g for 12 and 23, the difference is 11. To calculate the shift I used the table (11) and found out the difference between the first multiple that was given and the first multiple of 11.

To do this I worked out the difference between the second number and the first number, e.g. 6 to 18=12. Then I worked out whether I was going to up or down with my times tables. Once I realised that it was my 12 times tables I then figured out that the difference was 6.

Glendenning Public School pupils sent in their solutions. Divya and Sapumalee wrote to say:
We found out that the easiest and clearest way to solve these types of questions was to first find out the times table. We subtracted or added numbers in numerical order to the given numerals to find out the times table. We then worked out the pattern and solved our answer.

Sanna said:
To find the number of shifts, subtract the bigger number from the smaller number.
For an example:  4, 9, 14, 19, 24
The difference in the numbers is 5  ( 9 - 4, 14 - 9 ....). Hence the times table is 5.
To find the shift you have to get the first number of the sequence to be the
first number of the times table. Example  4 , 9 , 14, 19 , 24;  5, 10, 15 , 20 , 25;  You are shifting down by one (5 - 4 = 1 ).
Answer - Times Table: 5, Shift: Down by 1.

Mia and Taniya said:
To minus the two numbers closest to each other. For example for 3, 5, 7, 9; 3 - 5 = 2; 5 - 7 = 2; 7 - 9 = 2;
This would be 2 times tables shifted up by 1.

Sirat, Gurmehar and Ranvir wrote:
For example for 12, 19, 26, 33, 40:
First we have to get the first number (12) and then you find a number that can add to the first number to make the second number (19). Table = 7.
Then you have to add the first number (12) with itself and then minus a number (5) to make the second number (19).
Shifted by: 5 You have to remember that when you add to get your answer shifted by you have to write: down. When you minus to get your answer shifted by, you have to write: up.
Solution  12+7= 19;  12+12= 24-5= 19: Table: 7, shifted by (up): 5

Simar wrote:
To figure out the times table you have to subtract the first two numbers. When the times table number is found you have to find the number it's been shifted by. To find that out we have to subtract the first number by the number we found earlier or vice versa. Then you just have to write what you got in the place where you have to.
For example, if we have the sequence. 11 21 31 41 51;
21 - 11 = 10; 11 - 10 = 1.
Table 10, Shifted up by 1.
I know this works because I have tried it many times and it works for all the problems.ðŸ‘ŒðŸ˜Šâœ”

Scout from James Allen's Prep School sent in the following:

My method to solve goes like this:
Find the difference between the numbers given to find the accurate multiplication table. Try and notice the shift between the sequence and the multiplication table.
For example: If we started with these numbers: 1, 3, 5 ... it would be the times table of two because the difference between the terms in the sequence is 2.
This is shifted up by 1 from the original multiplication table of 2 (starting at zero) because the difference between 0 and 1 is 1. This means it is shifted up from the original times table of 0, 2, 4”¦

Elaina from John Colemon Elementary School in the USA wrote:

What I do is I always go to the second number in the times table, and I see what I can divide it by, and still get a whole number. If I do get a whole number, I go back to the first number and divide the first number by the same amount I divided the second number by, and if I can a whole number, and they match up you should get the right answer according to my calculations.

Thank you all for these explanations and solutions.