Times tables shifts
In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?
Problem
The numbers in the five times table are:
5, 10, 15, 20, 25 ...
I could shift these numbers up by 3 and they would become:
8, 13, 18, 23, 28 ...
In this activity, the computer chooses a times table and shifts it.
Can you work out the table and the shift each time?
Can you explain how you worked out the table and shift each time, and why your method will always work?
Levels 1 and 3 include tables up to 12.
Levels 2 and 4 include tables up to 20.
On levels 1 and 2, the numbers will always be the first five numbers in the times table.
On levels 3 and 4, the numbers could be any five numbers from the shifted times table.
You may be interested in the other problems in our Number Patterns Feature.
Getting Started
What is the same about numbers in a times table and the numbers in the shifted times table?
On level 3, how about putting the numbers in order?
Student Solutions
Thank you to everybody who sent in their ideas about this activity.
Viren shared these tips with us:
I look for the pattern and relations between numbers. In case of Level 1 and Level 3, by looking at lowest and highest numbers I figure out the times tables they might belong to. And then I work out what shift (up/down) is applicable.
In case of Level 2 and Level 4, the big numbers look intimidating but I work out the difference between numbers first, then figure out which number they are multiples of.
Good ideas, Viren. I wonder why working out the difference between the numbers is useful for finding the times tables that they might belong to?
Harry from Illogan School in Cornwall, UK sent in this method for the Level 4 problems, explaining how to use the difference to work out the times table:
This time I have numbers 123, 59, 171, 219 and 91
1) Put the numbers in order and find the differences
59 (+32) 91 (+32) 123 (+48) 171 (+48) 219
2) Find the highest number that goes into all the differences
Factors of 32 = 1, 2, 4, 8, 12, 16 and 32
Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48
So 16 in this case. This gives you the times table that has been shifted
3) Write out the 16x table, which will help you find the shift
In this case it is either 16 minus 5 or 16 plus 11. You can check it works for each number you were given
4) There will always be two answers because you can either add, or minus from the number you were given to get to a number in the 16 x table
This is really clearly explained, Harry!
Shaunak from Ganit Manthan, Vicharvatika, India described a similar solution:
This problem can be split into 2 parts: Working out the Table and Working out the Shift.
Working out the Table:
I observed the differences between the sets of consecutive numbers in the shifted times table.
In level 1 and 2, all these differences are the same.
In level 3 and 4, all these differences are different. The table has to be the greatest possible, so we have to find the HCF (Highest Common Factor) of the differences.
If the given sequence is 37, 17, 22, 52, 77, then the differences are 20, 5, 30, and 25. The HCF of these differences is 5, so the table is 5.
This method works because taking any two numbers in a times table and subtracting them will give a difference which is a multiple of the table. This happens because a times table is an arithmetic progression, which means a constant number (the table) is repeatedly getting added to the previous number, creating a sequence.
If the table is 5, then creating the sequence looks like this: 5, 5 + 5, 5 + 5 + 5, 5 + 5 + 5 + 5, and so on.
If you take two numbers from the times table, you will see that the table is repeatedly getting added to the smaller number to obtain the bigger number. Therefore, the difference between two numbers in a times table is a multiple of the table. This having been said, we can say that the HCF of the differences will give us the greatest possible table.
Let us consider the example stated above. Arranging the numbers in ascending order, we get 17, 22, 37, 52, and 77. 5 is being added to the first number to get the second number (17 + 5 = 22), 15 is being added to the second number to get the third number (22 + 5 + 5 + 5 = 37) , 30 is being added to the third number to get the fourth number (37 + 5 + 5 + 5 + 5 + 5 + 5 = 52) , and 25 is being added to the second number to get the third number (52 + 5 + 5 + 5 + 5 + 5 = 77).
Working out the Shift:
I first observed the first number of the shifted times table and the closest multiple of the times table to this number.
If the shifted times table is 30, 50, 70, 90, 110, then the table is 20, because all the differences are 20. I looked at the first number, 30, and figured the closest multiple of 20 is 20.
If the multiple is smaller than the first number of the shifted times table, then the difference of the first number of the shifted times table and the multiple is the shift up.
Considering the above case, we can see that 20 is smaller than 30. So the difference between 20 and 30, which is 10, is the shift up.
If the multiple is greater than the first number of the shifted times table, then the difference of the first number of the shifted times table and the multiple is the shift down.
If the shifted times table is 2, 5, 8, 11, 14, then the table is 3. The closest multiple of 3 to 2 is 3. We can see that 3 is greater than 2. So the difference between 3 and 2, which is 1, is the shift down.
We can observe that the shifted times table is also an arithmetic progression, just like the times table. The differences have the same HCF, but the first, second, third, fourth, and fifth have a common difference from their closest multiple.
Thank you for sending in these ideas, Shaunak, this is a really clear explanation of how the shifted times tables work and how they can be solved.
Veyhant from Ganit Kreeda, Vicharvatika, India sent in these pictures explaining their solution for each of the four levels. These pictures can be clicked on to make them bigger:
We also received similar solutions from Nikhil, Abhyudh, Viyaan, Aditi, Mridula, Pushan, Larisa, Laira, Shravani, Anika, Kiaan, Kaira and Shreehari (Group 1) and from Ahana, Sehar, Saanvi, Dhanvin, Aariz, Ananthjith, Vivaan, Sai, Pranathi, Paavani, Utkarsh, Tavish and Dhruv (Group 2) from Ganit Kreeda, Vicharvatika, India. Thank you all for sending in your ideas - you've worked really hard on this problem. Their full solutions can be seen by clicking on these links: Group 1 Ganit Kreeda, Group 2 Ganit Kreeda.
Thank you as well to Jeffery and Ava from Australia and Luna from CHPS in Australia who also sent in some similar solutions to this problem.
Teachers' Resources
Why do this problem?
This problem will help learners become more familiar with multiplication (times tables) facts, by encouraging them to think about number properties and therefore gain a deeper understanding of our number system. It also makes the inverse relationship between multiplication and division explicit. The interactivity will capture learners' curiosity so that they are motivated to find solutions and, in this way, they will be doing meaningful calculation practice.
In conjunction with the other tasks in this group, this task also offers the chance to focus on any of the five key ingredients that characterise successful mathematicians.
Possible approach
Explain to the group that you're thinking of a times table and ask them if they can work out which it is. Write these numbers on the board as you say them: 3, 6, 9, 12. What about 4, 8, 12, 16? 45, 50, 55, 60? How do they know?
Keep going until the class is confident and fluent at working out the times tables. To avoid shouting out, learners could write their answers on mini whiteboards.
Explain that you will now give the class some random numbers from a times table rather than the first four numbers. Write up, for example, 60, 20, 100, 50. Discuss that these are all in the 2, 5 and 10 times tables, but we're only interested in finding the largest possible times table, so we'll say these are numbers in the 10 times table.
Now show the interactivity from the problem and alert the children to the fact that it does something slightly different (but don't tell them what!). Generate a set of numbers using Level 1 and give the class a short time to discuss with their partner what they think the computer has done. Do the same a couple more times, without any whole-class sharing, but giving pairs a little time to refine their ideas. Then bring the class together and discuss what they think is going on. Link what they say to the terminology of "table" and "shift" that the computer uses. Emphasise that the table should always be the largest possible, and the shift should always be less than the table.
Ideally, each pair would now work at a computer or tablet to develop a method of finding the table and shift with ease. If that isn't possible, generate a dozen or so examples at appropriate levels, and write them on the board for the class to work on. Learners could also work in pairs and create examples for their partners to work out.
Once pairs are finding the table and shift easily, bring the class together. Generate a new example, using level 2 or 3, and ask a pair to talk through their thinking as they work towards the solution, but ask them to stop short of actually giving the answer. The rest of the class could write the answer on mini whiteboards once they've heard enough to work it out. Repeat, giving other pairs the opportunity to share their thinking.
Possible questions
What is the same between numbers in a times table and numbers in the shifted times table?
What can you learn from the difference between any two numbers in a shifted times table?
How do you find the shift once you've worked out the table?
Possible extension
The Shifting Times Table problem is a Stage 3 version of this task and the interactivity includes an extra level, which includes numbers in the 21-50 times tables. The tasks in this group would also provide useful follow-up to this one.
Possible support
Children may find a table square helpful as they work on this activity.