Shifting times tables
Can you find a way to identify times tables after they have been shifted up or down?
Problem
Shifting Times Tables printable worksheet
The numbers in the four times table are
I could shift the four times table up by 3 and end up with
What do you notice about the differences between consecutive terms in each sequence?
The interactivity displays five numbers from a shifted times table.
On Levels 1 and 2 it will always display five consecutive terms from the shifted times table.
On Levels 3 and 4 it could display any five terms from the shifted times table.
Use the interactivity to generate some sets of five numbers.
Can you work out the times table and by how much it has been shifted?
Once you are confident that you can work out the times table and the shift quite easily, here are some questions to consider:
What can you say if the numbers are all odd?
What about if they are all even?
Or a mixture of odd and even?
What can you say if the units digits are all identical?
What if there are only two different units digits?
What can you say if the difference between two numbers is prime?
What can you say if the difference between two numbers is composite (not prime)?
Can you explain how you worked out the table and shift each time, and why your method will always work?
You may also be interested in the other problems in our Dynamic Explorations Feature.
Getting Started
For the Level 3 and 4 problems, start by rearranging the numbers so that they are in order. Then look at the pairs of numbers that are closest together.
Student Solutions
Well done to everybody who managed to work out which times tables were being shown, and how they had been shifted. Abdullah from Doha College in Qatar noticed:
All times tables are linear sequences. Therefore, in order to find out the original times tables, you have to see the difference in each term of the sequence. That is the original times tables. Then, check how much lower the original times table number is in comparison to the times table, and how much higher it is. You then have the answer.
For example:
7, 10, 13, 17
The sequence has intervals of 3. Therefore it is a times table of 3. 7 is 1 more than 6 and 2 less than 9.
Thank you, Abdullah, this is very clearly explained.
Turker from Mark Twain Intermediate School in the USA had some similar ideas, and began to think about our questions:
The difference between the consecutive terms in a sequence of shifted times tables tells me what times table it is. Ex: 4,6,8,10…,112,114,116… The difference between the consecutive terms is 2. This means it's a times table of 2. Ex: 3,5,7,9…113,115,117…. The difference between the consecutive terms is also 2 but it is shifted up by 1 and shifted down by 1. This means that this table is still counting by twos meaning it's a table of multiples of 2.
If all the numbers are odd then the table is counting by an even number.
If all the numbers are even then it is also counting by even numbers.
If it is a mixture of even and odd numbers, then the table is counting by an odd number.
If all the (units) digits are identical then it will result in a table counting by even numbers.
I wonder which even number it would be counting in?
Leo from RCHK in Hong Kong also answered our questions:
What can you say if the numbers are all odd?
If the numbers on screen are all odd, it means that the table number will be even and the number that the table is shifted up or down by will be odd.
What about if they are all even?
If all the numbers on screen are even, the table number and shift numbers will all be even as well.
Or a mixture of odd and even?
If the numbers on screen are a mix of odd and even, the table number will be odd. The shift numbers will either be odd or even, but never two of the both. For example, if the shifted up number is odd, then the shifted down number must be even.
What can you say if the units digits are all identical?
If the unit digits are identical, it means that the table number is a multiple of 10.
What if there are only two different units digits?
If there are only two different unit digits, it means that the table number is a multiple of 5 but not a multiple of 10.
What can you say if the difference between two numbers is prime?
The table number will be a prime number
What can you say if the difference between two numbers is composite (not prime)?
If the difference is composite, the table number is a composite number.
Can you explain how you worked out the table and shift each time, and why your method will always work?
In levels 1-2, we found the table number by just simply finding the difference between the first two numbers, which will find the table number. After we find the table number, just divide the first number by the table number and find the remainder. The remainder will be the shifted up number and the shifted down number will be the table number minus the shifted up number.
In levels 3-4, we found the table number by finding the two numbers that have the least difference and finding the difference of those two numbers. The difference is the table number. We then would find the lowest number and divide the lowest number by the table number with remainders and the remainder will be the shifted up number. To find the shifted down number, simply just minus the shifted up number from the table number.
I agree that if the difference between two numbers is prime, the table number will be prime. But I don't necessarily agree that if the difference between two numbers is composite, the table number will be composite. I wonder why these two cases are different?
Cecilia, Arwen, Kerry, Hunter from Rainworth State School in Australia thought hard about each of the questions, and sent in the following explanations:
What can you say if the numbers are all odd?
If the numbers are all odd, the times table is even.
What about if they are all even?
If the numbers are all even, then the times table is also even.
Or a mixture of odd and even?
If it is a mixture of odd and even, then the times table is odd.
What can you say if the units digits are all identical?
The number set is going up by a multiple of ten.
What if there are only two different units digits?
When there are two different units in the ones place, then the times table is a multiple of five.
What can you say if the difference between two numbers is prime?
When the difference between two numbers in the sequence is prime, then that means that difference is the times table, because it is not divisible by anything else.
What can you say if the difference between two numbers is composite (not prime)?
When the difference between two numbers in the sequence is a composite number then there is a chance that the difference is not the times table because it could be a multiple of the times table.
Can you explain how you worked out the table and shift each time, and why your method will always work?
Our Mathematical Theory:
Our mathematical theory is that when you figure out the times table, the up and down shifts add up to that number. For example, if the times table is four, and the first number in the sequence is 5, then you go to the closest multiple of four below that number, which is 4. 5 minus 4 is 1. Then to figure out the shift up, from 5, you go to the closest multiple of 4 which is above 5, which is 8. 8 minus 5 is 3. 3 + 1 = 4, therefore it is the 4 times table.
This theory is like friends of ten (number bonds to ten) because, the times table number is like the ten in friends of ten. Then the two shifts adding to the table, are like the two numbers adding to ten.
From level 3/4 you have to find the two lowest numbers in the sequence (may not be next to each other) and find the difference between those two numbers. Then with the second smallest and the third smallest number, you find the difference between those. Then you find the highest common factor of the differences. That should be the times table. Then you use this method to find out what it is shifted by (This method can also work for level 1 and 2 but isn’t necessary). This is an example using this theory.
Thank you for sharing these ideas with us - I agree with everything you've said here!
Advaya, Shivashree, Kanaa, Reeve, Ananya, Tiana, Ishaan, Veer, Amay, Avic, Aheedutt, Ela, Ashrith, Shreehari, Kimaya, Mrunmayee, Aarav, Ruhi, Amey, Siddharth, Reyansh, Vihaan, Vivaan, Rudra, Aaryavir, Rudravir and Swara from Ganit Kreeda in Vichar Vatika, India worked hard on this activity. Take a look at Ganit Kreeda's full solution to see their ideas.
Thank you as well to the following students, who sent in similar solutions: Danny from WMS in England; Charisse from China; Charlie, Colin and Jayden from Renaissance College in Lebanon; Thomas from Witton in England; Dhruv from The Glasgow Academy in Scotland; Sophie from CHPS; and Adrian, Konnor, Kyran, Stefan and Yuji from Renaissance College Hong Kong.
Teachers' Resources
Why do this problem?
This problem encourages students to think about the properties of numbers. The use of an interactivity provides an engaging "hook" to stimulate students' curiosity and draws them into the structure of linear sequences and straight line graphs. It also provides a natural language, that of the "times table" and "shift" for talking about remainders and modular arithmetic.
Possible approach
This printable worksheet may be useful: Shifting Times Tables
The solutions are available here.
"I'm thinking of a times table. I wonder if you can work out which it is? $6, 12, 18, 24$" (writing the numbers on the board as you say them.)
Now show the interactivity from the problem, and alert the students that it does something slightly different (but don't tell them what!). Generate a set of numbers using Level 1 or 2, 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" used in the interactivity.
Emphasise that the table should always be the largest possible, and the shift should always be less than the table. This example could be used to bring these ideas out:
Possible suggestions that might emerge:
But we are interested in
Group students in pairs at a computer or with a tablet and challenge them to develop a strategy to find the table and shift with ease for Levels 1 and 2. Once they can confidently answer Level 1 and 2 questions, they can move on to Levels 3 and 4 where they are given random terms from the shifted times table instead of the first five terms. While students are working, circulate and listen out for students who have developed useful strategies that they can share with the rest of the class.
If computers are not available for students, use the interactivity to generate a dozen or so examples at appropriate levels, and write them on the board for the class to work on. Students could also work in pairs and create examples for their partners to work out, or work on the examples on this worksheet.
Once students are confident at finding the times table and the shift, ask them to work on the following questions:
- What can you say if the numbers are all odd?
What about if they are all even?
Or a mixture of odd and even?
- What can you say if the units digits are all identical?
What if there are only two different units digits?
- What can you say if the difference between two numbers is prime?
What can you say if the difference between two numbers is composite (not prime)?
Finally, bring the class together to discuss these questions and then generate a Level 4 example. Invite students to explain how they would tackle it.
Here is an account of one teacher's approach to using this problem.
Key questions
What is the same between numbers in a times table and numbers in the shifted times table?
Possible support
Perhaps start with the Factors and Multiples Game to practise working with multiples and factors. This could then be followed up by looking at the problem Remainders.
Possible extension
Here are some follow-up resources that may build on students' thinking about this problem: