Finding a square of a 2-digit number in your head

Recently we looked at how to square two-digit numbers ending in 5 in your head.  What about the other square numbers?  Well, there’s a quick way to do that in your head as well.

Let’s look at 372.
First of all round the 37 up or down to the nearest 10.
You added 3 to get 40, so you also need to subtract 3 to get 34 (37-3=34)
Multiply these to numbers together 40×34 = 4 x 34 x 10 = 1360 (4×30=120,  4×4=16, 120 +16 = 136, 136×10=1360)
Finally, add on the square of the number you added and subtracted (here it was 3 and 3×3=9 so add 9) 1360+9=1369.
So 372 = 1369

Here’s another example: 622
Round this down to 60, and then because you subtracted 2, you also have to add 2 so you get 60×64 (6 x 64 x 10). 3 x 60 = 360, 6 x 4 = 24, 360 + 24 = 384, 384 x 10 = 3840
Remember to add the square of the number you rounded down by. 22 = 4 so add 4.
3840 + 4 = 3844
So 622 = 3844

If you’ve enjoyed these tricks then you might enjoy the Secrets of Mental Maths course from The Great Courses. It’s fascinating!

Related post: How to multiply any 2-digit number by 11 (in your head!)

How to square 2 digit numbers ending in 5 (in your head)

This is another trick that my family already knew, but I only learnt it recently so I’m sharing it for anybody out there who doesn’t already know.

Squaring 2 digit numbers is easy if they end in 5.

5×5 = 25, so the last 2 digits of the answer will always be 25.

To find the first digits of the answer, take the first digit of the number you are squaring and multiply it by the next ten up.

Eg 65 x 65

The first digit is 6. The next ten up is 7, so to find the first digits of the answer you multiply 6×7.

6×7 = 42

The last two digits will always be 25 so 652 = 4225

352 = 1225  (3×4 = 12 and the last 2 digits are 25)

852 – 7225 (8×9 = 72 and the last 2 digits are 25)

It’s especially useful for squaring something-and-a-half eg 7.5 x 7.5. 7 x 8 = 56. The last two digits will be 2 and 5, but here you are using ½ not a whole number, so it won’t be 25 but .25 so the answer is 56.25.  3.52 is 12.25 and so on.

If you enjoyed this trick, you might also enjoy this post on how to multiply 2 digit numbers by 11 in your head.

Multiplying 2 digit numbers by 11

Earlier this year I discovered a trick for multiplying 2 digit numbers by 11.  I taught this to some of the children I tutor to help them work out 11×11 and 11×12 which they had been struggling with but then as it works for all 2 digit numbers they have enjoyed baffling their family and friends with their ability to perform calculations such as 54 x 11 in their head.

Apparently it’s not as big a secret as I had thought (well, everyone in my family already knew how), but I’m going to share it anyway just in case some of you don’t already know how…

First of all you split the digits of the two digit number, and these become the first and last digits of the answer. Eg with 54 x 11, the answer is going to start with 5 and end with 4. To get the number in the middle you just add the two digits together. 5 + 4 = 9, so put 9 in the middle. Your answer is 594.

35 x 11 = 3 [3+5] 5   that is 385
42 x 11 = 4 [4+2] 2   that is 462.

That’s all very well, but what if the two digits add up to more than 9?  That’s easy too. You partition that number, the units become the middle digit and the tens get added to the first digit.

86 x 11 = 8 [8+6] 68 [14] 6 → uh oh!  It can’t be 8146. That’s ok – split the 14. The 4 becomes the middle digit, and the ten (1) gets added to the first digit → 8+1=9, so the answer is 946.

Happy calculating!

Related posts: finger tables, how do you know if a number is divisble by…?How to square a 2 digit number ending in 5 in your head

D is also for… Division

D is for...Sometimes in maths it’s really useful to be able to look at a number and tell quickly what numbers it is possible to divide it by. There are a few tricks you can use to tell whether one number is divisible by another , so I’ll share them with you.

 

How can you tell if a number is divisible by 1?

It’s simple. All numbers are divisible by 1.

How can you tell if a number is divisible by 2?

This is simple too. If it’s an even number, it’s divisible by two; if it’s an odd number, it’s not.

How can you tell if a number is divisible by 3?

Add up all the digits until you get a single digit number, If it is 3, 6 or 9 then the number is divisible by 3. For example: 462         4 + 6 + 2 = 12           1+2 = 3. So 462 is divisible by 3. 729          7 + 2 + 9 = 18          1 + 8 = 9. So 729 is also divisible by 3.

How can you tell if a number is divisible by 4?

Look at the last two digits. Halve them. If you get an even number then it’s divisible by 4. For example   13,564         the last two numbers are 64. Half of 64 is 32 which is an even number so 13,564 is divisible by 4.

How can you tell if a number is divisible by 5?

This one’s a bit easier. If it ends in a 5 or 0 then it’s divisible by 5.

How can you tell if a number is divisible by 6?

If it’s an even number, and it’s divisible by 3 (see how can you tell if a number is divisible by 3) then it’s also divisible by 6.

How can you tell if a number is divisible by 7?

This one is a bit tricky, so bear with me. Take off the last digit and double it. Take it away from the rest of the number. If the answer you get is divisible by 7, then the whole number is divisible by 7. If you’re not sure then take off the next digit and repeat the process. This method definitely needs an example!

833             take off the last digit (3) and double it = 6

83 – 6 = 77

77 is divisible by 7 so 833 is also divisible by 7.

3192          take off the last digit (2) and double it = 4

319 – 4 = 315

Not sure whether 315 is divisible by 7, so take off the last digit (5) and double it  10

31 – 10 = 21 which is divisible by 7 so 3192 is divisible by 7.

How can you tell if a number is divisible by 8?

Look at the last three numbers. Halve them and halve them again. If you get an even number then it’s divisible by 8.

How can you tell if a number is divisible by 9?

Follow the procedure for telling if a number is divisible by 3. If the single digit you get is 9, then the number is divisible by 9. In the example given in how to tell if a number is divisible by 3, 729 is divisible by 9, but 462 is not.

How can you tell if a number is divisible by 10?

If it ends in a 0 then it’s divisible by 10.

How can you tell if a number is divisible by 11?

Alternately subtract then add the digits. If the answer is a multiple of 11 (including 0) then the number is a multiple of 11.  This one needs an example too!

6425936 –> 6-4+2-5+9-3+6 = 11, so 6425936 is divisible by 11.

How can you tell if a number is divisible by 12?

If it is divisible by both 3 and 4 (see above for how to tell) then it’s divisible by 12.

Related posts: C is also for…  E is also for….

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Teaching the Times Tables

I’ve been tutoring maths for a number of years now. I’ve tutored boys and girls. I’ve tutored individuals and small groups. I’ve tutored children of all ages from very different social backgrounds. But they have all had one thing in common: none of them knew their times tables, and this was really hindering their progress in maths.

Of course I told them that they needed to know their tables off by heart, but their parents and teachers had already told them this. If it was that easy they would have learnt them already. So this year I have made it my mission to get all the children I tutor to learn all of their times tables.

To start with I created a desire to learn them. I made a colourful chart to show progress, and offered rewards of stickers for each of the tables that they learnt. But not just any old stickers – exciting, shiny ones that made their eyes light up when they saw them. The boys especially liked these football ones from Superstickers.

Now I had children who were desperate to learn their times tables. What next?

We took the tables one at a time and started by chanting them. When we had chanted them forwards a few times, we did them backwards, then odd numbers only and even numbers only to get used to the idea of knowing them out of order. After that it was a case of practise, practise, practise. The trick was finding enough different ways to practise the same thing so that the children didn’t get bored with it.

I made some sets of cards with the questions and answers so that we could play pelmanism, and these proved very popular. I encouraged the children to read aloud the question as they turned each card over, and to work out what answer they needed to match before turning over the next card. We also used the same cards to play snap, and a race against the clock game to match all up all of the question cards with their answers – trying to be faster each time.

Although the children loved all of these games, I was very aware that I couldn’t rely on the same sets of cards forever without the children thinking “Oh no – not those again!” and losing motivation. I looked around for some new ideas and found some lovely products on Sue Kerrigan’s let me learn website.

The turn table cards were recommended to me by the trainer on a dyslexia course I attended. They are designed for multi-sensory learning and are really good fun to play with. On one side of the card they have a question eg 2×3 and a picture of an array to show children what 2×3 looks like and to give them a visual clue. On the other side is the answer. The children say the question and answer aloud (hearing their own voice) and then turn over the card to see if they are correct. There is a video of how to use them here . I usually use them with one child at a time, focusing on one set of tables at a time, using them as shown in the video, and then doing races against the clock to beat their own personal time. However I have also used them with a group of children each working on a different set of tables. One group of girls I worked with recently, who were all working on the same set of tables, made up another game to play with these cards which they found great fun: all of the cards were put answer-side-up in the middle of the table. I called out a question and they had to grab the card they thought showed the correct answer. They turned the card over to see if they were right, and if they were, they repeated the question and answer and kept the card.  If they were wrong they replaced the card. The winner was the girl with the most cards when they had all been grabbed. All of the children I have used these cards with have really enjoyed it, and I’m sure there are many more games that can be invented using them.

I found the maths wrap while I was browsing the site, and just thought I would give it a go. It’s used for learning tables “in order”, but is great for kinaesthetic learners. Across the top is a strip with numbers 1 to 12. At the bottom is space to put a strip of one of the tables, each of which contains all the answers but jumbled. You have to chant the tables aloud, hunting for the correct answer along the bottom strip and then wrapping the string around the correct number each time. When you have finished you can turn it over to look at the pattern marked on the back. If the children have got all the answers correct, the pattern made by the string will match the pattern printed on the back of the card. When I bought it, I thought it might be one just for the girls, but actually the boys have enjoyed using it just as much. One of my Year 5 boys said “Every child should have one of these. They’re really cool!” I even had texts from two mums, because their sons had been talking so much about how much fun it was that they wanted to know where I got them from so that they could get them as stocking fillers.

As we progressed through the tables we looked at how few they had left. By using counters to demonstrate that for example 2×3 was the same as 3×2, we were able to colour code each new set of tables to show which ones they already knew and which ones were still to be learnt. They learned the easy ones (2x, 5x and 10x) first, which made the chart look less bare, and earned them some shiny stickers pretty quickly. Then they did 4x (easy because it was double 2s). 3x came next (tricky but the colour coding showed that they already knew 2, 4, 5 and 10 x3, so there where only half of them still to learn). Then 6x was easy because it was double 3s. By the time we came to the tricky ones like 7x, the progress chart was looking quite full, and the colour coding showed that they already knew 2, 3, 4, 5, 6 and 10x 7, so all that was left was 7×7, 8×7 and 9×7. Suddenly the sevens didn’t seem so scary and the motivation continued.

Of course it took a long time, although considering the fact that I only see these children once a week it took less time than I expected. In September two of my boys didn’t know any of their times tables, not even 2x or 10x. They now know all of them. Not only do they know them off by heart, but they are able to apply them in all areas of maths, for example working with equivalent fractions. They immediately recognise numbers that are in their times tables which means their skills in division have improved. Their mental arithmetic skills have improved because they can multiply 6 by 7 straight away, instead of having to count up 7 lots of 6 on their fingers, so they have more time to think about what the questions are asking them to do with the information. They have both moved up a maths group at school and their confidence is higher. One of them said to me recently that he used to hate maths, but that he really loves it now. And that’s why I really love my job!

For maths and English tutoring in the north Birmingham, Sandwell and Walsall areas, visit www.sjbteaching.com. For links to other interesting education related articles, come and Like my Facebook page.

Related post: Teaching Number Bonds    A Multisensory Approach to Reading