Counting on & Square numbers

Z has known how to count up to 100 for a while... I'm not sure where she learnt it - from her sisters probably! And has recently randomly started doing addition on her fingers, e.g. she'll come to me while I'm cooking and say "4 plus 3 is 7!" while holding up her fingers to show me... So I thought I should probably put some effort in too and help her addition skills improve. 😅

Firstly, she hadn't got the hang of counting on when doing addition - i.e. she would count the fingers on one hand from 1 before counting on the fingers from the other hand, rather than just start at 4 and count on from there. So I asked her "What's 7 plus 7?" Of course, she couldn't do it on her fingers because she didn't have 7 fingers on each hand!

So we took out the multilink cubes and she chose two colours - red and black - to represent each number in my questions. She counted out 7 red cubes and 7 black cubes, then (starting from 1 again!) counted them all to 14. I covered the red pile with my hand and told her we know this pile is 7, so we don't need to count it again! And modelled counting on with the black blocks to get to 14.

We did another couple of questions like this with the blocks (e.g. 8 + 4, 6 + 9, so answers between 11-20) and I would cover the larger number with my hand after she'd counted out the blocks so she could count on with the smaller number. Then I would uncover the pile and she could count them all to check her answer was correct.

Then I asked her if she needed to count out the blocks for the bigger number at all, because I was covering the pile anyway! And she said no, but where should she put the number then? I lightly squeezed the top of her head with my hand and said to put the number in there (I found when teaching the twins when they were younger that this physical action helped them!).

I asked her what 8 + 5 was. Which was the bigger number? 8. So where does it go? In your head (*squeeze for emphasis)... Without prompting she then counted out 5 blocks. I touched her head to remind her 8 was in there then she used the blocks to help count on to 13. 😄

Final stage: I asked her if she could do bigger numbers like 15 + 4 then? And could she use her fingers to help in case she didn't have any blocks? She touched her own head (15 in here...) and counted on to 19 using 4 fingers. Last question, 22 + 8? Same method, she worked out it was 30! MashaAllah. 💪 When Papa came home later she was happy to show him she could now do maths with bigger numbers! And I was happy to see she'd retained what I'd taught her earlier. 😂💕

----------

While I had the blocks out, I decided to use them afterwards for the twins as an introduction to square numbers.

They have a multiplication grid poster in their bedroom and F had asked that morning why the numbers in a diagonal line were coloured differently to the rest. M straightaway answered because they were in a pattern of 1 x 1, 2 x 2, 3 x 3... So we went downstairs and I wrote these out to 5 x 5 on the whiteboard, with the answers, and an array of dots for each one to explain why they were called square numbers.

Fast forward to these blocks again - so several hours later - I asked the twins to use them to make as many square numbers as they could while I put A down for her nap.

MashaAllah they made these (and were also able to tell me they had used 91 blocks because there were 9 left over!) so I thought I'd quickly recap area while here...

E.g. how many cubes are in the square for 6 x 6? 36. So if I had a field and one side of the field was 6m, what would the area of the field be? They quickly answered 36 (and I filled in the m² units for them - I'm not interested in units at the moment so didn't want to dwell on them at this point). I asked similar questions for other square numbers in the context of word problems such as areas of walls, books, rooms, boxes, etc. Then to be sneaky I threw in a question on perimeter. They both fell for it. 😂 But when I repeated the question emphasising the word perimeter they laughed at themselves and were able to give the correct answer. MashaAllah. 😁😁

----------

While we're on the subject of maths, I'm going to put this here as a nice reminder of how the twins have different personalities and ways of thinking... Some of their answers to a worksheet they did the next day...

F, trying her hardest, but sometimes not seeing the wood for the trees:

M, being calm, laid-back and not making life difficult for herself:

😂👍

Maths Update: Times Tables (2x, 5x, 10x)

We've been focusing on multiplication and the times tables facts for the past couple of weeks. The NC for Year 1 introduces the 10x, 5x and 2x tables, which the girls are familiar with now, and the NC for Year 2 mentions them again - so I wanted to concentrate on quicker recall to help with their mental maths.

We began by recapping multiplication theory i.e. What does 2 x 5 mean? It means counting the number 2 five times. This is the same as 5 x 2, counting the number 5 two times - all demonstrated physically with cubes first then as dots on the whiteboard.

Over the week they then did a mixture of worksheets, oral questions, online games (we've signed up to education.com as someone organised a cheap HE lifetime deal!) and simply chanting both as facts (2 x 1 is 2, 2 x 2 is 4, etc) and as sequences (2, 4, 6, 8, etc) forwards and backwards as and when fit with our schedule - to give them plenty of practise and repetition in a variety of ways.

These were mainly to do with the 2x and 5x tables since, after a quick verbal quiz, they showed they understood the pattern for multiplying by 10, i.e. the number becomes 1 place value bigger - an easy trick being to put a 0 at the end (language is important here! I made sure not to say "add a zero", since mathematically adding 0 to a number means it stays the same! And I emphasised it's the place value which changes, which is an important building block for when they move onto decimal numbers in the future inshaAllah). They're able to recall/work out the 10x table up to 10 x 12, as well as calculate random whole numbers x 10, e.g. 24 x 10 = 240, 523 x 10 = 5230, etc. at a good speed, mashaAllah.

I printed out a multiplication square each for them each, up to 12 x 12, which I'm planning for them to keep until they memorise all their times tables... So a few years inshaAllah! I glued the square into a piece of coloured card, to help it last longer, and so there's also space to keep a log of which times tables are tested when. When they're able to quickly recall a multiplication fact, they can colour the corresponding square in - so the goal is to have the whole sheet coloured in. By quickly, the aim is for them to have memorised the fact rather than count it out each time, so I guess under 3 seconds? Instantly, ideally! But they're still young so inshaAllah with more repetition over the years, the speed of recall will only increase. (The girls asked me if I knew my times tables to which I said yes... When did I start learning them? Probably at around their age and I remember having a sheet like them at school to colour in too! So how long did it take for me to finish colouring them all in? I suppose around Year 3 or 4? So around 7 or 8? To which they both laughed because why did it take me 3 years just to learn this tiny square of numbers? F then said she'd definitely finish it all before she was 7. Lol. 😂)

You can download the multiplication square we used here, though a quick Google search will probably turn up many others!

As a craft activity to visually represent the multiplication families, we used pony beads to make "bracelets" for the 2x, 3x, 4x and 5x tables. So for the 2x table, they needed to use 2 beads of the same colour together, then 2 beads of a different colour, then 2 beads of a different colour, etc. etc. until they had a chain of 12 sets of 2 beads together. Then I helped them tie the elastic off into a bracelet. For the 3x table they used 3 beads of the same colour together, for the 4x table 4 beads and for the 5x table 5. So they could see visually the difference in size between 12 x 2, 3, 4 and 5 and I showed them how they could use the relevant beads as a kind of number line to help them count the answer to e.g. 3 x 6 or 7 x 5 etc. The activity itself was good fine motor practise as well as a lesson in concentration! M made the 2x, 3x and 4x bracelets in the same amount of time as F took to do the 5x as she kept getting distracted. 😅

We also played some card games to help practise faster recall, similar to the card games we made up for learning the number bonds to 10: Snap, where instead of saying "Snap!" for a matching suit they needed to say the answer to the multiplication fact for the card on top (e.g. if we were practicing the 2x table and the 10 of clubs was played followed by the 4 of clubs, the first person to shout the answer to 2 x 4 and touch the cards would win them), and also just a speed competition whereby I would shuffle the cards then hold up the one on top - the first person to say the answer (for whichever times table we were practising, e.g. if we were doing the 5x table and I held up the 6 of diamonds, the first one to shout "30!") would win the card, the winner being the person with the most cards at the end.

I made a quick worksheet of word problems, based on the 2x, 5x and 10x tables, set in Chuggington - as this is the current favourite TV show for the girls! Using their interests is an easy way to get them excited and keep their focus for a task. 😂

You can download the worksheet here.

Finally, I linked it all to division as the inverse of multiplication, ie 2 x 3 = 6 so 6 ÷2 = 3... demonstrated with physical objects first before using drawings and lastly just number sentences. With this knowledge they were then able to quickly answer missing number problems for multiplication (eg 5 x ? = 35) as well as division problems (18 ÷ 2 = ? 60 ÷ ? = 6) while practising their 2x, 5x and 10x tables at the same time.

Understanding the clock face

We've covered telling the time to the hour and half past the hour on both analogue and digital clocks. Before moving on to quarter past/to and the intervals between, I wanted the girls to understand and be familiar with the fact each of the numbers 1-12 on the clock represents 5 minutes/seconds. Rather than simply tell them this, I thought it would stick better if they worked it out themselves!

They already knew there are 60 minutes in 1 hour from becoming familiar with digital clocks in our everyday life (e.g. the clock on the oven which we've been asking them to read since they were 3!) so I began by asking: If there are 60 minutes in an hour and it takes 1 hour for the minute hand to go all the way round the clock, how many minutes fit into each of those big numbers 1-12? Sounds like a mouthful, but not so complicated illustrated on a clock at the same time. 😂

They understood they needed to share the 60 minutes between the 12 numbers, so it was division, but they couldn't work it out in their heads! I asked them if they wanted to use the blocks to help and they said yes. 😏

I took out the numbers 1-12 from our Melissa & Doug jigsaw and asked the girls to arrange them to look like a clock... So they knew they needed to put them in a circle but it was a challenge to get it looking right lol. They started out with a triangle, then one commented the 6 needed to be in a straight line under the 12, and eventually they ended up with a wonky oval.

Next, I gave them the tub of multilink cubes and asked how many did they need to have one for each minute? 60... And what did they know about their cubes and the different colours? There are 10 of each colour... So how many colours did they need to use? 6! There are two of them, so how many colours did they need to get each? 3! MashaAllah. They put the 60 cubes in the centre of their clock.

Now for the sharing part... If they put the same amount of cubes on each number, how many cubes would each number end up with? I prompted them by saying to first try giving 1 cube to each number.

They then continued sharing out the cubes so they all had 2, then 3, then I stopped them and asked them to estimate how many more times they thought they'd be able to go round the clock... One guessed 8, the other 11. 😂 Ok. So I let them go round again then stopped them again to see if they wanted to change their estimate. They both laughed and one said she thought only one more time after this so she should have estimated 2 to begin with - then the other agreed and exclaimed she thought they were all going to end up with 5 cubes each. 👍

We equated this back to the original question: if there are 60 minutes in an hour, how many minutes are there in between each number on the clock? 5! Then we counted in 5s round the clockface to check.

Finally, I told them to look at the real clock so they could count how many seconds there are in a minute - count how many times the second moves for it to go all the way round the clock, using 12 as a starting point. I asked them if there are 60 seconds just as there are 60 minutes, how many seconds do they think are in between each number? 5! Then we counted 5 seconds between two of the numbers to check.

Later on in the day, after a break, they completed the above worksheet to consolidate what they'd learnt. Taken from both *** and *** [edit to add titles]

After a few days, one of the twins asked if it was five past six because the big hand was on the 1... I said yes. Throughout the day she kept asking if it was so-and-so past, getting it right; I asked how she knew and she said because she was counting in 5s for each number! Masha'Allah. Within the next couple of days her sister had caught on too - so now they can both tell the time to the hour, half past and x minutes past the hour. 💪

More work with 10s

Just another self-reference post to show progress...

We did some more work on 2-digit numbers and adding/subtracting 10s. One of the girls found this a little more difficult than the other (the other was fine with an explanation then onto the worksheet) - I needed to model it for her a little more both with the multilink cubes, the abacus and writing colour coded examples on the whiteboard - so she could see visually that when adding/subtracting by 10 only the Tens are affected and the Units don't change at all. (Letts Make It Easy... Maths & English (Age 5-6) & Collins Easy Learning - Mental Maths (Ages 5-7))

We also did some work on the 10x table, by which I gave them the worksheets (Schofield&Sims - KS1 Problem Solving 2 & Letts Make It Easy... Maths & English (Age 5-6)) and the multilink cubes sorted into sticks of 10, read through the first question together as an example then left them to it... MashaAllah they were finished both sides within 10 minutes and without using the cubes at all! One of them laughed at me that I thought it would take them longer and they both laughed at me for taking the cubes out. 🙈 I tested them orally to check understanding and yes, they had either memorised or could quickly calculate their 10x table up to 10 x 10. 👌 Out of curiosity, I asked them what was 11 x 10? They weren't sure until I wrote it down for them to see, at which one of them got it straightaway. So I continued, then what was 20 x 10? That same twin understood it was 200 and 30 x 10 was 300 and so on; the other didn't get it - but that was just extra for my own curiosity. Not really something expected of them at this age! MashaAllah, I'm still super proud of them both. 💕

Counting in 2s, 5s and 10s (and 3s!)

The NC for Year 1 lists counting on or back in 2s, 5s and 10s - the next step after counting confidently in 1s. So I thought I'd challenge the girls and see how they coped with the idea.

As they can count up to 100, I started with counting in 10s as it seemed the simplest. The twins are both logical and visual in their learning, so it was simple to just write the numbers 1-10 down the whiteboard and then next to each number write the multiple of 10. After the first few, the pattern became obvious and they were able to tell me what needed to go next in order to get to 100 in 10s. They found it amusing how simple it was since I began by telling them they were going to do something really difficult by counting in 10s! I think the humour helped it to stick in their heads too. 😊 We did a few chanting games altogether, alternating filling in what went next orally by going round our circle of 3, and then they pretty much had counting in 10s sorted.

Another day, we looked at counting in 2s. I introduced the concept using their multilink cubes, so they had 10 cubes each which they counted and then had to group into 2s. Then we counted the cubes together in 2s and confirmed there were still 10. We did this with a few different toys over the next couple of days, e.g. small animals, Duplo, etc., increasing the total to 20 objects. One of them commented counting in 2s was just missing out a number, which I thought was a good observation!

That comment prompted me to print them out a couple of 100 Squares so they could see the pattern visually, as I thought it might help cement the idea in their heads: 

It's pretty self-explanatory what we did. 😛 I needed to supervise just to keep them on track but they managed the task ok. After the first few lines of 2s they noticed the pattern was just lines going down the page, so I prompted them to predict what the next line would look like. After about half the square, they were sure they just wanted to colour straight lines to fill in the rest, so I let them. 😂 Then we counted through the whole square in 2s, using their finger on the squares and making sure they counted one then missed one, just to check it was all correct. Then they did the 10s, which was quick and easy, and again we counted through them just to check. We also pointed out that all numbers end in 0 when counting in 10s.

After a break of a week or so, we repeated the process with 5s over a few days: doing it physically by grouping objects and pictures, chanting and singing and playing games verbally, then colouring in the pattern on a 100 square. This was helpful again in seeing all the numbers ended with 5 or 0 and alternated between 5 and 0.

Because I'd printed the sheets with 2 squares on them, we needed something to do on the bottom 100 square - so I asked them what they wanted to try counting in... just for fun. Thankfully, they said 3s! They needed to concentrate a bit more for this as we hadn't done any previous work on 3s. It was a little more challenging at first, but they soon got the hang of counting one then missing two so it was good concentration practise to get to 99! They were also happy to see the pattern at around the halfway mark and were able to use it to correct themselves if they made a mistake. 👍 No expectation for them to count in 3s at this stage! But I thought the activity was a good learning experience in stretching the mathematical side of their brains. 😁