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. 😂💕

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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. 😁😁

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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: Adding/Subtracting 2-digit numbers

After revising partitioning, we recapped how to add 2-digit numbers mentally, i.e. partition into Tens and Units, add the Tens, add the Units, add both parts together. They also had the option of adding the Tens from one number to the other, then counting up the remaining Units, e.g.

28 + 43 =

20 + 40 = 60
8 + 3 = 11

60 + 11 =

60 + 10 = 70
0 + 1 = 1

70 + 1 = 71 so 28 + 43 = 71

OR

28 + 40 = 68
68 + 3 = 71 so 28 + 43 = 71

We concentrated on the first method first, to get them used to partitioning the numbers. Then we moved onto the second method as something more efficient.

After the mental practise, I introduced the column method to them as an efficient way of writing their working out down. It's the same principle, partitioning and adding each place value one at a time, but written in a compact way, e.g.

16 + 43 = 

   16
+ 43
   59

We started with numbers which didn't carry over into the next column, then once they were comfortable with the method I explained how to write the extra Tens underneath the Tens column - and to make sure they included them in their answer! e.g.

45 + 36 =

    45
 + 36
    81
    1

We also spoke about using a quick mental method to check their answers.

We did all the above through explanation and examples on the whiteboard first, then answering questions from a photocopied worksheet using the method we were focusing on at the time.

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Next, we recapped subtraction of 2-digit numbers via partitioning and counting down, e.g.

38 - 14 =

38 - 10 = 28
28 - 4 = 24 so 38 - 14 = 24

Where the numbers stayed within the Tens, then using numbers in the Unit column which meant counting down across Tens, e.g.

52 - 26 = 

52 - 20 = 32
32 - 6 = 26 so 52 - 26 = 26

After some practise doing this mentally, I then introduced the column method to them - but for numbers which wouldn't require carrying, e.g.

88 - 43 =

    88
  - 43
    45

I decided that was enough for now, so they could get used to the method, and we would look at carrying from the Tens column at a later point inshaAllah!

The above was all done, one method at a time on separate days, first on the whiteboard and then in the context of numbers by working through practise pages from one of the workbooks we have.

Finally, I gave them some word problems to work through so they could use these skills in the context of money. They had the choice to use either a mental or a written method, whichever they found easier.

Adding two 2-digit numbers

The prerequisites for this activity were to be able to add together multiples of 10 in their head and know how to count on using a 100 square.

I decided to focus on mental methods for adding 2-digit numbers as I feel it's important they understand the basics before moving on to the quicker (and arguably easier) standard written methods.

The method I taught them to use is pictured above, which I went through on the whiteboard using coloured pens for simplicity. First, I wrote out an example question with the Tens in red, the Units in purple and the equation signs (and subsequent answer) in blue: 14 + 48 =

I told them the first step was to look at the Tens. What are the Tens? 10 and 40. So what is 10 + 40? 50. I wrote this sum underneath in red, to keep the colour co-ordination.

Next, look at the Units. What are they? 4 and 8. Which Unit is bigger? The 8.

So keep the Tens total (50) and add on the bigger Unit (8) to get 58. I wrote this underneath, colour co-ordinated.

How many Units are left to add on? 4. So I wrote 58 + 4 =

They could then use their 100 squares to find 58 and count up 4 more squares to calculate the total. 62. So I wrote 62 as the answer to both 58 + 4 and the overall question of 14 + 48.

Reading through, this looks like it took a long time to explain! But it really didn't, and with a few more examples and me reminding them of the steps the girls soon got the hang of this method. This is my personal method of choice for mental addition, which is the reason I taught it to them. As they get older and their mathematical understanding strengthens, insha'Allah, then I'll introduce the alternative methods for them to choose from. 😊

After two examples using numbers of my choice (one where the Units total to within 10, another where they didn't), I asked the twins to choose numbers at random for me to use in the questions - otherwise I wouldn't have used 49 + 94 as an example since it goes into the Hundreds! But Alhamdulillah, this was the last example we did together and they understood it fine. 👍

The next day - while leaving the examples on the whiteboard for reference - I gave them the pictured double-sided worksheet to do (taken from Schofield&Sims - KS1 Problem Solving 2). We did the first side (5 questions) together, with me scaffolding their thinking by reminding them of the steps to the method, and I helped with the first couple on the next page too. Then I let them try the last few by themselves and they were both able to do the final two questions completely independently, MashaAllah. 💕

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. 💕

Adding Multiples of 10

We spent the previous week or so recapping and strengthening their addition/subtraction skills by way of worksheets, number lines and mental maths - but always questions with 1-digit numbers (either +/- 1-digit, or +/- 2-digits).

This week, I want to focus on questions involving 2-digit numbers.

We began by getting the multilink cubes out and sorting them by colour into lines of 10. We did some simple addition sums using each stick to represent 1, e.g. 2 + 4 = 6. Then I asked them how many cubes were in each stick? 10. So what we actually did was 20 + 40 = 60. (The girls are already confident counting in 10s, which I think is a prerequisite for this activity). As we did each example in multiples of 10, I wrote the sum on the whiteboard using different colours to highlight the 0 as a placeholder.

Once they seemed like they had the hang of it, I wrote a question on the board without the colour coding and asked them to use the Multilink sticks to help them calculate the answer. MashaAllah, this whole bit of teaching took them less than 10 minutes to grasp.

Finally, I gave them the above two pages to complete. I sat with them as they did the first sheet (CGP Key Stage One Maths - The Question Book) then left them to do the top half of the second sheet (Letts Make It Easy... Maths & English (Age 5-6)) individually. For the second half of the sheet, I guided them in how to answer the first question, showing their working out, then let them answer the rest themselves.

Number Bonds to 20

Building upon our work on learning the number bonds to 10 (posted about here and here) and their understanding of partitioning, we then expanded this knowledge to calculate and so memorise the number bonds to 20 - with the overall aim of improving their mental maths skills.

I began by giving them a quick oral quiz on which numbers "matched" with each other to give a total of 10, i.e. "What do you add to 2 to make 10?" We then revised what they knew about partitioning numbers into tens and units by going through a couple of examples together on the whiteboard - stopping on the number 20.

I then explained to them how they could use the number bonds to 10 to calculate what number they needed to make a total of 20, because 20 is simply 10 more than 10! I gave them a couple of examples on the whiteboard (e.g. see picture above) then quizzed them to check their understanding.

Finally, I gave them some pages photocopied from Letts Make It Easy... Maths & English (Age 5-6) and Letts Monster Maths - Maths (Age 5-6) to recap and reinforce.

The bottom of the first page (left) was a good opportunity to teach them the importance of showing their working out, in terms of keeping track of what they're doing and working things out step by step, in order to pick up on where they might have made a mistake.

Word Problems with the PJ Masks!

My girls are all into PJ Masks at the moment... It comes on TV on TinyPop or you can find episodes pretty easily on YouTube (you can even find some dubbed in Arabic!). So I decided to utilise this by making up some simple word problems using the characters from the show.

Each sheet has 5 questions on it (a manageable amount for their age, I think). The first and second sheet have 1-step problems, the first focusing on addition and the second on subtraction. The third sheet has 2-step problems with a mixture of addition and subtraction.

We worked through them using cubes to physically represent the numbers. You could do it with a number line or whichever method you prefer though. We also wrote down each question as a number sentence (e.g. 5 + 2 =) before writing the answer, so they could get used to seeing how word problems translate into numerical problems. And we didn't do all the sheets in one go! We spread them out over a few days in between standard addition/subtraction practise just using numbers.

You can download the worksheets here. As always, feel free to edit as you wish - so you can change the numbers accordingly to make them more or less challenging. 😀

Number Bonds to 10 (Part 2)

Following on from their mini maths investigation into which two numbers can add together to make 10, I made up a few simple games to further practise these bonds using a pack of standard playing cards.

For all of these, the first step is to take out the 10s, Jacks, Queens, Kings and Jokers. Ace always represents 1. When I say "the matching card", what I mean is the one with the number which added together would make 10, e.g. A matches with 9, 2 with 8, 3 with 7, etc. etc.


1) A variation of "Snap!" [2 players (& dealer)]

I was the dealer and had all the clubs, A - 9. The girls had one red suit each, i.e. diamonds A - 9 and hearts A - 9. They had their cards face up in front of them on the table, put in numerical order to make it simpler (they could have held them in their hands, but as we were using a standard pack of full sized cards they were a little tricky for them to manipulate so I decided this way would be easier). I had my cards shuffled into a random order. The choice in suits was to differentiate between the dealer's cards and the players' cards.

The game was simple: I would place a card down in the middle; the first person to put the matching card on top and shout "Snap!" would win the card. Whoever had the most black cards at the end was the winner.

If you're only playing with one child this is easily adapted into a game the parent can play in as the second player by having the black cards shuffled and face down in the centre. Each player can then take turns to flip one card over into the middle.


2) Beat the Clock [1 player]

You only need two suits for this, one black and one red. I chose clubs and hearts. 😊 You also need a stopwatch.

Place the black cards face up on the table. We started with them in numerical order to make it easier.

Shuffle the red cards and hold them face down in one hand.

Start the clock! Flip one card in your hand over at a time and place it on top of the matching card on the table. The aim is to do this as fast as possible, then make a record of your time at the end.

The first time we did this, the girls both averaged 2 and a half minutes! They took their time flipping the card over and thinking about how many more were needed to make 10, sometimes checking on their fingers before placing it down... They didn't really have a concept of time, but I let them be slow as I wanted them to see themselves improve the next time. 😜 After they each had a go I modelled how to do it as fast as possible, whilst still pausing to say out loud what I was doing: "First card, 4; 4 plus 6 is 10; 4 goes on top of the 6. 2; 2 plus 8 is 10; 2 goes here," etc.) I wrote all our times on our whiteboard and said we'd try this again tomorrow to see if we could improve. And they did! We've been playing this game daily for a few days now and they're now averaging 40 seconds to complete it (my best so far is 10 seconds, so you have some idea of expectations!).

We play by having the twin who isn't using the cards to be in charge of the stopwatch. This has also definitely helped their understanding of time and how long 1 second actually is!

You can make the game a little more challenging by having the cards on the table in a random order, or in a square kind of shape rather than a line, so the kids can practise their skim reading skills at the same time. Depending on your child, they might notice that when the cards on the table are in numerical order then the cards they place on top are simply in backwards numerical order - so this helps stop them "cheating" by using this fact (although if they notice this fact themselves, well done to them as this can be a teaching point in itself!).


3) Flashcards [1+ players & dealer]

This is similar to the first game, but removes a little of the competitive element if you want to. 😛

Each player has one suit each. The players have their cards face up in front of them (in their hands or on the table) and in numerical order to make them easier to find (they don't need to be in order, of course, but this could be a good discussion point as to why they might want them that way and an introduction to strategic thinking!). The dealer has their cards shuffled into a random order.

The dealer holds up a card, one at a time, and the players need to find the matching card and hold it up. They can shout out the answer once they have the card, or they can stay quiet so it's not as clear who was faster! You can also choose to have the faster one take the dealer's card and compare who has the most at the end, or just congratulate them for getting the right answer and continue playing. It's up to you and what mood your kids are in at the time!

You could even play this with the dealer just holding up random cards and the kids shouting out the matching number as fast as they can, but I like the fine motor skills practise and skim reading practise the kids get from having their own suit of cards to physically find the answer with. Similarly, you could get the same effect from the dealer using something physical, e.g. 10 cubes in an opaque bag, instead of cards. The dealer takes out a random number of cubes and shows them to the children (dealer can count them out or leave the children to count them together/quietly to themselves for practise); the children have to work out how many cubes are left in the bag and hold up the card with the right number.


4) A variation of "Old Maid" [3 players+]

Put a card of your choice back into the pack to represent the Old Maid, e.g. one of the Queens or a Joker. Shuffle the cards and deal them equally to each player (I joined in with this game so we had three people playing - wouldn't really have been fun otherwise!).

First, each player looks at their cards privately and discards any pairs of matching cards. The goal is to get rid of all your cards and not be left with the Old Maid at the end. Alternatively, you could play this without the Old Maid (i.e. Queen/Joker) and just see which player is the first to get rid of all their cards, as this makes the game shorter (the winner doesn't feel left out!) and you don't really have a loser (which avoids temper tantrums amongst some kids!).

Once the matching pairs have been discarded, each player holds their remaining cards in their hands facing them so the other players can't see what they have (ideally you would fan them out in one hand, but this was a little tricky for my 4 year olds so I just let them hold them however).

You then take it in turns to go round the circle and take one card from the person next to you. If you get a matching pair, you discard it. Keep going until a) someone discards all their cards and is the winner or b) someone is left with the Old Maid at the end and is the loser. We play the 2nd version using the Joker - whoever is left with the Joker at the end needs to tell a joke/do something funny to try and make the others laugh. 😂


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That's all for now! If I think of or find any more games, I'll update this post with the summary. 😊

Number Bonds to 10 (Part 1)

We've covered basic addition using the standard methods (blocks, fingers, number lines, etc.) and the twins understand the principles, but I wanted their mental arithmetic to be quicker... So decided now was a good time to start getting them familiar with the number bonds to 10, i.e. which pairs of numbers add together to make the number 10.

I gave them each a piece of A4 paper with 10 squares (each the size of a Multilink Cube) down one side, a pencil, 10 cubes of one colour and 10 cubes of another. They had to see how many different ways of making 10 they could find, the two colours being to make the process visually easier.

Both of them decided to start with 5 of each cube and I modelled how to write the number sentence 5 + 5 = 10 on the whiteboard for them to copy onto their paper: handwriting practise as well as a way of keeping a record of their findings along with reinforcement of how sums are displayed. 😄

Now came the tricky part! I removed all the blocks of the second colour and modelled how they didn't have to have 5 of the first colour. Maybe there could be one less, or a couple more, or even none! So I had removed all the blocks completely. So what did they want to try next? I tried to keep it quite open so they were doing more exploring and thinking for themselves.

One decided to make the first colour one more, and ended up doing the task quite systematically. She found all 11 combinations quicker than her sister and it was easy to explain to her how she hadn't missed any out. The other took a more random approach, but she was determined and I could see she was really thinking about which combinations she hadn't tried yet. When she got to her 8th possibility she wasn't sure if she could find any more, so I read through her work in numerical order, using the first number in the sum as a reference, and purposefully paused at the possibilities she missed - she caught on quickly what she needed to try next and that she had 3 more sums to write.

With this activity, it was also nice to see that they remembered some of the addition work we'd done previously, as even before they'd filled in the squares with the second colour cube, they would already guess (most of the time correctly!) at how many more cubes were needed.

Because they worked through finding out what the number bonds were for themselves, rather than just being told them or being made to memorise them parrot-fashion from flashcards, I think it made the pairs stick in their heads more easily. After a break, we practised some mental arithmetic again and they were both noticeably quicker in finding the answers. 😊

Addition Practise

This isn't a very inspirational post - more a personal reference for myself to see what we were doing at this point in the year!

Some independent addition practise using multilink blocks as counters, keeping within number bonds to 10. Using the totalling up method for the first half and introducing the idea of counting on for the second.

I let them choose which color paper they wanted their questions on, then Papa marked their work. 😂