Problem Solving - where do I start?

Problem Solving - is this a phrase that you have heard many times in conjunction with your childs maths work but are actually unsure what it means?

I would imagine that you are not alone. In the good old days, where teaching was more traditional this phrase would generally mean giving children some pre-written word problems that were linked to some of the work they were doing. These were very staged and mostly unrealistic. However, these days problem solving means something entirely different and requires and a very different way of looking at maths.

Over the past few years maths teaching has developed and changed and there is now a strong belief of teaching for understanding. Basically we really want your children to understand what they are learning and how it works, rather than just to know a formula and how to put the numbers through it. Part of this teaching for understanding is problem solving. These days we want to teach children through problems and discussion. By doing this your child has to think about what they know and apply this to the problem and can often extend their own learning and what they know by working through a problem. Initially, this can be quite a difficult thing to understand and get our heads around, as parents. But once you start to get into it this can be great fun and a learning curve for all. The great thing about problem solving is it can be a learning adventure for everyone at any level.

There are a wide range of methods for using problem solving to teach. Children can solve collaboratively or independently. New ideas can be taught or old ones consolidated. Problems can extends a child's knowledge of a particular concept or just give them the vision to understand what they thought they knew.

I thought that today I would introduce problem solving to you as a parent  / tutor and start showing you how this can be used to help your children.

As I mentioned above, problem solving is very open ended and, I believe, the part that we, older people, struggle with the most is the fact there are no answers! We were brought up on the concept of maths being right and wrong. Problem solving is all about exploring and experimenting with numbers, a very different way of looking at things.

So, here is today's activity.

Explore the numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0.

This is an incredibly open ended problem which makes it a fab problem as there is so much that can be done with it but also one where lots of encouragement could be needed as it's quite easy to be lazy on  this problem.

When it comes to doing this problem with your child you may be lucky and have a child that is used to this kind of thing and will just get stuck in and play with those numbers. I would expect though, that most of you will have children that will look at you blankly and not know how to start. Below I have given you some question that will hopefully get the problem started and then just go from there. You will need paper and pens for all the jottings you might do.

When you introduce this problem keep it very simple and try not to give too much info and see what your child does. If they are not sure how to start, use some of the questions to help. The idea behind the questions is that they will spark other ideas and thoughts and hopefully you and your child, together, will  start to explore the numbers without the aide of my questions. The questions are there purely to get you going enough to get the ideas and conversation flowing with these numbers. I have only given questions and not broken this down into levels this week as it is so open ended. Go with the flow, start with the very simple questions whatever level your child and just keep nudging the conversation deeper and deeper depending on what level your child is. I have added in some more difficult question to really push that thinking if some children are struggling to do this on their own.

Number
What's the biggest number you can make using all of these digits?
What's the smallest number you can make using these digits?

Counting
Can you count with them?
Can you count forwards? backwards?
Can you count every other number? What are those numbers? What about the ones you missed out?
Can you join one of the numbers to each of the other numbers and make a sequence? What sequence did you make? Can you count further than you made?

Adding
Can you add any of the numbers together?
Can you add the first number to each of the other numbers? What happens to them? What about if you add the second number to each of the others? the third? etc.?
Can you add all the numbers together?
What happens if you add 1, 2 and 3 together? Now add 2, 3 and 4, what do you get? What about 3, 4 and 5? Continue.........look at all the answers.
Can you add any of these numbers together to make 10? Do any of the other numbers add together to make 10? How many different ways can you make 10? What about doing the same for 11, 12 etc.

Subtracting
Can you start at 9 and take each of the the numbers before it away? eg 9 - 8 - 7 -  6.......... (you may get 'no you can't' or you may get 'yes' and be able to start exploring the negative numbers they go into - particularly good with 9, 10 and 11 year olds)
Which numbers can you pair up so that you can take one away from the other and stay with positive numbers?
Can you take more than one of the numbers away from any of the numbers? What's the highest numbers of these numbers that you can take away from one of these numbers?

Multiplying
Can you multiply any of these numbers together?
What happens if you multiply the numbers all by the last number: zero?
Can you multiply more than 2 of these numbers together?
Can you multiply all these numbers together?
What happens if you multiply all the numbers by one of the numbers? Now do this again with another of the numbers?
Can you multiply the  numbers together in groups of three? Are there any patterns? What about if you multiply them together in groups of 4?

Dividing
Can you divide any of these numbers by one of the other numbers? Which ones? Did you develop a system to help you work this out methodically? If not, would this help?
Can you find a number that all of these numbers divide by? Can you find a number that none of these numbers divide by?
Can you put all the numbers together except one. The ones that are together should make a number that divide by the one not included (eg. 123467890 divided by 5)

Playing
Can you find another way to explore / play with these numbers?

Have fun and enjoy this weeks problem. I would love to hear about some of the discussions you have with this problem and how far into number exploration it took you.

Number Bonds

Number bonds: a phrase that can bring confusion to many! What are they? How do they work? How can we help our children with them?

From the time your child starts school you will hear the phrase 'number bonds', possibly from your child but more often from their teacher and in information sent home from school. Surprisingly this information often works on the assumption that we all know what number bonds are! Sadly we don't.

Number bonds are the addition pairs that make a particular number so, for example, if you were looking at the number bonds to 10 you would be looking for all the pairs of numbers that add up to 10:

0 + 10
1 + 9
2 + 8
3 + 7
4 + 6
5 + 5
6 + 4
7 + 3
8 + 2
9 + 1
10 + 0

The most frequently looked at sets of number bonds are the bonds to 10 and 20. Once children have understood and discovered these ones it is assumed that they understand how these patterns work and should be able to work the rest out as and when needed. However, there have been many times when teaching the older primary children when I have brought up the subject of number bonds and I get blank looks from students. This is mainly because number bonds are covered in Infants ( Kindergarten and 1st Grade) and often not revisited as there is so much more curriculum to cover in the older years. As a maths specialist, I feel they should be revised at least once a year, in the later years. The knowledge of these patterns and how to use them in all aspects of number is a brilliant skill to have and many children lack it. You will often find that the first ti e children do this, they write down the sums randomly and it's only when looking at them that they start to understand the patterns involved - the ones that are obvious to us!

As a parent/guardian/ tutor this is one area you can really help your children. It's such a fun topic to teach and can bring amazing rewards, not least your child's confidence with numbers.

So, this week, play with the number bonds and have some fun looking at the patterns and how they link across our number system.

Preschool

What: number bonds to any number up to and including 10 if they're ready. Start with 3 and move on from there.

How: start by using some blocks (or something similar that will do the same job: pencil crayons, cutlery, apples.........) and ask your child to find 3. Once they've done that hold up one block and ask them to find the other blocks you need to make 3. Once they have done this say the sum '1 block and 2 blocks makes 3 blocks'. After making sure your child has understood this and you've repeated it several times then do the same but with 2 + 1. Finally show the two different sums next to each other using the blocks and encourage your child to talk about what he / she sees. He / she might repeat the sums and tell you what each number is. Hopefully the will also notice that the sums are the same but written the opposite way round.

If three is easy then your next step is to progress up the numbers and do something similar to above. I wouldn't do much more than one number and all it's bonds in a session, if you're going to move forward to another number then do it the next day. If you want to write the sums down you can do but it's not essential. The important thing is starting to recognise the bonds for each number and to see the patterns with how they work.

5, 6 and 7 year olds

What: number bonds to 10, 20, some of them may be able to extend this to 30 and 40. Another extension is to do the whole tens number bonds to 100 (10 + 90, 20 + 80.......)

How: start by using some blocks (or something similar that will do the same job: pencil crayons, cutlery, apples..........) and ask your child to find 10. Once they've done that hold up one block and ask them to find the other blocks you need to make 10. Once they have done this say the sum '1 block and 9 blocks makes 10 blocks'. After making sure your child has understood this and you've repeated it several times then do the same but with 2 + 8. Continue doing this until you have all the sums. If you have enough equipment it would be great to have all the sums laid out on the floor / table. If you can't do this then as each one was 'discovered' write it down (or get them to) so they're all written down together. Then encourage your child to talk about what he / she sees. He / she might repeat the sums and tell you what each number is. They need to start to notice that the each number is each sum is one less / more than the previous and some of the sums are the same but the opposite way around. Once you've had a discussion I would finish for the day and let the info sink in. As a follow up activity next day ask your child if they can write all the number bonds to ten. If they can, great. If not, then just briefly go over some of the areas they may find hard. The most common problem in repeating this is children know there's a pattern but can't remember what it is. If they need a nudge then write the first two sums for them and see if they can continue.

If ten is achieved then your next step is to progress up the numbers with number bonds to 20, 30 and higher if you're both enjoying it and it's understood. I wouldn't do much more than one number and all it's bonds in a session, if you're going to move forward to another number then do it the next day.

7, 8 and 9 year olds

What: number bonds to 10, 20, 30. Then extend using the 10's and 100's.(10 + 90, 20 + 80..........., 100 + 900, 200 + 800....., 1000 + 9000, 2000 + 8000.........) You may find that you can also start looking at all the number bonds to 100 too.

How: If you're starting with the number bonds to 10, which you probably should as revision, then follow what I have written for the 5,6 and 7 year olds.
Once you've covered the basic number bonds  to 10, 20 and 30 etc. then you need to move onto the bigger numbers. Initially there is little alternative but to do it with paper and pencil. I try to make this more exciting by either using a white board and pen, or a huge sheet of paper and coloured pens or lots of sheets of paper; one for each sum and then put together in order...........you know what your child likes so use whichever of these works to make it more interesting for them.

By this point your child should have a good idea of what number bonds are and what the patterns are. What you are doing now is showing them how these bonds work across the whole number system and not just with the small numbers. So, maybe using one piece of paper get all the bonds to 10 written down, on the next sheet of paper get your child to write all the (10's) bonds to 100, then on the next sheet, the bonds to 1000 etc. Go as far as your childs knowledge of the number system will let you. Then look at all the sheets of sums and discuss the pattern and how it works.

The next step is to look at all the number bonds to 100. My favourite way of starting this is to give them 3 or 4 sums to work out and then discuss how they're working it out and if there is an easier way. So, ask your child to find what number they need to add to 25 to make 100. Then try 52 and then try 93. While this is happening just watch and see how your child does this. I suspect there will be a lot of finger use! Once they have done these three sums talk to them about how they did them, if they found it difficult or if they found them easy. If they found them difficult you need to show them there is an easy way. Ask them to look at each pair of numbers and for each pair add the units numbers together and then add the tens numbers together. What they should discover is that the units add up to 10 and the tens add up to 90. The next step is to try and encourage your child to explain why this is and then show them how they can use this knowledge to work out the number bonds easily. eg. 45 + ? = 100
So, I need to add something to my units number to make 10 and something to my tens number to make 90.
If I add 5 to the units number I have 10 and if I add 50 to my tens number I have 90. So I need to add 55. Check the sum works :)
Initially your child may need some paper to record these numbers on but as you practice then, hopefully, he / she will be able to do it in their heads. A good way to practice is to every so often just ask 2 or 3 sums or set a nightly challenge of 3 sums one night, 4 the next etc. You could time them and set the challenge of improving their time each night.

9, 10 and 11 year olds

What: revision of all number bonds mentioned above. Then moving forward with bigger numbers 10000 + 90000, 20000 + 80000..........., 100000 + 900000, 200000 + 80000.........and then number bonds to 1000.

How: With these children I suspect you will find that a lot of them don't really know the number bonds but with some quick revision they will soon pick up on the patterns. Use the info above and look at all the multiple of 10, 100 and 1000 bonds, they are fun and quick and easy to do and you may be able to get up to and beyond 1000000.

Once you've had fun with the big numbers then revise how they work out the number bonds to 100. Again, you may find they can do it but have never really noticed the pattern mentioned above of the units making 10 and the tens numbers making 90. If they haven't found this, try and help them to discover it. Once these are secure move onto the number bonds to 1000. It's a great skill to be able to find pairs of numbers that make 1000 and, if you know the patterns it's not difficult. As with above, I would give them 3 sums to work out, so ask them to find the number that you add to 295 to make 1000, then 507 and then 989. Once they have done this write the sums down and discuss how they did it and if they can find any patterns that will help them.Hopefully they will notice that the units add to 10, the tens add to 90 and the hundreds add to 900.
eg. 368 + ? = 1000
So, I need to add something to my units number to make 10, something to my tens number to make 90 and something to my hundreds number to make 900. So 8 + 2 makes 10, 60 + 30 makes 90 and 300 + 600 makes 900. So the number I am adding is 632.
As you practice you could turn this into a little competition to try and increase the speed and efficiency of working this out. You could race each other or do sums against own speed or within a certain time.

Whatever you do, have fun :)

Holiday Maths

So, the children are on holiday from school and, as a parent, you are keen to keep at least some of what they learnt last year working in their brains. What to do?

Some parents like the idea of buying workbooks or printing out work sheets and every week sitting your child down and doing some work. As a teacher, my recommendation is not to do the above but to do what I would call incidental maths. Children are on holiday from school to have a break from work and to let everything they have learnt sink in and be processed. So, sitting them down and working through books doesn't give children the break they need. However, doing some incidental maths can keep those skills consolidating and give your child a break from the more formal work. It also teaches them that learning can be fun and takes the pressure off both of you having to sit and do school work when 'everyone else is having fun!!'

What is incidental maths? Incidental maths is maths that you can make happen as you go about your every day life. For example, you may be sitting at the dinner table and everyone has a drink. As you're eating dinner why don't you ask a few questions about the drinks? How much water do you think is in your glass? Why? So, if you put all our glasses together how much water do you think we'd have altogether? If I drank half my water, how much would I have left?

This is quite a difficult thing to describe as every child is at a different level and every child's day is different. As a parent it is your chance to really have some fun with the maths and just let it happen within the day, without it being a big thing. You will just have to grab the opportunities and make them work for you and your child.

Below I have described some of the more obvious scenarios that could happen to most of us to get you going and inspire you to do more. I have tended towards the higher level of student as sometimes it is harder to think of more challenging activities. I hope those of you with younger children can find ways to make the suggestions simpler. I would love to hear some of the things that you do end up doing as your incidental maths.

Number work

Traveling - It was this many miles from a to b and then we did this many miles, how many have we traveled in total? We are traveling at 60 mph and have 6 miles left to do, how long will it take us? The journey is x miles long, how long will it take us to complete traveling at 60mph?
Car registration plates - how many different numbers can you create with what you see. Write down the next 6 plates you see and use the digits to create the biggest number you can or the smallest. can you take the smallest away from the biggest? Or add all 6 numbers together, or find the smallest of the six and take it away from the biggest of the six.
Numbered lamp posts, bollards etc. - Have you seen anything on a local walk with numbers that you can add, take away, multiply, spot the pattern in?
House numbers - add them, take them away, swap the numbers around to make the biggest / smallest etc.

Measure / Capacity

Drinking - how much liquid have you drunk today, let's keep a record and add it up? How much is everyone drinking at the table? How much is left if two of us drink everything in our glasses? How much wine is Mummy drinking? What is the capacity of her glass?
Water Play - mine have a tea set they play with in the bath so more questions about how many cups would fill the teapot? or the pan? How many spoonfuls fill the cup? How many cups fill the bucket? How much water does it take to water the plants, how can you work it out?
Suitcases - how big is my suitcase? Does it follow the airlines regulations? If not, by how much is it out? Whose suitcase is the biggest? Heaviest? What makes it the heaviest?

Money

Foreign Currencies - what is the exchange rate? How does it work? So, how much will you get in the local currency if you exchange 10 pounds / dollars? When shopping encourage your child to spend their money and either just count it out or work out how much it is costing them in English / American money using the exchange rate. They could also add up how much you are spending as you go around the store, or they could add up by rounding up or down each item so they have an estimate of how much your shopping will cost. Alternatively you could give them 10 pounds / dollars and ask them to get the food for dinner with that money, adding up as they go as they can not over spend.

Shape work

Shape spotting - what shapes can you see? Do they fit together? Why do you think it is that particular shape?

Time work

Day to day - we are leaving in 10 minutes, so when the big hand gets to........ Lunch will be at 12:30, can you tell me when it is that time? If it takes 45 minutes to get to Denver and it's 9:30 now, what time will we get to Denver?
Calendar - Have a calendar displayed with the monthly activities on so you can discuss how many days until........,  how many weeks until........, when you are next going swimming.

Data Handling

Long journeys - give them a piece of paper and a pencil, can you record how many different coloured cars you can see in the next 5 minutes? How many different kinds of animals can you see in the next 30 minutes? Can you now sort what you have seen and tell me which animal was the most common, least common etc.?
Holiday - Record what we do each day on holiday and then create a table showing how many shops we went into, how many times we went swimming, how many museums we visited, how many restaurants we went to?

What's the time Mr Wolf?

The Faces Of The Clock

The Big Hand is busy

But the Small Hand has power.

The large one counts the minutes.

But the Little One names the hour.

When both Hands stand at the top together,

It's sure to be Twelve O'clock. But whether

That's twelve at noon or twelve at night

Depends on if it's dark or light.

Teaching time can, literally, feel as if you are going round and round in circles. Every child learns about time in their own time and not necessarily when you would expect. I have always found that you are far better to work at teaching time little and often than trying to achieve a lot in a small amount of time. If you gently plug away at this in the background children will eventually start to make sense of it all and then be able to use time within their every day lives.

This week I thought it would be fun to get crafty again. So, your first activity is to make a clock.


You will need:
A paper plate
Two long thin pieces of cardboard (cardstock)
A split pin (or a bent paperclip!)
Whatever you want to use for decoration: felt pens, crayons, sticky numbers, jewels, stickers.........

What you need to do:

• Start by writing or sticking on the numbers 1 - 12 around the outside edge of the back of your paper plate. 
• Using the two pieces of card make your big hand and little hand and stick them in the middle with the split pin. 
• You may want to add small lines between the numbers to show the minutes 
• Decorate! 

Once you have your clock you can then do some fun time telling activities. Start by doing a very brief review on telling the time at your child's level and then do some of the games I have added to the bottom of this article to practice that skill. Please, this is a little and often subject, so I would teach and do an activity for no more than 10 minutes and then choose another activity to do the following day for 10 minutes. Again, the next day do 10 minutes. Then leave it for a few days and come back to it. You will find the knowledge has either stayed put and you can move forward or there is a little confusion and you will need to do some more activities at the same level to consolidate.

Normally I differentiate by age. For time I am just going to list the skills that need to be taught starting with the easiest and ending on the hardest. This is because every child learns time at a different pace of learning and does not always fit within the typical parameters for this. I would suggest everyone starts at the beginning and works their way down the list of skills, gradually building them up. For those with children really needing to start at the beginning this is something that will happen slowly, please don't feel you need to be at the bottom of the list by the end of the summer!!

The last two skills are using knowledge of time rather than telling the time. When your child can confidently tell the time they then need to be able to use this knowledge to survive every day life - hence the final two skills.

1. o'clock - can your child confidently read the whole hours on the clock not in chronological order? eg. 1 o clock, 2 o clock, 3 o clock.......

2. Half past - can your child confidently read the clock when it is half past the hour? eg. half past 2 (2 thirty), half past 6 (6 thirty) etc

3. Quarter past and quarter to - can your child confidently read the clock when it is quarter past or quarter to the hour? eg. quarter past 4 (4 fifteen), quarter past 7 (7 fifteen), quarter to 6 (5 forty five), quarter to 10 (9 forty five)

4. 5 minute intervals - can your child confidently show how many minutes past each big number represents? eg. 1 means 5 minutes past, 2 means 10 minutes past, 3 means 15 minutes past - they can show this by counting round the clock using the hands on the clock to help. (You may want to mark the minutes on your clock at this point)

5. Minutes and hours - can your child confidently read how many minutes past a particular hour it is? eg. It is 10 minutes past 4, It is 45 minutes past 6...........

6. Minutes to the hour - can your child confidently read how many minutes to the next hour it is? eg. 20 minutes to 8, 10 minutes to 9, 5 minutes to 2.......

7. Can your child confidently read the television guide and work out when a program will finish?

8. Can your child confidently read a train or bus timetable and work out how long a certain part of the journey may be or what time you would arrive at a particular destination?

Activities

What's the time Mr Wolf?
One of you close your eyes while the other sets the clock to a certain time. The one that has set the time then asks 'What's the time Mr Wolf?' and the other person opens their eyes and reads the time.

Drawing the time
Find something to write on and write with that is a novelty eg. chalk on the driveway, whiteboard pen on the window or mirror, paint........ Draw a clock face and then ask your child to draw a specific time on the clock. If it's white board pens this works really well because you can rub the time out and reuse the clock face.

Dice Time
Use two dice. Choose each time whether to throw one or two dice. When they have been thrown add the total and then your child has to show the time on the clock using the number shown on the dice as the hour. eg. dice show 10 then 10 o clock needs to be created. If you are also using minutes throw the dice again and use that number to show where the minute hand needs to go and then your child needs to read the time they are showing on the clock. eg. a 10 was thrown first and then 2. The 10 is for the hour and the 2 is where the big hand needs to be so the time would read 10:10 or 10 past 10.

Challenge
Have somewhere to keep score. Ask your child to show a particular time on the clock. If they get it right they get the point, if they get it wrong you get the point. You may find your child then wants to swap round and give you some times to show on the clock - they love challenging us adults. Give it a go and have some fun.