# Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

## Problem

*Consecutive Numbers printable sheet*

*I wonder how often you have noticed numbers that follow one after another: 1, 2, 3 ... etc.? Sometimes they appear in reverse order when a countdown is happening for a launch of a rocket. But usually they happen in order going up, like when you read through a book and notice the page numbers. These kinds of numbers - whole numbers that follow one after another - are called consecutive numbers.**This investigation uses the idea of consecutive numbers and gives us other numbers to explore. You may very well discover things that no-one else has discovered or written about before, and that's great!***This is how it starts.** You need to choose any four consecutive numbers and place them in a row with space between them, like this:

4 5 6 7

When you've chosen your consecutive numbers, stick with the same ones for quite a while, exploring different ideas before you change them in any way. Now place $+$ and $-$ signs in between them, something like this:

4+5-6+7

4-5+6+7

and so on until you have found all the possibilities. Are you sure you've got them all? You should include one using all pluses and one that includes all minuses.

Now work out the answers to all your calculations (e.g. 4+5-6+7=10).

Next, try other sets of four consecutive numbers and look carefully at the sets of answers that you get each time.

Are you surprised by anything you notice?

It is probably a good idea to write down your 'noticings'. This can lead you to test some ideas out by starting with new sets of consecutive numbers and seeing if the same things happen in the same way.

You might now be doing some predictions that you can test out...

Finally, it is good to ask the question "I wonder what would happen if I ... ?"

You may have thought up your own questions to explore further. Here are some we thought of:

"What would happen if I took the consecutive numbers in an order going down, instead of up?"

"What would happen if I only used sets of three consecutive numbers?"

"What would happen if I used more consecutive numbers?"

"What would happen if I changed the rule and allowed consecutive numbers to include fractions or decimals?"

"What would happen if I allowed a $+$ or $-$ sign before the first number?"

This problem was chosen as a favourite for the NRICH 10th Anniversary website by Bernard Bagnall. Find out why Bernard selected it in the Notes.

## Getting Started

Having chosen your four consecutive numbers you place $+$ and $-$ signs in between them.

You have three empty spaces, so how many possible combinations are there for arranging the $+$ and $-$ signs?

Are you sure you have found them all? How will you know?

## Student Solutions

Kayley from Arnhem Wharf Primary School in London sent in this very good solution to the task:

When you use an even number of consecutive numbers, your answer will be even because there are two odd numbers and two even numbers in the equation. An odd number plus or minus another odd number equals an even number; an even number plus or minus an even number equals an even number; therefore the two answers (odd +/- odd and even +/- even) are even numbers and if you add it all together it will still be an even number.

I noticed that if you use the sequences +--, --+ or -+- with four consecutive numbers, your answer will be the same, no matter what consecutive numbers you use, if both sets numbers are ordered in the same way (e.g. from smallest to biggest). A set of three consecutive numbers will have odd answers if the starting number is even because an even number add or subtract an odd number will be odd, then when you add or subtract another odd number the answer will be odd.

I also found out that if there is a set of three consecutive numbers, and the starting number is odd, all the answers will be even because an odd plus/minus an even number is odd and if you plus/minus an odd number, the answer will be even.

These pictures show her work, the red writing being her thoughts. Larger versions can be viewed here: First page and Second page

Iris, Hannah, Hayden and Tawana from Fenstanton and Hilton Primary School said:

We worked out the calculations for 1, 2, 3, 4 as our consecutive numbers. These are the calculations we worked out:

1+2+3+4= 10; 1+2+3-4= 2; 1+2-3-4= -4; 1-2-3-4= 4;

1-2-3+4= -2; 1-2-3-4= -8; 1-2-3+4= 0; 1-2+3+4= 6*Are you surprised by anything you noticed?* We noticed that all the numbers are even, which would make sense because if you are adding consecutive numbers there will always be two odd numbers and two even, which means they will always add up to an even number.*What would happen if you had the numbers going in descending order instead of ascending order?* We found out that when we put the numbers in descending order they were all still even and that some of the numbers were there same but mostly different.

We got:

4+3+2+1=10; 4+3+2-1=8; 4+3-2-1=4; 4-3-2-1=-2

4-3-2+1=0: 4-3+2+1=4; 4+3-2+1=6; 4-3+2-1=2

James and Polly from the very small Meavy CE Primary School sent in this excellent solution:

Our answer to the consecutive numbers challenge:

30-31-32-33=-66 and 30+31+32+33=126 are some of the questions only using adding and subtraction using the numbers 30, 31, 32, 33. You can only get eight different calculations:

1. + + +; 2. + + -; 3. + - +; 4. + - -

5. - + +; 6. - + -; 7. - - +; 8. - - -

We found out that the answers you always get are 0, -2 and -4 using - - + (in any order).

If you put something in the middle of the eight answers, once you have put them in order, the gap between the answers will be like a mirror image.

We did the problem four times so that we could make sure that our answer was right. These were the groups of numbers we used:

The numbers we started with | The eight answers | The gaps between the answers |

1,2,3,4 | 10,6,4,2,0,-2,-4,-8 | 4,2,2,2,2,2,4 |

7,8,9,10 | 34,18,16,14,0,-2,-4,-20 | 16,2,2,14,2,2,16 |

20,21,22,23 | 86,44,42,40,0,-2,-4,-46 | 42,2,2,40,2,2,42 |

30,31,32,33 | 126,64,62,60,0,-2,-4,-66 | 62,2,2,60,2,2,62 |

James thinks that the middle number in the gaps is double the start number and the outside numbers are double the second number. Polly thinks the middle number in the gaps is double the start number but the outside numbers are double the first number and add two. Both of our suggestions are right.

Mrs Bromley a teacher at Skelton Primary School, York sent in Joseph's work, saying,

This morning we really enjoyed investigating consecutive numbers using your resource.

She remembered that Joseph said:

I decided to investigate odd and even numbers. I was surprised that even amounts of numbers give an even answer. I predicted that using three consecutive numbers would always give odd answers but I was proved wrong! Next, I would like to look at what happens if you start with an odd or even number.

You can see Joseph's work more clearly as .jpg files here and here.

Thank you all for these really helpful solutions which may encourage others to pursue the same task.

## Teachers' Resources

### Why do this problem?

This problem has the capacity to interest young and old alike. It offers an element of surprise which makes learners curious to investigate further and want to explain. I have used it with a wide range of attainment levels, and new things keep on being found out. It is a fantastic context in which to look for patterns, explain why these patterns occur, and as a result, to gain a deeper understanding of our number system. It offers opportunities to work together by sharing results and making decisions as to which consecutive numbers to look at next.

### Possible approach

To introduce the problem, discuss what consecutive means, and invite pupils to suggest a starting number. Let them select the three operations and take four or five examples, but don't discuss how many possibilities there are at this stage. Give children time to find other possibilities and encourage them to explain how they know they have them all.

Once all eight have been agreed on, let learners choose other sets of consecutive numbers to investigate. It is sometimes useful to suggest to the class that they are being detectives and seeking out links, relationships and reasons. Listen out for expressions of surprise, which you could proble a little by asking learners why they are surprised and how that makes them feel towards the task.

Most children find some connections between the eight answers that they find in each case. This could be that all the answers are even or the fact that some results appear with every group of four consecutive numbers. Encourage pupils to explain why in each case.

### Key questions

### Possible support

On the odd occasions that pupils needed support I have found that putting a number of pupils together to work as a sharing group is helpful.

### Possible extension

I have found that all the students who have been involved in this investigation have got very excited as various observations are made, patterns seen and questions asked. The most enjoyable times for me have been hearing ten year olds using their own form of algebra and coming to some powerful realisations about why every one has a $0$, $-2$ and $-4$.

The problem has also been the starting point for some pupils to be able to ask "I wonder what would happen if ...?", as suggested at the end of the problem itself. For consideration of negative numbers as well, look at Consecutive Negative Numbers.

### For more extension work

# Times Tables Shifts

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

## Problem

The numbers in the five times table are:

5, 10, 15, 20, 25 ...

I could shift these numbers up by 3 and they would become:

8, 13, 18, 23, 28 ...

In this activity, the computer chooses a times table and shifts it.

Can you work out the table and the shift each time?

Can you explain *how* you worked out the table and shift each time, and why your method will always work?

Levels 1 and 3 include tables up to 10.

Levels 2 and 4 include tables up to 20.

On levels 1 and 2, the numbers will always be the first five numbers in the times table.

On levels 3 and 4, the numbers could be any five numbers from the shifted times table.

Level 1Level 2Level 3Level 4

? | ? | ? | ? | ? |

The shift is always less than the times table.

Table Shifted updown by

*You may be interested in the other problems in our **Number Patterns Feature**.*

## Getting Started

What is the same about numbers in a times table and the numbers in the shifted times table?

On level 3, how about putting the numbers in order?

## Student Solutions

Thank you to everybody who sent in their ideas about this activity.

Viren shared these tips with us:

I look for the pattern and relations between numbers. In case of Level 1 and Level 3, by looking at lowest and highest numbers I figure out the times tables they might belong to. And then I work out what shift (up/down) is applicable.

In case of Level 2 and Level 4, the big numbers look intimidating but I work out the difference between numbers first, then figure out which number they are multiples of.

Good ideas, Viren. I wonder why working out the difference between the numbers is useful for finding the times tables that they might belong to?

Harry from Illogan School in Cornwall, UK sent in this method for the Level 4 problems, explaining how to use the difference to work out the times table:

This time I have numbers 123, 59, 171, 219 and 91

1) Put the numbers in order and find the differences

59 (+32) 91 (+32) 123 (+48) 171 (+48) 219

2) Find the highest number that goes into all the differences

Factors of 32 = 1, 2, 4, 8, 12, 16 and 32

Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48

So 16 in this case. This gives you the times table that has been shifted

3) Write out the 16x table, which will help you find the shift

In this case it is either 16 minus 5 or 16 plus 11. You can check it works for each number you were given

4) There will always be two answers because you can either add, or minus from the number you were given to get to a number in the 16 x table

This is really clearly explained, Harry!

Shaunak from Ganit Manthan, Vicharvatika, India described a similar solution:

**This problem can be split into 2 parts:** Working out the Table and Working out the Shift.

**Working out the Table:**

I observed the differences between the sets of consecutive numbers in the shifted times table.

In level 1 and 2, all these differences are the same.

In level 3 and 4, all these differences are different. The table has to be the greatest possible, so we have to find the **HCF (Highest Common Factor)** of the differences.

If the given sequence is 37, 17, 22, 52, 77, then the differences are 20, 5, 30, and 25. The HCF of these differences is 5, so the table is 5.

This method works because taking any two numbers in a times table and subtracting them will give a difference which is a multiple of the table. This happens because a times table is an arithmetic progression, which means a constant number (the table) is repeatedly getting added to the previous number, creating a sequence.

If the table is 5, then creating the sequence looks like this: **5, 5 + 5, 5 + 5 + 5, 5 + 5 + 5 + 5**, and so on.

If you take two numbers from the times table, you will see that the table is repeatedly getting added to the smaller number to obtain the bigger number. Therefore, the difference between two numbers in a times table is a multiple of the table. This having been said, we can say that the HCF of the differences will give us the greatest possible table.

Let us consider the example stated above. Arranging the numbers in ascending order, we get 17, 22, 37, 52, and 77. 5 is being added to the first number to get the second number (17 + 5 = 22), 15 is being added to the second number to get the third number (22 + 5 + 5 + 5 = 37) , 30 is being added to the third number to get the fourth number (37 + 5 + 5 + 5 + 5 + 5 + 5 = 52) , and 25 is being added to the second number to get the third number (52 + 5 + 5 + 5 + 5 + 5 = 77).

**Working out the Shift:**

I first observed the first number of the shifted times table and the closest multiple of the times table to this number.

If the shifted times table is 30, 50, 70, 90, 110, then the table is 20, because all the differences are 20. I looked at the first number, 30, and figured the closest multiple of 20 is 20.

If the multiple is smaller than the first number of the shifted times table, then the difference of the first number of the shifted times table and the multiple is the shift up.

Considering the above case, we can see that 20 is smaller than 30. So the difference between 20 and 30, which is 10, is the shift up.

If the multiple is greater than the first number of the shifted times table, then the difference of the first number of the shifted times table and the multiple is the shift down.

If the shifted times table is 2, 5, 8, 11, 14, then the table is 3. The closest multiple of 3 to 2 is 3. We can see that 3 is greater than 2. So the difference between 3 and 2, which is 1, is the shift down.

We can observe that the shifted times table is also an arithmetic progression, just like the times table. The differences have the same HCF, but the first, second, third, fourth, and fifth have a common difference from their closest multiple.

Thank you for sending in these ideas, Shaunak, this is a really clear explanation of how the shifted times tables work and how they can be solved.

Veyhant from Ganit Kreeda, Vicharvatika, India sent in these pictures explaining their solution for each of the four levels. These pictures can be clicked on to make them bigger:

We also received similar solutions from Nikhil, Abhyudh, Viyaan, Aditi, Mridula, Pushan, Larisa, Laira, Shravani, Anika, Kiaan, Kaira and Shreehari (Group 1) and from Ahana, Sehar, Saanvi, Dhanvin, Aariz, Ananthjith, Vivaan, Sai, Pranathi, Paavani, Utkarsh, Tavish and Dhruv (Group 2) from Ganit Kreeda, Vicharvatika, India. Thank you all for sending in your ideas - you've worked really hard on this problem. Their full solutions can be seen by clicking on these links: Group 1 Ganit Kreeda, Group 2 Ganit Kreeda.

Thank you as well to Jeffery and Ava from Australia and Luna from CHPS in Australia who also sent in some similar solutions to this problem.

## Teachers' Resources

Why do this problem?

This problem will help learners become more familiar with multiplication (times tables) facts, by encouraging them to think about number properties and therefore gain a deeper understanding of our number system. It also makes the inverse relationship between multiplication and division explicit. The interactivity will capture learners' curiosity so that they are motivated to find solutions and, in this way, they will be doing *meaningful* calculation practice.

In conjunction with the other tasks in this group, this task also offers the chance to focus on any of the five key ingredients that characterise successful mathematicians.

#### Possible approach

Explain to the group that you're thinking of a times table and ask them if they can work out which it is. Write these numbers on the board as you say them: 3, 6, 9, 12. What about 4, 8, 12, 16? 45, 50, 55, 60? How do they know?

Keep going until the class is confident and fluent at working out the times tables. To avoid shouting out, learners could write their answers on mini whiteboards.

Explain that you will now give the class some random numbers from a times table rather than the first four numbers. Write up, for example, 60, 20, 100, 50. Discuss that these are all in the 2, 5 and 10 times tables, but we're only interested in finding the largest possible times table, so we'll say these are numbers in the 10 times table.

Now show the interactivity from the problem and alert the children to the fact that it does something slightly different (but don't tell them what!). Generate a set of numbers using Level 1 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" that the computer uses. Emphasise that the table should always be the largest possible, and the shift should always be less than the table.

Ideally, each pair would now work at a computer or tablet to develop a method of finding the table and shift with ease. If that isn't possible, generate a dozen or so examples at appropriate levels, and write them on the board for the class to work on. Learners could also work in pairs and create examples for their partners to work out.

Once pairs are finding the table and shift easily, bring the class together. Generate a new example, using level 2 or 3, and ask a pair to talk through their thinking as they work towards the solution, but ask them to stop short of actually giving the answer. The rest of the class could write the answer on mini whiteboards once they've heard enough to work it out. Repeat, giving other pairs the opportunity to share their thinking.

#### Possible questions

What is the same between numbers in a times table and numbers in the shifted times table?

What can you learn from the difference between any two numbers in a shifted times table?

How do you find the shift once you've worked out the table?

#### Possible extension

The Shifting Times Table problem is a Stage 3 version of this task and the interactivity includes an extra level, which includes numbers in the 21-50 times tables. The tasks in this group would also provide useful follow-up to this one.

#### Possible support

Children may find a table square helpful as they work on this activity.

# Domino sets

How do you know if your set of dominoes is complete?

## Problem

When you buy a set of 0-6 dominoes they often come in cardboard boxes - and those boxes sometimes don't last very long!

What if you were given lots of dominoes in a bag?

Before you started playing it might be a good idea to find out if you have a full set!

How would you go about it?

How could you be sure?

What if someone gave you some 0-9 dominoes?

How many do you think there would be in a full set?

If you do not have any dominoes, you might find our interactive Dominoes Environment useful.

*You may like to try **Amy's Dominoes** as a follow-up to this task.*

## Getting Started

How many are there with blanks?

Are there the same number of dominoes with ones?

Could you organise the dominoes in some way to show you have/haven't got them all?

## Student Solutions

Well done to everybody who had a go at this activity. This task can be quite fiddly so we were sent several incorrect solutions, as well as a lot of correct ones! One of the solutions we received was more suitable for Amy's Dominoes, so we've uploaded that there instead.

The children from Linkwood Primary in Scotland sent in the following ideas:

We started with the doubles from 6 to 0. Then we thought about all the other dominoes that would have a 6. Then we did the same for 5, 4, 3, 2 and 1. When we had all of the dominoes we counted them all and found we had 28. We also noticed that for the dominoes with 6 there were 7, for 5 there would be 6 and so on.

Good ideas - this method was used by a lot of children!

We were sent in several similar solutions by the children at Brentwood Preparatory School in England. Bella said:

I used a set of 0-6 dominoes and discovered that the number of dominoes was always 1 more than the value of the face. e.g: there are 7 dominoes with a 6 on them, 6 with 5 dots, 5 with 4 dots .......

To check you have all the dominos, you could start with all the doubles and sort them by their faces, eg: 6-6 to 6-0.

In total there are 28 dominoes and I also found that, if you had 0-9, you would have 55 using the pattern that I found.

Chloe said:

You could arrange the dominoes in columns starting with a 0 on the left side of the domino and then 0-6 on the right side. In the next column you would have a 1 on the left and 0-6 on the right side. In every new column the left side would be increased by 1. The final column would have 6 on the left side. If all the dominoes are there you know that you have a full pack, but if there are some missing you know there is not a full pack.

Dhruv from The Glasgow Academy in the UK sent in the following picture, which can be clicked on to make it bigger. This the same arrangement that Chloe described in the previous solution.

Thank you all for sending in these very clear explanations, and well done for noticing that the arrangement will make a triangular shape.

Matilda and Eliza from Rainworth State School in Australia sent in this solution for the 0-9 dominoes:

Our solution is for the 0-9 dominoes. We lay the dominoes out in rows in order of the number on the top of the domino. When we finished laying them out, the pattern looked like a staircase because each one of the dominoes in the row below had been already used in the one above that. We counted the pattern - it went 1+2+3+4 and so on until +10. This equals 55 dominoes. We checked we were correct by counting the dominoes one by one. Our domino set was like the one described in the problem, the box was broken and not all the dominoes were there so we had to use dominoes from another set and we even had to draw one.

This is a very clear solution, and I like the way you've been creative with making a full set of dominoes when you couldn't find one. This is an excellent example of why we might need to be able to check how many dominoes we are expecting to find in a full set!

We received some excellent solutions from the following children at Ganit Kreeda, Vicharvatika, India: Ananya, Shivashree, Shreehari, Kanaa, Ruhi, Mrunmayee, Aarav, Shravani, Pushan, Ishan, Adithya, Aniket, Anirudh, Vikrant and Avyuktth. Take a look at Ganit Kreeda's full solutions to see their ideas.

Shreehari sent in the following video explaining some of these ideas:

Shaunak from Ganit Manthan, Vicharvatika, India also sent in a video to explain how they approached this task:

Thank you both for sending in these clear explanations of how you approached this task.

If you'd like to learn more about Shaunak's method, you might like to have a look at Shaunak's full solution to see an explanation of how the general version of this task can be solved using algebra.

Some children took a different approach and thought about how often each number would come up across a full set of dominoes. Ayden from St John's College in Cardiff, Wales said:

If you bought a pack of 0-6 dominoes, and they came in a bag, and if you wanted to check if you had a full set, you could count the number of dominoes with a particular number on one side, and then see if it adds up to 7, because there are 7 sets of each number. With this method, you can find out exactly what domino you are missing. For example, if you are counting for the number 3, and there are 6 dominoes, and 3 with 0, 1, 2, 3, 5, and 6, then you know that the 3/4 domino is missing. This is a more efficient, and quicker method than counting all the dominoes individually, but if you chose to do that, then you would know if there is a full set by having 28 dominoes in total.

In a set of 0-9 dominoes, you could use the same method mentioned above to find out if you have a full set. However, if you choose to count the dominoes individually, you would know if you are correct by having 55 dominoes in total.

We received a lot of similar solutions from the children at Park Hill Junior School in the UK. Isaac, Felix, Alba and Hugo had a different way of thinking about this task:

There should be 8 of each number so you count 8 sixes, 8 fives, 8 fours, 8 threes, 8 twos, 8 ones and 8 blanks.

The key to finding the missing ones is:

Line them up in rows or columns of its number.

OR:

There should be 28 dominoes in a pack so you count them.

4x7=28

This way of thinking about it is really interesting - I wonder why there are 8 of each number? And I wonder why that means there will be 28 dominoes in a pack?

## Teachers' Resources

Why do this problem?

When we use structured apparatus such as Cuisenaire rods, dominoes, Numicom etc. we often assume that children understand more about the structure than perhaps they do. This activity encourages children to explore the structure in their own way - and you may be surprised at how many different ways there are. You could use this activity in one of two ways, depending on your children. The most obvious way is to focus on the different patterns that can be built up, but you could also use the activity to focus on the explanation or justification of the children's description or argument. You might like to have a go yourself before offering this activity to the children...

Possible approach

You will need several sets of dominoes (you could make laminated card sets from this print out), one set between three or four children. If the dominoes are usually stored in a box, transfer them to a bag so that the children don't have a sense of a box being full.

Ask the children to tip the dominoes out onto the table and have a look at them, perhaps having an opportunity to play with them and arrange them into different patterns if this is the first time they have seen them.

Suggest that they may or may not be full sets and that before we can use them for a game we need to know if any are missing. Could they help you by finding out? Allow some time for this.

When a group gives you a definitive answer ask them how they know and challenge them to arrange the dominoes on the table in some way to show you that they have/have not got a full set. Again allow some time for this.

When, and if, appropriate you could suggest that the children move around the classroom to have a look at what other groups have done. You may then want to offer a little more time for those groups who haven't made much progress to have another go, having seen others' responses.

Groups will respond in different ways. In increasing order of sophistication, groups may:

- play a domino game where they match dominoes into a long train or loop, for example:

- make piles or groups of dominoes with numbers in common, for example:

- arrange the dominoes into a pattern or array, for example:

In each case, ask how the pattern/structure they have chosen helps. The most sophisticated explanations will use the idea of all possible pairings of the numbers 0-6.

The interactivity here may be helpful for whole class discussions of this activity and/or for learners to use as they work on the task, if real dominoes are not available.

Key questions

How many are there with blanks?

Are there the same number of dominoes with ones?

Tell me how you're arranging them. Why did you choose that way?

Could you organise them in a different way?

How does this show you that you have/haven't got them all?

Possible support

Reducing the number of dominoes can make a pattern easier to spot. Offer a subset, say 0-4, and allow time to play and look for patterns before asking if they have a full set.

Possible extension

What if you had a set of 0-9 dominoes (download a set here). How many do you think you would have? Do you need to make a set? How else could you record your ideas?

The task Amy's Dominoes is a natural extension to this activity and can be solved using the dominoes as a resource, or without them by more confident children.