Mathematical challenges for all pupils - Southwark Schoolsschools.southwark.gov.uk/assets/attach/2239/Mathematical challenges... · can solve problems by ... The development of this

Mathematical challenges for

all pupils

Problem solving and reasoning lessons

for KS1 and KS2

Developed by: Sonia Miguez Jorge, Claire Gillespie, Hannah Wood

and Diane Andrews

November 2016

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Introduction

This document was developed by a small working party* which included practicing classroom

teachers, maths subject leaders and a primary maths consultant. The lessons have been

trialled by teachers in Southwark schools.

There is currently a strong emphasis on problem solving and reasoning.

The main aims of the National Curriculum (2014) for mathematics are to ensure that all

pupils:

become fluent in the fundamentals of mathematics

can solve problems by applying their mathematics

can reason mathematically by following a line of enquiry

The development of this document was influenced by the document ‘Problems and Puzzles’, a

numeracy strategy resource, but has been brought up to date and aligned with the National

Curriculum (2014).

Its aim is to provide a bank of problem solving lessons that will offer support for incorporating

problem solving and reasoning in KS1 and KS2.

About the problems

Problem solving lessons have been developed for each term for all year groups. Progression

throughout the year and across year groups is built into the lessons.

All problems have a similar format: the problem is introduced to the whole class, the children

solve it independently (or in pairs or small groups) and then discuss their findings during the

plenary.

There are teaching objectives for each lesson including mathematical content and problem

solving and reasoning skills. These objectives can be shared with the pupils in an age

appropriate way.

Key vocabulary to be developed is also included with suggestions of useful resources and

visuals to support the teaching and learning.

Suggestions for follow up problems are included. These problems build on the strategies and

skills developed in the main problem.

How to use the problems

The problems are ideally taught following a unit of work where the mathematical skills and

knowledge have been taught. For example, Monster (Y2) could be used at the end of a unit on

money when children have the prerequisite mathematical skills for tackling the problem. The

problems allow children to use and apply the skills that they have been learning in a problem

solving context.

All of the problems need to be introduced to the children to ensure that they have the

strategies for working on them on their own. The ‘getting started’ section includes a short

mental/oral starter activity followed by an initial activity that can be discussed and tackled

as a class. It is important not to use this as a script but to adapt it to meet the needs of the

class and individuals.

The independent tasks can be attempted on an individual basis but there are benefits to be

derived from having children working in pairs or groups of three. This promotes mathematical

discussion and sharing of ideas, and encourages children to co-operate.

The independent tasks have different levels of challenge indicated by *, **, ***.

These are only suggestions and the aim is that they are adapted to suit the needs of the

children. There is also a suggested task for SEN pupils and an additional challenge question

provided for those children that need it.

It is advised to have mini-plenaries throughout to share strategies and ideas and to ensure

that children are on track.

The plenary section aims to bring children back together and to give an opportunity for

discussing the final results as well as explaining and justifying methods and strategies.

It is also an opportunity to develop reasoning skills further.

The problem solving lessons can be used flexibly. Some of them have scope for children

working on larger scale problems than a 45- 60 minute lesson allows.

The follow up problem aims to build on skills taught in the main problem solving lesson. It

could be introduced the following day or the following week so that the strategies learnt can

be applied.

It is recommended that the follow up lesson provides less scaffolding than the main lesson

and should be adapted in response to AfL.

Conclusion

It is important to teach problem solving and reasoning skills on a regular basis.

The problems in this document are intended to support this and the principles can be applied

to other problems throughout the term.

However, it is vital to develop reasoning skills and to provide opportunities for mathematical

discussion in every mathematics lesson. This can be achieved through effective questioning,

providing opportunities for discussion and setting activities in a meaningful and engaging

context.

* Members of the working party:

Sonia Miguez Jorge, maths subject leader and class teacher @St Francis RC Primary

School, Southwark;

Claire Gillespie, assistant head and class teacher @ St Francis RC Primary School,

Southwark;

Hannah Wood, formerly maths subject leader and class teacher @ St John’s and St

Clement’s School, Southwark;

Diane Andrews, maths consultant @ Count on me consultancy

With thanks to those teachers who have trialled the problem solving lessons and provided

useful feedback.

November 2016

Getting Started/Main Teaching Input:

Use a washing line with number cards 1 to 12 to count forwards and backwards. Ask children: What is one more than 4? How did you work it out? Repeat with other numbers, including

bridging 10. Hide one number and ask: Which number have I hidden? How do you know?

Show children three pins numbered 1-3. Ask a child to roll the ball and knock down as many as

they can. Total the pins that have been knocked down by recording in a number sentence e.g. 1 + 2 = 3. Explain that the score is three.

Then pose the problem: What if I knocked down the pins with the numbers 1 and 3? What would the score be? Talk to your partner.

Take a response and model recording e.g. 1 + 3 = 4

If we knocked down all three pins, what would the score be? Talk to your partner.

Children to record on mini-whiteboards. Take a response and ask them how they worked it out. Model recording e.g. 1 + 2 + 3 = 6

Show the problem ‘Four-pin bowling’ and read ‘Which pins must Joshua knock down to score exactly 5?’ Point out to the children that we now have four pins. Children to talk to partners

to find a solution and record on mini-whiteboards. Collect a response and model recording e.g. 4 + 1 = 5.

Read the rest of the problem ‘Find 2 different ways to score 5 ...’

Discuss resources that might help them solve the problem e.g. number tracks, counters.

Independent Tasks: (including possible levels of challenge)

** Ask the children to find two different ways to score 5; find two different ways to score

6; find two different ways to score 7. Children record solutions as modelled, using resources to support and, if needed, using a recording sheet

e.g. + = 7 or + + = 7

* Adult led group. Children to initially approach problem practically using pins (or similar).

Adult to model the recording of scores on small whiteboard. Children respond to oral question(s): What if I knocked down the pins with the numbers 2 and

3? What would the score be? Model recording. Children then find totals of two pins, recording

on sheet provided

e.g. 3 + 2 = 4 + 1 =

*** Children to complete the task and record as modelled. Extend by asking: What is the

largest possible score that you could make? Children explain how they know and record using a

number sentence.

SEN: Provide five numbered skittles/pins (1-5) or similar. Children put them in numerical order from 1-5. Hide one and ask: Which pin have I hidden? How do you know? Repeat with other

numbers. Consider photographs as evidence.

Additional Challenge Question:

Use five pins numbered 1-5. Ask children to find three ways of making a score of 8.

Plenary/Drawing Together:

Collect solutions to the original problem. Ask children: What resources helped you solve the problem? What did you find challenging?

Pose the additional challenge question and ask children, in pairs, to find solutions where the

total is 8, recording on mini-whiteboards. Take responses and record their number sentences.

Follow-up Problem:

Pose a similar problem e.g. ‘Catching fish’. Use fish numbered 1-5. Ask children to catch fish with a total of 6, 7, 8 and 9. Find two (or three) ways to make each total.

Adapt the problem, as above, and in response to AfL.


Use a blank counting stick to count in ones, forwards and backwards, from 0 to 10 (using the

stick as a number line). Then roll a dice to generate a new starting number and count on in ones from that number e.g. 6 to 16. Repeat with another starting number. Ask children to predict

the end/final number and explain how they worked it out.

Show children three number cards, numbered 3, 5, 2. Ask children to choose two of the

numbers and add them together. Record on mini-whiteboards. Ask children: Which two cards did you choose? What total did you make? How did you work it out?

Take a response. Model a possible strategy, such as counting on from the biggest number, and record using a number sentence e.g. 5 + 2 = 7. Ask: Did anyone make a different total?

Ensure that the three totals are found and discuss why 2 + 5 will give the same total as 5 + 2.

Show children three different number cards, numbered 6, 5, 4. Ask children to work with a partner to find as many totals as they can by adding two of these

numbers and record on their mini-whiteboards. Take responses. Ask a child to model recording using a number sentence e.g. 6 + 5 = 11. Ask: How did you work it out? Did anyone make a different total? Ensure that the three answers are found. Ask children: Were these numbers more challenging to add up? Why?

Show the problem ‘Pick a pair’ and read ‘Pick a pair of numbers. Then write an addition number sentence using these numbers. Add the numbers together to find the total.’ Discuss the vocabulary used and point out to the children that we now have four number cards.

Children to talk to partners to find a solution. Collect a response and model recording e.g. 4 + 3 = 7.

Read the rest of the problem: ‘Pick a different pair of numbers...’ Ask children: Now that we have four number cards, do you think there will be more totals or

fewer totals? Why? Discuss resources that might help them solve the problem e.g. number

tracks.


** Ask the children to solve ‘Pick a pair’, explaining that there are 6 totals. Record solutions

using number sentences as modelled.

* Adapt the problem so that all totals are within 10, for example: 2, 3, 4, 6. Provide recording

sheets if needed.

e.g + =

*** Ask the children to complete the task and record as modelled.

Extend by giving them an additional number card. Ask children to predict if there will be more totals or fewer totals. Children to then find all totals.

SEN: Provide four cards/snowflakes, numbered 1-4. Ask children to find counters to match the numbers. Consider photographs as evidence. Ask children: How many counters are there

altogether? How will we find out? Count all counters, using one to one correspondence (1-10).

Additional Challenge Question: Pose the problem: Now take one number away from the other. How many different answers can you find? Predict first. Were you right? Children to explain how they know and record using

number sentences.


Collect solutions to the original problem and ensure that there are no repeats. Ask children: What resources helped you solve the problem? What did you find challenging?

Model a systematic recording and explain that this helps us to ensure that we don’t repeat or

leave out any of the solutions.

Follow-up Problem:

Pose a similar problem e.g. ‘Tony Take Away’: Tony loves to subtract numbers. He has four cards numbered 14, 10, 5 and 3. He chooses two of his number cards and writes a take-away

number sentence. How many different answers can he get?



Count forwards and backwards in multiples of two. Drop 2p coins into a jar and children count in twos. Children close their eyes and listen to the

sound of 2p coins dropping into the jar, counting in twos in their heads as they hear them drop. Ask children: How much money is in my jar? How many 2p coins are in my jar? How did you work

that out?

Ask children: Why do we use money?’ ‘What coins do we use? Discuss the use of the symbol for

pence (p). Show children coins to £1 (1p, 2p, 5p, 10p, 20p, 50p, £1), e.g. using large display coins or on

IWB.

Pose a problem: If I bought an apple for 3p, which coins could I use to pay for it exactly?

Children to talk to partners to find a solution. Take a response and model recording e.g. 1p + 1p + 1p = 3p. Invite a child to come and pay 3p for an apple using these coins.

Ask children: Is there another way to make 3p? Take a response from children and model

recording e.g. 1p + 2p = 3p. Invite another child to pay 3p for an apple using these coins. Ask children: Why don’t we use a 3p coin? Address this common misconception.

Show the problem ‘Lottie’s Lollipops’ and read ‘Lottie bought a lollipop. It cost 6p. She paid for it exactly. Which coins did she use?’

Children to talk to partners to find a solution and record on mini-whiteboards. Collect a

response and model recording e.g. 2p + 2p + 2p = 6p. Invite a child to pay for a lollipop using these coins.

Read the rest of the problem ‘There are five different ways to do it. Find as many ways as you can.’ Discuss resources that might help solve the problem.


** Ask children to find as many combinations as they can, using coins to support. Record

solutions as modelled.

* Adapt question by replacing 6p lollipop with a 4p lollipop. Ask children to find the 3 possible

solutions. Provide coins and, if necessary, children can record using the coins. Extend by asking children to find at least one way to pay for the 6p lollipop and show using coins.

*** Ask children to complete the task and explain how they know that they have found all the

possible ways. Extend with: What if the lollipop cost 7p? Would there be more ways or fewer ways to pay for it? Children solve the problem.

SEN: Ask children to pay for a 4p lollipop using only pennies. Extend asking children to pay for a 5p and then a 6p lollipop, using only pennies. Record with photographs.

Additional Challenge Question: What if you bought a bag of lollipops for 20p? You paid for it exactly using only the silver

coins. What coins would you use? How many combinations are there?


Collect solutions to original problem. Ask children: Which way used the most coins? The fewest coins?

Model systematic recording. Ask children: How do we know we have found all the possible ways? Are all the ways different? How many ways did we find? Establish that there are five ways to make 6p.

Follow-up Problem:

Pose a similar problem e.g. ‘Pippa’s Purse’: Pippa has 7p in her purse. What coins does she have?

There are 6 ways to make 7p. Find as many different ways as you can.

Encourage children to record systematically, as modelled in previous lesson. Adapt the problem, as above, and in response to AfL.


Recall number bonds to 10. Play number ping-pong; teacher to model with a pupil. Teacher says/bats one number and the pupil responds with the corresponding number to make a total

of 10. Then play as a class. Teacher says/bats a number and children bat back the number to give a total of 10. Repeat with bonds to 20.

Show children three buckets numbered 1-3. Ask a child to throw two bean bags into buckets of their choice.

Total the buckets with the bean bags in by recording in a number sentence e.g. 3 + 2 = 5. Explain that the score is five. Then pose the problem: What if I threw the bean bags into the buckets with the numbers 1 and 3? What would the score be? Talk to your partner.

Take a response and model recording e.g. 1 + 3 = 4 Ask: What if I threw both bean bags into bucket number 3? Take a response and record:

3 + 3 = 6

If we had three bean bags, and we threw one into each bucket, what would the score be? Talk

to your partner. Children to record on mini-whiteboards.

Take a response and ask them how they worked it out. Model recording e.g. 3 + 2 + 1 = 6

Show the problem ‘Bean-bag buckets’ and read ‘Dan threw 3 bean-bags. Each bean bag went in a bucket. More than one bean bag can go into a bucket.’

Point out to the children that we now have four buckets, numbered 1, 2, 3, 4 and three bean-

bags. Remind the children that more than one bean bag can go in a bucket and that all bean bags must be in a bucket. Ask: What could be one of Dan’s scores? Children to talk to partners to find a solution and

record on mini-whiteboards.

Collect a couple of responses and model recording e.g. 2 + 2 + 2 = 6; 4 + 4 + 1 = 9

Read the rest of the problem together.

Discuss resources and strategies that might help them to solve the problem e.g. number tracks, counters, empty number lines.


** Ask children to find three ways to score 6, find three ways to score 7 and find three ways

to score 8. Remind them that more than one bean bag can go in a bucket and that all bean bags

must be in a bucket. Record solutions as modelled, using resources to support and, if needed, using a recording sheet.

e.g. + + = 6

Extend with: What is the highest score Dan can get? What is the lowest score Dan can get?

* Adult led group. Children to initially approach the problem practically using buckets (or

similar). Adult to model the recording of scores on small whiteboard. Children respond to oral question(s): What if my bean bags went into buckets 1, 2 and 3? What would the score be? Model recording. Children find totals of three buckets, recording on the

sheet provided

e.g. 3 + 2 + 2 = 4 + 3 + 1 =

*** Complete the problem as written, recording independently.

SEN: Provide ten numbered buckets (1-10) or similar. Children to put the buckets in numerical order from 1-10. Hide one bucket and ask: Which bucket have I hidden? How do you know?

Repeat with other numbers. Consider photographs as evidence.

Additional Challenge Question: If the buckets were numbered 3, 4, 5 & 6, could you find three ways to score 12? What other scores could you get? Record solutions, as modelled.


Collect the three possible solutions for scoring 6 (then 7 and 8), modelling a systematic recording. Ask children: What resources/strategies helped you solve the problem? What did you find challenging?

Ask: What was the highest possible score? The smallest possible score? How do you know? What other scores did you find for Dan? How many were there? Take responses and record

their number sentences on the whiteboard.

Follow-up Problem:

Pose a similar problem e.g. ‘Leap to 10’. A frog needs to cross the pond, leaping on three

different numbered lily-pads (six lily pads, numbered 1-6). Find three ways of making 10. Could you make 11? 12?

Adapt the problem, as above, and in response to AfL, including the use of appropriate resources.

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Count forwards and backwards in multiples of five. Drop 5p coins into a jar and children count in fives. Children close their eyes and listen to the

sound of 5p coins dropping into the jar, counting in fives in their heads as they hear them drop. Ask children: How much money is in my jar? How many 5p coins are in my jar? How did you work that out?

Ask children: What coins do we use in this country?

Show children all coins to £2, e.g. using large display coins, or on IWB. Discuss the silver coins and explain that these are the ones which will be used in today’s problem. Pose a problem: I bought this toy in a charity shop. It cost 20p and I paid for it exactly with silver coins. Which coins do you think I used? Children to talk to partners to find a solution

and record on mini-whiteboards. Collect a child’s response and model a possible recording e.g. 10p + 10p = 20p.

Invite a child to come and pay 20p for the toy using these coins.

Ask children: Is there another way to make 20p? Collect responses and model systematic

recording e.g. 20p 10p + 10p = 20p

10p + 5p + 5p = 20p 5p + 5p + 5p + 5p = 20p

Ask children: How do we know that we have found all possible combinations?

Show the problem ‘Monster’ and read ‘Alesha bought a monster using only silver coins. It cost her 45p.’ Referring to the previous problem (20p toy), ask children: Do you think there will be

more combinations or fewer combinations? Why do you think that?

Read the rest of the problem: ‘There are nine different ways to pay 45p exactly using only

silver coins. Find as many ways as you can.’


** Ask children to find as many combinations as they can, using coins to support if needed.

Encourage systematic recording, as modelled.

* Adapt question by replacing 45p monster with a 30p toy. Children to find as many

combinations as they can. (There are 6 possible combinations.) Provide coins and, if necessary, children can record using the coins. Extend by asking children to find at least one way to pay

for the 45p monster and show using silver coins.

*** Ask children to complete the task, recording systematically. Children to explain how they

know that they have found all the combinations. Encourage them to check that no combination

has been repeated.

SEN: Children pay for a 30p toy using only 10p pieces. Extend by asking children to pay for a

40p toy and then a 50p toy using only 10p pieces. Encourage counting in tens. Record with photographs.

Additional Challenge Question: Children respond to question: What if the monster cost 50p? Do you think there will be more

combinations or fewer combinations to pay using only silver coins? Why do you think that? How many different ways to pay for the 50p monster are there? How do you know that you have

found all combinations?


Collect solutions to the problem. Record solutions on strips of card/paper and display. Ask children: Which combination used the most coins? The fewest coins?

Invite a child/children to rearrange the strips such that they demonstrate systematic

recording. Ask children: How do we know we have found all the combinations? Are all the combinations

different? How will we check?

Discuss how systematic recording helps us to check that all the combinations have been found.

Follow-up Problem:

Pose a similar problem e.g. ‘Coins in my purse’: I have 2 silver coins in my purse. How much money do I have? There are ten possible solutions. Find as many different solutions as you can.

Encourage children to record systematically. Adapt the problem in response to AfL e.g. by reducing the number of types of coins 20p, 10p,

5p (six possible solutions).

Additional challenge question: What if I had three silver coins in my purse?


Count forwards and backwards in steps of two from 1 to 19. Ask children: What do you notice

about these numbers? Establish that they are odd numbers. Ask children: If we carried on counting would we say the number 29? How do you know? Children to talk to partners. Collect

feedback. Repeat with numbers: 32, 45 and 50.

Ask children: Is it true that the total of two odd numbers is always even? How do you know?

Children to talk to partners and record on mini-whiteboards. Collect feedback and establish that it is true. Model examples e.g. 3 + 5 = 8. Consider the use of Numicon to support children’s understanding / explanations.

Pose a problem: Two turtles laid some eggs. Each turtle laid an odd number of eggs. Altogether

they laid 12 eggs. How many eggs did each turtle lay?

Children to talk to partners to find a solution and record on mini-whiteboards. Collect a child’s response and model a possible recording e.g. 5 + 7 = 12. Discuss other ways of

recording, for example using pictures.

Ask: Are there any other possible solutions? Collect responses and model a possible systematic

recording: e.g. 12 = 11 + 1

12 = 9 + 3 12 = 5 + 7

Ask: How do we know that we have found all possible combinations?

Ask children: What if you added three odd numbers? Would the total be odd or even? How do you know? Children to talk to partners and record on mini-whiteboards. Collect feedback and

establish that the answer would always be odd. Model examples e.g. 1 +3 + 5 = 9; 5 + 5 + 5 = 15 Consider the use of Numicon to support children’s understanding / explanations.

Show the problem ‘Bird’s eggs’ and read together ‘Three birds laid some eggs. Each bird laid an odd number of eggs. Altogether they laid 19 eggs. How many eggs did each bird lay?’

Refer back to turtle problem and ask children: Do you think there will be more solutions or fewer solutions? Why do you think that?

Children to talk to partners to find a solution. Collect a child’s response and model a possible recording e.g. 19 = 17 + 1 + 1.

Ask children: What resources could help you solve the problem? Remind children of available

resources and methods of recording including drawing pictures.


** Ask children to find as many solutions as they can, using resources to support if needed.

Encourage systematic recording, as modelled.

* Adapt question by replacing 19 eggs with 11 eggs. Children to find as many solutions as they

can. Provide resources and, if necessary, children can record using pictures. Extend by asking children to find at least one way to find a solution to the original problem using 19 eggs.

*** Ask children to complete the task, recording systematically. Children to explain how they

know that they have found all the solutions. Encourage them to check that no combination has been repeated.

SEN: Children to sort odd and even numbers to 10 .Then find different solutions to the following problem: Two turtles laid some eggs. Each turtle laid an odd number of eggs.

Altogether they laid 10 eggs. How many eggs did each turtle lay? Children to use resources to

support and record solutions.

Additional Challenge Question: What if there were three birds and altogether they laid 24 eggs? But this time each bird laid

an even number of eggs. Do you think there will be more solutions or fewer solutions? Why? Solve the problem.


Display the ten possible solutions to ‘Bird’s eggs’ and ask children: Have I recorded them systematically? How do you know? Can you see any patterns? Are all the solutions different?

Discuss how systematic recording helps us to check that all the solutions have been found. Ask: Did anyone have a different way of recording the solutions?

Discuss the use of pictures, diagrams and recording tables.

Follow-up Problem:

Pose a similar problem e.g. ‘Three Monkeys’ (problem 31 in ‘Problems and Puzzles’). Provide limited scaffolding for this follow-up problem but refer back to the ‘Bird’s eggs’

problem. Adapt this problem as above and in response to AfL, including the provision of appropriate resources.


Count forwards and backwards in multiples of 2 and then 3 to the 12th multiple.

Pose a problem: There are some bicycles in the toyshop. I counted 24 wheels. How many bicycles are there?

Children to talk to their partners to find the solution and record on mini-whiteboards. Take a response from the children and ask: How did you work that out? Ensure children relate

this problem to division.

Repeat with a similar problem: In another shop there are some tricycles. I counted 27 wheels. How many tricycles are there? Take a response from the children and ask: How did you work

that out?

Pose a more challenging problem: However, in my local toyshop, they sell both bicycles and

tricycles. I counted 15 wheels altogether. How many bicycles and how many tricycles are there?

Children to talk to their partners to find a solution and record on mini-whiteboards.

Take a response and ask: How did you work that out? Did anyone work it out differently? Did anyone find another solution? How do we know that we have all the possible solutions?

Establish that the two solutions are: 3 bicycles (6 wheels) and 3 tricycles (9 wheels) or 6 bicycles (12 wheels) and 1 tricycle (3 wheels).

Address any misconceptions that may arise.

Model a possible recording to find the solutions using a table. Draw children’s attention to pairs of numbers that total 15.

Show the problem ‘Spaceship’ and read together, emphasising that there are at least two of each type of alien. Discuss resources that might help them solve the problem.


** Ask the children to find two different answers to this problem using a method of their

choice. Encourage them to record, as modelled, using the blank table you have provided for them.

* Adapt the problem by replacing 23 legs with 16 legs. Children to find at least one answer

(there are two possible answers/solutions). Use resources to support e.g. Unifix grouped in twos and threes. Extend by asking children to find one solution to the core problem.

*** Ask the children to find two different answers to this problem. Record, as modelled,

using the table provided. Extend by asking: What if there were 34 legs altogether. How many Tripods were there? How many Bipods were there? Find at least three different

answers/solutions?

SEN: Children use only Bipods (multiples of 2). Ask: How many Bipods are there, if there are

10 legs? 14 legs? 20 legs? Use resources to support and consider recording with photographs.

Additional Challenge Question: Some Quadrapods, which have 4 legs, flew in from planet Zeno with the Tripods. There were

41 legs altogether. How many Tripods were there? How many Quadrapods were there? Find three different answers/solutions?


Refer back to the original problem, ‘Spaceship’ and collect solutions. Establish that the two solutions are 3 Tripods (9 legs) and 7 Bipods (14 legs) or 5 Tripods (15 legs) and 4 Bipods (8

legs). Ask children: What helped you solve this problem? What did you find challenging? Discuss how

recording in a table can help us to find all the solutions.

Follow-up Problem:

Pose a similar problem e.g. ‘At the toy shop’ (problem 23 in ‘Problems and Puzzles’). Provide limited scaffolding for this follow-up problem but refer back to the ‘Spaceship’ problem. Adapt this problem as above and in response to AfL, including the use of appropriate

resources.

Multiples

of 2

Multiples

of 3

2 3

4 6

6 9

8 12

10 15

12 18


Use a counting stick to count forwards and backwards in multiples of 4. Through doubling, connect multiples of four and eight. Then count forwards and backwards in multiples of 8.

Remind (briefly) the children of the story of Jack and the Beanstalk.

Tell them that the Giant was three metres tall when he was seven years old. Ask children: How tall is 3m? Can you think of anything that would measure 3m? Take responses.

Explain that the giant’s height doubled every year until he was 10 years old. Ask them to work

out how tall he was when he was ten. Children to talk to partners and use mini-whiteboards to record their answers. Take a response and ask: How did you work that out? Discus strategies

for doubling.

Show them how to record the results in a table.

Ask: Why do you think tables are used to record work?

Agree as a class that results in columns are easier to check.

Age

(years)

Height

(metres)

7 3

8 6

9 12

10 24

Show the problem ‘Jack’s Magic Beans’ and read together. Establish that the beanstalk is doubling in height every day. Ask children: How will you measure your height? What resources will you use? What units of measurement will you use? Remind children to predict first and then answer the question

‘When will the beanstalk be taller than you?’


** Ask the children to solve ‘Jack’s Magic Beans’, recording in a blank table you have provided for them e.g:

* Adapt the problem by asking children to work out the height of the beanstalk on Friday.

Record in a table provided. Extend problem by asking children to find the height of the beanstalk on Saturday and then Sunday.

*** Complete the task and record in a table. Extend by posing the problem: How tall would you be if you doubled your height every year until you were ten?

SEN: Ask children to measure using non-standard units, such as cubes, the height of a beanstalk (real or pictorial). Tell children that the beanstalk will double in height overnight. How tall will it be tomorrow? Repeat with different beanstalks and introduce standard units,

if children are ready.

Additional Challenge Question: Pose the problem: When the beanstalk gets to 512cm it shrinks to half its height every day. How many days will it take to get back to 1cm tall?


Ask children to share their conclusions providing them with a sentence starter ‘The beanstalk will be taller than me on ...’ Ask: What helped you solve the problem?

Display a completed table for this problem and discuss how the table helped them to record their results. Convert the measurements to mixed units.

Follow-up Problem:

Pose a similar problem e.g. ‘Big Baby’. The giant’s baby measured 60cm in length when he was born. During the first year of his life he grew 10cm; during the second year of his life he

grew another 20cm; during the third year of his life he grew another 30cm and so on. How old will he be when he is twice as tall as he was when he was born? How tall will he be

when he is your age? How tall will he be when he is 10 years old?

Adapt the problem in response to AfL, including the provision of appropriate resources.

Day Height (cm)

Monday 1

Tuesday


Display two polygons e.g. a square and a right-angled triangle. Ask: Tell me something that is the same about these two shapes. Children to talk to partners.

Take a range of responses. Discuss using appropriate mathematical vocabulary, referring to key vocabulary cards as

necessary. Now ask: Tell me something that is different about these two shapes. Children talk to partner

then feedback as above. Encourage children to identify any lines of symmetry in the shapes and develop the language symmetrical and non-symmetrical.

Pose a problem: Tom placed 5 squares and made this shape -

Ask: Is this polygon symmetrical or non-symmetrical?

Children to talk to partners and explain their choice. Establish that the polygon is symmetrical and ask children to identify the line of symmetry.

Ask: Is there another symmetrical polygon that Tom could make with 5 squares?

Ensure children understand that the squares should be placed whole side to whole side, like

this: and not like this:

Children work with a partner to sketch a possible polygon on their mini-whiteboards and identify line(s) of symmetry. Share with the class.

Show the problem ‘Polly’s Polygons’ and read together, ensuring children understand that Polly is making polygons using 6 squares. Discuss the use of square tiles and mirrors to support the

problem.


** Ask the children to solve the problem ‘Polly’s Polygons’, finding as many different

symmetrical shapes as they can. Use resources to support as necessary. Record on centimetre squared paper and mark the lines of symmetry.

* Ask the children to solve the problem using 5 squares, finding as many different shapes as

they can (symmetrical and non-symmetrical). Initially find possibilities using squares of paper/square tiles and then record on centimetre squared paper. Extend with: Identify which

shapes are symmetrical and which shapes are non-symmetrical.

*** Ask the children to solve the problem ‘Polly’s Polygons’, finding all possible different

symmetrical shapes. Record on centimetre squared paper and mark all lines of symmetry.

Children explain how they know that they have found all possibilities, including those polygons which have 2 lines of symmetry.

SEN: Children use practical resources to create different polygons using 5 squares. Encourage

them to check that no shapes have been repeated. Use language of ‘same’ and ‘different’.

Additional Challenge Question: Using the polygons that have been created, children predict which has the smallest perimeter

and which has the largest. Then find the perimeter of each polygon.


Ask children: What helped you to solve the problem? Is there anything that you found challenging?

Show two polygons, one of which is a rotation of the other, e.g.

Ask: Are these polygons the same or are they different?

Children to talk in pairs, explaining their reasoning, then feedback to the class. Show all the possible polygons that could be made from 6 squares. Ask: Which polygon, made from 6 squares, has the smallest perimeter? Largest perimeter? How do you know?

Children to talk to partners. Take feedback.

Follow-up Problem:

Pose a problem e.g. ‘Farmer Jean’. Farmer Jean has 20m of fencing. She wants to make a rectangular pen for her pigs. Find all the possible rectangular pens that she could make using the 20m of fencing.

Adapt the problem in response to AfL, including the provision of appropriate resources.


Count forwards and backwards in multiples of 4 to the 12th multiple. Ask related multiplication and division questions.

Pose a problem: Betty the bird has fewer than 19 eggs in her nest. She counted her eggs in fours. She had 1 left over. How many eggs could she have?

Children to talk to partners to find a solution and record on mini-whiteboards. Collect children’s responses and ask them how they worked it out. Model a possible recording

e.g. 4 + 1 = 5 8 + 1 = 9

12 + 1 = 13 16 + 1 = 17 Do we need to continue further? Why/why not?

Next, she counted her eggs in fives. She had 3 left over. How many eggs could Betty have?

Children to talk to partners to find a solution and record on mini-whiteboards.

Collect children’s responses and model a possible recording e.g. 5 + 3 = 8

10 + 3 = 13 15 + 3 = 18

Ask children: So, how many eggs does Betty have? How do you know? Establish that there is

only one solution which is 13 eggs.

Model how this could be recorded in a table.

Discuss how the table makes finding the solution easier.

Show the problem ‘Susie the snake’ and read together. Discuss resources that could help solve

the problem.


** Ask children to solve the problem ‘Susie the snake’, explaining that there is only one

solution. Provide children with a blank table for recording. Extend with: What if Susie had between 20 and 40 eggs?

* Adapt the question. Susie the snake has up to 20 eggs. When she counted her eggs in threes

she had 1 left over. When she counted her eggs in fives she had 1 left over. How many eggs has

Susie got? Explain that there is only one solution. Provide children with a blank table for

recording and resources (e.g. cubes), if needed. Extend with original problem.

*** Ask children to solve the problem ‘Susie the snake’, explaining that there is only one

solution. Extend with: Susie the snake has up to 30 eggs. She counted her eggs in fours. She had 3 left over. She counted her eggs in sixes. She had 1 left over. How many eggs has Susie

got? Children to find all possible solutions.

SEN: Susie the snake has some eggs (fewer than 20). When she counts them in fives there is

one left over. How many eggs could she have? Use resources (e.g. cubes/counters) to support and consider recording with photographs. Find

at least one solution.

Additional Challenge Question: Maggie the magpie has up to 30 eggs. She counted her eggs in threes. She had 2 left over. She counted them in fours. She had 1 left over. She counted them in fives. She had 4 left over.

How many eggs does Maggie have?


Collect the solution to the original problem. Show a table of results and establish that Susie had 19 eggs. Ask children: What helped you solve this problem? What did you find challenging? How did your method of recording help you find the solution?

Follow-up Problem:

Pose a similar problem e.g. ‘Robbie’s Rabbits’. Robbie has some rabbits (fewer than 40). Robbie

keeps his rabbits in hutches. When he puts 6 in each hutch there are 2 rabbits left over. When he puts 5 in each hutch there are 3 rabbits left over. How many rabbits does he have? Find

two solutions to this problem. Refer back to ‘Susie the snake’ problem.

Adapt this problem as above and in response to AfL, including the provision of appropriate resources.

4 + 1 = 5 5 + 3 = 8

8 + 1 = 9 10 + 3 = 13

12 + 1 = 13 15 + 3 = 18

16 + 1 = 17 20 + 3 = 23

Do we need to record

this last calculation?

Why? Why not?


Display three number cards 2, 3 and 6. Ask the children: Can you make a three-digit number? Is it possible to make more than one? How many can you make?

Children record on mini-whiteboards and then feedback with possible answers.

Ask: How do you know you’ve found all the possibilities? Have you checked for repeats?

Explain that by working systematically, all 6 possibilities can be found. Model a systematic recording e.g. 236, 263, 326, 362, 623, 632

Pose a problem: Chibu has eight cards and all his cards are different. There is a number from 1 to 8 on each card. Chibu has chosen three cards that add up to 12. What could they be? Try

to find more than one possible solution.

Children talk to partners to find solutions and record on mini-whiteboards. Collect one solution e.g. 6 + 5 + 1 = 12 and ask: How did you work it out? What was your strategy?

Agree that we could start with the largest possible number, mentally work out the complement

to 12, and find two numbers that would total this complement, for example: 8 + 3 + 1 = 12

Collect all solutions found and ask: Do we have all the possible solutions? How can we be sure?

Have any solutions been repeated?

Discuss how 6 + 5 + 1 is the same as 5 + 6 + 1 in this problem.

Explain to the children that, in order to be sure that all the possibilities have been found, we need to work systematically.

Ask: How could we be systematic? Take responses and model a possible systematic recording

e.g. 8 + 3 + 1 = 12 7 + 4 + 1 = 12

7 + 3 + 2 = 12 6 + 5 + 1 = 12

6 + 4 + 2 = 12 Show the problem ‘Card tricks’ and read together.


** Ask the children to solve the problem ‘Card Tricks’, explaining that there are seven

possibilities. Children find all possibilities, recording systematically, as modelled, or using a systematic recording of their choice. Extend with: What if Chico chose four cards that add up to 24? Do you think there would be more or fewer possibilities? Why? Prove it.

* Adapt the problem: Chico’s cards are all different. There is a number from 1 to 8 on each

card. Chico has chosen three cards that add up to 16. What could they be? Children find all

(five) possibilities. Use resources e.g. number cards, to support and record as number sentences. Extend with: Find at least one solution to the problem ‘Card Tricks’.

*** Ask the children to solve the problem ‘Card Tricks’. Children find all possibilities,

recording systematically, as modelled, or using a systematic recording of their choice. Extend with: What if Chico chose five cards that add up to 20? Do you think there would be more or fewer possibilities? Why? Prove it.

SEN: Chico has cards from 1-9. They are all different. He has chosen two cards that add up to 10. What could they be? Try to find more than one solution. Use resources (e.g. Numicon

and number cards) to support. Consider recording with photographs and model using a number

sentence.

Additional Challenge Question: Pose a problem: Chico says: ‘It is possible to make any total from 10 to 26 with four cards’. Is

he right? Prove it.


Collect the solution to the original problem. Establish what the seven solutions are. Ask children: What helped you solve this problem? What did you find challenging? How did your method of recording help you find all of the possibilities?

Model a systematic recording.

Follow-up Problem:

Pose a similar problem e.g. ‘Diane’s Dice’. Diane rolls four dice. She scores a total of 18. What numbers could she have rolled? There are eight different possibilities. Try to find them all.


Have any of the possibilities

been repeated? Have we worked

systematically?

Getting Started / Main Teaching Input:

Display a number sequence such as 1, 4, 7, 10 ... Ask children: What would the next number in

the sequence be? How did you work it out? Children to talk to partners. Collect children’s responses and agree a rule that connects the numbers in the sequence. Ask them: Would 19 be

in the sequence? How do you know? Repeat with other number sequences.

Pose a problem: In my local ice-cream parlour there are two flavours of ice-cream, vanilla and

chocolate. I can choose two scoops of ice-cream. What two scoops do you think I will choose?

Children to talk to partners to find a solution. Collect children’s responses and show the

possibilities on IWB (or similar). Establish that there are three possibilities- point out that strawberry/vanilla and vanilla/strawberry are the same combination in this problem as the

scoops are side by side.

Ask children: Can you think of another way to record this? Children to talk to partners. Take

responses from children and establish that there are different ways of recording, including pictures, diagrams and the use of initial letters. Model a possible recording using initial letters

of the flavours, showing the three possibilities e.g. V + V, V + C, C + C

Pose problem: What if my ice-cream parlour sold three flavours of ice-cream, vanilla, chocolate and strawberry? I can still choose two scoops. Would there be more possibilities? Or fewer?

How do you know? Take responses.

Ask children to predict the number of combinations they think there will be and with a partner

find all of the possibilities, recording on mini-whiteboards. Take responses of the different combinations they have found.

Ask children: How do we know we have found all of the possibilities? Establish that there are

six possibilities with three flavours. Model a possible way to organise the recording systematically by keeping one of the variables

constant e.g. V + V C + C S + S V + C C + S

V + S

Discuss with children the use of a table for recording results:

Number of flavours

Number of combinations

1 1

2 3

3 6

4

5

6

Show the problem ‘Cobi’s Cones’ and read together. Establish five possible flavours of ice-cream e.g. vanilla, strawberry, chocolate, butterscotch and lemon.


** Ask children to solve the problem ‘Cobi’s Cones’, by first finding the number of combinations for four and then five flavours. Record results in a table, as modelled. Extend with: Make a

decision about Cobi’s statement and decide whether he is right or wrong, explaining why.

*Ask children to find the number of combinations for four flavours using resources to support,

if needed. Record this result in the table provided. Then find the number of combinations for five flavours, recording results in the table. Then read ‘Cobi’s Cones’ and establish whether he

is right or wrong.

*** Ask the children to solve the problem ‘Cobi’s Cones’, by first finding the number of

combinations for four and then five flavours. Record results in a table, as modelled. Make a decision about Cobi’s statement and decide whether he is right or wrong, explaining why. Extend with: Identify and describe any patterns in the table of results and predict the number of combinations for six flavours.

SEN: Use resources to find the number of combinations for four flavours, recording using objects or pictures. Encourage children to check that no combinations have been repeated. Use

language of ‘same’ and ‘different’.

Additional Challenge Question: How many combinations would be possible if there were ten different flavours to choose from? How will you find out?

Plenary / Drawing Together:

Discuss with children their findings and establish together that Cobi is wrong. Invite children

to show their work and ask them how working systematically helped them to find all possibilities. Show a completed table of results and ask children: Can you see a rule that connects the number of combinations? Children to talk to partners and then feedback responses. Ask

children to apply this rule to find how many combinations there would be for six flavours. Ask children: How did you work that out?

Follow-up Problem:

Pose a similar problem e.g. ‘Gino’s Gelati’. Gino has an ice-cream shop that sells four different flavours. He sells his cornets with three scoops of ice-cream. How many different combinations

could you buy from Gino’s Gelati?

Adapt the problem in response to AfL, including the provision of appropriate resources. Provide

limited scaffolding but refer back to ‘Cobi’s Cones’.

Look for patterns in the table of

results.

Can you predict how many

combinations we will have with

four flavours?


Count forwards and backwards in multiples of 7 and then 9 to the 12th multiple.

Pose a problem: In my garden there are some ladybirds with 9 spots. I counted 63 spots in

total. How many ladybirds are there? Children to talk to their partners to find the solution. Take a response from the children and ask: How did you work that out? Ensure children relate

this problem to division.

Repeat with a similar problem: In my neighbour’s garden, I saw ladybirds with 7 spots. I counted

56 spots. How many ladybirds are there? Take a response from the children and ask: How did you work that out?

However, in the local wildlife garden, we can find ladybirds with 4 spots and ladybirds with 5 spots. I counted 37 spots altogether. How many ladybirds with 4 spots and ladybirds with 5

spots are there? Children to talk to their partners to find a solution and record on mini-whiteboards. Take a response from the children and ask: How did you work that out? Ask children: Did anyone find another solution? How do we know that we have all the possible solutions?

Establish that the solution is three ladybirds with 4 spots (12 spots) and five ladybirds with 5 spots (25 spots). Address any misconceptions that may arise.

Model a possible recording to find the solutions using a table.

Show the problem ‘Zids and Zods’ and read together. Establish that they will only be one solution to the first part of this problem. Ask children: Which multiples will you be using?

Multiples

of 4

Multiples

of 5 4 5

8 10

12 15

16 20

20 25

24 30


** Ask the children to find the answer to the first part of the problem recording, as modelled,

using a table. Extend with: What if Zids have 5 spots, Zods have 7 spots and there are 70 spots altogether? Find the solution.

* Adapt the problem: At the wildlife garden, red ladybirds have 4 spots and yellow ladybirds

have 5 spots. Altogether I counted 36 spots. How many red ladybirds are there and how many yellow ladybirds are there? Children to record using the blank table provided. Extend with:

Find the solution to the first part of ‘Zids and Zods’. Provide blank table and multiplication

grid, if needed.

*** Ask the children to find the answer to the first part of the problem recording, as

modelled, using a table. Extend with: What if Zids have 5 spots, Zods have 7 spots and there are 140 spots altogether? Find as many solutions as you can.

SEN: In the wildlife garden, there are some red ladybirds with 5 spots. I counted 30 spots. How many ladybirds are there? Repeat with other multiples of 5. Use resources to support.

Additional Challenge Question: Some Zeds (with 7 spots) joined the Zids (with 4 spots) and Zods (with 9 spots). I can count

70 spots altogether. How many Zids are there? How many Zods? How many Zeds? Ask children

to find all possible solutions.


Collect solutions to the original problem. Establish that the solution is: 3 Zids (12 spots) and 4

Zods (36 spots). Ask children: What helped you solve this problem? What did you find challenging? Discuss how

recording in a table helps us to find all the solutions. Show the additional challenge question and discuss with children how they might approach this problem.

Follow-up Problem:

Pose a similar problem e.g. ‘Busy Basketball’. In a basketball match 88 points where scored.

Team A’s score was a multiple of 7 and Team B’s score was a multiple of 9. What was the result to the match? Is there more than one possibility?

Provide limited scaffolding for this follow-up problem but refer back to ‘Zids and Zods’. Adapt this problem as above and in response to AfL, including the provision of appropriate resources.


Display these three digits: 4, 7, and 5 and ask: How many three-digit numbers can you make using each of these digits once? Children create as many 3 digit numbers as they can, recording

on mini-whiteboards (457, 475, 547, 574, 745, 754). Ask: Which is the biggest number that you can make? Which is the smallest number that you

can make? Children talk to partners. Take responses and ask: How did you work that out?

Ensure children relate this to place value.

Pose further questions e.g. How many different numbers can you make in total? How do you know that you have found all

of the possibilities? Discuss systematic recording. Which number is closest to 500? Which number is the largest even number? Which is the

largest multiple of 5? What is the difference between the largest and smallest number that you made?

Pose an additional question: What is the digit sum of each of the numbers you made? Establish

that the digit sum is 16 (because 7 + 5 + 4 = 16). Ask: Can you find another number with a digit sum of 16? Children to offer suggestions e.g. 916

Show the problem ‘Three Digits’ and read together. Show the children an abacus and explain that the abacus works in a similar way to the place value grid. Ask: What is the maximum number of beads that can be placed on any of the sticks? Take a response from the children and ask: Why can’t you put 10 beads on a stick?

Read the problem again and highlight the fact that there are only 25 beads and that all beads must be used. Ask: If there are 25 beads and three sticks what is the smallest number of

beads you can have on a stick? Children to discuss with partners. Take feedback asking for a

clear explanation.

Demonstrate to children that if we put 9 beads on each of two sticks we will have used up 18 beads (9 + 9 = 18), which means that we only have 7 beads left over (25 – 18 = 7).

Establish that the smallest number of beads that we can have on any stick is 7 and the largest is 9, so the only digits we can use to solve this problem are 7, 8 & 9.

Ask children to use the digits to make a three-digit number with a digit sum of 25. Take one

example (e.g. 988) and agree that this is a correct possibility. Clarify that, in this problem, you can repeat the digits in order to find numbers with a digit sum of 25. If necessary, re-read the problem. Ask: How many three digit numbers do you think you will be

able to make with 25 beads? Predict first and then solve the problem.


** Ask the children to find all possibilities and arrange them in order, starting with the

smallest number. Extend with: Explain how you know that you have found all of the possibilities. Which number is closest to 850? Which is the largest even/odd number? What is the

difference between the largest and the smallest number?

* Adapt the task: Provide children with 25 beads and an abacus (or 25 counters and a place

value grid) and ask them to find all possibilities. Then ask them to record the numbers, ordering them from smallest to largest. Ask: How did you order these numbers?

*** Ask the children to find all possibilities and arrange them in order, starting with the

largest, and explain how they know that they have found all possible numbers. Extend with: If you use more beads there will be more possibilities? True or false? If you use fewer beads there will be fewer possibilities? True or false? Convince me!

SEN: Ask children to use three different digit cards to make a three-digit number and record this number on a place value grid (ensure that they can read the number that they have

made).Show the number on an abacus. Then ask them to make a different three-digit number with the three cards, recording as before. Ask: Which number is bigger? How do you know?

Additional Challenge Question: What number of beads allows you to make the maximum number of possible three-digit

numbers? How will you find out? How will you convince me that you are correct?

NB this is an extended task. Recording in a table will help children to see a pattern (of

triangular numbers).


Collect the solution to the original problem, establishing that there are 6 possibilities. Ask: How do we know that we have found all the possible three-digit numbers? Discuss

systematic recording. Pose the question: If we had 27 beads, how many different numbers could we make? Predict first and then solve. Establish that there would only be one solution, 999. Repeat with 26 beads,

predicting first and then finding all (three) possibilities. Ask: So do more beads give more possibilities or fewer possibilities?

Invite children who solved *** task to share their findings and conclusions when using fewer beads.

Follow-up Problem:

Pose a similar problem e.g. ‘Nadia’s new number plate’. Nadia is buying a new car and is having a personalised number plate made. She chooses the letters NJC (her initials) and then has to

choose a three-digit number. Her favourite number is 24 (her door number) so she chooses a three-digit number with a digit sum of 24. How many possible number plates could she make?

Adapt this problem in response to AfL, including the provision of appropriate resources. Provide

limited scaffolding but refer back to ‘Three digits’ problem.


Recall multiplication and division facts for a range of times tables up to 12 x 12, including identifying square numbers.

Pose a problem: Alesha has picked 18 oranges, which she needs to put in baskets. There must be an equal number of oranges in each basket. How could she do this?

Children to talk to their partners to find a solution. Take a response from a pair of children and ask: What helped you to work that out?

Discuss, encouraging the use of precise mathematical language such as factor, factor pair and multiple.

Ask children: Did anyone find another solution?

Through questioning, ensure children understand that she could put 9 oranges in 2 baskets (9 x 2) or 2 oranges in 9 baskets (2 x 9) and that 2 and 9 are both factors of 18.

Collect and record all (six) possible solutions. How do we know that we have all the possible solutions? How could you record your results to

be sure of this?

Take responses from the children and model a possible systematic recording.

e.g. 3 x 6 6 x 3

2 x 9 9 x 2 1 x 18 18 x 1

Show the problem ‘Adam’s Apples’ and read together. Discuss resources that might be useful.


** Ask the children to solve the problem ‘Adam’s Apples’, working systematically and

recording results in a clear and organised way. Ask children to explain how they know that they have found all the possibilities. Extend with: Consider why, in this case, there is an odd number of possibilities and explain.

* Adapt the problem: Adam picks 20 apples, which he needs to put in bags. There must be an

equal number of apples in each bag. How many different ways could he do this? Provide

resources, e.g. multiplication grids. Children to solve problem, recording in a clear and organised way. Extend with: What if Adam picked 25 apples? Do you think there would be more or fewer possibilities? Children solve problem and say whether their prediction was correct.

*** Ask the children to solve the problem ‘Adam’s Apples’, working systematically and

recording results in a clear and organised way. Ask children to explain how they know that they have found all the possibilities and why, in this case, there is an odd number of possibilities. Extend with: Find other number of apples which have an odd number of possibilities.

SEN: Adam picks 12 apples, which he needs to put in bags. There must be an equal number of apples in each bag. Give children 12 counters/apples and pose the question: If Adam has 4 bags,

how many apples will go in each bag? Children record with practical resources (consider taking photographs). Model recording: 4 groups of 3 = 12. Repeat with 3 bags, 2 bags etc.

Additional Challenge Question: What if Adam picks 29 apples? In how many different ways could he put them in bags?

Having solved the problem, children explain why there are only two possibilities. Extend by finding other numbers which only have two factors.


Pose Adam’s problem using 25 apples and collect all possibilities. Record using factor pairs. Establish that there are only three ways that Adam can put the 25 apples into bags.

Discuss why there is an odd number of possibilities using language of factors and factor pairs. Ask: What other numbers have an odd number of factors? Establish that square numbers

always have an odd number of factors. Collect solutions to the original problem (36 apples).

Pose the additional challenge question and establish that 29 is a prime number and that prime numbers only have two factors.

Ask: What helped you solve today’s problems? Children discuss with a partner and feedback.

Follow-up Problem:

Pose a problem e.g. Peter’s Primes: Peter says that all prime numbers (except 2) are the sum of two consecutive numbers. Is Peter correct? How do you know? Convince me!

Adapt the problem in response to AfL, including the use of appropriate resources.


Count forwards and backwards in multiples of 12 to the 12th multiple. Then count forwards in multiples of £1.20.

Pose a simple problem: An apple costs 40p. I spent exactly £2.00 on apples. How many did I buy?

Children to talk to their partners to find the solution. Take a response from the children and ask: How did you work that out?

Pose a more challenging problem: I am having a party and want to make a tropical fruit salad. A pineapple costs £1.20. A mango costs 70p. I spent exactly £10 on fruit. How many pineapples

and how many mangos did I buy?

Children to talk to their partners to discuss how they might approach the problem and then

record on mini-whiteboards. Encourage children to make a list of multiples of £1.20 and a list of multiples of 70p. Take feedback.

Model using a table to record.

Children discuss with their partners how the table will help them to

solve the problem. Draw out strategy of looking for pairs of numbers that total £10.00 (£7.20 + £2.80).

Establish that I bought 6 pineapples and 4 mangoes.

Show the problem ‘Farida’s Fish’ and read together. Discuss resources that might be useful.

Multiples of

£1.20

(pineapples)

Multiples

of 70p

(mangoes)

£1.20 70p

£2.40 £1.40

£3.60 £2.10

£4.80 £2.80

£6.00 £3.50

£7.20 £4.20

£8.40 £4.90

£9.60 £5.60


** Farida paid exactly £15.00 for some fish. How many of each fish did she buy? Children

find the solution to this problem recording, as modelled, using a table to record multiples of £1.80 and £1.40. Extend with: What if Farida paid exactly £20 for some fish? How many of

each fish did she buy?

* Adapt the problem: A clown fish costs 50p and a parrot fish costs 40p. Farida spent exactly

£3.00 on fish. How many of each fish did she buy? Children find the solution. If necessary,

provide a blank table for recording. Extend with: What if Farida spends exactly £6.00 on fish? How many of each fish did she buy?

*** Ask the children to solve the second part of the problem ‘Farida’s Fish’, finding both

solutions for £20 and recording, as modelled, using a table.

SEN: Children count forwards in multiples of 5 to the 10th multiple and then in multiples of 50 to the 10th multiple. A clown fish costs 50p. I spent exactly £1.50 on fish. How many clown fish did I buy? If necessary, provide children with 50p coins. Repeat with other multiples of 50p.

Additional Challenge Question: Farida also bought some rainbow fish, which cost £1.20 each. She spent exactly £25 on the

three types of fish (goldfish, angel fish and rainbow fish). How many of each kind of fish did she buy?


Return to the original problem, ‘Farida’s Fish’.

Display a table showing multiples of £1.80 and £1.40. Firstly, consider the solution when Farida spends exactly £15 and then when she spends exactly £20. Discuss the use of the table. Ask children: What did you find challenging? What helped you solve the problem?

Show the additional challenge question and discuss with children how they might approach this

problem.

Follow-up Problem:

Pose a similar problem e.g. ‘Spendthrift’ (problem 79 in ‘Problems and Puzzles’). Provide limited scaffolding for this follow-up problem but refer back to ‘Farida’s Fish’. Adapt this problem as above and in response to AfL.


Display a simple number sequence e.g. 6, 12, 18, 24 ... Ask: Tell me the next number in the

sequence. How do you know? Take a response. Ask children: What the rule is for the sequence in words? Children to talk to partners. Take responses. Ask: What is the 10th term in the sequence? How do you know? Take feedback.

Display a second number sequence e.g. 1, 7, 13, 19 ... Ask: Tell me the next number in the

sequence. Take a response and asked how they worked it out. Ask: Can you tell me a number in the sequence that is greater than 30? How do you know?

Children to talk to partners. Take responses. Ask: Would 67 be in the sequence? How do you know? Feedback.

Ask children: What the rule is for the sequence in words? Children to talk to partners. Take

responses. Ask: What is the 10th term in the sequence? How did you work it out? Feedback.

Pose a problem: Cara’s school has new triangular dining tables. Three children can sit around

one table: Four children can sit around two tables:

How many children can sit around 3 tables?

Provide whiteboards and pens so children could draw the diagrams if needed. Take feedback, encouraging a child to explain how they worked it out.

Ask: How many children can sit around 4 tables? 5 tables? 6 tables?

Children to talk to their partners to discuss how they might approach the problem. Encourage

them to consider how they could record their results in an organised way. Take feedback.

Model using a table to record, eliciting appropriate headings. Children discuss with their partners any rules and patterns that they identify

in the recording table. Take responses. Ask: How could the table help you to find the next term in the sequence and

the relationship between the number of tables and the number of chairs?

Take a range of responses encouraging children to explain the rules in words

(e.g. each time you add one table, you add one chair; the number of chairs equals the number of tables plus two) and using symbols (e.g. C = T + 2)

Show the problem ‘Paddy’s party’ and read together, ensuring children understand that Paddy’s tables are square. Explain that they can use any method to help them solve the problem, e.g.

drawing diagrams, recording in a table.


** Ask the children to solve the problem ‘Paddy’s party’, recording in a table, as modelled.

Children identify the rule and explain in words. Encourage them to look at the relationship between the number of tables and the number of chairs. Extend with: How many children can sit around 10 tables? Encourage children to use the rule to solve the problem.

* Adapt the problem: Fiona is organising a party for her little brother, Paddy. The children

will sit around square tables. Four children can sit around one table; six children can sit around

two tables. How many children can sit around 3 tables? 4 tables? What happens to the number of chairs when we add a table?

Children use a recording table, provided for them if needed. Encourage children to look for patterns. Extend with original problem: How many children can sit around 6 tables?

*** Ask children to solve the problem ‘Paddy’s party’, recording results in a table, as modelled.

Extend with: How many children could sit around 10 tables? 20 tables? Can you find a rule to find the nth term in the sequence?

SEN: Paddy is celebrating his birthday party. He has 2 square tables. How many children can he fit around them? What if he had three tables? Use resources (e.g. 2D shapes) to support.

Additional Challenge Question: Pose a question: 26 children are coming to Paddy’s party. How many tables will they need?

Encourage children to use the inverse to the previously identified rule.


Go back to original problem ‘Paddy’s party’. Discuss solution and table of results. Ask a child to share the rule in words. Ask children: How can we express that rule using algebra? Children

to talk in pairs, then feedback to the class. As a class agree that the rule is C = 2T + 2. Ask: How can we use that rule to find out how many children can sit around 25 tables? Ask: How can we use that rule to find out how many tables Paddy will need to accommodate 16

children? As a class agree that we have to use the inverse.

Follow-up Problem:

Pose a similar problem e.g. ‘Hannah’s handshake problem’: Hannah is having a party. If there

are four children at the party and everyone shakes hands with everyone else, how many handshakes would there be? What if there were 10 children at the party? Encourage children

to record results in a table, look for patterns and a rule; describe the rule in words and algebraically. Adapt the problem as above and in response to AfL, including the provision of appropriate

resources.

Tables Chairs 1 3

2 4

3 5

4 6

5 7

6 8


Display a Venn diagram with two intersecting circles labelled ‘square numbers’ and ‘multiples of

3’. Children to suggest numbers that would go in each section of the diagram, including the empty set.

Then play ‘Guess my rule’. Display a Venn diagram with 2 intersecting circles that have their

labels hidden e.g. ‘multiples of 9’ and ‘even numbers’. Children to offer numbers and teacher positions them in the correct part of the diagram.

Children to talk to partners, discuss the properties of the numbers and attempt to identify the rule.

Once children think they have identified the rule, they suggest a number and where it fits in the diagram e.g. number 18 in the intersection, number 49 in the empty set. As a class, agree

the labels on the Venn diagram.

Show the problem ‘Michael’s Money Bags’, and read the problem together. Ask children: Can you explain the problem? What do we know already? What do we have to find out?

Children talk to partners. Take feedback. Establish that the aim is to make any amount of money from 1p to 15p using the unopened money bags.

Explain that they can use any method to help them solve the problem, e.g. drawing diagrams, recording in a list, addition number sentences or a combination of all of these.


** Ask the children to solve the problem ‘Michael’s Money Bags’, recording their results in a

clear and organised way, using a method of their choice. Children to identify the pattern/rule

that connects the amount of money in each bag and explain it in words. Ask: Predict how much money would be in a fifth bag. What are the different amounts of money

you could make with five money bags? Extend with: How much money would be in the 10th bag? How did you work it out?

* Adapt the task by providing 15 pennies and four money bags. With adult support establish

that we need to put 1p in the first bag and 2p in the second bag and ensure they understand why. Children to then tackle the rest of the problem in pairs and record using a method of their choice. Extend with: Can you see a pattern? What is that pattern? Predict how much money would be in a fifth bag.

*** Ask the children to solve the problem ‘Michael’s Money Bags’, recording using a method

of their choice. Children to identify the pattern/rule that connects the amount of money in each bag and explain it in words. Then predict how much money would be in a 10th bag and explain

how they worked it out. Extend with: Can you identify a rule that connects the number of the bag to the number of

pennies in that bag? Encourage the use of a table to identify relationships and rules. If

necessary, draw children’s attention to the fact that the third bag contains 4p (2² = 2 x 2= 4),

the fourth bag contains 8p (2³= 2 x 2 x 2 = 8).

SEN: Give the children three money bags containing 1p, 2p, 4p. Children to work out all the

possible totals from 1p to 7p. Extend by giving them a bag with 8p in and ask children to find as many totals as possible.

Additional Challenge Question: Ask children to find a rule that would enable them to find the amount of money in the nth bag

and express this algebraically, using index notation.


Go back to original problem ‘Michael’s Money Bags’. Discuss the solution and recording methods

used. Children share the pattern/rule they identified, in words. Ask children: What helped you to solve the problem? Is there anything that you found

challenging? Take feedback. Ask: How would you find the amount of money in the 5th bag? 6th bag? 10th bag?

Children talk to partners and then give feedback. Discuss doubling methods but also encourage

children who solved *** extension to share their findings and express the rule that connects the number of bag to the amount of money in that bag.

Extend by discussing index notation and together find a rule for the nth term ( 2𝑛−1).

Follow-up Problem:

Pose a similar problem e.g. Sonia says that she can make any number from 1-20 using only the numbers 3, 5 and any of the operations (including brackets). Is she right? Prove it! Extend with: Claire says that she can make any number from 1-20 using only the number 3 and

any of the operations (including brackets). Is she right? Convince me! Adapt the problem in response to AfL, including the provision of appropriate resources.

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Mathematical challenges for all pupils - Southwark Schoolsschools.southwark.gov.uk/assets/attach/2239/Mathematical challenges... · can solve problems by ... The development of this