problem solving with patterns pdf

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Number patterns

Here you will learn about number patterns, including how to find and extend rules for sequences, input/output tables and shape patterns.

Students will first learn about number patterns as part of operations and algebraic thinking in 4th and 5th grade. They continue to build on this knowledge in middle school and high school.

What are number patterns?

Number patterns are groups of numbers that follow rules. They can use input/output tables to create sequences .

Two types of sequences are arithmetic and geometric .

• An arithmetic sequence is a list of numbers where the same amount is either being added or subtracted every time.

Each sequence has a starting number, a rule and terms (the numbers that make up the sequence).

For example,

Rule: Subtract 6 each time.

There are also patterns, between the terms of two or more arithmetic sequences.

Both sequences start at 0. If you multiply the left column by 3, you get the terms in the right column.

This is because using the rule +9 is three times more than the rule +3.

Step-by-step guide:   Arithmetic sequence

• A geometric sequence is a number pattern where the rule is multiplication or division.

Rule: Multiply the previous term by 5.

Rule: Divide the previous term by 3.

Step-by-step guide:   Geometric sequence formula

Step-by-step guide:   Sequences

This page will highlight rules that involve whole numbers only.

[FREE] Number Patterns Worksheet (Grade 4 to 5)

Use this quiz to assess your grade 4 to 5 students’ understanding of number patterns. 10+ questions with answers on 4th and 5th grade number pattern topics to identify areas of strength and support!

• Input/output tables are tables that are used to show two sets of numbers that are related by a rule. The rule can be one step or multi-step, but has to work for each relationship shown in the table.

What is the rule for the table below?

To find the rule, look for the relationship between the input and the corresponding output.

Notice, 11 is being added to each input to get the output, so the rule is ‘add 11. ’

Step-by-step guide: Input/output tables

As students learn to work with number patterns, they can learn more about generalizing patterns by working with shape patterns.

• Shape patterns are any set of polygons, 3D shapes, letters or symbols that follow non-operational rules. There are repeating shape patterns and growing shape patterns .

A repeating pattern has a core that repeats over and over again.

The core of the pattern above is:

The core can be used to extend the pattern.

The next shape in the pattern would be:

because the last part of the core shown is:

• A growing pattern has parts that stay the same, but other parts that change.

• Changing – the left column starts with 0 and increases by 2.
• Staying the same – the 1 cube on the top right.

Step-by-step guide: Shape patterns

Common Core State Standards

How does this relate to 4th grade math and 5th grade math?

• Grade 4 – Operations and Algebraic Thinking (4.OA.C.5) Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3 ” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
• Grade 5 – Operations and Algebraic Thinking (5.OA.B.3) Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3 ” and the starting number 0, and given the rule “Add 6 ” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

How to use number patterns

There are a lot of ways to use number patterns. For more specific step-by-step guides, check out the pages linked in the “What are number patterns?” section above or read through the examples below.

Number patterns examples

Example 1: geometric sequence.

What are the next three terms in the pattern?

3, 6, 12, 24, 48…

Identify the rule.

Rule: Multiply the previous term by 2.

2 Use the rule to extend the pattern.

Multiply the last term by 2 to extend the pattern to the next three terms.

3 State and explain any patterns within the terms.

Except for the first term, all the terms are even.

The first term is odd, but multiplying it by 2 (even), makes the next term even.

From then on it is always an even term times 2 \rightarrow \text { even } \times \text { even }=\text { even }.

Example 2: compare arithmetic sequences

Compare the following sequences:

0, 2, 4, 6, 8… \, and \, 0, 12, 24, 36, 48…

Identify the rule of each sequence.

Find a pattern between the related terms of each sequence.

Look at the related terms for each sequence. What do you notice?

Both sequences start at 0. If you multiply the left column by 6, you get the terms in the right column.

Use the pattern to write a comparison statement.

The terms in the second sequence are 6 times the terms in the first sequence.

This is because using the rule +12 is six times more than using the rule +2.

Example 3: input/output table – identify the rule

What is the rule for the table?

Look at the relationship between the input and the corresponding output.

Looking from each input to output, see if you notice an obvious relationship. If not, use subtraction to find the difference between each input and the corresponding output.

Decide if the rule is add/subtract or multiply/divide.

Each output is 27 more than the input, so the rule is addition.

Write the rule.

Rule: Add 27 to the input.

Example 4: input/output table – find a missing value

Find the missing value in the table.

From each input to output, the difference is not the same, so the rule is not addition or subtraction.

Try to find a relationship that involves multiplying or dividing. Since the relationship from input to output is decreasing, try division.

Rule: Divide the input by 7.

*Note, the rule ‘Multiply the input by \, \cfrac{1}{7} \, ’ is also correct.

Use the rule to find the missing value(s).

70 \div 7=10

The missing value in the table is 10.

Example 5: 3D shape pattern, repeating

Create a rule for the pattern and find the next shape.

Identify the core – the part of the pattern that repeats.

This is a repeating pattern with cubes that go purple, purple, orange, purple, red, purple.

Notice that purple is repeated multiple times within the core. Always look at all the shapes given to confirm the pattern core.

Use the core to find and justify the next part in the pattern.

The next shape is:

Example 6: shape pattern, growing

Identify what is changing and what is staying the same.

• Changing – 0 triangles on the top and left side, 1 triangle on top and left side, 2 triangles on top and left side, 3 triangles on top and left side.
• Staying the same – The first triangle at the bottom is always there.

Create a rule based on Step 1.

Start with 1 triangle. Add 1 triangle to the top and 1 triangle to the left side each time.

Use the rule to find and justify the next part in the pattern.

The next part in the pattern is:

because 1 triangle is added to the top and 1 triangle is added to the side each time.

Teaching tips for number patterns

• There are many printable number patterns worksheets that can be used, but also give students the opportunity to create their own patterns. Then let their classmates find the rules and extend them. This helps keep students interested and deepens their understanding of patterns by challenging them to create and giving them opportunities to engage in discourse around patterns.
• Support students who are not yet confident in all operations by providing useful tools such as number lines, hundreds boards or counters. Completing calculations should not be a burden when working with patterns.
• Focusing on justifying pattern rules is as important as identifying them. Justification is not only a necessary mathematical practice, but doing it can teach students how to write better rules. Thinking about why a rule works, draws attention to the general parts of a pattern – how they are changing or staying the same, which is the basis of a valid mathematical generalization.

Easy mistakes to make

• Making an operational error Since finding the rule and extending the pattern require a calculation, this leaves room for mistakes to be made. Always double check your work and take time to think about if your answers are reasonable.
• Thinking there is only one way to write a growing shape pattern rule Often there is more than one way to describe what is changing and what is staying the same in a growing shape pattern. Encourage students to look for more than one way to describe their rule, as this is a helpful stepping stone to equivalent expressions.

Practice number patterns questions

1. What is the next number in the pattern?

4, 20, 100, 500, 2,500…

Multiply the last term by 5 to extend the pattern.

2. Starting number: 68

Rule: subtract 6 each time.

Which statement is true about the terms of the sequence described above?

All the terms are even

All the terms are odd

The terms alternate between even and odd

The first term is even and the rest are odd

Extend the sequence based on the starting number and the rule.

Since 6 (even number) is being subtracted each time, and even – even = even, all the terms are even.

3. What is the rule for the table?

Add 20 to the input

Multiply the input by 6

Multiply the input by 8

Add 25 to the input

Looking from each input to output, see if you notice an obvious relationship.

If not, use subtraction to find the difference between each input and the corresponding output.

Try to find a relationship that involves multiplying or dividing.

Since the relationship from input to output is increasing, try multiplication.

Rule: Multiply the input by 6.

4. Find the missing value in the table.

Each output is 12 less than the input, so the rule is subtraction.

Rule: Subtract 12 from the input.

Since the input is missing, thinking about what number subtracted by 12 will equal 83 :

The missing value in the table is 95.

5. What is the next shape in the pattern?

This pattern goes e, c, S, 8, o, e, 8 and then repeats.

6. What is the next part in the pattern?

Identify what is changing and what is staying the same:

• Changing – one purple in the middle and one green on either side, two purple in the middle and two green on either side, three purple in the middle and three green on either side.
• Staying the same – There is always a purple x with a green x beside it and below it at the top.

Rule: Start with a purple x with a green x to the right and below it. Repeat, adding the 1 st term again, but to the left and down one .

because it adds the first term again, but to the left and down one (shown in purple outline).

Number patterns FAQs

No, since students are most familiar with whole number operations, the standards start with these numbers. However, in middle school, sequences grow to include integers and other types of rational numbers. This also continues in later grades to include sequences with complex numbers.

Fibonacci’s sequence (or fibonacci numbers) and triangular numbers (pascal’s triangle) are two sequences that are commonly explored in upper level mathematics. Step-by-step guide : Triangular numbers (coming soon)

While shape patterns do not typically include fractions or decimals, they could be used in a way that does not involve calculations (for example in the repeating pattern \, \cfrac{1}{5} \, , \, \cfrac{1}{8} \, , \, \cfrac{1}{5} \, , \, \cfrac{1}{8} \, , \, \cfrac{1}{5} \, , \, \cfrac{1}{8} \, … ).

Yes, there are sequences that involve square numbers, cube numbers, and other more complex operations.

The next lessons are

• What is a function
• Laws of exponents
• Scientific notation
• Explicit formula
• Recursive formula
• Quadratic sequences

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Patterns Questions

Patterns questions and answers are given here to help students understand how to solve patterns questions using simple techniques. Solving different pattern questions will be beneficial for a quick understanding of the logic in patterns. In this article, you will learn how to solve patterns in maths with detailed explanations.

What are Patterns in Mathematics?

In mathematics, a pattern is a sequence of numbers that are formed in a particular way. Every pattern contains a specific rule. For example, the sequence of even numbers is a pattern since each number is obtained by adding 2 to the previous number.

i.e., 2, 4, 6, 8, 10, 12, 14,….

Here, 2 + 2 = 4

8 + 2 = 10 and so on.

Also, check:

• Number patterns in Whole numbers
• Algebra as pattern

A pattern can be of numbers or figures, which means we can also observe patterns in a sequence of similar figures. Let’s have a look at the solved problems on the various number and figure patterns.

Patterns Questions and Answers

1. Identify the pattern for the following sequence and find the next number.

2, 3, 5, 8, 12, 17, 23, ____.

2, 3, 5, 8, 12, 17, 23, ____

The pattern involved in the given sequence is:

12 + 5 = 17

17 + 6 = 23

23 + 7 = 30

Therefore, the next number of the given sequence is 30.

2. Observe the below figure and identify the missing part.

Consider the question figure, where the design in each part will be obtained by rotating the previous design by 90 degrees in the clockwise direction.

So, the missing part will be option (b).

Hence, the complete figure is:

3. Write the next three numbers of the following sequence.

173, 155, 137, 119, 101

The pattern in the given sequence is:

173 – 18 = 155

155 – 18 = 137

137 – 18 = 119

119 – 18 = 101

So, the next three numbers can be written as:

101 – 18 = 83

83 – 18 = 65

65 – 18 = 47

Thus, the sequence is 173, 155, 137, 119, 101, 83, 65, 47.

4. Observe the following figure and choose the correct option.

Each part of the square box contains a triangle inscribed in the circle in the given figure. Also, the triangle in the next circle is the vertical image of the previous one.

Similarly, in the second row, the triangle in the missing part will be an image of the previous one.

Thus, the missing part is option (a).

5. What is the formula for the pattern for this sequence?

11, 21, 31, 41, 51, 61, 71

The numbers in this sequence are written as:

11 + 10 = 21, 21 + 10 = 31, 31 + 10 = 41, and so on.

This can also be expressed as:

11 = 10 + 1 = 10 × 1 + 1

21 = 20 + 1 = 10 × 2 + 1

31 = 30 + 1 = 10 × 3 + 1

41 = 40 + 1 = 10 × 4 + 1 and so on.

From this, we can write the formula for the above pattern as: 10n + 1, where n = 1, 2, 3, etc.

6. Find the pattern in the sequence and write the next two numbers.

10, 17, 36, 73, 134,…

Given sequence is:

10 = 1 3 + 9

17 = 2 3 + 9

36 = 3 3 + 9

73 = 4 3 + 9

134 = 5 3 + 9

So, the next number = 6 3 + 9 = 216 + 9 = 225

Again, the next number = 7 3 + 9 = 343 + 9 = 352

Therefore, the sequence is:

10, 17, 36, 73, 134, 225, 352.

7. Observe the pattern given below. Find the missing number.

In the given figure, we can observe that the sum of the four numbers is equal to the number written in the middle of the shape.

That means,

11 + 22 + 33 + 44 = 110

16 + 24 + 32 + 40 = 112

? + 23 + 34 + 12 = 114

? = 114 – 23 – 34 – 12 = 45

Therefore, the missing number is 45.

8. What is the next number of the following sequence?

20, 18, 21, 16, 23, 12, 25, 8, 27, 4, 33, ?

Given sequence:

20, 18, 21, 16, 23, 12, 25, 8, 27, 4, 33

Let us find the difference between two consecutive numbers of the sequence to identify the pattern.

18 – 20 = -2

21 – 18 = 3

16 – 21 = -5

23 – 16 = 7

12 – 23 = -11

25 – 12 = 13

8 – 25 = -17

27 – 8 = 19

4 – 27 = -23

33 – 4 = 29

Here, we can see that the differences are the prime numbers.

So, the number number = 33 – 31 = 2

9. Estimate the next number of the following sequence.

1, 2, 6, 15, 31, ?

Let’s write the difference between consecutive numbers.

2 – 1 = 1

6 – 2 = 4

15 – 6 = 9

31 – 15 = 16

Here, 1 = 1 2 , 4 = 2 2 , 9 = 3 2 , 16 = 4 2 .

Thus, the next number of the sequence will be obtained by adding 5 2 , i.e. 25, to the previous number.

Therefore, 31 + 5 2 = 31 + 25 = 56.

10. What will be the next number of the given sequence?

1, 5, 12, 22, 35, ?

Let’s write the difference between consecutive numbers.

5 – 1 = 4

12 – 5 = 7

22 – 12 = 10

35 – 22 = 13

The difference between these numbers follows a pattern that 3 is odd to the previous difference.

So, the next number will be obtained by adding 13 + 3, i.e. 16 to 35.

Hence, the next number = 35 + 16 = 51.

Practice Problems on Patterns

• Find the correct number to complete the pattern given below.

20, 21, 23, 26, 30, 35, 41, ___.

• Find the next number in the sequence, 12, 21, 23, 32, 34, 43.
• Write the missing numbers in the following.

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Patterns in Problem Solving

Do you detect a pattern here? Solve the problems by making a table. Keep going with More Patterns in Problem Solving .

View aligned standards

Related guided lesson.

Multiplication 1

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Manipulatives in Maths - A Classroom Guide for Teachers

Mathematical manipulatives are touted as essential tools for learning, but let's be honest—we've all experienced that moment of dread when we hand them out. Suddenly, your carefully planned lesson turns into chaos: One pupil starts building a fortress with the base ten blocks while another is hiding all the shiny counters.

Yet, despite these challenges, manipulatives play an important role in maths education. They bridge the gap between abstract concepts and tangible understanding, helping pupils grasp basic number sense. In fact, the National Curriculum emphasises their importance across all key stages, recognising that hands-on learning is vital for developing maths fluency, reasoning, and problem-solving skills.

So, how can we take advantage of these tools without losing control of the classroom? Let's explore the world of maths manipulatives—what they are, why they matter, and how to use them effectively in your primary school lessons.

What are manipulatives?

It can sound complicated, but manipulatives are simply hands-on tools that make abstract mathematical concepts concrete and visual . They're the building blocks, quite literally in some cases, that help pupils wrap their heads around tricky number ideas through good old-fashioned play, exploration, and modelling.

These learning aids come in all shapes and sizes, from the humble counter to the more elaborate Cuisenaire rods . Their key purpose? To give pupils something tangible to manipulate as they grapple with mathematical concepts. Whether it's using multilink cubes to understand place value or fraction circles to visualise parts of a whole, manipulatives help bridge the gap between 'maths on paper' and 'maths in real life'.

Common manipulatives you'll find in primary classrooms include:

Multilink cubes

Cuisenaire rods, base ten blocks, bead strings.

• Balance scales

Clock faces

Digit cards, hundred squares.

These tools align perfectly with the National Curriculum's aims of developing mathematical fluency, reasoning, and problem-solving skills. By allowing pupils to physically interact with mathematical ideas, manipulatives help build a strong foundation for more complex concepts down the line. They're not just toys or distractions—they're powerful learning tools that can transform how your pupils understand and engage with maths.

Why are they important?

Over the past two decades, research has consistently shown the positive impact of using manipulatives in the classroom. A 2013 report published in the Journal of Educational Psychology identified "statistically significant results" when teachers used manipulatives compared with when they only used abstract maths symbols. This highlights the role that manipulatives play in supporting conceptual understanding and facilitating the progression from concrete to abstract thinking.

Alignment with CPA approach

The NCETM agrees that physical manipulatives should play a central role in maths teaching. "Manipulatives are not just for young pupils, and also not just for those who can't understand something. They can always be of help to build or deepen understanding of a mathematical concept."

This approach aligns perfectly with the concrete-pictorial-abstract (CPA) progression. Once children are confident using manipulatives or 'concrete' resources, they can then move onto pictorial representations or the 'seeing' stage. Here, visual representations of concrete objects are used to model problems. This stage encourages children to make a mental connection between the physical object they just handled and the abstract pictures , diagrams or models that represent the objects from the maths problem.

Enhance problem solving

But manipulatives do more than just support understanding—they're powerful tools for enhancing problem-solving skills. By allowing pupils to physically manipulate and visualise mathematical concepts, they can more easily devise strategies to tackle complex problems. This hands-on approach often leads to those 'aha!' moments we all love to see in our classrooms.

Support engagement

Moreover, manipulatives play an important role in fostering engagement and motivation. Let's face it—maths can sometimes seem dry and abstract to young learners. But introduce some colourful counters or interlocking cubes, and suddenly you've got a room full of eager mathematicians. This increased engagement is key to developing a positive attitude towards maths, which in turn supports long-term learning.

This deep understanding allows pupils to move beyond mere memorisation of facts and procedures, towards true mathematical fluency—where they can apply their knowledge flexibly and efficiently across a range of contexts.

In essence, manipulatives are not just helpful additions to our maths teaching toolkit—they're essential components in building a comprehensive, engaging, and effective mathematics education.

Types of manipulatives in primary mathematics

In this section, we'll break it common types of manipulatives into bite-sized pieces, just like we do for our pupils.

Physical manipulatives: the classics

These are the tangible, grab-them-with-your-hands resources that have been the backbone of maths classrooms for years. They're the ones that inevitably end up stuck between classroom seats and occasionally in someone's shoe.

Below is a list of common physical manipulatives in the classroom:

Ideal for teaching place value, addition, and subtraction with regrouping.

Fraction tiles

Excellent for comparing fractions and understanding equivalence.

Great for exploring 2D shapes, symmetry, and area.

Images: Wikipedia.org

Versatile tools for counting, measuring, and understanding volume.

Fantastic for developing number sense and exploring number relationships.

Essential for basic counting, sorting, and introducing simple addition and subtraction.

Useful for teaching multiplication, division, and fractions.

Image: Pinterest

Helpful for developing number sense and practicing skip counting.

Useful for probability exercises and generating random numbers for various activities.

Great for pattern recognition, matching, and basic addition facts.

Essential for teaching time-telling and understanding intervals.

Images: Pinterest & Pinterest

Useful for place value activities and forming large numbers.

Excellent for identifying number patterns and supporting multiplication and division.

Virtual manipulatives: a new kind of tool

Manipulatives have gone digital! These are interactive, online versions of our physical favourites. Think of them as the maths equivalent of e-books.

Some popular virtual manipulatives include:

Online number lines

These number lines are zoomable, clickable, and free of the uneven lines that are often result of our hand-drawn versions.

Digital base ten blocks

All the functionality without the risk of losing pieces under desks.

Interactive fraction tools

Slice and dice up pieces in any way imaginable.

Whether physical or virtual, the best manipulative is the one that helps your pupils understand the concept at hand. Whether that's a handful of multilink cubes or a fancy online simulator, if it's making those mathematical lightbulbs flicker on, you're on the right track!

Implementing manipulatives in the classroom - let them play!

Whether you have a bumper pack of manipulatives, a shared bank of resources or your very own DIY versions, it's important to teach children how to use them independently. Here are some best practices for integrating manipulatives effectively into your lessons:

• Introduce gradually : Bring in manipulatives one at a time. If you don't have enough for each child, set up a 'maths table' where pupils can take turns exploring. This works particularly well with younger years where 'choosing tables' are common.
• Allow for exploration : Give children a chance to play with and explore the manipulatives before using them for instruction. Through this exploration, they can start to imagine how the resource might be useful.
• How could you use this?
• How might this help you when adding or subtracting?
• Why do you think they're different sizes - what could that represent?
• Model usage : Once children are familiar with a resource, introduce a simple maths problem and ask them to use the manipulatives to solve it. Model the problem-solving process step-by-step, then guide children through it.
• Scaffold learning : Start with highly structured activities, then gradually reduce support as pupils gain confidence. For instance, begin with direct instruction on how to use base ten blocks for place value, then move to guided practice, and finally independent problem-solving.
• Year 1: Using counters or number lines to support addition and subtraction within 20.
• Year 2: Use fraction tiles to help pupils recognise, find, name and write fractions of a length, shape, set of objects or quantity.
• Year 3: Utilising place value charts (physical or digital) so pupils can recognise 3-digit numbers (100s, 10s and 1s).
• Integrate into lesson plans : Don't treat manipulatives as an add-on. Instead, weave them into your lessons as essential tools for understanding. Plan specific points in your lessons where manipulatives will be most beneficial.
• Support diverse learners : Manipulatives can be particularly helpful for English Language Learners (ELLs) and pupils with learning disabilities. They provide a universal language of mathematics that transcends verbal communication barriers.

Images: The Average Teacher

Manipulatives across Key Stages 1 and 2

Next, let's breakdown more examples of manipulatives in the classroom by Key Stage.

Key Stage 1 (Years 1-2): Laying the foundations

In these early years, it's all about getting hands-on with numbers and shapes.

• Number and Place Value : Introduce counters, number lines, and base ten blocks. Pupils can observe how 10 ones form a 'ten stick', helping them grasp place value concepts.
• Addition and Subtraction : Utilise multilink cubes for hands-on learning. Pupils can physically join or separate cubes to represent addition and subtraction operations.
• Fractions : Fraction tiles can be effective tools for teaching fractions. They provide a visual and tactile representation of concepts like 'half' and 'quarter'.
• Geometry : Employ geoboards for creating 2-D shapes. Pupils can then be asked to match these shapes on a 3-D surface to enhance spatial understanding.

Key Stage 2 (Years 3-6): Progressing with Purpose

As our mathematicians-in-training grow, so does the sophistication of our manipulatives. We're not ditching the basics, just building on them.

• Multiplication and Division : Array cards and Cuisenaire rods are useful for these operations. For multiplying by 6, pupils can line up 6 rods of 4 to visualise the concept.
• Fractions, Decimals, and Percentages : Fraction circles can be used alongside decimal place value charts. The 100 square is effective for teaching percentages.
• Geometry : The geoboard is a helpful tool for teaching perimeter, area, and symmetry concepts in a hands-on manner.
• Statistics : Data can be represented using multilink cube bar charts or human pictograms, making statistics more engaging for pupils.

CPA Journey: From Concrete to Pictorial to Abstract

Remember, our end goal is for pupils to solve problems without relying on physical props. Here's how we might progress:

• Concrete : Pupils physically manipulate objects to solve a problem. For example, using counters to work out 5 + 3.
• Pictorial : They draw a picture or diagram to represent the problem. Our 5 + 3 might become five circles and three circles.
• Abstract : Finally, they use mathematical symbols and numbers alone. "5 + 3 = 8."

The beauty of this approach? Pupils can always 'go back' a stage if they're struggling with a new concept. Stuck on an abstract problem? Draw a picture! Need more practise? Grab those counters!

Remember, every child's journey through these stages is unique. Some might race through, others might linger longer at certain points. The key is to ensure they have a solid understanding at each stage before moving on.

An example of moving from the concrete, to pictorial, to abstract stages.

Manipulative manners

Once you have introduced your resources, speak as a class and explain that they should come up with a set of rules for how they are treated and used. Giving children ownership over the manipulatives as well as the respect to make their own rules will make them feel accountable and lessen the likelihood of negative behaviours when using manipulatives. Write the rules up as a class and display them so they can be referred to.

Storing manipulatives

NRICH recommends children having access to manipulatives “Give open access to all the resources and allow the children free reign in choosing what to use to model any problem they may be tackling. I would make sure that children of all ages had this access from 3 to 11 years old and beyond.” While this is exactly what teachers would like to replicate in their classrooms, not all classes learn in the same way and this isn’t always achievable due to space, budgets and children’s prior experiences of manipulatives.

Once you have introduced a manipulative, decide as a class where you should store it . You know what works best for your class, so consider different options such as communal drawers, a maths table, individual packs or a collection of manipulatives for each table. Set clear rules around using and treating manipulatives to ensure they are not broken or lost. Additionally, you could create a monitor for each resource so the children can take ownership and make sure they stay tidy and accounted for.

Creating a classroom culture that uses manipulatives will aid children’s fluency and help develop their ability to solve problems, reason mathematically and share! If manipulatives are introduced in a considered and gradual way, with clear boundaries from an early age, children should see them as part of everyday learning and they will not be a novelty. They will be seen as tools instead of toys — and hopefully no more multilink towers!

For the community, by the community

Maths — No Problem! Community Event Conference, 28 November at The Royal Society, London.

Lily Lanigan

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Every Problem, Every Step, All in Focus: Learning to Solve Vision-Language Problems With Integrated Attention

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