Seven powerful maths manipulatives for all ages

Many educators are advocating for better access to manipulatives across the school years and beyond, including in tertiary and community education programs. Learn more about the importance of manipulatives here: 

• Using Concrete Manipulatives in Mathematical Instruction
• The Concrete, Representational, Abstract (CRA) Approach
• Developing Your Use of Manipulatives in Maths Teaching

This list of concrete manipulatives for the maths classroom has been created in consultation with primary and secondary maths teachers, learning specialists and mathematicians. It has focussed on materials that are versatile across all ages. 

There are many other wonderful manipulatives that are valuable for exploring, playing with and understanding maths concepts. These are simply a set of favourites which we have found to provide a range of playful, positive and powerful opportunities.

For each item, we've included some suggested activities to play with 🛼 

1. Natural items

Gumnuts, pebbles, twigs and leaves. From counting collections to patterning and more, natural materials collected by students bring ownership, wonder and natural beauty to the maths learning experience.

If you're looking for more ways to leverage nature and the outdoors in maths instruction, we recommend this podcast interview with Sam Millar, who established a whole school Outdoor Learning Program. We also love the book Messy Maths: A Playful, Outdoor Approach for Early Years, which is full of exciting possibilities for harnessing the outdoors for meaningful maths across the junior school curriculum.   

2. Bottle caps

Creating collections from existing items is a wonderful way for students to get involved in the formation of resources they will use when exploring maths problems.  

An added benefit of using these as ‘counters’, is that the objects can be bigger and easier to pick up and manipulate, especially for smaller hands.

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A whole class, year level or school can be involved in a bottle cap or button collection. A particular class or year level could head up the collection, sorting and counting of items. They could use the counting collections routine to count and create collections of particular sizes for everyday use.

Thanks to Marissa Cashmore for drawing our attention to using bottle caps in this way.

3. Two-sided counters 

Given the binary aspect of many maths concepts, the use of two-sided counters transcends the curriculum and they are fast becoming one of our favourite manipulatives. Explore patterning, addition and subtraction, algebra, probability and binary data collection (e.g. yes/no). 

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Here are a couple of ways we like to get started with two-sided counters:

• Lulu - A Counter Game from Hawaii
• What’s red worth?

4. Square tiles

Square tiles can be used as counters but also provide special opportunities to connect arithmetic with geometry when exploring problems like Square Building or playing Subtract a Square.

Electing a transparent variation (picture below) makes them a preferred option for games and activities played on number charts.

Examples include Exploring place value and number patterns and playing Nim on a Hundred Chart.

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5. Dominoes

There is so much fun to be had with dominoes from kindergarten and Prep years through to adult. Getting started by having students check that they have a complete of dominoes is itself a task of mathematical value, as students sort and organise them in different, systematic ways.

Doug Clarke has shared lots of playful domino games and challenges which make great maths warm ups. For instance:

• Highest Sum: Find a partner to play with. Face all dominoes face down. Play begins with both players turning over one domino at the same time. They each add up the dots on their domino and the player with the highest sum takes and keeps both dominoes. If the sum is the same, they take one each. 
• Lowest Difference: Similar to Highest Sum, in this more challenging variation players work out the difference between the two sides on their domino. The player with the lowest difference takes both dominoes. 
• Domino Detectives: The leader/teacher has a secret domino. Individually, or in pairs, students place their full set of dominoes face up. The leader/teacher gives clues to help students figure out which domino in their set is the same as a secret domino. Clues are announced and time is given for students to eradicate dominoes. E.g. for a domino with 4 dots and 2 dots
  • 'The mystery domino is not a double'.
  • 'The sum of my dots is six'.
  • 'I do not contain any blank sides'.
  • 'The difference between my two parts is two'.

There are many domino puzzles and challenges to enjoy. One of our favourites is Domino Squares

6. Wooden cubes

There are many kinds of cubes that bring unique functions to the maths manipulative space (e.g. unifix cubes, multi-link cube). But if we had to pick one, it would be the humble wooden cube - lesser known yet versatile and full of creative potential. 

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Wooden cubes can be used for estimating and counting, addition and subtraction (especially 'difference'), height, length and volume, growing patterns, cubed numbers and more.

Here are some problem-solving activity suggestions using wooden cubes: 

• Four Cube Houses
• Which View?
• Painted Cube

For more wooden cube challenges, check out videos 2 and 3 of Marissa Cashmore's manipulatives workshop. 

7. Wooden pattern blocks

Last but not least (this is our all time favourite manipulative) are pattern blocks. Explore counting and arithmetic, fractions and proportional reasoning, shapes, linear and area measurement, symmetry and transformations, and more. All this and the joy that comes from playing and modelling mathematics with wooden pattern blocks,

Here are some of our favourite tasks for introducing wooden pattern blocks to students: 

• Start with Play: Give students time to play with and see what's possible with pattern blocks. Apart from the creative and mathematical stimulation, they will likely create artefacts that you can use as a springboard for a future activity or lesson.
• Pattern Block Triangles: An example of mathematical beauty in action, this lesson connects counting and addition practice to geometric shapes. The task itself involves a slow build from easier to more challenging work. Recommended for Year 1 through to adults.
• Forty Faces: A playful challenge that invites students to use addition and multiplication to connect number concepts to their artefacts. Recommended for Year 2 through to adults. 
• Symmetry Challenge: Create pattern block artworks and notice/learn about the symmetry present. Use A5/A4 sized, rectangular mirrors to confirm lines of symmetry. Explore transformations and rotational symmetry. 

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