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DOTS + TODS™
DOTS+TODS™ counters are Oakfield Learning's teacher designed alternative to maths manipulatives such as double sided or two-colour counters and algebra tiles. They can help pupils ages 9+ to develop a deeper understanding of the basic concepts of algebra and how to add, subtract, multiply and divide positive and negative integers. Their interlocking design provides a clear concrete representation of a zero pair.
HOW DO THEY SUPPORT THE MASTERY CURRICULUM USING THE CPA APPROACH?
FURTHER READING: HOW TO HELP CHILDREN UNDERSTAND POSITIVE AND NEGATIVE INTEGERS, DIRECTED NUMBERS AND ZERO PAIRS
Set of 100 DOTS+TODS™ (50 of each ) made from durable, thick plastic easy to grasp and simple to use
What are DOTS+TODS™
White DOTS Counters (+1)
Interlocking Zero Pair Design
White DOTS Counters (+1)
Each white DOT counter is used it represents a positive one (+1).
To represent the value positive 4 (+4), four white dot counters would be used.
(+1) + (+1) + (+1) + (+1) = +4.
If two more white DOT counters are added to the group altogether the white DOTS would then represent the value +6.
Black TODS Counters (-1)
Interlocking Zero Pair Design
White DOTS Counters (+1)
Each black TOD counter represents a negative one (-1).
To make the value of negative four, four black TODS counters would be used.
(-1) + (-1) + (-1) + (-1) = -4
To make the value negative six, two more black TOD counter would need to be added to the group.
Interlocking Zero Pair Design
Interlocking Zero Pair Design
Interlocking Zero Pair Design
To be able to understand the rules of adding, subtracting, multiplying and dividing positive and negative integers (also called directed numbers), it is essential to have a secure understanding of zero pairs. What are Zero Pairs?
Our DOTS+TODS have been designed to interlocking design to make a clear representation of a zero pair.
(+1) + (-1) = 0 (-1) + (+1) = 0
How do they support the maths mastery curriculum?
CONCRETE
PICTORIAL
PICTORIAL
PICTORIAL
PICTORIAL
PICTORIAL
ABSTRACT
PICTORIAL
ABSTRACT
CPA Approach
Concrete > Pictorial > Abstract
The CPA approach is a method used to help scaffold learning. It is an essential tool for maths mastery.
When being introduced to an abstract mathematical concept that can be particularly hard to grasp like numbers below zero, physical objects (often referred to as maths manipulatives) help by modelling a problem. Pupils can interact with representations such as our DOTS+TODS™ to them to develop a deeper understanding of directed numbers and basic algebra.
When they are confidently using concrete manipulatives, pupils will be able to use drawings, pictures or diagrams of the concrete objects or use diagrams which support learners by still seeing what is happening whilst being less reliant on concrete materials.
Finally, once they have mastered mathematical concepts using concrete and pictorial representations, they will be able to apply what they have learnt and use more efficient methods such as abstract symbols and numbers in order to solve a problem .
Concrete Representations
When zero is often thought of as 'nothing', how is it possible to represent a value of less than nothing using something?
DOTS+TODS™ are physical representations that help pupils to make sense of numbers above and below zero. They can also be used to support basic conceptual understanding of algebra.
They can be used to help develop a secure understanding of what happens when directed numbers are added, subtracted, multiplied and divided. Using DOTS+TODS™ can help pupils to make sense of the rules involving positive and negative numbers such as:
+ + = + - - = + + - = - - + = -
Pictorial Representations - Number Lines
We recommend using DOTS+TODS™ in conjunction with number lines to help pupils with the transition of using pictorial representations to support the move from concrete to abstract.
In our tutorial videos we often display number lines alongside our concrete modelling with DOTS+TODS™.
Progressing to pictorial representations is essential when using larger numbers where would be impractical and time-consuming to use concrete resources.
In addition to horizontal number lines, vertical number lines are also incredibly useful, especially when first introducing pupils the concept of negative numbers. Thermometers could be used as a concrete model before moving on to pictorial representations.
More Pictorial Representations
Real-life experiences like temperatures, can really help children to grasp mathematical concepts.
DOTS+TODS™ are a based on a commonly used analogy of making sandcastles (also called piles and holes).
A sandcastle/pile (representing a value of positive one) can be made from from what has been dug out when making a hole (which represents a value of negative one). If the contents of a sandcastle/pile (+1) are added to a hole (-1) it fills the hole leaving a flat surface (0). These pictorial representation are useful to further explain a really abstract concept like zero pairs.
Of course a maths trip to the beach may be a fun way to explore this in an interactive way; perhaps a more practical alternative would be to draw diagrams representing piles and holes like in the image above.
Piles and Holes
In our search to find pictorial representations for adding and subtracting directed numbers using the concept of piles and holes that is used in many schools , we came across James Tanton's 'Exploding Dots' concept.
There are lots of videos but we found these two particularly helpful. In this first video he explains how to draw piles and holes. In the next video, James Tanton demonstrates how the concept of 'dots and 'anti-dots' or 'dots and tods' act as an alternative visual representation of piles and holes and zero pairs.
DOTS+TODS™
As part of the CPA approach, in order to understand maths at a deeper level, it is essential to use tangible concrete resources such as DOTS+TODS™ first so pupils can gain experience and understanding. Physical resources also helps to engage pupils in the learning and make maths fun!
When they are ready to progress to using pictorial representations, pupils could draw on whiteboards or in their maths books just like James Tanton does in his videos.
Some believe that concrete resources are not necessary for older pupils, however this is not the case. You might like to find out more about why maths manipulatives should still be used in secondary school by reading this article.
Abstract Representations
After developing a deep conceptual understanding using the concrete DOTS+TODS™ and pictorial representations, pupils should be able to use only abstract numbers and symbols which will now have more meaning.
They will be able to use these as a much more efficient way to solve problems involving directed numbers and algebra.
What are Zero Pairs?
What are Zero Pairs?
What are Zero Pairs?
Whilst pupils may think zero just represents 'nothing' and has 'no value' as it represents 'no quantity or amount' it certainly isn't worthless and in fact it can make a big difference! It may be interesting for them to discover that 0 hasn't always been recognised as a integer with its own symbol. This article gives some interesting historical facts about 'WHO INVENTED THE ZERO'. Did you know that zero is an even number?
It is vital that pupils realise the importance of zero as a place holder between decimal numbers and whole numbers and positive and negative integers. Until now, maths resources such as two-colour or double sided counters have commonly been used to explore zero pairs. We believe our unique interlocking DOTS+TODS™ give a much clearer practical and visual way to explore how negative and positive integers combine and form a 'pair' that have a 'zero' value.
Zero pairs are made when two opposite numbers with the same digit can be added together to make zero. One of the numbers needs to be positive and one has to be negative. For example: +1 and -1, +5 and -5, +10 and -10.
The white DOT counter (with a value of positive one / +1) can combined with a black TOD counter (that has a value of negative one /-1 to represent the following:
0 + 1 - 1 = 0
So,
(+1) + (-1) = 0
and
(-1) + (+1) = 0
What are Zero Pairs?
What are Zero Pairs?
This video give a detailed explanation of the meaning of zero and zero pairs. It also demonstrates some activities that can be done with DOTS+TODS™ to help pupils investigate and understand that no matter how many counters they use, as long as there is the same number of positive one (+1) DOTS as negative one (-1) TODS, they will always pair up and to represent a value of zero.
This is helpful when pupils are first introduced to adding and subtracting positive and negative numbers; they often make a common mistake.
Here is an example: (+2) + (-2) = either +4 or -4
Because DOTS+TODS™ are concrete resources that pupils can physically interact with, they can actually see and understand see why both of these answers are incorrect. As each time one a DOT is added to a TOD they make combine to represent zero, the two white DOT counters would 'pair up' with the two black TODS making two sets of zero pairs which altogether represents the value 0. As nothing plus nothing is equal to nothing (0 + 0 = 0)
So the correct answer is: (+2) + (-2) = 0
More tutorial videos can be found below and on our YouTube channel.
For more ideas of how to help children to understand positive and negative integers, directed numbers and zero pairs you may like to read this article.
To find out more about where negative and positive integers and directed numbers fit within the National Curriculum for England, read the section below.