1. Bridging between Concrete, Pictorial and Abstract

Because Bar Models closely resemble classroom manipulatives (such as dienes and blocks), they make the jump from using physical objects, to representing maths problems pictorially, far less daunting.

They are also incredibly flexible: to begin with we can incorporate images of objects or present the bar as a ruler with units of measurement, for example. Gradually we can remove the pictures and children will still understand that the numbers and length of the bars represent a physical value.

2. Explaining the inverse relationship between addition and subtraction

Bar Models make solving missing number questions simple – if a child knows two of the values in a question, they can always calculate the other. This understanding also helps them recognise number bonds in different forms, such as when the “equals” sign is in a different position, giving them the confidence to tackle unfamiliar questions.

3. Proving Commutativity

Commutative law states that, in addition and multiplication sums, we can swap the two numbers around and the answer will still be the same (i.e. that “8 + 4” is equal to “4 + 8”). We can use bar models to prove this to the children, as they can see that the bars are the same length, whichever way around the parts are.

4. Addition: Aggregation

Bar Models are particularly powerful at showing that when two (or more) numbers come together, they make another number. They are helpful at explaining that part + part = whole, and are a nice stepping stone onto the more abstract “cherry” part-whole model.

5. Addition: Augmentation

Rephrase questions involving Bar Models to make them less intimidating, such as asking how long the bar will be when another part is added onto it. (We can also use Bar Models to explore the inverse operation, by asking how long the bar would be if part was taken away, or how much of the bar would remain.)

Tip: Bar Models can be very effective when used in conjunction with number lines, as both representations present horizontally.

6. Subtraction: Calculating the difference

As the Bar Model is a spatial representation, it enables children to visualise the physical difference between two values which, in turn, helps them work out how much they need to add onto the subtrahend bar in order to make it the same length as the minuend bar above. This can be done by counting up to the next multiple of ten, then by adding the tens, and then any ones.

7. Partitioning numbers in different ways

When it comes to partitioning numbers – especially into tens and ones – the Bar Model is an invaluable tool. Ask the children to draw how many different ways they could partition a two digit number into tens and ones. (They could use physical Dienes blocks for support.)

8. Representing larger numbers

Unlike many other pictorial representations, where each “one” is represented by an individual object, Bar Models are able to efficiently represent three and four digit numbers. By reusing this rare spatial representation throughout their mathematical journey, we help children to spot patterns and similarities between topics, as they tackle increasingly complex problems.

9. Making comparisons (and tackling word problems) 

Because they represent values spatially, Bar Models explicitly show the difference in size between two values; this can be useful for proving that one number is more or less than another.

Children can draw Bar Models to help visualise word problems or explain their workings, right through to KS4 – especially for problem solving questions that ask whether there is enough of something, or how much of something is required.

10. Introducing fractions

By using a representation the children are already familiar with, we can make the jump to learning about fractions less daunting as children know they’ve seen something similar before.

Bar Models also encourage children to use the correct terminology when talking about fractions from the start. They immediately recognise that halves are two equal parts of a whole, and that quarters are four equal parts of a whole.

BONUS TIP: Show your pupils how to freehand draw Bar Models fairly accurately, so that they can use them to estimate answers to addition, subtraction and number bond questions. By helping children predict what the approximate value will be, Bar Models are really useful for supporting children to quickly check their work by flagging any obvious mistakes.

So here are our 3 top tips to boost NumBots engagement over the school break:

1. Set an (achievable) Activity Goal

By equipping children with a platform like NumBots – which makes it easy and enjoyable to do maths at home – even homework averse families should keep on track with regular maths practice.

Use “Activity Goals” to set how long you expect pupils to play for. We suggest 12 minutes per week, as families are more likely to engage if the time set sounds achievable.

2. Encourage families to install the mobile app on all their devices

Whether staying at home, visiting family, or going to the beach, pupils can play NumBots anywhere in the world, via the mobile app – which is free to install on as many devices as you like.

As well as downloading it onto their children’s tablet, encourage parents to install the NumBots app on their own smartphones, so their children can play whenever the mood strikes!

Simply download the NumBots app from the device’s app store.

3. Show pupils how to download their own stage certificates

During term time, pupils are motivated by the certificates you reward for completing stages in Story Mode.

Make sure your pupils know how to download the stage certificates themselves, at home, so they can keep collecting them over the school break:

1. 1. Go to “Badges”.
   2. Click on the completed stage’s badge.
   3. Click “Download Certificate”.

BONUS IDEA:

Tell your pupils to set their sights on earning a bronze, silver or gold trophy in Challenge Mode. For example, “earn a silver trophy on all the subitising Challenges” or “a bronze trophy on the Number Bonds to 10 Challenge”.

Motivated by the trophies, they will then explore the rest of the platform.

(Maybe even promise a prize for those who achieve a trophy when they return to school!?)

Like everything we do at Maths Circle, this upgrade is not just about aesthetic improvements. We went back to the beginning and asked ourselves:

1. What do our users need from Challenge Mode?

2. How does it support sound pedagogical practice?

Have a look at Challenge Mode’s improvements:

Check out the new and improved Challenge Mode!

Challenge Mode now unlocks when children pass Iron (Stage 2) level 26 in Story Mode.

There are 30 Challenges in total which each belong to one of these four topics:

• Subitising (new)

• Number Bonds

• Adding

• Subtracting

Each Challenge focusses on one set of facts at a time – aligning with the work of Number Sense Maths.

During Challenge games, children attempt to answer as many questions as they can correctly in one minute. Unlike in Story Mode, which is highly visual to support learning, the questions in Challenge Mode are normally abstract (i.e. they don’t involve pictorial representations or mathematical models), in order to consolidate learning and improve efficiency of recall. Therefore children shouldn’t be encouraged to play a Challenge game until they’ve passed the levels teaching that topic in Story Mode.

Children replay Challenge games in order to beat their high scores, climb the leaderboards* and earn trophies and certificates.

The best bits:

• Quick and easy to practise the topic you want to improve.

• Children can self-identify where their weaknesses are in a positive way, by seeing which Challenges they are yet to earn a trophy for.

• Excellent for playing in class or against other players in your organisation – children are excited to appear on the leaderboards*.

• Grown-ups can track children’s performance and see how much they’ve improved at each calculation strategy.

• Personalised certificates build self-confidence and enthusiasm for mathematics.

They’re our top five favourite things about Challenge Mode. What are yours?? Go try it out and let us know!

* The leaderboards are only visible to children and grown-ups in your organisation (they’re not public) and they display players’ robot names, not their real names. While NumBots leaderboards normally encourage healthy, motivational competition, some teachers prefer not to allow children to see each other’s high scores. In this case they can hide the leaderboards from pupils by logging into their teacher account here, then going to “Settings & Admin” (in the left hand menu).

What exactly is a rekenrek?

The rekenrek contains two parallel bars; ten beads are threaded onto each bar.

The beads sit on the right hand side of the rekenrek when they are not being used. The user pushes the beads they require to the left when they are using them.

How is it different to other abacuses?

The key difference is that on each bar of the rekenrek, the first five beads are red and the last five are white. This transforms the rekenrek from a counting tool, to a subitising tool!

As well as being able to feel and visualise the number of beads, children can see how each number relates to 5 and 10. (For example, if they are asked to “Show 7” on the rekenrek, the children will know to push the five red beads and 2 of the white beads to the left, rather than count out 7 beads one by one.)

Children are encouraged to push the beads to the left as a group, rather than one bead at a time, as this moves pupils on from counting in ones and improves their number sense.

What are they used for?

As well as improving children’s understanding of each number, the rekenrek is particularly useful for establishing the base-10 system and explaining addition methods which refer to it.

Furthermore, using the “concrete” manipulative helps children to understand physically how whole numbers can be made up of two parts (part + part = whole).

The parallel bars of beads also enable children to compare two numbers. As a result, the rekenrek can be used in the classroom to support discussions about commutativity, more than/less than, calculating the difference between two numbers and subtraction.

How are rekenreks used in NumBots?

Given the many ways that rekenreks can support children learning maths, we decided to incorporate them into Story Mode on NumBots. The rekenrek levels have now joined all our other virtual manipulatives, including classroom favourites such as dienes and bar models.

Check a few of our favourite rekenrek levels out for yourself:

• Iron – Level 23
• Copper – Level 12
• Steel – Level 7