Although the content in this blog has been written for Year 6 teachers, Year 5 teachers, in fact all teachers, are more than welcome here!

Dear Year 6 teacher,

Well, what a year so far. Not the year you pictured or planned I imagine? A year which would normally bring school journeys, end of year productions and, of course, SATs…

### End of Key Stage 2 assessments

This letter to you was supposed to be delivered right at the beginning of the Spring term, when we thought that we knew what the assessment scheduling would look like. In fact, I wrote this originally: *‘Whether we think they should or shouldn’t take place, we know now that it is very likely that our Year 6 pupils will sit their maths SATs tests. I shall be careful here not to say that they definitely will – just in case!’ *Just call me Mystic Meg!

What do we know now?

On page 55 of the ‘Restricting attendance during the national lockdown’ guidance document found here, there are details about the fact that statutory assessments planned for summer 2021 have been cancelled. You, of course, will know about this already!

### Our moral purpose

So does this cancellation mean that we shouldn’t worry about any form of assessment? Of course not I hear you cry! It is necessary that we continue with formative assessment methods whether we are teaching the children face to face or remotely (see Kate Kellner-Dilks wonderful blog, ‘Where is the overlap between learning at home and online learning?’ here), as well as forming summative judgements.

Irrespective of this announcement and this year’s measures, I am sure that you will agree that whether SATs were to take place or not, we want to ensure that our Year 6 children are as ready for secondary school and KS3 maths as they can be. In addition to this,

schools will likely to be reporting to parents in the summer term. On page 55 of the document mentioned above it says:

*‘We know that schools will continue to use assessment during the summer term to inform teaching, to enable them to give information to parents on their child’s attainment in their annual report and to support transition to secondary school. We strongly encourage schools to do this, using past test papers if they wish.’*

### USING assessments to inform planning

It is, in my opinion, really useful to continue to use SATs papers to inform planning for the rest of this year. With the cancellation of the tests, this also gives you more time. Having taught in Year 6 for a number of years, the May deadline, was often in the back of my mind in terms of securing learning and coverage. **We now have some time back to prioritise and teach concepts deeply**.

But in a year where teachers are needing to consider missed learning and gaps in knowledge, what should the priorities be? In this first letter to you, I have considered some of the key arithmetical concepts.

It may be that your Year 6 children have recently completed practice SATs papers and so these may have helped you to identify concepts which need to be revisited, either in small groups or as a class, or highlighted concepts which have not yet been taught. It is worth taking time to **diagnose **incorrect answers and to **dig deeper**.

- Has a mistake been made that the child could correct on their own if they were asked to look at the question again?
- Or is the error due to misconceptions or a lack of understanding of the concept being tested?

If this concept has been taught since September, this may be a priority to revisit.

If the concept hasn’t been taught yet this year, it will give useful insights into pupils’ prior knowledge and starting points. Looking at the ESSENTIALmaths and the White Rose Maths long term plans, this may include domains such as ratio and proportion and geometry as learning sequences yet to teach. I will add a note here that during this current period of lockdown, teachers may want to consider which priorities, in the context of remote education, be taught successfully. Personally, I would wait to teach ratio and proportion when we have the children back in the classroom if at all possible.

### Learning worth revisiting and securing

So what learning might it be worth revisiting either in fluency sessions, maths meetings or 10 minute revision sessions in terms of your pupils’ **arithmetical proficiency**?

A detailed analysis of the 2019 arithmetic paper highlighted how important secure place value knowledge is and how many of the questions require this understanding (particularly when combined with understanding of multiplication facts).

For example, questions such as:

- 9 x 41,
- 1210 ÷ 11 and
- 5/6 x 540.

When tackling questions such as these, children who have efficient methods will have more time to deal with those calculations that are more complex.

### Building from secure learning

One particularly useful strategy (and especially this year) is to use low entry point models and worked examples to build from. As Hendrick and Heal (2020) point out, *‘students’ prior knowledge about a particular domain is really the key element in terms of how their learning will progress’*.

It is important not to assume children’s confidence with previously taught content. Using low entry point models provides an opportunity to explore prior knowledge and experiences and then build out from there.

For example, let’s take the calculation mentioned previously – 9 x 41.

If we start from ‘9 x 4 =’ you are able to see the groups (both of 9 and of 4) so can check understanding of commutativity (9 x 4 = 4 x 9). You can also see the product, count it if necessary, and therefore make connections to the language of multiplication.

You are also easily able to make the connection to understanding of division – one which often even Year 6 children find hard to articulate.

### Prioritise place value understanding

It is easy to assume that children recognise the connection between 9 x 4 = 36 and 9 x 40 = 360, with 9 x 40 being ten times greater than 9 x 4. I would urge you not to assume. These links may well have been made in previous years but lack of security in this understanding can significantly impact arithmetical fluency so it is worth the time now.

For some pupils, it might be that changing the language from 9 x 40 to 9 x 4 tens can secure that understanding of 4 **tens **being tens times greater than 4 and then 36 **tens **being ten times greater than 36.

By using the base facts and understanding of place value, the calculation 9 x 41 then becomes 9 x 40 + 9 with no need for a formal written method.

### From here you can go almost anywhere!

What if the fact was not a known fact? How could you calculate it? Perhaps using commutativity, so 4 x 9, or using a known benchmark fact, e.g. 10 x 9 – 4 or the associative law, so 2 x 2 x 9.

What if the fact was not a known fact? How could you calculate it? Perhaps using commutativity, so 4 x 9, or using a known benchmark fact, e.g. 10 x 9 – 4 or the associative law, so 2 x 2 x 9.

You could consider how the model would change to show 9 x 5, 9 x 3, 19 x 4 (using 10 x 4 and 9 x 4)… For your more confident children, there is much depth to be found in exploring alternative strategies.

However, the place I would suggest that you must spend some time is making connections by building from base facts.

It is also important to make links to decimals, again not assuming that children will recognise the connection to 9 x 0.4 or 0.9 x 0.4.

Using place value counters or giving the tens in the image above a value of 0.1 instead, will help children to see these relationships.

Similarly, with the calculation 5/6 x 540, if children can find relationships between the numbers – 540 and 6 in this case – and make connections with a simpler example, 5/6 of 54, this calculation becomes a lot easier and more accessible.

This understanding will also help children when solving calculations using formal long multiplication. For example, for the calculation 3468 x 62…

Can children articulate how each partial product is found, so 2 x 8, 2 x 60, 2 x 400 etc. Do they make the common error when multiplying by the 6 tens in the multiplier (62 in this case)?

So 6 **tens **x 8 is 48 **tens **or 480, rather than misrepresenting the value of the 6 and instead recording 6 x 8 = 48.

### Prioritise rehearsal of language

Focusing on language is equally important when using long division.

I’ll make a confession here – I didn’t really understand long division when I was first taught it at secondary school. I couldn’t remember the procedure that we were meant to follow and I never really got my head around it until I started teaching it using magnetic base 10.

Since my school days (longer ago than I care to think about!), I have heard of a mnemonic – dangerous monkeys swipe bananas – which, while useful to remember the process (divide, multiply, subtract, bring down), does little to help me to understand. What is divide? Why am I subtracting? What is the ‘bring down’ part all about?

Now it may be tempting, especially in a year where we know that we have much content to cover and missed learning to make up for, to skip straight to teaching long division procedurally but **it is vital that children experience the ‘whats’ and the ‘whys’** of long division.

Building time in to model calculations using place value counters or base-10 and allowing children to explore these models and articulate exactly what is happening (without the help of monkeys and bananas) will really help this learning to stick. **It is likely that Year 6 children may have missed rehearsal, or even teaching of long division in Year 5 and so a little longer spent here will be time well spent.**

As mentioned previously, starting with full worked examples (such as the image below) and then exploring models with familiar divisors will mean that children can focus on the structure of the procedure. The following images illustrate how a speaking frame and models could be used to explore the calculation 3016 ÷ 13.

A useful scaffold here would be to provide lists of the multiples required (multiples of 13 in the example above) so that calculating these is not the focus. Once children are more confident with the method, these can then be removed.

### Carefully consider the outcomes from assessments

As with long division, when multiplying and dividing fractions, which is new learning in Year 6, spending time exploring the structure of the mathematics before teaching the tricks, or even better, children generalising about examples and discovering the tricks for themselves, will help the learning to stick.

When working with a Year 6 teacher recently, we analysed arithmetic papers for the class and two areas that few children had answered correctly were:

- multiplying pairs of fractions and
- dividing a fraction by a whole number.

The teacher went on to tell me that when the children had their papers returned to talk through chosen strategies and misconceptions, little attention had been paid to these questions as they had not been taught. The teacher wanted to explore these fully in the spring term rather than present the tricks and flicks, which could be tempting to do in order for these questions to be answered correctly. Similar to the monkey and bananas long division procedure above, the tricks may be remembered for a short amount of time but won’t demonstrate secure understanding and are likely to be forgotten or misapplied in the future.

### Prioritise reasoning within arithmetic

Again, dependent on your long term plan, percentages may not have been introduced in Year 5 if they were towards the end of the spring or planned for in the summer term so it will be important to consider this for your class.

In the 2019 arithmetic paper, there were 4 questions which required pupils to find percentages of amounts and the average percentage of pupils answering these correctly ranged from approximately 82% (for 20% of 3,000) to 53% (for 36% of 450).

Now I know that this isn’t about preparing for a test. However these questions provide useful guides for us. When calculating with percentages, it is not one size fits all in terms of a method and so the use of concrete models to explore and play about with how to move from what is known to new facts in order to build multiple strategies and demonstrate true understanding is so important.

The beadstring is an excellent resource for this.

*If the beadstring represents £100, then 100% is what?*

*What is each bead worth?*

*Which section would represent 50%? And 10%? And 1%?*

We talked earlier about connections, and this is an ideal place to draw connections to children’s understanding of fractions and part whole.

*What if the beadstring represented £300?*

*What would each bead be worth now?*

*How does that affect what else you know?*

Using the sentence stem, **‘If I know this, then I know…’** encourages children to be flexible with using what is known. The beadstring also exposes connections to number lines and supports children to consider the magnitude of percentages they are working with.

In 2018, there was a question asking children to find 99% of 200.

I have seen countless practice papers where children have found 10%, found 1%, multiplied both by 9 and then combined. While not incorrect in their method, many then made errors somewhere along the line in their calculations. They are following a procedure.

But if we consider the position of 99% relative to 100%, we can now start to be more efficient in our chosen methods. Consider these three:

Another method that could be employed is the scaling method.

*If 99% of 100 is 99, how could this help us?*

This brings us back to encouraging those connections. If this strategy hasn’t been used, displaying it as a worked example, discussing, and then playing with it may be a useful way of adding to a child’s strategy toolkit.

As children become more fluent in approaching arithmetic questions, ensure that connections are made to other representations that they may come across. For example, do they understand **why **5/6 **x** 540 could also be represented as 5/6 **of** 540?

What other examples could they give? Is 9 x 4 the same as 9 of 4?

When faced with the question: 1 3/4 x 10, will your children connect this to the arguably easier 1.75 x 10? Answers of either 17 1/2 or 17.5 were acceptable.

The following blogs: Analysis of 2019 KS2 Maths SATs Arithmetic paper (Part 1) and Analysis of 2019 KS2 Maths SATs Arithmetic paper (Part 2), written by Rachel Rayner, are well worth reading, or re-reading, as they provide useful analysis of the 2019 arithmetic paper.

### Further professional development opportunities

For more insights into achieving mathematical age-related expectations in Year 6, join us for our spring term digital CPD which begins on Wednesday 10th February and includes 2 live webinars and 4 pre-recorded sessions. More details and booking information can be found here.

### References:

Hendrick C and Heal J (2020) Just because they’re engaged, doesn’t mean they’re learning. Available at https://impact.chartered.college/article/just-because-theyre-engaged-doesnt-mean-learning/ (accessed 15th December 2020)

https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/950510/School_national_restrictions_guidance.pdf (accessed 11th January 2021)

Contains public sector information licensed under the Open Government Licence v3.0

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