Reactivate – assess – teach – rehearse… Year 4 problem solving with fractions

    Published: 06 October 2020

    I actually wrote the vast majority of this blog in January, before the words ‘unprecedented’, ‘socially-distanced’ or ‘lockdown’ became commonplace in our vocabulary. But interestingly, even though so much has happened, I haven’t had to make too many changes! What I have done though, is add in some verbs that I hope WILL become commonplace in the classroom – ‘reactivate – assess – teach – rehearse’. Oh… and I also changed the title.

    So what I have done is drop these words in across my original blog to illustrate that this is how we would have taught before and this is how we should be teaching still. As Siobhan King writes in her Back On Track in Maths blog from July 2020, ‘be confident and uncompromising in your pedagogy’.

    Ok, so from this point forward, anything that I have added to the original blog is in purple

    Reactivate and assess

    There are 30 children in your Year 4 class. It’s Monday. Before break time, the children were learning how to use ‘the whole’ and the number of equal parts to find fractions of quantities. Just after lunch, the children were asked to solve the following problem to assess their current understanding (this assess was already there!):

    2/3 of 27

    Group A (3 children) gave the answers 23, 7 and 3.

    Group B (6 children) gave the answer of 9.

    Group C (21 children) gave the correct answer of 18.

    Following the children providing their answers, a short discussion took place around their strategies and models.

    Also, just to clarify here, the groups above are not based on any kind of ability grouping – the children are merely ‘grouped’ based on the answers they gave to this particular question to allow discussion of the ‘what now’.

    In the maths lesson on Tuesday, the children will be using their skills from Monday to solve problems such as:

    Joel collects marbles with his brother.

    Joel has 18 marbles and owns 34  of the collection.

    How many marbles do the brothers have altogether?

    So what do we do to develop the reasoning and problem solving skills of the whole class within this context, given the responses to the assessment question above?

    Assess - teach

    Group A appear to have a misunderstanding about the concept of finding a fraction of a number. A quick conversation with them reveals that this is indeed the case. They have created numbers using the digits in the problem but were unable to talk about parts and wholes or equal groups – cornerstones in the understanding of fractions of amounts. Some pre-teaching will be key for them in accessing the learning tomorrow. We have 10 minutes in the afternoon that we can make use of for this. The children will consider this model, similar to models used in this morning’s lesson but taken from a Year 1 learning sequence:

     

    Table

     

    They will lay their cubes on blank models and move them to share into equal groups. Explicit links will be made to the familiar bar model and familiar strategies for division. Language developed will include equal groups, parts, the whole, sharing and grouping, half, third and quarter. This language will already be up on the working wall so could either be referenced or taken down for the children to use.

     

    Table

     

    For example:

    • The whole is 12.
    • When finding thirds, the whole is shared into three equal groups.
    • The whole, 12, can be shared between 3 parts. There are 4 in each part.
    • When finding 1/3, you will have one of the 3 equal parts.
    • 1/3 of 12 = 4.

    The children will take the vocab cards and the fraction models used and pop them back up on the working wall, ready to reference tomorrow.

    Teach - assess

    So…it’s Tuesday. Now what?

    When teaching today, bear in mind that (potentially) 9 of the children (Groups A and B) have an understanding of unit fractions but may not be so secure with non-unit fractions so this will be a key piece of learning to focus on while teaching the reasoning. Let’s briefly go back to an earlier model from Year 4 to draw previous learning out of the long term memory (reactivate). For discussion during the model, they are going to be sitting in flexible pairings to allow all children to take part in rich discussion. Group A can bring their language from the pre-teach yesterday to the equation (a nice bit of ‘assigning competence’ (Babcock)).

    To begin with, multi-link cubes can be used to explore the structure of fractions. For example:

     

    Graphic image

    What fraction of the cubes are red? What fraction are blue?

    How do you know?

    Children need to be using precise language here, for example:

    • There are 5 cubes so each one is worth one fifth.
    • Four fifths are red because 4 out of 5 cubes are red.
    • 1 fifth is blue because 1 out of 5 is blue.

    Repeat this with:

     

    Graphic image

     

    So… let’s start building a problem. To give all of the children ‘a way in’, provide a statement to represent:

    Sally buys four fifths of the shop’s apples.

    Build up a model together. Reason together. Support the children with finding a starting point.

    How many equal parts are we working with? How do you know? Where’s the clue?

    Highlight the key word here using a louder intonation when read aloud:

    Sally buys four fifths of the shop’s apples.

    Have the class repeat this back, making the ‘fifths’ more prominent.

    Refer the children to the working wall at this point, where there are some bar models showing a range of fractions (as in the examples from the pre-teach above). The language from the pre-teach will also be there for both the adults and the children to use.

     

    Fractions table

     

    Group A will notice (or be prompted to notice) that this as a familiar model from yesterday’s pre-teaching of unit fractions. There may be a new piece of language in ‘fifths’.

    Repeat this strategy with the next part of key information to build into the model.

    What else do we know?

    We know that… Sally buys four fifths of the shop’s apples.

     

    Fractions table

     

    Now add some more information:

    Sally buys four fifths of the shop’s apples. She bought 20 apples.

    What else do we know?

    We know that… Sally bought 20 apples.

    Build this new information into the model, making the link to the four fifths explicit.

     

    Fractions table

     

    There is no ‘answer’ yet because the children were presented with statements, not a problem to solve. This allows the focus to be on the structure and not on who can get to the answer quickest.

    Now the children will discuss what the question could be. They may suggest things such as:

    • How many apples were left in the shop?
    • What fraction of the apples were left in the shop?
    • How many apples were there in the shop before Sally did her shopping?

    This will allow attention to be drawn to whether each question involves finding a part or finding a whole.

    Sally buys four fifths of the shop’s apples. She bought 20 apples.

    How many apples were in the shop at first?

    What do we already know? What do we not know?

    We know that…

    • Sally has four fifths of the apples and that is 20.
    • one fifth of the apples must be 5 because 20 ÷ 4 = 5 (This is where the modelling is crucial because the children may at this point do 20 ÷ 5 because they are working with fifths.)
    • fifths are 5 equal parts so every part must be worth 5.

    We don’t know…

    • the whole number of apples at the start.

     

    Fractions table

     

    Make conscious connections here by drawing on links to known number facts (5 x 5), repeated addition or counting in multiples to bring in familiar learning.

    Sally buys four fifths of the shop’s apples. She bought 20 apples.

    How many apples were in the shop at first?

     

    Fractions table

     

    There were 25 apples in the shop at first

    Make the strategy explicit to the children:

    • Model ‘the statement’ (what you know)
    • Tackle the problem (what you don’t know) and take small steps to find a solution

    A decision can then be made as to who is ready to tackle a similar problem (to rehearse) following the teacher model and who will need some further guided teaching. Those working independently should be referring to the working wall which shows bar models for halves, quarters, thirds and fifths, alongside the sentence starters of, ‘We know…’ and ‘We don’t know…’.

    Problems to tackle could include:

    Joel collects marbles with his brother. Joel has 18 marbles and owns 3/4 of the collection.

    How many marbles do the brothers have altogether?

    or

    27 goals were scored over the weekend, with two thirds of the goals occurring on Saturday.

    Use this information to fill in the goals below:

     

    Table

     

    or

    Opal class have 30 pupils, of which 2/3 are boys.

    Ruby class have 25 pupils, of which 3/5 are boys.

    Which class has more boys?

     

    The TA, armed with key vocabulary (e.g. whole, equal parts, numerator, denominator) to use and possible fraction scaffolds to choose from, will focus on the children working independently. Attention will be drawn to the working wall and the initial teacher model to promote discussion around ‘what’s the same and what’s different?’ The prompts used throughout the model can also be used here… ‘What do we know?’ and ‘What do we need to find out?’

    Meanwhile, the teacher will work with the children who need support with finding a starting point to get them into their problem. For their initial problem (Joel and his brother), they will be given a choice of scaffolds to use – bars broken up into thirds, quarters and fifths. This will support with a starting point and also make an explicit link back to the pre-teaching from yesterday. This group will be carefully monitored by the teacher to allow misconceptions to be picked up and tackled in the moment but with an appropriate expectation for the children to work on the problem independently.

    Just a quick note here about the impact of school closures in summer 2020…

    Following that initial reactivation question (or based on a transition conversation with the Year 3 teacher), it may have become clear that the teaching that this learning builds upon was missed or had been poorly retained. If this had been the case, planning for the direct teaching would need to take this into account.

    Year 3 sequences or steps could be used to ensure that children are building from secure foundations. They will need to have explored and made sense of unit and non-unit fractions for example. Destination Questions from the Year 3 learning sequence could be used to reactivate previous learning to see how well children can reason around fractions prior to planning the teaching. For example:

    Sally has 20 pets.

    Two fifths are fish. One fifth are hamsters.

    The rest are cats.

    How many of the pets are cats?

    Further rehearsal could then take place outside of the maths lesson over a period of time through fluency sessions which focus around reasoning to problem solve and finding that starting point. (This rehearsal was already here!)

    So back to the original lesson structure…

    Strategies employed in this lesson to differentiate were as follows:

    • Diagnostic assessment for learning
    • Pre-teaching
    • Drawing on previous learning to pull from long term memory (reactivation)
    • Concrete and visual modelling using small step reasoning
    • Peer-to-peer discussion
    • Split-start independent and guided learning
    • Scaffolds on which to hang the maths problem
    • Strategic adult support

    These are only some of the strategies that can be used to differentiate in the primary classroom. For more of a look at how to use scaffolding to enable access to learning that children may not be able, or willing, to reach without it, have a read of Charlie Harber’s blog, ‘Differentiation in maths - scaffolding or metaphorical escalators!

    Resources and upcoming training to support teaching approaches detailed in the blog:

    Fluency Session Materials

    Back on Track: Mathematics

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