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I can explain the use of the commutative and associative laws when multiplying three or more numbers.

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Licence

This content is © Oak National Academy Limited (2023), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Key learning points

You can change the order of the factors or group them in different ways and the product remains the same.

Common misconception

Pupils only ever calculate an equation from left to right.

Ensure pupils are given time to explore where the commutative law might be best used and where the associative law might be better used. Pupils should be expected to reason why they have chosen to adopt a strategy.

Keywords

Commutative - The commutative law states that you can write the values of a calculation in a different order without changing the calculation; the result is the same. It applies for addition and multiplication.
Associative - The associative law states that it doesn't matter how you group or pair values (i.e. which we calculate first), the result is still the same. It applies for addition and multiplication.

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Hi, my name's Mr. Peters and in this essence today we are gonna be thinking about developing our understanding of the commutative and associative law and how we can apply these when multiplying three numbers together.

If you're ready, let's get started.

So by the end of this session today, you should be able to say that I can change the order of the factors or group them in different ways and the product each time would remain the same.

Throughout the session today, we've got a couple of key words we're gonna be referring to.

The first one is factor.

You have a go at saying them after me.

The first one, factor, second one, product, the third one, commutative, and the fourth one, associative.

Factors are whole numbers that exactly divide another whole number.

The product is the result of multiplying two or more values together.

The commutative law states that you can write the values in a calculation in a different order and the result will stay the same.

This applies for both addition and multiplication.

And the associative law states it doesn't matter how you group or pair values in a calculation, it doesn't matter which pair we calculate first as the result will be the same throughout as well.

This also applies to both addition and multiplication.

So this lesson today is broken down into two cycles.

The first cycle will be to express and represent three factor problems, and the second cycle will be manipulating three factors.

If you're ready, let's get going.

Throughout the session today, we're gonna have both Izzy and Alex to join us, and they're gonna be sharing their thinking and their questions throughout to help us develop our understanding too.

So we can see here that in supermarkets, strawberries are often displayed in open trays, and in each one of these trays we can see that the strawberry punnets are placed into rows and columns.

In these examples, there are three rows and there are four columns.

If there were two trays of strawberries out on display, how many punnets would there be altogether? Hmm, how do you think we could represent this as a calculation? That's right, we can say it's three multiplied by four multiplied by two.

Let's see where these numbers come from then.

Well, the three is represented by a number of rows in a tray.

The four is represented by a number of columns in the tray.

And then finally, the two represents the number of trays that we've got altogether.

Let's think about this in a different context now.

In the dairy aisle, boxes of eggs are stored on a shelf.

There are four boxes left on the shelf.

What calculation would help us to find the total number of eggs on the shelf? Here's an example of the boxes.

Take a a moment for yourself to have a think.

That's right, we could represent this as two multiplied by three, multiplied by four.

Let's again see where these numbers come from.

Well, in one tray there are two rows, so the two represents the number of rows, and then in one tray there's also three columns, isn't there? So the three represents the number of columns, and as we know, there are four boxes on the shelf on there.

So the four represents the number of boxes that there were.

Okay, time for you to check your understanding now.

Have a look at our picture, what does each number represent? The two represents the number of rows in one tray, doesn't it? The three represents the number of columns in one tray and the five represents the number of trays that there are altogether.

Well done if you got that.

Okay, let's revisit the context of the strawberry punnets.

We could use cubes to represent this where one cube would represent one punnet of strawberries.

Let's see how we can build this up together.

Let's take a look at the blue layer on the top.

We know that this blue layer has three rows and it has four columns.

Then we also know that there are two layers here.

So we've got the top layer, which would represent the first tray of strawberries, and then we've got the bottom layer, which would represent the second tray of strawberries.

So we can represent this as three multiplied by four, multiplied by two.

The three represents the number of rows in one tray, the four represents the number of columns in one tray, and the two represents the number of layers, or in this case, the number of trays that we've got.

In terms of calculating this, we can say that three multiplied by four multiplied by two is equal to 12 multiplied by two 'cause three times four is equal to 12.

12 multiplied by two is 24.

And therefore we can say that there are 24 punnets of strawberries on display.

Have a look at our cubes this time.

What's the same and what's different about them? Well, that's right.

It's exactly the same shape as what we've just used, isn't it? However, this has been separated into three different colours this time.

So each layer looks slightly different.

Let's work through this and think about how we could calculate it this time.

Hopefully we can see that in one layer, for example, the blue one, there are two rows and there are four columns.

So we could represent this as two multiplied by four.

And how many layers are there all together? That's right, there's three layers this time, aren't there? So we can represent this as two multiplied by four multiplied by three.

The two, as we know, represents the number of rows.

The four represents a number the columns, and the three represents the number of layers this time, doesn't it? So we could say that two multiplied by four multiplied by three is equal to eight multiplied by three.

And again, that would be equal to 24.

Hmm, it's exactly the same answer as what we had before, wasn't it? And let's have a look at this last example as well.

What'd you notice this time? That's right, the number of cubes, again, has stayed the same, hasn't it? And also the number of layers has changed, hasn't it? We've got four layers this time.

Let's see how we could represent that.

Let's have a look at the blue layer.

We can see that it has two rows and three columns, and there are four layers with exactly the same layout as the blue layer.

So we could represent this again as two multiplied by three multiplied by four.

The two represents the number of rows, the three this time represents the number of columns, and the four represents the number of layers this time as well.

Again, two multiplied by three multiplied by four is the same as saying six multiplied by four because the two times by three makes the six, and therefore six fours are 24.

So total amount would be 24 cubes.

So let's look over these three again then, have a look at them, what'd you notice? That's right, you may have noticed they've been coloured into different layers, haven't they? But also the amount of cubes stayed the same each time.

And the calculations we did were reordered and actually the product remained the same each time, didn't they? So we can say that when we're multiplying, no matter what order we place the factors in, the product will always remain the same.

Okay, a quick check for understanding here then.

Say the sentences and write the equation to match the cubes.

One layer has two rows and three columns.

There are five layers altogether.

We can write this as two multiplied by three, multiplied by five.

Okay, time for you to have a go at practising now.

Can you look at each one of these cubes and fill in the stem sentences below, including writing the equation at the bottom? Once you've done that, have a look at these different contexts and write the expression, which would represent each one of these problems. Good luck with that and I'll see you back here shortly.

Okay, let's run through these together.

In the first cube, there were four rows, there were two columns, and there were three layers.

We can write that as four multiplied by two multiplied by three, and that was equal to 24.

In the second one, there were three rows, there were two columns, and there were four layers.

Again, we can represent this as three multiplied by two multiplied by four this time, and that would still make 24.

And then the last one there were four rows, three columns, and two layers this time.

So four multiplied by three multiplied by two is also equal to 24.

Okay, and then for each one of these examples, the first one, there are five chocolate chips on each cookie and there are six cookies on each tray, and there are two trays.

So we can represent this as five for each chocolate chip, six for the number of cookies that we've got, and then two for the number of trays that we've got altogether.

In the second one, there are six petals on each flower, and there are five flowers in each vase.

And then there are two vases.

So we can represent this as six multiplied by five multiplied by two.

The six represents the number of petals, the five represents the number of flowers, and the two represents the number of vases.

And finally, if a horse is fed six carrots a day, and there are two horses in each stable, and there are five stables in the barn, how many carrots are fed each day? Well, we could represent this as six for the number of carrots, two for the number of horses in each stable, and five for the number of stables in each barn.

Well done if you've got all of those.

Okay, on to cycle two now, manipulating three factors.

Okay, so let's have a look here.

We've got a block here made up of cubes.

And if we want to find out the total number of cubes, how could we represent this as an equation? Alex seems to think there's gonna be more than one way of doing this, take a moment to have a think for yourself.

Well, here's one way we could tackle it, isn't there? Let's use our stem sentence to help us here.

Maybe you could say it with me.

One layer has three rows and one layer has five columns.

There are four layers altogether.

So we could write this as three multiplied by five multiplied by four, couldn't we? This would be one way to find the number of cubes, which in this case would be 60.

Alex is saying we could say this as three times five four times.

Let's have a look at a different way.

Have think this time, how could you represent this? Again, say the sentence then with me.

One layer has four rows, one layer has five columns, and there are three layers altogether.

So we could represent this as four multiplied by five multiplied by three.

And again, we know that's now going to be 60, don't we? And again, we could say this as four multiplied by five three times, four times five three times.

Can you have a go at saying that? And here's one more example as well.

Have a look at each layer this time, how many rows and columns are there? Okay, let's use our stem sentence then.

One layer has four rows, one layer has three columns, and there are five layers altogether.

So we can represent this as four multiplied by three multiplied by five.

Again, that would equal 60 altogether.

And we could also say this as four times three five times.

We've got four lots of three, and we've got that five times.

So let's go back and look at all these three again together.

What was it that we noticed? Well, hopefully you've noticed by now that the product was the same for each one of these equations, wasn't it? The product each time was 60.

There were 60 cubes in total in each amount.

And Alex has noticed that the factors were placed in different arrangements, weren't they? Again, in the first one we have three, five, and four in that order.

The second one we had four, five, and then three.

And then the third one we had four, three, then five.

So we can begin to generalise this.

We can say that if you change the order of the factors, the product remains the same.

Could you have a go at saying that? Let's have a look.

Three multiplied by five multiplied by four is equal to 60, four multiplied by five multiplied by three is equal to 60, and four multiplied by three multiplied by five is equal to 60.

The factors keep changing position, but the product remains the same each time.

When the order of the factors change and the product remains the same, we call this a special law.

The law is called the commutative law, can you say that? So the commutative law tells us that when the order of the factors change position, the product remains the same.

Well done if you spotted that.

Okay, time for you to check your understanding now.

True or false, the commutative law means you can swap the position of the first two factors in an equation only? Take a moment to have a think.

Okay, and we know that's false, don't we? Have a look at our justifications.

Which one of these helps to support your argument? That's right, it's the second one, isn't it? You can have any number of factors and swap the position of those, and the product will remain the same.

It's not just the first two factors you're allowed to swap around.

So we've got a cuboid again, and we've got three expressions which represent how we could find the total number of cubes in this cuboid.

Alex said, \"Did you know that we don't actually have \"to multiply from left to right each time either?\" So far we've always started with a number on the left and then multiplied it by the number in the middle and then multiplied that by the number on the end.

We don't actually have to work like this.

Izzy is saying, \"What do you mean?\" Well, let's have a look.

Alex says, \"When multiplying, you can choose a pair \"of numbers to multiply first.\" So let's have a look.

So in this example, we can multiply the three by the five, first of all, and then multiply that by the four, and we can use a pair of brackets to show which pair we multiply together first.

So if we multiply the three by the five, first of all, that would give us 15, and then we can multiply that by four, which would give us the 60, wouldn't it? However, we could also do this another way.

We could multiply the five by the four, first of all, and then multiply that by the three.

So let's put some brackets around the five and the four, first of all.

We know that five times four is equal to 20, and then if we multiply that by three, that would be equal to 60 again.

Izzy's pointed out the product has remained the same no matter what pair of factors we multiplied first.

Hmm, that's really interesting.

Which pair did you prefer to multiply first? We're going to look at another example now, but before we do that, have a quick look at how the layers on this cuboid have been laid out.

Now have a look how the colours have been used to show the layers on this cuboid.

How could we work this one out? Well again, we've got four multiplied by five multiplied by three this time.

The factors have changed position because the four represents a number of rows, the five represents a number of columns, and the three represents a number of layers, doesn't it? So as Alex is saying, we could multiply the four by the five, first of all, and we'll put some brackets around that, and then multiply that by the three.

That's a similar calculation to what we did before, and we still got the 60, didn't we? However, using the same cuboid in the same layers, it doesn't matter which order we put these factors in, does it? Okay, so even though the four represents the rows, the five represents the columns and the three represents the layers, we could find a pair of these to multiply first, and it won't matter, we'll still get the same product.

So we could multiply the five by the three first this time.

That would mean multiplying the number of columns by the number of layers, first of all.

So let's put the brackets around that.

Five multiplied by three is equal to 15.

And then we'd have to multiply that by the four, which would give us a 60 again.

So again, the product has remained the same even though we've multiplied a different pair of factors first.

We can see that more clearly here.

We can see how we've grouped my factors into pairs by using my brackets, and the product remained the same each time.

Izzy's wondering, \"Does that always work \"for any three numbers that multiply together?\" Alex says yes, let's have a look at another example.

This time he's chosen four multiplied by six multiplied by two.

And again, in the first example, he's multiplied the four by the six, first of all, that gives us 24, and then multiply that by two to give us 48.

And the second example, he's multiplied the six by the two, first of all, which gives us 12.

Multiply that by the four, that also gives us 48.

So to summarise, when you multiply any three numbers together, the product will remain the same no matter what pair of factors you multiply together first.

This also has a special name.

This is called the associative law, could you say that? Well done, so we can use the commutative law and the associative law to make our calculations easier.

And Izzy's saying that as well.

She's saying she can already see how this is gonna help her to calculate more efficiently when working with numbers in her head.

Okay, time for you check your understanding now.

Look at the equation.

Which numbers would you multiply together first, A, B, or C? Take a moment to have a think.

That's right, it would be B, wouldn't it? We'd need to multiply the seven and the five here first 'cause it's surrounded by the brackets, wouldn't it? The brackets are showing us which numbers we are gonna multiply first.

Okay, and onto our last task for today then.

What I'd like you to do is to use the commutative law to show as many different equations as you can to represent and calculate how to find the total number of cubes in this cuboid.

And then for task two, what I'd like to do is match the expressions that are equal.

Good luck with those and I'll see you back here shortly.

Okay, let's see how you got along.

You may have represented it as three multiplied by four multiplied by five.

The three in this case with represent the number of columns, the four would represent the number of rows, and the five would represent the number of layers.

You may have done three multiplied by five multiplied by four.

Again, changing the order this time, but getting the same product.

You may have done four multiplied by three multiplied by five, or you may have done four multiplied by five multiplied by three.

You could have done five multiplied by four multiplied by three, and you could have also done five multiplied by three multiplied by four.

Do you notice I worked that out systematically? I started with the three being the first factor, and then exhausted all the options that I could have for that.

Then I used the four as the first factor and exhausted all the options.

And then I used the five as the first factor and exhausted all the options as well.

Hopefully you can manage to come up with those as well.

Okay, and then matching the expressions, here we go.

Two multiplied by four multiplied by one is equal to two multiplied by four 'cause the brackets around the four multiplied by one.

Four multiplied by two is in bracket, so that would be eight.

Eight multiplied by three, those two would match.

Here, the brackets first around the five multiplied by the eight, so that would be the 40, and then we multiply that by three.

Again, five multiplied by two is in the brackets here, so that would be equal to 10.

So that would be equal to 10 times eight.

Two multiplied by four is equal to eight.

That's in the bracket, so eight times eight.

20 multiplied by four, well, that again would be equal to 80.

So 80 times by eight.

And then finally 200 multiplied by four, that would be equal to 800.

So the last one would be matching 800 times by eight.

That's the end of our learning for today.

Hopefully, you're feeling a lot more confident about what the commutative law is and what the associative law is and how you can apply these to different calculations.

So to summarise our learning today, if you change the order of factors, the product remains the same.

This is known as the commutative law and can be applied to both addition and multiplication.

You can also multiply different pairs of factors and the product would remain the same.

This is known as the associative law, and again, applies to both addition and multiplication.

Thanks for joining me today.

Hopefully you've got something to take away with you and you can start applying that in your everyday maths.

Take care, and I'll see you soon.

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"lesson": "Explain the use of the commutative and associative laws when multiplying three or more numbers", "copyright": { "attributions": [], "licence": "OGL", "canEmbed": true } } }, { "id": "f4509599-71d0-9255-f568-4dcb05b66d87", "type": "Resource", "resourceType": "Quiz", "title": "Explain the use of the commutative and associative laws when multiplying three or more numbers", "quizType": "Starter quiz", "items": [ { "type": "Question", "question": { "text": "What multiplication fact does this array represent?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1706541420%2Froakt46hb8ggb9wgldyi.png&w=640&q=75", "imageKey": "f4509599-71d0-9255-f568-4dcb05b66d87/question/b4af00ffa559434d80c9272ad82e1f8b.png" }, "questionType": "Multiple choice", "correct": [ { "text": "5 x 7 = 35" }, { "text": "7 x 5 = 35" } ], "choices": [ { "text": "5 x 7 = 35" }, { "text": "6 x 7 = 42" }, { "text": "5 x 6 = 30" }, { 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"https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1706541421%2Flykxp7fpdo7w7qk2mj2w.png&w=640&q=75", "imageKey": "f4509599-71d0-9255-f568-4dcb05b66d87/question/affa56944adee8aabb264aa690b492f1.png" }, "questionType": "Multiple choice", "correct": [ { "text": "A" }, { "text": "D" } ], "choices": [ { "text": "A" }, { "text": "B" }, { "text": "C" }, { "text": "D" } ], "id": "114312d7-cf26-f4c8-3a50-85cdf65276a7", "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1" }, { "type": "Question", "question": { "text": "Complete the missing factor: 7 × ______ = 56" }, "questionType": "Short answer", "correct": [ { "text": "8" }, { "text": "eight" } ], "id": "d37e7430-bc37-0eee-4184-235d31252114", "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1" }, { "type": "Question", "question": { "text": "Which of these are not factors of 40?" }, "questionType": "Multiple choice", "correct": [ { "text": "9" }, { "text": "3" } ], "choices": [ { "text": "5" }, { "text": "10" }, { "text": "4" }, { "text": "9" }, { "text": "3" } ], "id": "1bf47bb0-e958-a99d-27f0-fa5bd61945ea", "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1" }, { "type": "Question", "question": { "text": "Which factor is missing from the factor bug?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1706541422%2Fwblemec7oodpc26rvpro.png&w=640&q=75", "imageKey": "f4509599-71d0-9255-f568-4dcb05b66d87/question/ff2890d7f0811cf33c28957f27ea164a.png" }, "questionType": "Short answer", "correct": [ { "text": "3" }, { "text": "three" } ], "id": "f380f3e9-a730-fedc-e2d0-bb385bb9670f", "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1" } ], "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1", "meta": { "gradeStageId": "ks2", "subject": "Maths", "grade": "Year 5", "topic": null, "unit": "Multiply 3 or more numbers (commutative and associative laws)", "lessonCount": "5", "age": [ 9, 10 ], "gradeId": "age_9_10", "gradeStage": "Key Stage 2", "lesson": "Explain the use of the commutative and associative laws when multiplying three or more numbers" } }, { "id": "3d079dea-1795-2037-eff0-67a0340f37b3", "type": "Resource", "resourceType": "Quiz", "title": "Explain the use of the commutative and associative laws when multiplying three or more numbers", "quizType": "Exit quiz", "items": [ { "type": "Question", "question": { "text": "Look at the image. What does the number 3 represent in this expression? 2 × 3 × 4", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1706541418%2Fe8nrnbztk4rfnr0rovgi.png&w=640&q=75", "imageKey": "3d079dea-1795-2037-eff0-67a0340f37b3/question/41f466b137e87550c82656ccd89d16ce.png" }, "questionType": "Multiple choice", "correct": [ { "text": "The number of columns in each tray." } ], "choices": [ { "text": "The number of columns in each tray." }, { "text": "The number of rows in each tray." }, { "text": "The number of trays." } ], "id": "6b666d33-7e3d-96d1-6903-28a5da2e75a6", "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1" }, { "type": "Question", "question": { "text": "One layer has three rows and four columns. There are two layers. What does the 3 represent?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1706541419%2Fysnyov5wfoifwibethky.png&w=640&q=75", "imageKey": "3d079dea-1795-2037-eff0-67a0340f37b3/question/c7e0464a64a2f5210f0f6bb101ce64b8.png" }, "questionType": "Multiple choice", "correct": [ { "text": "The number of rows in a layer" } ], "choices": [ { "text": "The number of columns in a layer." }, { "text": "The number of layers." }, { "text": "The number of rows in a layer" } ], "id": "ae7e8a59-c07f-291c-48a1-771fcafc0ff2", "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1" }, { "type": "Question", "question": { "text": "Select the equation that most accurately matches this representation.", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1706541419%2Fx9pix8ixlnv3mp8xqfyr.png&w=640&q=75", "imageKey": "3d079dea-1795-2037-eff0-67a0340f37b3/question/394335d6bbe2383f0071b06ff53e6e32.png" }, "questionType": "Multiple choice", "correct": [ { "text": "4 x 3 x 5" } ], "choices": [ { "text": "3 x 4 x 5" }, { "text": "4 x 3 x 5" }, { "text": "5 x 3 x 4" } ], "id": "b5cfdd1d-3b1a-5c88-b0ec-1ba4fde36a2b", "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1" }, { "type": "Question", "question": { "text": "Select the missing equation?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1706541419%2Fgvoiyi40msizjpb9b6pm.png&w=640&q=75", "imageKey": "3d079dea-1795-2037-eff0-67a0340f37b3/question/424e0b8da2a7e29e809cba0327827ae7.png" }, "questionType": "Multiple choice", "correct": [ { "text": "5 x 3 x 4" } ], "choices": [ { "text": "4 x 5 x 3 " }, { "text": "5 x 3 x 4" }, { "text": "3 x 3 x 5 " }, { "text": "3 x 5 x 4 " } ], "id": "2d96a9e0-f465-d8c2-8df0-7c733fd68e62", "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1" }, { "type": "Question", "question": { "text": "Look at the equation. Which two numbers would you multiply together first? 3 × (4 × 6)" }, "questionType": "Multiple choice", "correct": [ { "text": "4 and 6" } ], "choices": [ { "text": "3 and 4" }, { "text": "3 and 6" }, { "text": "4 and 6" } ], "id": "a887a21a-6c87-93cb-b981-357abd897624", "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1" }, { "type": "Question", "question": { "text": "Match the statements." }, "questionType": "Matching", "pairs": [ { "left": { "text": "This is known as the associative law." }, "right": { "text": "Multiply different pairs of factors & the product will stay the same." } }, { "left": { "text": "This is known as the commutative law." }, "right": { "text": "Changing the order of the factors, the product remains the same. " } } ], "id": "cc0bbe60-1cfc-06e8-28fe-f99cb0bfb25b", "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1" } ], "lessonId": "e0b0cd69-a044-7063-090d-f0bc966f59e1", "meta": { "gradeStageId": "ks2", "subject": "Maths", "grade": "Year 5", "topic": null, "unit": "Multiply 3 or more numbers (commutative and associative laws)", "lessonCount": "5", "age": [ 9, 10 ], "gradeId": "age_9_10", "gradeStage": "Key Stage 2", "lesson": "Explain the use of the commutative and associative laws when multiplying three or more numbers" } } ] }