Lesson video

Hi there, my name is Mr. Tazzyman and I'm gonna be teaching you a lesson today from the unit that's all about understanding and representing multiplicative structures.

Are you ready to start? Then let's get going.

This is the outcome for today's lesson then.

By the end, we want you to be able to say, "I can partition one of the factors in a multiplication equation using representations." These are the key words that you are gonna see.

I'll say them and I want you to repeat them back to me.

So I'll say, "My turn," say the word, and then I'll say, "Your turn," and you can repeat it back, ready? My turn, multiples, your turn.

My turn, partition, your turn.

My turn, distributive law, your turn.

My turn, adjacent, your turn.

Okay, this is the meaning of each of those words.

A multiple is the product of a number and an integer.

Partitioning is the act of splitting an object or value down into smaller parts.

The distributive law says that multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

For example, four multiplied by three, and you can see that the four has been partitioned into two and two there, is equal to two multiplied by three plus two multiplied by three, which is equal to 12.

Adjacent means next to each other.

In this lesson, it's used to refer to multiples that are next to one another.

This is the outline then for today's lesson, we're gonna start by looking at representing the adjacent multiples rule.

Then we're gonna look at representing the distributive law.

Jacob and Andeep are here to help us as well.

They're gonna discuss some of the maths that you'll see on screen, which should give us an insight into our learning.

Hi, Jacob, hi, Andeep.

Okay, everyone, we ready? Let's do this.

Lucas and Andeep look at a multiplication equation with a missing number.

Five times eight is equal to something.

"I know my eight times table, I know this one." "I'm not sure, but I do know that four times eight is equal to 32." There's a number line.

You can see we've got four lots of eight and we've ended on 32.

"That's how I know it," says Jacob.

"I add on one more multiple of eight." Five times eight is equal to 40 because 32 plus eight is equal to 40.

"I see, the fourth and fifth multiple are adjacent, so have a difference of eight." Lucas and Andeep try to write down their thinking as an equation.

So you've got five times eight, the expression to begin with.

"We worked out the fourth multiple of eight," so five times eight is equal to four times eight, "And then added on another multiple of eight to get to the fifth," plus eight is equal to 40.

Lucas and Andeep solve another equation by looking at adjacent multiples.

So this time we've got four times seven is equal to something.

"I know that the fifth multiple of seven is 35, so the fourth multiple will be seven less." So they've got four times seven is equal to five times seven, take away seven, which is equal to 28.

"Adjacent multiples of a number have a difference equal to that number." What a good generalisation from Jacob.

"We can easily calculate multiples of a number by adding on or subtracting that number from an adjacent multiple." Okay, let's check your understanding of the adjacent multiples rule.

Complete the missing numbers in the equation, and we've got a hint here from Jacob.

"Use the generalisation that adjacent multiples of a number have a difference of that number and the number line below." And the number line he's talking about is at the bottom there and that represents the idea that adjacent multiples of a number have a difference of that number.

Okay, pause the video and give that a go.

I'll be back to reveal the answers shortly.

Welcome back, here are the answers then.

Four times nine is equal to five times nine subtract nine, which is equal to 45 subtract nine, which is equal to 36.

Lucas uses a part-part-whole model to partition.

"I have partitioned five counters into four and one." "I'll represent this model using equations." Andeep has gone on to label each of those bits.

You've got the whole and the parts labelled, five, four, and one, and he's got the equations, four plus one is equal to five, five subtract one is equal to four, five subtract four is equal to one.

"Each counter is worth one, which I can label." So you can see the ones have been put onto each counter there.

"I'll rewrite the expressions and equation too." So now that Jacob has changed the value of each one of those counters to definitely being one, Andeep's gonna rewrite them.

Have a look and see what he does.

He's written an expression underneath each of the bits of that model.

We've got five times one, four times one and one times one.

Four times one plus one times one is equal to five times one.

"What if I change the value of the counters to two instead?" Hmm, "Then I'd need to change the expressions and equation again." So the expressions have now changed and there's the equation changing.

Now we've got four times two plus one times two is equal to five times two.

Jacob this time says, "What if I change the value of the counters to five instead?" There they go.

"Here's the expressions and equation amended again," says Andeep, he changed the expression and the equation.

Now our equation reads four lots of five plus one lot of five is equal to five lots of five, makes sense.

"Last one, what about if they were each worth eight?" There they go, they've changed to eight.

"Here's the expression and equations once more," says Andeep.

Now we've got four times eight plus one times eight is equal to five times eight.

"That looks familiar," says Jacob.

"It's the equation we started with," says Andeep.

Remember this number line? This is what they started with, but we've represented it using a part-part-whole model rather than a number line.

"We can partition adjacent multiples." Okay, let's check your understanding then.

Complete the part-part-whole model and the equation to represent five times six.

Pause the video here and have a go at completing those equations and that model.

Welcome back, let's reveal the answers then.

Well, the counters were each worth six, so they needed a six inside them.

The expressions accompanying each bit of that model were five times six with the top one, which you've been given already, that's the whole.

Then for the parts we had four multiplied by six and one multiplied by six.

That gave an equation that read as follows.

Four times six plus one times six is equal to five times six.

Is that what you've got? I hope so.

Okay, let's move on.

What's the same and what's different? These are three of the models that you've already looked at.

You need to identify what's the same and what's different about them, including looking at their equations as well.

Hmm, well, Andeep says, "The factor of five has been partitioned in the same way.

The structure of the calculation is the same for each." "The fifth multiple is the total of the fourth and first multiple." That's our adjacent multiples rule.

"The value of each counter is different in each model," and we can see that from the labels.

In the first one, each counter is worth one, in the second, each counter is worth two, and in the third each counter is worth five.

Okay, let's do the same process again.

What's the same and what's different? Look across the three examples this time.

What do you notice? Hmm, "Again, the factor of five has been partitioned in the same way, so the structure of the calculation is the same for each." You can see that in the part-part-whole models, we've still got five units having been partitioned into part four and a part of one.

"One of the equations is adding and the others are subtracting." Did you notice that? The first equation, four times five plus one times five is equal to five times five, is an addition, but in the second and third one they read differently.

Listen, five times five subtract one times five is equal to four times five.

Five times five subtract four times five is equal to one times five.

Andeep looks at a stacked number line showing five times eight compares it to the part-part-whole model.

You can see it there on your screen.

The part-part-whole model is on the left and the stacked number line is on the right.

I wonder if you could do some comparison or have a think about what those number lines actually refer to.

Well, here's Andeep's explanation.

"The top number line counts out the total number of counters in the whole and the bottom number line shows the corresponding multiple of eight." So if we were gonna say three multiples of eight, it would be 24 and you can see the three is above the 24, except there's some shading behind it as well.

I wonder what the significance of that is.

"The shaded areas show how the multiples have been partitioned using the adjacent multiples rule.

Five is equal to four plus one, so five times eight is equal to four times eight plus one times eight." Can you see there that there's some shading behind those number lines and it goes all the way up to four and 32? Then it turns to yellow.

Well, if you think about that, you can see that it corresponds to the part-part-whole model 'cause that has parts of four and parts of one.

Andeep and Lucas discuss another stacked to number line.

"This represents using adjacent multiples to find five times six." Andeep says, "What's the equation then?" What do you think? How would you write this out as an equation? Hmm, here it is.

Four times six plus one times six is equal to five times six.

"This also shows subtraction," says Andeep.

Five times six subtract one times six is equal to four times six.

"Five is equal to four plus one, so five times six is equal to four times six plus one times six." "Four is equal to five minus one, so four times six is equal to five times six minus one times six." Okay, time for your first practise task.

Complete the part-part-whole model using the equations.

So you've got two part-part-whole models there to try, A and B, and you've got some missing gaps that you need to fill in.

Don't forget to look at the counters as well and the expressions that describe each of those circles in the model.

For number two, I want to match the stacked number lines with the equation they represent.

There's three stacked number lines on the left.

You need to draw a line from those to one of the equations of the three on the right.

For number three, complete all three representations showing five times nine is equal to 45.

So we're describing five times nine is equal to 45 using these three representations, but remember we've got to use the adjacent multiples rule to do it.

Okay, pause the video here and give those three a go.

Good luck, think carefully.

I'll be back shortly with some feedback.

Welcome back, let's do number one first.

The missing encounters all needed seven in because they all had the same value.

That meant that the expressions with missing numbers read four times seven and one times seven and the equation at the bottom was four times seven plus one times seven is equal to five times seven.

Let's look at B then.

The expression missing numbers was four times nine for the whole and three times nine for the part on the left.

Any of the counters had nine in them because they were all the same value, nine.

The equation at the bottom read, three times nine plus one times nine is equal to four times nine.

Pause the video here for extra time to mark accurately.

Welcome back, let's match up for number two then.

The top stacked number line matched with the middle equation, the second stacked number line matched the top equation, and the bottom matched the bottom.

If we take the top one just to show you as an example, you can see you've got five lots of 10, which is the entire stacked number line there.

Subtract one lot of 10, which is the yellow part, is equal to four lots of 10, which is the pink part.

And here's number three.

The part-part-whole model had expressions of five times nine, four times nine and one times nine and each of the counters was worth nine.

For the stacked number line, on the top number line, it needed to read zero, one, two, three, four, and five, and then it was zero, nine, 18, 27, 36, and 45.

And lastly, the equation, four times nine plus one times nine is equal to five times nine.

Okay, now it's time to start the second part of the lesson.

It's the distributive law that we're going to represent this time.

Andeep revisits the part-part-whole model to partition.

"What if I redistribute one of the counters?" And he does just that.

Can you see one of the counters has moved from the first part to the second part? And it means that in the whole, we've now got a slightly different representation.

"This shows subtraction and addition.

I'll focus on addition for now." And he's written down three of the equations that he thinks this part-part-whole model shows, but he's only going to focus on three plus two is equal to five.

"What if each counter represents one again?" "Here's the new equation using expressions for each part." Three times one plus two times one is equal to five times one.

This is all very similar to what we were looking at when we were representing the adjacent multiples rule using a part-part-whole model.

"What if each counter represents two?" "Here's the new equation using expressions for each part." Three times two plus two times two is equal to five times two.

"What if each counter represents five?" "Here's the new equation using expressions for each part." Three times five plus two times five is equal to five times five.

What's the same and what's different then between these three examples? Have a look at each of them and think about that.

Andeep says, "The multiples have been partitioned in the same way.

The structure of the calculation is the same for each." "The fifth multiple is the total of the third and second multiple.

The value of each counter is different in each model." Okay, let's look at these two then.

What's the same and what's different between these two? "The first one represents the adjacent multiples rule." Can you see the partitioning there? One has been separated from the other part.

"One of the parts is only one multiple." "In this one, the partitioning is different, but we are still adding multiples of five together." All right, what's the same and what's different between these three? "All the part-part-whole models are the same." They are, you can see that, they're all exactly the same.

"The equations are all different featuring addition or subtraction of expressions," and you can hear that, listen.

Three times five plus two times five is equal to five times five, or five times five subtract three times five is equal to two times five, or five times five subtract two times five is equal to three times five.

Andeep looks at the stacked number line showing five times eight and compares it with the part-part-whole model.

"The top number line counts out the total number of counters in the whole and the bottom number line shows the corresponding multiple.

The shaded areas show how the multiples have been partitioned using the distributive law this time." Five is equal to three plus two, so five times eight is equal to three times eight plus two times eight.

All right, let's check your understanding.

Draw the part-part-whole model as a stacked number line that represents it.

Remember, it doesn't have to be a perfect stacked number line, but you just need to make sure that you've put in the two number lines and decided where you are going to split the shading up.

Okay, good luck.

Pause the video here and have a go.

Welcome back, did your number line look something like this, your stacked number line? You should have had two number lines on top of one another.

The first one going from naught up to seven, counting in ones, and the second one going from naught to 56 counting in sevens.

The split in the shading should have been marked going through five and 40 because we had five lots of eight and two lots of eight.

Here's the equation.

Five times eight plus two times eight is equal to seven times eight.

Jacob and Andeep interpret another part-part-whole model that uses the distributive law.

"Nine is equal to six plus three." Can you see the counters there? If you ignore the values, you can see that nine is equal to six plus three.

"So nine times three is equal to six times three plus three times three.

I think there are more ways of interpreting this part-part-whole model.

It shows other equations." What do you think, hmm? "Six is equal to nine subtract three, so six times three is equal to nine times three subtract three times three." Andeep's remembered subtraction as well.

"Three is equal to nine subtract six, so three times three is equal to nine times three subtract six times three." Jacob says he cannot remember seven multiplied by eight.

Anyone else find that times table tricky? I know I do sometimes.

"I'm not very good at remembering seven times eight, but I do know my eights up to the fifth multiple." How could he find the product of seven and eight using the distributive law? Hmm, "We can partition the multiples, which is also called the distributive law." So the act of actually partitioning the multiples up and using that understanding is the distributive law.

You are distributing the multiples.

"I'll use a part-part-whole model.

I'll partition the seven multiples of eight into five and two." There it is.

"The equation will be adding the expressions for the parts." So we've got seven multiplied by eight, and we've got five times eight, and we've got two times eight.

Those are the expressions that Jacob could now add together.

Five times eight plus two times eight is equal to seven times eight.

"Then, I can use the value of the expressions I know.

So he's swapped five times eight for its total value, which is 40, and then two times eight for its total value, which is 16.

40 plus 16 is equal to seven times eight, so 40 plus 16 is equal to 56.

Seven times eight is 56, a good way of remembering.

It's time for the second practise task now.

To begin with, I want you to complete the part-part-whole models and stem sentences.

You can see them below there, there's A and there's B to do.

Don't forget, there's a few counters with some missing numbers in as well.

For number two, you've got to complete the stem sentences for the stacked number line.

There are three there because we've got addition and two subtractions that are represented by the stacked number line.

For number three, I'd like you to roll a dice twice to write a multiplication expression.

Partition one of the factors using the distributive law and represent it as a part-part-whole model, a stacked number line, and an equation.

Okay, have a good go at those, good luck.

Pause the video and I'll be back shortly with some feedback.

Welcome back.

Let's look at number one to begin with, A.

All the counters should have had seven in because they were all worth seven, and our expressions underneath each of the circles should have read five multiplied by seven for the whole, three multiplied by seven and two multiplied by seven for the parts.

That left us with a stem sentence that read as follows, three is equal to five minus two, so three times seven is equal to five times seven minus two times seven.

For B, all the counters were worth nine.

The expressions were six times nine for the whole, four times nine and two times nine for the parts.

That then left you with a sentence that read as follows, six is equal to four plus two, so six times nine is equal to four times nine plus two times nine.

Okay, pause the video here for some extra time to ensure that you've got accurate marking.

Let's do number two then.

The first one should have read seven is equal to five plus two, so seven times seven is equal to five times seven plus two times seven.

For the second one, five is equal to seven minus two, so five times seven is equal to seven times seven minus two times seven.

And finally, two is equal to seven minus five, so two times seven is equal to seven times seven minus five times seven.

Okay, pause the video here to be able to mark those accurately.

For number three, here's what Jacob did.

He rolled a five and a seven, so he's written out a stacked number line there, the top numbers going from zero to five, counting up in ones and the bottom numbers going from zero to 35, counting up in sevens, and he's split his shading at the three mark.

He's written a part-part-whole model as well.

The counters were all worth seven and the expressions were five times seven for the whole, three times seven and two times seven for the parts, and he's written out an equation, or two equations I should say, as well.

Three times seven plus two times seven is equal to five times seven or five times seven subtract two times seven is equal to three times seven.

Well done, Jacob.

You might have got something similar here.

Pause the video so you could perhaps share some of your work with somebody else to give each other feedback.

That brings us to the end of the lesson then.

Here's a summary of our learning.

The adjacent multiple rule means that we can easily calculate multiples of a number by adding on or subtracting that number from an adjacent multiple.

The distributive law says that multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

This means a factor can be partitioned to solve multiplication problems efficiently.

My name's Mr. Tazzyman.

I enjoyed learning today and I hope you did too.

Maybe I'll see you again in another maths video soon.

Bye for now.