Lesson video

Hello there, my name is Mr. Tilston, I'm a teacher.

If I've met you before, it's lovely to see you again and if I haven't met you before, it's nice to meet you.

I really like math, so it's a real treat and a real pleasure and a real honour to be here with you today to teach you this math lesson.

If you're ready to begin, let's begin.

The outcome of today's lesson is this.

I can explain how making a factor 100 times the size affects the product.

You might have had some very recent experience of making a factor 10 times the size.

Can you remember how that affected the product? We've got two key words today.

If I say them, wIll you say them back? Are you ready? My turn, factor, your turn and my turn, product, your turn.

Can you recall what those words mean? Well, let's have a little reminder anyway.

A factor is a number which exactly divides another whole number.

So for example, three and four are both factors of 12.

A product is a result of two or more numbers multiplied together.

So in that example, three multiply by four is equal to 12.

The product is 12.

Our lesson today is split into two parts or two cycles.

The first will be making a factor 100 times the size and the second will be applying the concept in context.

Let's begin by thinking about making a factor 100 times the size.

In this lesson, you will meet Andeep and Jacob.

Have you met them before? They're here today to give us a helping hand with the maths.

Jacob and Andeep look at a multiplication, a very simple one.

It's two multiplied by three is equal to six.

"This is two groups of three" says Andeep, "which is equal to six." Is that how you saw it? Well, Jacob sees it slightly differently.

He says, "I think this is two, three times." Who do you agree with? Well, they could both be right.

"Both are correct and have the same product, because of commutative." That means we can swap over the factors.

So two multiplied by three is equal to six, but also three multiplied by two is equal to six.

That's called the commutative law.

"For today, can we use my interpretation to explore the lesson?" Yes, Andeep, "sure." Quick check time.

Look at the equation and its description by Andeep.

Can you represent this using base 10 one blocks? Draw or arrange them.

So, "this is two groups of four, which is equal to eight." Represent that please pause the video.

Let's see.

Well it could look a little like this.

That's two groups of four.

Jacob and Andeep look at a multiplication, same one as before, two multiplied by three is equal to six.

This time we're going to vary the representation Andeep says, "Let's model this with place value counters." Have you got place value counters? You could model along with us.

"It is two groups of three ones," says Jacob.

Can you see that? Two groups, each group has three ones.

"What would happen if we make three 100 times the size?" Hmm, what do you think? How could we represent that with place value counters? Well, we could do this, we could exchange.

So, "The value of each counter needs to be made 100 times the size" and that we've done.

So each one counter has changed value to become a 100 instead.

That gives us this equation.

Two multiplied by 300 is equal to 600.

There are now two groups of 300, which is equal to 600.

So can you see that? Two different groups each with 300 in, giving a total of 600.

Jacob and Andeep compare the equations.

We started with two groups of three, which is equal to six.

Then we scaled up a factor by making it 100 times the size.

There's two factors there, two and three and the one that we scaled up was the three.

So it wasn't three anymore, we made it 100 times the size and it became 300.

We then did exactly the same thing to the product.

We scaled up the product making it 100 times the size.

So instead of six, it became 600.

Six multiplied by 100 is equal to 600.

Let's do a check.

Fill in the missing labels that show what the factor and product have been scaled up by in these related multiplications.

So two multiplied by four is equal to eight.

It's a known times tables fact.

Two multiplied by 400 is equal to 800.

So can you see we've done some scaling, but can you say by how much? Pause the video.

what are we multiplying those numbers by? It's the same number and it's 100.

The factor has been made 100 times the size and the product has been made 100 times the size.

Jacob and Andeep look at a multiplication the same one.

"Last time we scaled up the group size by 100.

What would happen if we scale up the number of groups?" What do you think? So we've changed the factor.

"So that would mean changing two to be 100 times the size." So instead of two it becomes 200 Andeep say, "That's like rolling a three on a dice 200 times and counting the number of dots." So this is a dice, here's a three.

This is a three on a dice 10 times, and this is a three on a dice, 100 times.

One block of 100 rolls of three has 300 dots.

That's one block and this is two blocks.

So how many is that? "Two blocks of 100 rolls of three have 600 dots." "So 200 groups of three is equal to 600." Once again, we've multiplied one of the factors by 100, so we multiply the product by 100.

"We started with two groups of three, which is equal to six." So two multiplied by three is equal to six.

"Then we scaled up two by making it 100 times the size, 200." "That meant that the product was 600, which is also 100 times the size." So we did the same thing to the product as we did to one of the factors, multiplied it by 100.

Jacob and Andeep compare the equations from both sets of scaled up equations.

What's the same and what's different? Well "Different factors have been made 100 times the size in each equation." So can you see in the first equation, it's the second factor that's been made 100 times the size and in the second example it is the first factor that's been scaled up 100 times.

"The product is 100 times the size in both." That's what's the same, the product is 600.

"If one factor is made 100 times the size," "The product will be 100 times the size." This is a big generalisation that I want you to really focus on in this lesson.

So I'm going to say it with you this time.

Should we read it together? You ready? "If one factor is made 100 times the size," "the product will be 100 times the size." Okay, just you in three, two, one, go.

Brilliant.

True or false? If one factor is made 100 times the size, the product will be 100 times the size.

True or false? And can you explain why? Hmm, pause the video.

That's true.

If a factor has been made 100 times the size and it doesn't matter which of the two factors, then the value of each unit in the product is 100 times the size.

Jacob and Andeep explore further.

"Now we can use our times tables knowledge to find related multiplication facts." I know that three multiplied by five is equal to 15.

So I can calculate three multiplied by 500 and 300 multiplied by five." "Remember that to multiply a whole number by 100, you placed two zeros after the final digit to show the value is 100 times this size." So we're not adding two zeros, we are placing two zeros.

So five multiplied by 100 is equal to 500.

So we've scaled up that 100 times the size and we are going to do the same to the product.

So 15 multiplied by 100 is 1,500.

Three multiplied by five is equal to 15.

This time, let's scale up the other factor 100 times the size and again we'll do the same to the product, which once again is 1,500.

So it didn't matter which of the two factors we scaled up by making it 100 times the size, the product was also 100 times the size.

Jacob and Andeep complete a multiplication wheel.

"We need to know our four times tables here to find the products." I'm sure you've done lots of these before.

Each sector is showing four groups of a number and a product.

"For example, four groups of three is equal to 12." So the groups are in that middle circle.

Let's complete the rest.

Can you complete those products? Well here are the completed products.

"What if we took the factor four, that's the number of groups and made it 100 times the size.

So instead of being four, it's going to be 400." How is that going to affect the product? Hmm? What do you think? "Well, if one factor is made 100 times the size, the product will be 100 times the size as well.

So we need to multiply each product by 100" by placing two zeros at the end to multiply a whole number by 100, we can place two zeros after the final digit and here we go.

So each of those products has been made 100 times the size, because one of the factors, the four was made 100 times the size.

Andeep wants to take that a little further.

He says, "shall we now make the factor showing group size 100 times the size each time?" So that middle circle, what will change? Well again, "If one factor is made 100 times the size, the product will be 100 times the size." This time we're not changing the four, we're changing the other factors.

"Either factor still makes a product 100 times the size." So here we go.

So the factors have been made 100 times the size in the middle and the products have been made 100 times the size on the outside and you may notice that the products are the same as they were when we turned four into 400 by making that 100 times the size.

Let's have a little check.

What is the mistake in the jottings below? Five multiplied by eight is equal to 40.

So 500 multiplied by eight is equal to 40.

Hmm, that's not right, but can you explain why? And can you fix it? Pause the video.

Wasn't right was it? What had happened there? What had gone wrong there? I can see that one of the factors has been scaled up.

It's 100 times the size.

So that five has become 500, but the product hasn't been scaled up.

So this is what it should have been.

500 multiplied by eight is equal to 4,000, so we needed to make that product 100 times the size as well.

It's time for some practise.

Number one, complete the jottings below by completing the missing numbers using your knowledge of times tables and your understanding of making a factor 100 times the size.

Number two, complete the times tables facts and use this to complete the multiplications featuring a factor 100 times the size.

Number three, complete the related multiplication wheels and you'll notice the first one is just a six times table, the second one is a 600 times table, so think what we've done to that factor and in the third one it's the middle numbers that have changed and the same for B and remember all the way through this, if we make the factor 100 times the size, we have to make the product 100 times the size.

Pause the video and good luck with that.

Welcome back.

How are you getting on? Shall we have some answers? Let's have a look.

So number one, two multiply by six is equal to 12.

Now we've multiplied one of the factors of six by 100 to become 600.

So we have to multiply the product by 100 as well.

So 12 becomes 1,200.

This is B, 200 and 1,200 and C, D, E, and F and in all of those examples we've made one of the factors 100 times the size.

So we've made the product 100 times the size as well and number two, three multiplied by nine is equal to 27, so three multiplied by 900.

That's one of the factors been made 100 times bigger equals 2,700.

So that product has been made 100 times the size as well is B, C, D, E and F and once again, in all of those cases when one of the factors has been made 100 times the size, the product has been made 100 times the size as well.

The multiplication wheels.

The first one was a straight six times tables wheel, but then we made that six, that factor 100 times the size.

So these were the answers.

All of the products are now 100 times the size as well and then for the next one, we went back to six times, but we changed the other factors, we made them 100 times the size, but what do you notice? The products are the same as when we made the other factor 100 times the size in the previous example and here's B, we start with a straight seven times table and then we scale that seven times to make it 700 times.

So it's a hundred times the size.

So the products are a hundred times the size and here back to seven times, but we made the other factors 100 times the size.

So we make the products 100 times the size just like before and once again you'll notice the outer ring in the last two examples is the same.

They've got the same numbers.

You're doing really, really well and it's time to move on to the next cycle, which is applying the concept in context.

Jacob and Andeep look at a multiplication.

This is 600 multiplied by eight is it equal to, hmm.

Well, "Previously we've started with a times table," says Andeep.

"But we don't have one this time." Hmm, can you sort of see one that's buried within it though? Hidden within it? "So now we have to decide which times table it is best to recall to help here," which one resembles that the most? What do you think? Have a look.

Can you see a times table hidden within that? It's this one.

Six multiplied by eight is equal to 48.

That's our known times tables fact hopefully.

Now six multiplied by 100 is equal to 600.

So we've scaled up one of the factors making it 100 times the size.

We have to do the same with the product.

So 48 multiplied by 100 is 4,800.

It's time for a quick check for understanding.

Circle the most relevant times tables facts and then use it to find the missing number.

So we are looking to find the missing number in 500 multiplied by six is equal to something, which times tables factors is helpful.

Is it five multiplied by six is equal to 30? Is it six multiplied by five is equal to 30? Or is it three multiplied by 10 is equal to 30? Pause the video.

It was this one.

That's the one that most closely resembles it.

Five multiplied by six is equal to 30.

If five multiplied by six is equal to 30, we can use that.

We can scale up that factor making it 100 times the size, therefore we scale up the product making it 100 times the size and 30 multiplied by 100 is 3000.

Very well done if you've got that, you're on track.

Jacob and Andeep look at a problem involving equivalent expressions.

Here we've got four multiplied by 800 is equal to eight multiplied by something.

Hmm, what could we do here to find out the missing value? "Neither of these expressions are a product, so how can we solve this?" What do you think? What tips have you got? "This uses commutativity." That rule that we can swap over the two factors and still get the same product.

"I know that four multiplied by eight is equal to eight multiplied by four," that could be helpful.

Four multiplied by eight is equal to eight multiplied by four.

"I also know that either factor can be made 100 times the size for the product to be 100 times the size." It doesn't matter which.

So we could make the eight 100 times the size to make 800.

"The missing number is 100 times the size of four therefore, which is 400." "I see, both of these expressions give the same product," yes they do.

So four multiplied by 800 is equal to eight multiplied by 400.

In the same way that four multiplied by eight is equal to eight multiplied by four.

Even though different factors have been made 100 times the size.

There are four jars of marbles, how many marbles are there altogether? Hmm, "This time we need to work out what's being asked.

I don't think we have enough information." No I don't Andeep either.

Let's see, what could be helpful? What would it be helpful to know? "We need to know how many marbles are in each jar so we can calculate the total." What about this? There are four jars, each contains 300 marbles, that's better.

How many marbles are there altogether? We can work it out now.

"This involves multiplication because there are four jars of 300, which is four multiplied by 300." "Let's use a known times table to help us to solve this problem.

Which times table should we use?" What can you see hidden in there? "I think if we use four multiplied by three is equal to 12, if there were four jars of three marbles, there would be 12 marbles." "Each jar has 300 marbles, which is 100 times as many as three.

So the product should also be 100 times as many." So four multiplied by 300, we've scaled up that factor, making it 100 times the size, so we need to scale up the product making it 100 times the size, so that would be 1,200.

"There are 1,200 marbles altogether." Time for another really quick check for understanding.

Circle the times tables fact that will be most useful for solving the following worded problem.

There are five jars, there are 200 marbles in each jar.

How many marbles are there altogether? So which is the useful times tables fact? Is it five multiplied by five is equal to 25? Is it two multiplied by two is equal to four? Or is it five multiplied by two is equal to 10? Think about which one resembles it the most closely.

Pause the video Could you see a times tables fact within that? It's this one, five multiplied by two is equal to 10, that's our known times tables fact and well done if then went a step further and you multiplied one of those factors, that's the two by 100 and then multiplied the product, that's the 10 by 100 to give you the answer, 1000, 1000 marbles altogether.

It's time for some final practise.

Match the related pairs of calculations by drawing lines between them and then solve them.

Number two, find the missing numbers and remember that rule.

If you scale up one of the factors, making it 100 times the size, you're going to do the same to the product, make the product 100 times the size as well.

Number three, match the equivalent expressions by drawing a line between them.

So this is all about commutativity and number four, solve the following worded problems, which I will leave you to read.

Make sure you've read them carefully before you attempt them.

Very best of luck with that and I'll see you soon for some feedback.

Welcome back, how did you get on? Number one, you can use three multiplied by seven to help you solve 300 multiplied by seven, they're related.

Three multiplied by seven equals 21.

That's a known multiplication fact, hopefully and we've scaled up one of the factors, that's a three, making it 100 times the size.

So we've scaled the product, making it 100 times the size as well.

So that's 2,100 and these two are related.

You can use that to help solve it.

So four multiplied by eights equal to 32.

So four multiplied by 800 is equal to 3,200 and here are the other answers, well done if you've got those.

Number two, find the missing numbers.

For that first one, you might take the known multiplication fact that eight multiplied by seven is equal to 56 and then scale that product up.

So that's 5,600.

This is B, C, D, E, F, G and H.

Well done if you've got those and in all of those examples, it was helpful to find a known multiplication fact within.

Number three, match the equivalent expressions by drawing a line between them.

So this is where our knowledge of commutativity helps.

So two multiplied by 900 is equivalent to nine multiplied by 200.

3 multiplied by 1,200 is equivalent to 12 multiplied by 300.

600 multiplied by seven is equivalent to 700 multiplied by six.

1,100 multiplied by two is equivalent to 200 multiplied by 11 and 12 multiplied by 800 is equivalent to 1,200 multiplied by eight and solve the following worded problems, there are eight classes in a school, each class has 200 glue sticks at the start of the year.

How many glue sticks is this altogether? Did you use your known times tables fact of eight multiplied by two is equal to 16 as a starting point? Eight multiplied by two is equal to 16 and we've scaled up one of the facts, making it 100 times the size, we've scaled up the product, making it 100 times the size is 1,600.

1,600 glue sticks.

B, each child in a school needs five exercise books.

There are 300 children.

How many books are needed? Did you use the known multiplication fact five multiplied by three is equal to 15 as your starting point and then scaled up? Three multiplied by 100 is equal to 300.

We scaled up that factor, making it 100 times the size.

So we need to scale up the product, making it 100 times the size and that's 1,500, 1,500 books and C, six classes each raised 400 pounds for charity.

How much have they raised altogether? Did you spot the multiplication fact six multiply by four within that? That equals 24 and then we're scaling up one of the factors making it 100 times the size, that's a four in this case, so that becomes 400 pounds.

So we've got to do the same to the product that's going to be 100 times the size, making it 2,400.

So 2,400 pounds altogether.

We've come to the end of the lesson.

You've been amazing.

Today we've been explaining how making a factor 100 times the size affects the product.

So if one factor and it doesn't matter which is made 100 times the size, the product will be 100 times the size and you've investigated lots of examples of that including in context, so it can be applied in context by drawing upon known times tables facts.

So finding those times tables facts buried within it and making one of the factors 100 times the size.

It's important to analyse the numbers to select the most appropriate times tables fact to help.

You've been incredible today.

Give yourself a pat on the back.

It is thoroughly deserved.

I hope I get the chance to spend another maths lesson with you at some point in the near future.

Have a fabulous day, whatever you've got in store, and be the best version of you that you can possibly be.

Take care and goodbye.