Lesson video

Hello, I'm Mr. Tazzyman and today I'm going to teach you this lesson from the unit that's all about equivalence, compensation, and subtraction.

So sit back, be ready to listen and engage, let's do this.

Here's the outcome for the lesson then.

By the end, we want you to be able to say, "I can use equivalence and compensation strategies to solve subtraction problems in a range of contexts." These are the key words that you might expect to hear during the lesson.

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

So I'll say my turn and say the word and then I'll say your turn and you can say it back.

Ready? My turn, difference.

Your turn.

My turn.

Adjust, your turn.

Okay, it's all very well being able to say the words, but let's just double check we know what they mean as well.

The difference is the result after subtracting one number from another.

You can see a subtraction equation there.

Seven subtract three is equal to four, and in that equation, four is the difference.

Let's look at adjust then, when you adjust, you make a change to a number.

This is done to make a calculation easier to solve mentally.

This is the outline for today's lesson on using equivalent and compensation strategies to solve subtraction problems in a range of contexts.

To begin with, we are gonna look at some different problems from data.

Then we're gonna look at some difference puzzles.

These are two math friends that we'll meet throughout, Sofia and Lucas, hi Sofia, hi Lucas.

They're gonna help us by discussing some of the math prompts that you see on screen, so make sure you listen carefully to what they have to say because they're gonna be revealing a lot of the learning as we go through the lesson.

All right, are you sitting comfortably? Let's get learning.

Lucas and Sofia look at a bar graph featuring the four best hockey teams in an ice hockey local league.

You can see that there, a graph showing goals of four hockey teams. The red bars are the goals scored and the blue bars are the goals conceded.

We've got Skatey Spuds, Polar Bears, Lucky Pucky, and Sticky Sticks.

Along the Y axis, you can see that that's the number of goals.

It goes from zero to 50 and counts in intervals of 10.

Let's calculate the goal difference of each team.

"I don't think we need to do any calculation," says Sofia.

What has Sofia noticed? Hmm, have a look at that graph.

What do you think Sofia has noticed? I see, the difference is constant, it's 15.

Yes, each team's bars have a difference of 15.

You can see it there on Skatey Spuds and Polar Bears and Lucky Pucky and Sticky Sticks.

Remember, the bars are all different heights because they've scored or conceded different numbers of goals.

However, the difference between the numbers of goals they've scored and conceded is constant across all four teams. Okay, let's check your understanding so far.

What is the goal difference for the team in the graph below? We've got a graph here showing another team's number of goals scored and conceded.

It's the Penguin Players.

You need to calculate the goal difference for their team.

Pause the video and give it a go.

Welcome back.

The goal difference was 25.

You can see the red bar shows that they've scored 40 goals, but they've conceded 15, which has been shown by the blue bar.

40 is our minuend, 15 is our subtrahend.

The difference between 40 and 15 is 25, so the goal difference is 25.

Okay, let's move on.

The teams all play one another in the next two rounds of fixtures.

The results are listed below.

Skatey Spuds, three, Polar Bears, two.

Lucky Pucky, four, Sticky Sticks, one.

Lucky Pucky, one, Polar Bears, one.

Skatey Spuds, zero, Sticky Sticks, two.

What does three, two mean? So Sofia is looking at that first set of results and she's looking at Skatey Spuds against Polar Bears.

Skatey Spuds scored three and Polar Bears two.

So that's what the three and the colon and the two mean in this context.

Who has the best goal difference now? What a great challenging question.

"Let's work it out," says Sofia.

I'll start with Skatey Spuds.

Here is the initial goal difference.

So the graph there just features Skatey Spuds, but the data is taken from that first graph that we saw a little while back.

40 subtract 25 is equal to 15.

We already knew the goal difference was 15.

They scored three more but conceded two in the first match and can you see, Sofia is looking specifically at what Skatey Spuds scored and conceded against Polar Bears.

The overall score was three, two, but that meant that Skatey Spuds scored three but conceded two.

Then Sofia puts that as a jotting on her initial equation, which represents the goal difference before these fixtures.

She's added on three to the goals scored and added on two to the goals conceded.

They are the adjustments.

Now we've got 43 subtract 27, which is equal to 16.

So the goal difference has improved by one.

"Their goal difference was 16 going into the second match," says Sofia.

In the second match they scored zero and conceded two.

You can see it at the bottom there.

So we need to do those adjustments.

We add two to the number conceded, but there's no adjustment for the goal scored.

That's because they didn't score any.

We end up with 43 subtract 29 is equal to 14.

"Their goal difference now stands at 14," says Sofia.

Now for Polar Bears, same thing again but with different numbers because it's a different team.

50 subtract 35 is equal to 15.

That was their initial goal difference.

From the first fixture, there's an adjustment of plus two for the goals scored, but plus three for the goals conceded because they lost.

52 subtract 38 is equal to 14.

Next fixture they drew one, one.

So they need to adjust both the goals scored and goals conceded by one.

They end up with 53 subtract 39, which is also equal to 14.

The goal difference didn't actually change between those two fixtures.

Okay, Lucas steps in and says, "I'll do Lucky Pucky." 45 subtract 30 is equal to 15.

That's their initial goal difference before these results, Lucas has adjusted the goals scored by adding four and adjusted the goals conceded by adding one because Lucky Pucky won four, one.

Now it's 49 subtract 31, which is equal to 18.

Next fixture was one all, so he's adjusted both the goals scored and goals conceded by adding one.

50 subtract 32 is equal to 18.

Finally, Sticky Sticks.

There's the initial equation with the adjustment from the first result, 35 subtract 20 is equal to 15, one extra added to the goals scored, four extra added to the goals conceded because Sticky Sticks lost four, one.

Now it's 36 subtract 24, which is equal to 12.

Let's look at the last fixture.

Well in the last fixture, Sticky Sticks won two, nil and didn't concede any.

So the goals scored has been adjusted by adding two, but the goals conceded has remained the same.

38 subtract 24 is equal to 14.

So their goal difference is now 14.

Lucas and Sofia write out the equations for each team and then redraw the bar graph.

So you can see in that table on the right, you've got all four teams in one column and then you've got their most recent goal difference equation in the right hand column, which gives the difference for each of the teams. Skatey Spuds, you can see that they've scored 43 but conceded 29.

The red bar shows that they've scored 43 and the blue bar shows they've conceded 29.

The difference is shown by the difference between those two bars.

There's Polar Bears, scored 53, conceded 39, so they've got a goal difference of 14.

Lucky Pucky scored 50, conceded 32, giving them a goal difference of 18.

Sticky Sticks scored 38, conceded 24, giving them a goal difference of 14.

At the start, they all had 15 as their goal difference.

Now, Lucky Pucky have the greatest goal difference.

Okay, it's time for your practise task then.

You are gonna complete a similar task to Sofia and Lucas.

Below is a bar graph showing the goals scored and goals conceded for four teams in a local football league.

Use the bar graph to complete the table.

So you've got four teams, Happy Feet, Jaguar Juniors, Bolly Bears, and Goal Hangers.

You've got to fill in the table which has boxes for how many they've scored, how many they've conceded, and then a box where you can calculate what the goal difference is.

Here's number two then, the results of two sets of fixtures are shown below.

Complete the table by writing in the equation for goal difference.

So you've got to do a similar task to the one that's just been modelled.

They are all the results.

You've got the first round of fixtures at the top and the bottom has the second round of fixtures.

What's their goal difference after these? For number three, you're gonna use that information you calculate in the previous question to draw out a new goal difference graph.

Okay then, pause the video here.

Enjoy and good luck.

I'll be back in a little while for some feedback.

Welcome back.

Let's have a look at number one to begin with then.

Happy Feet scored 35 and conceded 35, giving them a goal difference of zero.

Jaguar Juniors scored 45 and conceded 30, giving them a goal difference of 15.

Bolly Bears scored 25 and conceded 10, giving them a goal difference of 15.

Finally, Goal Hangers scored 45 and conceded 35, giving them a goal difference of 10.

Pause the video here if you need extra time to mark that.

Here's number two then.

We had two sets of fixtures from which we needed to calculate the new goal difference for each team.

Happy Feet, well, they won two, nil and then they lost three, one.

Here's the jottings for their new goal difference then.

At the top you can see the equation was their original goal difference before these results, that was 35 subtract 35, which is equal to zero.

Then in the first match they won two, nil, so the number of goals scored was adjusted by adding two.

That gave a new goal difference of 37 subtract 35, which is equal to two.

In the second result, they lost three one.

So we've adjusted the goals scored by adding one and adjusted the goals conceded by adding three.

That gives them 38 subtract 38, which gives them a goal difference of zero again.

There it is.

Jaguar Juniors ended up with 50 subtract 36, which is equal to 14.

Bolly Bears with 27 subtract 14, which is equal to 13.

And Goal Hangers were 52 subtract 39, which is equal to 13.

If you need a little bit more time to mark those, then pause the video now.

Okay, here's number three then.

So we had our goal difference.

We've calculated it in question two.

The bar graph should have looked something like this.

There's Happy Feet, both bars being 38.

Jaguar Junior scored 50 and conceded 36.

Bolly Bears scored 27 and conceded 14.

And Goal Hangers, they scored 52 but conceded 39.

There's the bar graph.

Again, pause the video if you need some more time to mark that.

Okay, it's time to move on to the second part of the lesson then.

Now we're gonna be looking at some puzzles involving difference.

Lucas and Sofia have a look at some digit cards laid out in an upside down pyramid.

What do you notice? Have a look at those numbers.

Think about what we're learning about today and can you spot a pattern or see any connections? Lucas says, "The number below is the difference between the two above.

Oh yes I see, 10 subtract eight is equal to two, for example.

Similarly, 11 subtract eight is equal to three and three subtract two is equal to one.

Lucas and Sofia try and tackle a difference puzzle.

They need to arrange the numbers one to six in the configuration shown below so that each number is the difference of the two numbers above.

They've got one, two, three, four, five, and six.

Sofia says, "The numbers will decrease as you go down each row because it's subtraction." So let's start with placing one at the bottom and five and six at the top left and right.

That's good reasoning, Sofia.

I think we need a greater difference going down.

So let's put two in because it's the next smallest after one.

And remember, they've already used one.

That leaves four and three in the middle row where they fit.

So you can see in this pyramid, you've got six subtract two is equal to four, so four is underneath six and two and you've got five subtract two is equal to three.

The difference between five and two is three.

So three is underneath the two and the five.

Finally you've got the difference between four and three is one, so one is at the bottom.

Okay, your turn.

Sofia thinks there might be more than one solution.

Can you find another? So can you arrange these cards in the same configuration so that the number below the two above is the difference between them? Pause the video and give it a go.

Welcome back.

Did you manage to find the solution? Well, Sofia says, "I'm gonna try six and one in the top row because they're the only two numbers that give a difference of five.

Three can't go on the middle row or top row because it would leave a difference of three, so it would have to be repeated.

It must be at the bottom.

Two and five have a difference of three, so two must go in the middle row leaving top left for four." You can see that it works.

Now it might be that some of you have mirror images of this, but that still counts.

Okay, ready to move on? Let's go for it.

Lucas sets out to create a new puzzle for Sofia.

He takes the same difference puzzle and adjusts one of the numbers with addition.

So you can see he's written plus four on the number three there, that's his adjustment.

"For this to work, which other numbers will I need to adjust in the same way?" I see, so if he adds four to that three, that will become seven, but that will mean that the rest of the puzzle doesn't work.

So what other numbers will need to change in order to make it work still? What do you think? Hmm.

"Well, if I adjust the three by four, then the difference between the two numbers above will also need to be four more, so I'll need to increase five by adding four." It changes to nine.

"I'll also need to adjust the four by adding four to make sure the one at the bottom is correct.

Because of that, I will adjust the six by adding four to make sure that the new difference of eight is correct." Lucas then shuffles the new cards up and gives them to Sofia to solve the puzzle.

"So two and one are still the same, so go in the same position.

If I'm subtracting two, then it must be the greater numbers in the top row." There we go, well done, Sofia.

"What was the adjustment?" Says Lucas.

What a great question.

"You added four to get one of the new numbers." "Correct, it was this one," says Lucas and he points out that the three had been adjusted by adding four, which started the whole thing off.

This time, Sofia sets up a puzzle for Lucas.

She adjusts one of the numbers with subtraction.

Can you see she's written subtract two on the number five there? For this to work, which other numbers will I need to adjust in the same way? So this time we are thinking about an adjustment, which is subtraction.

If she takes a five and subtracts two, giving her three, which other digit cards will need to adjust? What do you think? Hmm.

There's one solution.

We've got the three that's changed already.

Then we've got one and two in the middle row and a four top left.

"I think there's a second way," though, that's what Sofia reckons and she is correct.

There's another way of doing it.

A four, zero, and three in the top row.

Four, three in the middle and a one at the bottom.

Okay, your turn, let's check your understanding.

We've got the original puzzle laid out on the left and then we've got a new puzzle laid out on the right.

What adjustment was made? Pause the video and have a think.

Welcome back.

Okay then let's see if you managed to work out that adjustment.

It was subtract one.

Six subtract one is equal to five, four subtract one is equal to three, one subtract one is equal to zero.

The other cards remained unchanged.

Okay, let's do the second practise task.

For number one, I want you to arrange the number cards below into the configuration so that each number is the difference of the two numbers above, so it's the same as the puzzles that you've just been looking at.

We've got different numbers though for each one, for A and B.

For number two, for each of the following, work out the new numbers using the adjustment.

Can you find more than one solution? You've got two A and B there, and then we're gonna do number three.

Create your own puzzle for a learning buddy to solve.

You can use the numbers below to work from and adjust or create an entirely new one if you wish.

Write out the numbers onto scrap paper and then shuffle them.

Pause the video here, enjoy those tasks.

and I'll be back in a little while for some feedback.

Good luck.

Okay, welcome back, let's look at number one to begin with.

Here in A, with the solutions that you could have had, on the left, you can see you've got 12, 4, 10, eight, six in the middle and two at the bottom.

and on the right hand side there, bottom right, you can see you've got eight, 12 and two on the top row, four and 10 on the middle row and six on the bottom.

Pause the video here if you need some more time to check those carefully.

Here's B then.

We've got 16, 24 and four in the top row, eight and 20 in the middle row and 12 in the bottom row.

Or you could have had 24, 8, and 20 in the top row, 16 and 12 in the middle row, and four in the bottom row.

Again, pause the video if you need more time to mark those carefully.

Here are the answers for number two then.

For two A, there was an adjustment of takeaway three and you can see that then had a knock on effect to all the other cards.

We decided that the first adjustment was made on the number four, but you might have chosen any of those four cards that were changed.

As Sofia says, "Here's one solution and there were more." Okay, here's B, that was an adjustment of plus two, but again, it could have been either the six, the four, or the one that were adjusted, that still, each of those would've come out with that new puzzle on the right.

And here's number three.

This is the one that was completed and given to Sofia by Lucas.

We had 16, six, and nine in the top row, 10 and three in the middle, and seven in the bottom row.

I hope you managed to enjoy creating your own and giving it to somebody else to calculate.

Okay then let's summarise our learning today.

Understanding the underlying structure of subtraction is helpful when it comes to solving problems in a range of contexts.

Being able to mentally calculate adjustments to the minuend, end subtrahend, and difference according to the context is a useful skill across a wide range of situations, including the use of statistical data and solving puzzles.

My name is Mr. Tazzyman.

I've enjoyed teaching you today, I've really enjoyed this unit and I hope that I'll see you again soon in another maths lesson from Oak.

Bye for now.