Lesson video

Hello, my name is Mr. Tazzyman and I'm going to be teaching you today's lesson from the unit that's all about equivalence, compensation and addition.

I hope that with some of the learning that you experience today, you'll have a better understanding of some of the underlying structures for addition problems that you face in maths.

Here's the outcome for today's lesson then.

I can solve addition calculations mentally by using known facts.

These are some of the words that you are going to hear and you are going to use during your learning.

I'm going to say them and get you to repeat them back to me.

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

Ready? My turn, equation.

Your turn.

My turn, adjustment.

Your turn.

My turn, whole.

Your turn.

My turn, part.

Your turn.

Here are the meanings to those words then.

An equation is used to show that one number, calculation, or expression is equal to another.

An adjustment is a change to a number using either subtraction or addition.

This can be done to a part or a whole.

The whole is all the parts or everything, the total amount.

A part is a piece or section of the whole, and the bar model at the bottom there shows this relationship.

This is the outline then.

Solve edition calculations mentally by using known facts.

We're going to start by thinking about the fact that we always need to analyse and adjust.

Then we can look at missing numbers, inequalities and justifications.

We're starting with analyse and adjust and here are two friends we'll meet along the way.

Sofia and Jacob, they're going to give us some prompts, some hints, some tips, and maybe even reveal some answers for us.

Ready to start? Okay, let's do this.

Jacob and Sofia look at an addition question.

0.

35 plus 54.

65 is equal to an unknown.

"Ooop, this has decimal fractions! I'll use columns here.

I know it well." "Wait! We can be more efficient." What has Jacob spotted? "We need to analyse the numbers closely." "I've spotted something.

Look at the decimal fractions." What do you think it might be that Jacob has seen? "I know that 35 and 65 are complements to 100, so I know that decimal fractions added together are equal to one." 0.

35 plus 0.

65 is equal to one.

I can use this known fact to help me.

"I get it.

A part of our known fact can be adjusted." "Exactly.

If we adjust one part and the other part remains the same, we have to adjust the whole by the same amount." You can see on their jottings there that they've made an adjustment by adding 54 to one part, so they've had to add 54 to the whole.

"That was much more efficient," says Sofia.

They've got an answer of 55.

"I think I could do that mentally with jottings." Okay, it's your turn.

Analyse the parts and adjust one of the parts to find the missing number.

Pause the video here and have a go at that.

Welcome back.

You might have started with this known fact.

0.

27 plus 0.

73 is equal to one.

Then you need to adjust the second part by adding 16.

Therefore, you need to adjust the whole by adding 16 as well.

Giving you an answer of 17.

Jacob and Sofia look at another addition question.

0.

25 plus 54.

42 is equal to an unknown.

"The decimal fractions aren't complements to one.

Columns it is!" "Wait! Let's analyse closely again." Remember, it's always worth looking at those numbers really closely to see if you can do it mentally with jottings.

How would you complete this one mentally? "I can calculate 42 plus 25 in my head without bridging 100.

It's 67, so that means 0.

42 and 0.

25 are equal to 0.

67.

I can use this known fact to help me." "Now to make an adjustment to help me solve the first equation." So he's added 54 to the second part, then he adds 54 to the whole, so he gets 54.

67.

"Again, much more efficient! Impressive speed." Okay, it's your turn.

Analyse the numbers and see if you can calculate the addition mentally using adjustment.

Remember to start by working out a quick known fact to use.

Pause the video and have a go.

Welcome back.

You might have started with this known fact.

0.

18 plus 0.

41 is equal to 0.

59.

It's possible that you even began by adding 18 and 41 or 41 and 18 in order to get this known fact.

Then you might have adjusted 0.

41 by adding 16 and therefore also adjusted the whole in order to maintain a balanced equation by adding 16 to get 16.

59.

Sofia solves more addition problems like this.

What's the same and what's different? So we've got three sets of jottings there.

Can you see any similarities? Can you see any differences? "The whole is the same in each.

It's 55.

The parts are different in each.

One-hundredth has been redistributed between the parts of our known fact." "But the redistribution hasn't affected how we have adjusted to get the solution.

The redistribution has just affected the known fact that we use." Sofia solves more addition problems like this.

What's the same and what's different this time? "One part remains the same, 54.

65.

One of the parts is increasing by one each time.

But it's not the part that we are adjusting from." "Because one of the parts is one more each time, the whole is one more." Okay, what's the same and what's different this time? Have a good look at each of those sets of jottings.

"The part of the known facts we are adjusting from is decreasing by one-hundredth each time.

That means that the whole of our known fact is also decreasing by one-hundredth each time.

They're not decimal fraction complements to one, but they nearly are, so they are easy to calculate." What's the same and what's different this time.

Have a good look.

"One of the parts is increasing by 10 each time." "The other part of 0.

35 is remaining the same each time.

Consequently, the whole is being adjusted by 10 more each time." Jacob and Sofia compare two calculations.

"Can we adjust one part to help?" What do you think? "Let's analyse the parts to see." "In the first equation, the decimal fractions are complements to one.

That means we can use adjustment easily." 0.

57 plus 0.

43 is equal to one.

We can adjust that by adding 32.

So the whole needs to add 32 added to it as well, giving 33.

"In the second equation, the decimal fractions would bridge one if we added them.

So I think I'd use a written method to add these." There's the written method, so the answer is 33.

22.

Jacob has chosen to use column edition.

Jacob and Sofia tick the expressions that they think could be calculated mentally using adjustment.

What do you think? Have a look at those six expressions.

Which ones do you think would be good for adjustment rather than just using a written method straight away? "All those ticks have decimal fractions that are complements to one.

The others don't offer known facts that are easy to calculate." "Respectfully, I disagree! This one will need bridging, so I don't think I'd use adjustment here, but I think that I could calculate a known fact mentally so I could use adjustment on this one." "This calls for a race! You use adjustment and I'll try columns." The race is on.

"Go!" Jacob's already finished.

Sofia's only worked out the hundredths.

They've both got the same result though.

Jacob and Sofia face another addition.

This time featuring the larger numbers.

321,000 plus 79,000 is equal to an unknown.

"Larger numbers!" "We can still analyse." "I know 21 plus 79 is equal to 100.

That means 21 thousands added to 79 thousands is 100 thousand.

So we have a known fact to use and then we can adjust the part and the whole." Add 300,000 to that part.

So the whole also needs 300,000 added to it.

That gives 400,000.

Jacob and Sofia look at another addition question.

321,000 plus 41,000.

"I can't see complements to 100,000 here, so I think it'll be columns." "I disagree! Analyse the parts." How could you complete this mentally? We can still use a known fact.

"I know that 21 added to 41 is equal to 62.

I can use that fact, but with one thousands instead of ones." 21,000 plus 41,000 is equal to 62,000.

Add 300,000 as an adjustment and do the same for the whole.

We get 362,000, no columns in sight.

Okay, it's your turn.

Can you complete this calculation mentally by analysing the parts and adjusting? Pause the video here and have a go.

Welcome back.

Here's the known fact that you might have decided to use.

26,000 plus 32,000 is equal to 58,000.

You might have got that by starting with 26 plus 32.

You then need to adjust that first part by adding 200,000.

Keep the other part the same, which means that the whole needs to be adjusted by adding 200,000 as well in order to make sure there's a balanced equation.

That gives an answer of 258,000.

Sofia has to tick the expressions that she thinks she could calculate efficiently using adjustment.

What do you think? Look at those three expressions.

Which of those are best for adjustment? Which would you not use adjustment for and maybe use a written method like columns? "In this one I can see complements to 100,000 because I know 89 plus 11 equals 100.

Here, I can quickly calculate that 21 plus 35 equals 56 and use this known fact to help.

The bottom one is trickier because I'll need to bridge a hundred-thousand.

I know this because 56 plus 78 bridges 100.

I won't tick this one." Okay, time to check your understanding again.

Why might this calculation be harder to use adjustment with? Have a think about that.

Pause the video and I'll be back shortly to tell you what we thought.

Welcome back.

Here's what Jacob said.

"To use adjustment here, I would have to bridge a hundred-thousand." Okay, time for your first task.

For number one, I want you to use a known fact and adjustment to solve these mentally.

Use jottings if needed.

For number two, below are some addition expressions.

Analyse the parts and tick the expressions you think could be solved mentally using adjustment.

Write an explanation about your decisions.

We've got 789.

59 plus 0.

41, 20.

67 plus 789.

59, 789.

24 plus 0.

41.

For number three, it's the same task, but the numbers are different.

236,000 plus 64,000, 18,000 plus 231,000, 236,000 plus 178,000.

For number four, you've got to complete the following addition in two ways.

Firstly, calculate mentally using adjustment.

Then use a written method such as column addition.

Which did you find to be more efficient? Remember here, when you've done those two methods, do check that the answer you've got for each is the same, to make sure that you are doing each correctly.

But also be reminded that part of this is about finding a more efficient method, a more efficient journey to the answer, not just getting the answer right straight away.

Okay, good luck, enjoy.

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

So pause the video here.

Welcome back.

Let's look at number one.

Be ready to mark.

Here was A, we had 24, 124, 125, 124.

99 and 223.

99.

You'll see that a lot of those expressions had featured a lot of similarities which may have helped with the next ones.

Here's B.

300,000, 400,000, 399,000, 349,000, and 949,000.

Let's move on to number two then.

Jacob has ticked the top expression because it features a decimal complement.

0.

59 plus 0.

41 is equal to one.

"This means adjustment is easy here.

We just need to complete 789 plus one, which is 790.

For the bottom expression I can easily calculate a known fact.

0.

24 plus 0.

41 is equal to 0.

65.

The unticked expression needs bridging so it's harder to use a known fact here." Here's number three.

"The top expression features a complement to 100,000 which is 36,000 plus 64,000.

For the middle expression, I can easily calculate a known fact using different units.

I know that 18 plus 31 is equal to 49, so I know that 18 thousand added to 31 thousand is 49 thousand.

The unticked expression needs bridging across 100,000, so it's harder to use a known fact here." Okay, number four.

A was 157.

B was 400,000.

"I found adjustment to be more efficient because it was quicker and the numbers both featured complements so it was easy to do." Is that what you found? Okay, ready to move on? Let's go.

It is time for the second part of the lesson, missing numbers and inequalities.

Jacob and Sofia look at a very important word, because.

"This word helps us to make justifications." "What are they?" "Explanations of why we know something is correct or true." "Explanations with evidence?" "Yes, absolutely." Jacob solves an addition using adjustment and justifies his answer.

He has a known fact already.

29,000 plus 27,000 is equal to 56,000.

229,000 plus 27,000.

Well, he knows there's been an adjustment of 200,000, so there is an adjustment of 200,000 to the whole as well.

So that gives 256,000.

"It is 256,000 because one of the parts is 200,000 more so the whole also needs to be 200,000 more." Sofia has a go at the same task.

29,000 plus 27,000 is equal to 56,000.

19,000 plus 27,000 is equal to.

She makes an adjustment of 10,000 by decreasing by 10,000 to one of the parts.

So she also subtracts 10,000 from the whole giving her 46,000.

"It is 46,000 because one of the parts is 10,000 less, so the whole needs to be 10,000 less as well." Jacob has a go at justifying with decimal fractions.

0.

24 plus 0.

76 is equal to one.

0.

24 plus 29.

76 is equal to.

Well, there's an adjustment of 29 made to the second part, so the whole also needs to be adjusted by adding 29.

We get 30.

Jacob says, "It is 30 because one of the parts is 29 more so the whole also needs to be 29 more." Sofia has a go at justifying with decimal fractions.

0.

315 plus 0.

317 is equal to 0.

632.

0.

315 plus 0.

017 is equal to.

There's an adjustment of three-tenths made, subtracting it from the second part and also subtracting it from the whole.

So we get 0.

332.

And she says, "It is 0.

332 because one of the parts is three-tenths less so the whole also needs to be three-tenths less." Okay, your turn to have a go at something similar.

Complete the missing number and the stem sentence justification.

We've got 54,000 plus 15,000 is equal to 69,000.

Then we've got 34,000 plus 15,000 is equal to an unknown and you've got the sentence on the right to complete as well.

Pause the video and have a go.

Welcome back.

Let's see how you got on.

So the adjustment was 20,000 subtracted.

That gave an answer of 49,000 and your sentence might have read as follows.

It is 49,000 because one of the parts is 20,000 less so the whole also needs to be 20,000 less.

Sofia and Jacob want to work out which symbol should go between two expressions being compared.

What do you think? Have a look at those expressions.

We've got 15.

14 plus 0.

86, 15.

14 plus 0.

85.

Then we've got 15.

14 plus 0.

66 and then 15.

14 plus 0.

86.

Then we've got 15.

14 plus 0.

86 and 11.

14 plus 0.

86.

Jacob says, "Do I need to find one value for each expression?" "No, I think we can analyse the parts and use justification here." Sofia tries the first one.

What do you think? "This should have a greater than symbol because the second part of the first expression is one-hundredth greater." So she's really analysed those parts well and she's used that to give her the solution.

Next, Jacob has a turn.

What do you think? "This should have a less than symbol because this time the second part of the first expression is two-tenths less." They both look at the last one.

What do you think? "It's different this time because the change is in the integers and not the decimal fractions." "This should have a greater than symbol because the first part of the second expression is four-ones fewer." Okay, your turn.

Let's check your understanding.

Without calculating the value of each expression, can you state which symbol should go between these two expressions? Use a justification to explain your answer.

We've got 13.

27 plus 0.

73 and the second expression is 13.

37 plus 0.

73.

Okay, pause the video and have a go.

Welcome back.

Sofia says, "This should have a less than symbol because the first part of the first expression is one-tenth less." Is that what you saw? Is that what you got? I hope so.

Here's your second task.

For number one, you need to calculate the missing numbers and justify your answer.

Use the stem sentence below to help.

So you can see that there's a stem sentence at the bottom of the screen which you can fill in for your justifications.

For number two, write the correct symbols between the expressions below and justify your answer.

Can you see a pattern? For number three, a worded problem.

Jacob and Sofia play an arcade game with two different levels.

They want to know who has scored more points.

Below are their scores in a table.

Who scored the most points? Justify your answer.

Okay, pause the video here and I'll be back in a little while with some feedback.

Good luck and enjoy.

Welcome back.

Let's look at number one to begin with.

We'll start with A and B.

For A, the answer was 379,000.

Your justification might have sounded like this.

"It is 379,000 because one of the parts is 20,000 less, so the whole also needs to be 20,000 less." For B, 725 was the answer.

"It is 725 because one of the parts is 400 more, so the whole also needs to be 400 more." For C, 497,000 was the answer.

"It is 497,000 because one of the parts is 40,000 more so the whole also needs to be 40,000 more." For D, the answer is 42.

69.

"It is 42.

69 because one of the parts is 36 more so the whole also needs to be 36 more." Another related known fact is that four times 10 is equal to 40, but this question is additive and not multiplicative.

So some of you maybe made that mistake, have a check.

"That means four times 10 is equal to 40 isn't relevant here." Okay, let's look at number two then.

All of the symbols in number two were less than symbols.

The expressions are increasing by one unit each time, but the place value of the increasing unit is also getting greater.

And here's number three.

"Lucas one! The parts for each of us were very similar, but he had 40 more here, so his whole was also 40 more." Did you manage to spot that? Okay, let's summarise our learning today.

When looking at addition calculations, it is easy to go straight to using a written method.

However, there are more efficient methods available with closer inspection of the parts of the calculation.

Sometimes adjustment can easily be used if there are complements within the parts, or if a quick mental addition can produce a useful known fact to work from.

When comparing addition expressions, you don't always have to calculate the value.

My name is Mr. Tazzyman and I've really enjoyed learning with you today.

I hope that I'll be able to see you in another maths lesson.

Bye for now.