﻿Hello, I'm Mr. Connell, and I'm very pleased to be your teacher today. Do you have a pencil and some paper ready? Great. Let's get started. This is the practise activity that Mrs.Furlong left you with. Did you have a go? Did you use the generalisation? Let's say it together. If the minuend is kept the same, and the subtrahend is increased or decreased, the difference decreases or increases by the same amount. If you haven't had a go at this practise question, try it now. Another important point that we must remember is that the value of the expressions on each side of an equal symbol, must be the same. We'll come back to that later. Let's see what's underneath those splurts. Where did you start in question A? As the generalisation said, that the minuend is kept the same. I decided to check the middle equation first. To find the minuend, I need to do the inverse of subtract 300. That's it. I need to add 300 to the difference. So 72,065 add 300 is equals to 72,365. So the minuend is the same in all of the equations, right. On the bottom equation, compared to the middle equation, the difference has increased by 100. So the subtrahend needs to decrease by 100. Did you get 200 as well? You might also have solved it by looking at the hundreds column. In the top equation compared to the middle one, the subtrahend has increased by 300. So the difference needs to be decreased by the same amount. It needs to be decreased by 300. So that's 71,765. Well done, if you found that as well. Let's see what was under those blue splurts now. I'm not going to assume that the minuends are all the same. It's always good to check. Going to look at the bottom equation first. To work out the minuend I need to do the inverse of subtract 0.34. So, 12.231, add 0.34 is equal to 12.571. The minuends are not all the same. Glad I checked. Let's look at the middle equation now. I'm going to subtract 0.34 from 12.671. Could you do it mentally? The thousandths stay the same, seven hundredths minus four hundredths is three hundredths. Six tenths minus three tenths is three tenths. So it should have 12.331. Now compare the top equation with the middle equation. The minuend is kept the same. The difference has decreased by five hundredths. So the subtrahend in the top equation needs to increase by five hundredths. 0.34 plus five hundredths. It must be 0.39. Did you get that? I hope so. Look at these expressions. What symbol should go in the circle? Let's not rush it. Look at the numbers. Think about the strategies and generalisations that you've been practising , like the same sum strategy. Pause the video to have some thinking time. Did you see any connections between the addends? No. I didn't see anything obvious either. But do we have to work out the exact sum of each expression to work out which one is the greater? Well, let's use our number sense. We could estimate the sum of each expression. We could round each addend to the nearest whole number. 23.63 we'd round to 24, 25.34 rounds to 25. So 24 and 25. It's a bit less than 50. Isn't it? Well, on the right-hand side, 35.28 would round to 35, 29.16 would round 29. Well, I can already see that that's more than 50. So 30 and 20 would be 50, so that's greater than 50. So I can pretty quickly say that 23.63 add 25.34 is less than 35.28 add 29.16. Look at the expressions again. What else do we notice about the addends in the expressions? Did you spot both of the addends on the left are less than the addends and the expression on the rights? So another way of looking at this is first it's compare the largest addend on each side. 25 is less than 35. Now compare the smallest addends on each side. 24 is less than 29. So again, we can say, that the expression on the left is smaller than the expression on the right. Now look at these expressions. What symbol should go in the circle this time? Are there any connections between the addends? No, I can't see any helpful links either. We'll try estimating again. Start by rounding each addend to the nearest one hundred thousand. 473,621 rounds to 500,000 362,175 rounds to 400,000. Okay. What about on the right? 358,726 rounds up to 400,000 and 470,218 rounds up to 500,000. Oh, they're both the same! Okay. Let's try rounding to the nearest 10,000. What do we get this time? 470,000 add 360,000. Okay. What about the right hand side? Huh, It's the same again? Hmm. What'd you suggest we do next? Yeah, I agree. Let's round to the nearest 1000 see if that's any help. What did we get this time? You tell me, is this what you got? 474,000 add 362,000 and 359,000 add 470,000. Okay. That's a lot clearer now then isn't it. The left-hand side is greater than the right hand side. Left expression is greater than the right hand expression. Look at these expressions again. Is there another way to compare the expressions without calculating? Have a look at the largest addend on each side compare those. 473,621 is greater than 470,218. Look at the smaller addend on each side. 362,175 is greater than 358,726. So both addends in the expression on the left are greater than the addends on the right. So overall 473,621 add 362,175 is greater than 358,726 at 470,218. And we did it without having to work out the sums of the expressions. You're going to do this one for yourselves. Look at the numbers first, how you estimate by rounding, pause the video, come back. when you think you know the symbol. Hello again, what did you decide? How did you estimate? First I rounded to the nearest 10 thousand. The addends on the left, both round to 20,000, The addends on the right round to 20,000 at 10,000. So you might think that the expression on the left is the greater one, but rounding to the nearest 1000 gave a different solution. Did you try that? 16,358 rounds to 16,000. 18,763 rounds to 19000. 21,890 rounds to 22,000 and 13,231 rounds to 13,000. So the sums of my addends that I've rounded are 35,000 on the left. Oh, and 35,000 on the right. We're going to need to work out this one exactly. Is that what you did? On the left hand side, I used column addition. Did you? Here I set up my columns. Does it matter which addend goes on the top? No it doesn't. So, if we start in the ones columns, 8 ones add 3 ones is 11 ones. So 11 ones, I'm going to regroup into 1 ten and 1 one. I've got my regroup digits underneath the line. You might do something else. Do whatever your teacher tells you to do, and you do normally. In the tens column, we have five tens add six tens add one more ten that was regrouped. That's 12 tens, 12 tens regroups into 100 and 2 tens. In the hundreds column, we have three hundreds, seven hundreds add 100 hundred, eleven hundreds. 11 hundreds regroups into one thousand and one hundred. In the thousands column, we have 6,000 add 8,000 has one regroup thousand. So 15,000 which regroups into one ten thousand and five thousands. And then in the ten thousands column, we have one and one and one more. So we have three, ten thousands. So our sum is 35,121. On the right hand side, we could use column addition. Maybe you used another strategy. Did you do this, 21,819 is only 110 away from 22,000. So we can use the generalisation. If one addend is increased by an amount, and the other addend is decreased by the same amount, the sum remains the same. So 21,890, add 110 is 22,000 13,231 subtract 110 is 13,121. If I add those together, we get 35,121. Hang on. Both of those are the same. So our symbol in the circle is an equals symbol. Both sides of the equation are the same. You've seen this image in a previous lesson, when you were finding missing numbers to balance equations with additional expressions, just like the question where you just done. The value of the expressions on each side of the equal symbol must be the same. The equal symbol is really important in mathematics. Look at this missing number problem. In previous lessons, there were connections between the addends on each side of the buttons that made it easy to find the missing value. In this example, there aren't really any easy links. So sometimes you have no choice, but to calculate. Now, first step is to find the value of the complete expression on the left. 34 add 46 is equal to eighty. Our second step is to calculate the value of the missing number. We could think, what do I need to add to 72 to make 80? Or we could do 80 subtract 72. both of those. All right. So eight is the value that we need to balance the equation. Did you get that? Now look at this balanced equation involving decimal fractions. This time, the empty box is on the left. Can you see any easy connections between the addends on the right and the number 14.73 on the left?. Is there anything in the place value of the digits to help? Pause the video for some thinking time. What did you find? Nope. There were no connections to make the same sum it method easy. We're going to have to calculate again, aren't we? So our first step is to calculate the sum of the complete expression on the right. 32.52 add 8.36. Start in the hundredths columns, two hundredths add six hundredths is eight hundreds. In the tenths columns, five tenths add three tenths is eight tenths. In the ones column, two ones add eight ones is ten ones. We'll have to regroup ten ones into one ten, and zero ones. I've put my regroups one, ten underneath the line, you can put it wherever you want to, of course. And in the tens column, three tens add one ten is four tens. So our complete expression, our right-hand side is equal to 40.88. Now second step is to calculate the value of the missing number. Can you remember what we did last time? We need to subtract 14.73 from 40.88, to find out what that missing value is going to be to make the equation balance. We'll start in the hundredths column. Eight hundredths subtract three hundredths, is five hundredths. In the tenths, eight tenths subtract seven tenths is one tenth. In the ones column, we have zero ones subtract four ones. We'll need to regroup. We'll regroup one of the tenths from the tenths column into ten ones, which will leave us three tens in the tens column. So now I have ten ones subtract four ones is six ones and three tens subtracts one ten is two tens. That means that our missing number must be 26.15. How could you check that? That's right. You could add 14.73 add 26.15. Oh, I think you can do that mentally. What did you think? Yes, of course. We're right. Well done. You're going to do this one on your own. What will you look for first? Here is as little clue. Pause was the video and come back when you think you've solved the equation. Did you do it? Did you solve it? Did you decide that the numbers weren't kind for using the same sum rule? So you just had to calculate, What was your first calculation? Did you find the sum of the complete expression on the left? 34,725 add 43,269. So, in the ones column, five ones add nine ones is 14 ones, which we regroup into one ten and four ones. Two tens add six tens add the one ten is nine tens, seven hundreds add two hundreds is nine hundreds, four thousands add three thousands is seven thousands and three ten thousands at four ten thousands is seven ten thousands. So our sum on the left came to 77,994. Is that what you go? Well then if you did, let's see if we can solve to find that missing box then. What did you do next? That's right. You had to subtract 17,453 from 77,994. Now I'm just looking at those numbers. They look, it's quite a nice subtraction there. No regrouping. Great. So in the ones column, 4 ones subtract 3 ones is 1 one. 9 tens subtract 5 tens is 4 tens. 900 subtract 400 is 500 hundreds, 7 thousands subtract 7 thousands, no thousands and 7 ten thousands subtract 1 ten thousand is 6 ten thousands. So our missing number should be 60,541. How did you check it? That's right. I think both of those numbers on the right 60,541 and 17,453 should give us 77,994. It looks like you could do that mentally. Have a go. That's it. We were right. Brilliant, excellent work. I'm going to leave you with two practise questions to do before the next lesson. Remember to look for any connections between the addends that just could help you use the same sum rule. Also, think about how you could use rounding to estimate what the missing number should be. Finally, check your work for the important rule. Make sure the value of the expressions on each side of an equal symbol are the same. Thank you for all your hard work today.

In this lesson, we will balance equations where the compensation property of same sum cannot efficiently be applied.

In this lesson, we will balance equations where the compensation property of same sum cannot efficiently be applied. Resources provided by NCETM (CC BY-NC-SA: https://creativecommons.org/licenses/by-nc-sa/4.0)

Maths

Y6 Maths

Key Stage 2

Year 6

Balance Equations Where the Compensation Property of Same Sum Cannot Efficiently be Applied

Extending calculation strategies and additive reasoning

Number

THEME

UNIT

Adjusting addends

Same sum' with larger numbers

Same sum' with decimals

Balancing equations using the 'same sum' strategy

Balancing equations using compensation

Balancing equations: Does the order of addends matter?

Increasing an addend

Decreasing an addend

Solve calculations mentally by relating them to known facts

Find an unknown addend

Introduction to same difference

Same difference in context

Use the Language of Minuend, Subtrahend, and Difference

Transform calculations using the same difference

Practice: Transforming Calculations to Make Them Easier to Solve Mentally

Transform a subtraction calculation to make the written algorithm easier to apply

Practice: 'Same Difference' in Different Contexts

Balancing equations to find unknown values

Explore how the difference changes when only the Minuend is changed

Apply the generalisation about how the minuend and difference change to solve problems

Explore how the generalisation can be used as a mental calculation strategy using Known facts

Thinking flexibly

Comparing Strategies

The more we subtract, the less we are left with. The less we subtract...

Contexts where the Minuend is Kept the Same, and the Subtrahend Increases

Contexts where the minuend is kept the same, and the Subtrahend decreases

Further practice to reason about how the change in the subtrahend changes the difference

Explore problems in which the new difference must be found

Balance Equations Where the Compensation Property of Same Difference Cannot Efficiently be Applied

Lesson overview:Balance Equations Where the Compensation Property of Same Sum Cannot Efficiently be Applied

Core Content

Extending calculation strategies and additive reasoning: