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

﻿Hello, it's Mr. Whitehead. I'm ready for your math lesson. All you need to be ready is a pen or pencil and some paper and your practise activity from last lesson. Press pause, go and grab those bits, and then we can get started. Okay, ready? Can you hold up your activities for me? Let me take a look at the missing addends that you found to balance the equations a and b. Okay, let me show you how I approach the problems. Part a, I started by rounding those known addends to help find an approximate value for the missing addend. So, by rounding, I have three addends. The two to the right, I added together to teach all 80,000. Now I know that the addends to the left need to total the same sum, also 80,000. So with my known addend of 40,000, the unknown addend approximately is 40,000 as well. Next, using the two known addends from the right, I totaled them. I added them together to find a sum, 79,559. And I've now used the known addends from the left. I've subtracted it from the sum of the addends to the right. And by subtracting that known addend, the difference from the subtraction is the same as the unknown addend. It is the unknown addend, 41,703. Part b, a similar approach. So rounding, first of all, to estimate the value of the missing addend. 42 plus 23 is equal to 65. The left hand side expression needs to total 65 as well. The known addend of 15 and unknown addend of 50, approximately 50. Using that with the original addends, I did the addends from the left. They're both known. I know that on the right, the unknown addend plus the known addend need to equal 65.24. So once again, if I subtract the known addend from 65.24, the difference I'm left with is the missing addend, the unknown addend, 49.95. Okay, so last time the focus was on addition. And in this lesson, we're going to look at similar activities from last lesson, but with subtraction in mind. Let me show you a few symbols that we're going to be using today. Symbol at the top, really familiar with this. And this is the sentence that we've been using to explain what that symbol means. Can you read it aloud on three? One, two, three. The value of the expressions on each side of an equal symbol must be the same. How about the symbol underneath? Do you know what it means? If you do, what changes would you make to the sentence to match this symbol? If you don't know what it means, take a look at it. Notice it's similar to the equal symbol. It's about the same as the equal symbol, but not quite. This symbol is the approximately equal to symbol. And that's a lot to say. And if we're writing the words down, it's a lots of write down. So we can also think about it as is about. We're going to use this symbol today to compare expressions that are about the same, approximately equal. We will also use these two symbols today. Test on your visualisation skills. Can you picture an image that helps you to recall the difference between these two symbols to match up the words with the symbols? The symbol at the top is greater than, and the symbol at the bottom, less than. As I say, it's subtraction today, so we're going to be using the language of subtraction, minuend, subtrahend, and difference. Okay, here is the first problem. Two expressions, subtraction and a missing symbol circle. Press pause, have a go at finding the missing symbol. Ready? Did you find the missing symbol? How did you find it? Mental approach, written method to calculate the difference between the minuend and subtrahend? Did you use any rounding from last lesson to find an approximate difference for each of them? If you did, that's the focus of this lesson, continuing to use rounding to approximate. Let me show you how I approached it. Expression on the left, I rounded each of those numbers in the subtraction expression, the minuend and subtrahend, to the nearest 10. So 89 rounds to... And 21 round to 20, nearest multiple of 10. Nine 10s subtract two 10s is equal to seven 10s . Expression on the right, what would that round to? 112 rounding to 110, nearest multiple of 10, and 72 rounds to 70, nearest multiple of 10. 11 10s subtract seven 10s equal to four 10s , 40. So now I've got some approximate differences. I can compare them. Seven 10s compared to four 10s. Seven 10s is greater than four 10s. Using the approximate differences, I can solve the missing symbol problem. The expression on the left is greater than the expression on the right. Okay, for this second problem, I want us to have a think about using rounding to approximate in a slightly different way. Looking at the expressions, still subtraction, still looking for a missing symbol. Expression on the left. Now looking at these numbers, I wonder if this time we could round to the nearest whole number to make those expressions easier, simpler to work with. So 42.2 subtract 7.825, the approximately equal to expression would be 42 subtract eight. Now, although the numbers are easier to work with now, they look simpler. It's 42 subtract eight, but still going to take some mental work to find the difference. Another way to look at this problem would be to consider how big, what size is that difference roughly going to be, or can I say whether it's going to be greater than or less than a particular number? 42 subtract eight, if I think about that, I can say, "Well, the difference is definitely going to be "less than 40." The difference is going to be less than 40. Right hand side, what will be approximately equal to expression be if we round to the nearest whole number? Minuend will become 126 and the subtrahend will become seven. So once again, 126 subtract seven wouldn't be too challenging to work out the difference there, but I want us to think about it in a similar way. Could we say, "Well, the difference is definitely going "to be greater than or less than a particular number?" Now I've gone for this and wonder how it compares to what you're thinking right now. I've gone for greater than 100, but if I subtracted seven from 126, definitely would be a difference is greater than 100. So now that I've got those two approximate differences, 40 and 100, I can compare those and fill in the missing symbol. Read it out for me with the missing symbol, please. One, two, three. 40 is less than 100. And we can use that and refer back to the original problem. Our approximate differences of the left and right hand side expressions with the missing symbol filled in. Okay, third problem, but same minuends and subtrahends as on the previous slide. I want to show you a third way of finding a missing symbol without calculating the expressions. Let me draw your attention to the minuends. What can you tell me about 42.2 and 125.7? Tell me. Yes, 42.2 is less than 125.7. What can you tell me about the subtrahends? Tell me. You could say 7.825 is greater than 6.903, but I've gone for this symbol. Why is it okay for me to use this symbol? Those two values, those two subtrahends. They're about the same. They're approximately equal. So with that in mind that, give this sentence read for me on three. One, two, three. The subtrahends are approximately equal, so approximately the same amount is subtracted from the minuends, 42.2, 125.7, roughly the same amount is subtracted. So then let's think about the rough size of those differences. When we think about that, 42.2 is the smaller minuend, so the difference between it and the subtrahend will be less. I wonder if you could say that bottom sentence, but switch the minuend to 125.7. What else in the sentence would have to change that time? Give it go. 125.7 is a bigger minuend. So the difference between it and the subtrahend will be greater. So with that in mind, I'm able to say that the left hand expression is less than the right hand expression. Okay, so we have looked at three ways of using rounding and approximation to find approximately equal to expressions and amounts to explain the missing symbols. Approximation. Yes, it will work to use mental strategies and written methods to solve the missing differences to complete those expressions. Yes, that will work, but we thinking smartly, and we're looking at those numbers and using our rounding skills as well. Here are three for you to have a go at. Press pause. If you're tempted to use a mental strategy or a written method, that is okay, but then I challenge you to also use approximation to solve the problems too and see how that works. Press pause and come back when you're ready. Okay, ready? Here's the first problem. 96.5 subtract 8.57, 46.4 subtract 11.12, missing the symbol. What's it going to be? Hold up your paper. Let me have a look at how you've approached it. Can I see anyone that's used approximation, that's rounded those numbers? And then which type of approximation strategy have you used? Here's how I solved this one. 96.5 becomes 97. 8.56 is rounded to nine. 97 subtract nine. This time I've said that the difference is going to be greater than 80. I've not subtracted nine from 97, but I'm giving an approximate difference. It will be greater than 80. On the other side, I've rounded to 46 subtract 11. So once again, I haven't calculated the difference, the approximate difference using 46 and 11, but I have thought if I subtracted, the difference would be greater than 30. So now as I look at my 80 and 30 for the left and right hand side expressions, I can fill in my missing symbol, 18 is greater than 30. Is that how you solved it? Did you solve it differently? Did we get the same solution? We got the same solution. We found it in different ways. Next problem. How do you solve this one, the same as last time or differently? I've gone for a different approach for this one. 73 subtract 24. I've rounded, and that has become 70 subtract 20. Now this time, looking at those numbers, I've decided to subtract them. So I've got an approximate difference by subtracting two 10s from seven 10s equal to five 10s. On the right hand side, I've done the same and rounded to 180 or 18 10s and nine 10s. 18 10s subtract nine 10s is equal to nine 10s. Comparing those approximate differences, five 10s is less than nine 10s. Okay, third problem. Did something unexpected happen here when you were looking for the missing symbol? Let's have a look and see why. Did you round your left hand side expression to the nearest whole numbers? So what was your approximately equal to expression? 24 subtract six, which is equal to 18. So, the left hand side expression is approximately equal to 18. On the right, what'll be minuend and subtrahend round to? 21 subtract three. 21 subtract three is equal to 18. The right hand side expression is approximately equal to 18. So the missing symbol that we're looking for is equal to. The left hand side and the right hand side are equal to one another. I wonder, did anyone notice any patterns across the minuends in subtrahends. Have a look at it this way. Anything that you notice? And this is linking back to some of the previous lessons. Ah. The minuend. I've subtracted three from 24.267 to reach 21.267. There's a difference of three there. Between the subtrahend, there's also a difference of three. Now, previous sentence that we used, read it aloud with me on three. One, two, three. If the minuend and subtrahend are changed by the same amount, the difference stays the same. So perhaps some of you noticed the connection between the minuend and subtrahend as a different way of explaining how you noticed that the missing symbol would be the equals symbol. That's a really nice connection to the next part of the lesson. Now you've seen this before, and we used this to explain that the value of the expressions on each side of the equal symbol must be the same. You also looked at this last lesson with addition. Let's have a look at it in this lesson with subtraction in mind. Okay, so here's a problem for you. We need to balance this equation. The equal symbol is there. So the expression on the left, the value of that expression, and the value of the expression on the right need to be the same. This is subtraction, so that differences need to be the same. Press pause, and have a go at finding the missing minuend. Ready? Now, this time, it has been important to do some calculation and mental approach or a written method to solve the difference of the expression on the left. The minuend is there. The subtrahend is there, so we can subtract them and find the difference that matches that expression. 427 subtract 274. I've used a written subtraction. Is that what you've done, or did you solve it differently? Let's see if we have the same difference. Seven subtract four. So we have grouping to do here. 12 subtract seven. Three subtract two. 153. 427 subtract 274 is equal to 153. The difference there, 153, how does that help me with solving the missing minuend? I'm tempted to pop 153 into that space as well, but that's not looking right. If I now subtracted, the difference is going to be different to the left hand side, and the differences need to be the same. So that's not right. Let me have a think. 153. Blank space. In that blank space, subtract 385 needs to be equal to 153. So, well, in fact, if I add the difference of 153 to 385, that will give me my missing minuend. Another written addition for me. Three ones plus five ones. Five 10s plus eight 10s, 13 10s. 100 plus 300 plus another 100, 538, 538 subtract 385. The difference on the left and the difference on the right are equal. Now, I did say that a written subtraction or mental strategy was going to help you with this problem, but, again, I wonder if anyone noticed a connection between the left hand side and the right hand side expressions. Have a look at them like this. Anything that you notice? Maybe you did, maybe you didn't. The subtrahend has increased by 111. Now, I want the differences to be equal. And I know if I make some changes to the subtrahend, I need to make these same changes to the minuend to keep the difference the same. Fill in the gaps for me in this sentence on three. One, two, three. I've added 111 to the subtrahend, so I need to add 111 to the minuend to keep the difference the same, 538. So perhaps you spotted that connection there to help you balance the equation. Okay, with that in mind, here's one for you to have a go at. Press pause, balance the equation, keep the difference equal, then come back and we can have a look at it together. How did you get on? Have you found the missing minuend? Have you balanced the equation with the same value on the left and the right, the same difference on the left and the right? How did you find that missing minuend? Let me show you how I approached it. To begin with, I arranged the expressions vertically, one above the other, and my attention was drawn to the two subtrahends. Notice the tenths, hundredths and thousandths are the same, but the whole numbers are different. And as I looked at those whole numbers, five and 26, I noticed an increase of 21. Now from previous lessons, I know to keep the same difference across those two expressions, which is what we want. I need to make the same change to the minuend by increasing the minuend by 21 as well, 57.7, then the difference will be the same. The equation will be balanced. Give this sentence a read aloud with me on three. One, two, three. If the minuend and subtrahend are changed by the same amount, the difference stays the same. Now, perhaps you solved it in a different way, and I've got some jottings to show you, or my written subtraction, actually, that also can be used. The minuend and the subtrahend from the expression on the right, I can subtract to find a difference, 30.837. And I know that if I add that to the known subtrahend, I will find my missing minuend, 57.7. And we are at the end of our lesson. So, in a moment, press pause, take a photo, copy the activity down, so that you can have a go at it between now and the next lesson. Your teacher will review it with you next time. Thank you for joining me. See you again soon.

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

In today's lesson, we will look at similar activities from last lesson but with subtraction in mind, specifically the following symbols: = (equals),  ≈ (approximately equal to) , > (greater than), < (less than).

Maths

Y6 Maths

Key Stage 2

Year 6

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

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

Video

Lesson Complete