Introduction to same difference

Key Stage 2

Maths

Extending calculation strategies and additive reasoning

﻿Hello, it's Mr. Whiteted here, ready for this lesson. Can you go collect what you need for us to get started your practise activity from last lesson, a pen or pencil, and some paper. Press pause, grab what you need, then come back and press play. You have what you need? Let's get started. This is the practise activity that you were set at the end of the last lesson. Let's have a look at how you've done. Can you hold up your jottings, your workings out, your solutions and let me have a little look. Fantastic. Let's take a look at the first part. Can you read the equation with me? Ready? 25,560 add 23,400 equals 48,960. Underneath that, there's a second equation with an unknown value. Is it an unknown addend or an unknown sum? Right, it's an unknown addend. Now, looking at that second equation, I'm sure some of you thought I can use the inverse operation. I can subtract the known addend from the sum to find the missing addend. And I can use a written subtraction do that. And yes you can. But I want us to think about the learning from last lesson. The sentence at the bottom, read it with me. If the sum is changed by an amount and one addend is kept the same, the other addend changes by the same amount. So let's look closely at anything that is the same and anything that's different between those two equations. Did you notice this? The second addend, 23,400, it's increased by 100 to become 23,500. Let's look at the sum. Maybe not as easy to spot, but the sum has also increased. Also increased by 100. There were nine one hundreds, increasing by one, we have 10, one hundreds that we regroup as one thousands. So the thousands digit has increased from eight to nine. The addend has increased by 100, the sum has increased by 100. So what about the unknown addend? It's kept the same. So I said, some of you may well have used the inverse operation to find the unknown addend. Yes, that will work too, when we have the same value. However, the use of our sentence I hope you can see, really saved us some time and didn't need us to use a written method to solve that unknown addend. Part two, agree or disagree. Explain why, no calculating. First one, read it with me, 235 add 235 add 255 is, I'm visualising, greater than 235, add 245, add 255. Hold up for me to show you how you approached this and how you've shown me, whether you agree or disagree. Compare that to how I approached it. I made use of some colour again, to help me focus in on the left part of the inequality and the right part of the inequality, and then reordered them, so I had one above the other, and can you see what I spotted? Did you spot it too? Two of the addends have remained the same, 235 and 255. One of the addends has changed. It's increased or decreased? It's increased. By how much? Good, by 10. 235 increased by one 10, becomes 245. So how does this help me to say if I agree or disagree? Is the left-hand side greater than right hand side? Well, we can see that the green addend is greater than the pink addend. So it's not correct to say that the left-hand side is greater than the right-hand side. This is incorrect. I disagree. And there are my jottings, to show why I disagree. Now we could have used some calculating, even though it said not to, we could have used some calculating. We could have added the three pink addends, then added the three green addends. We could have taken those sums and subtracted one from the other to find the difference. And there's that difference of 10, between the two parts of the inequality. I don't know, it's helping me reach the same solution, but comparing to what I did to start off with, I felt like I've been able to explain myself more coherently using the pink, the green and spotting the increase to help me say that's incorrect, I disagree. The right-hand side is in fact greater than the left. The next part of number two, again due to doing calculating, why don't you follow the instruction and try not to? I used some colour again. I reordered each side of the inequality symbol. What I did, it helped me to see that actually, look at my arrows. On each side of the inequality symbol, there are the same numbers, 9 million is on each side, and so is 5,040,000. So right now I've got an inequality symbol. The left-hand side is greater than the right-hand side. How can that be? If the numbers are exactly the same, I disagree with this second inequality. It's not an inequality, both sides are equal. So I've actually got an equation. What's on the left equals what's on the right. Okay, final parts of number two. Once again still using some colour, really helping me to see the maths and really helping me to start thinking about whether I agree or disagree. Once again I have placed the left side above the right-hand side and I've looked closely at the numbers. Again, I can see that 64.258 is on the left and on the right. So let me look at 19.278 on the left, on the right I've got 19.277 and 0.001. Let me look closer at those two parts on the right. 19.277 and 0.001, well, that's equal to 19.278. So actually once again, what's on the left is equal to what's on the right. So I don't need the less than symbol, what do I need there instead? Absolutely, the left-hand side is equal to the right hand side. I've got an equation and I should be using the equal sign between those two parts. Who is ready for a challenge. Hold up your sheets so I can see how you approach the challenge, before we look at it together. Let's have a read. Read it with me, please. At the beginning of the term, there are 192 pupils in Key Stage two. Year four, 68 pupils, year five, 65 pupils, year six, 59 pupils. During the term, one new pupil joins year five, and three pupils leave year four. How many pupils are there in Key Stage Two, at the end of the term? Now, just as in last lesson, where we had the house point problems, it was helpful to use a bar model to help see the marks. I approached this problem in the same way. And I wonder if you opted for a bar model or perhaps a part whole diagram, or maybe some equations. I re-read the problem and started to represent it using the bar model. The whole bar represents the total number of pupils, 192. I then split my whole bar into two parts. One part for each year group. Year four representing 68 pupils. How many in year five? 65, and how many in year six? 59, I then represented that same information as an addition equation with three addends, one addend for each part of the bar, one addend for each group and a sum representing the combined number of pupils from those three parts, from those three addends. The next part of the problem talks about one new pupil joining year five, and three pupils leaving year four. And at this point I thought it might be more helpful for me to see the maths if I switched to a pothole diagram. So I represented the hole and the three parts, and then in pink, you can see where I've shown three leaving year four, year four being reduced by three and year five increasing by one. Now the impact that's going to have, I changed two addends, I decreased an addend, I increased an addend, what's going to happen to my sum? So I just visualised decreasing by three, three have left, the sum decreases by three. But then it increases by one, for the new pupil joining in year five. So actually I just need to decrease the sum by two. And I've represented that here, decreasing an addend by three, increasing an addend by one, decreasing the sum by two. If you represent your thinking like this, or in a different way, we have the same total 190 as the solution to the problem, but we've solved it and shown our thinking in different ways, does it matter? We maybe would think about efficiency, we maybe would think if there was a smarter way of working, but we can reach the same solutions in different ways. Over the next few lessons, we're going to be thinking about the idea of same difference. Now you would already have seen this idea when studying consecutive numbers. Consecutive numbers, numbers that follow one another. One and two, are consecutive, 10 and 11, are consecutive numbers, 15 and 16, and so on. The difference between consecutive numbers is always one. It's always the same. This also applies with consecutive even numbers, two and four, 10 and 12, 22 and 24. The difference between consecutive even numbers is always two. They always have the same difference. And likewise with consecutive odd numbers three and five, 13 and 15, 21 and 23. The difference between consecutive odd numbers is always the same, it's always two. As the lessons build, we're going to use this idea of same difference to support us with our mental calculations. For today, we're going to look at some images and really build our understanding of what same difference looks like. Let's take a look at these two blocks on the number line. What do you notice about where the blocks start? Which number do the blocks start? Zero, and where do they end? Two. Now we can talk about that space between zero and two, the length of those blocks with the language of subtraction, and we can represent it with a subtraction equation, two subtract zero is equal to two. The number two is showing us the difference between zero and two. The length of the blocks. How about this time? Where do the blocks start? Where did they finish? And once again, we can represent the length, the space between two and four with a subtraction equation, four subtract two is equal to two. Ooh what have you noticed about those two equations so far? Something that's the same is the difference. The difference between zero and two and the difference between two and four is two, same difference. How about this time? Can you have a go at writing down for me a subtraction equation to match what you can see on the screen right now? Write it down for me, then hold it up. Ready? Have you shown me six subtract four equals two? That difference is still the same. Difference is not changing. What do we have here? This time can you say it aloud with me? The subtraction equation to match what we can see. Ready? Eight subtract six equals two. Still the same difference. And this time choose whether you want to write it down or you can read it aloud with me in a moment. If you're writing it down, quickly write it on your paper the subtraction equation to match what we can see here. And ready on three to show me or to read this aloud. One, two, three, 10 subtract eight is equal to two. Same difference. As those blocks have moved along the number line, the difference has remained the same. The length of the blocks has remained the same but what's changed? The starting point and the finishing point of those blocks on the number line. Let's look at this in a different way. What's different here with this image? I can't see the individual blocks this time, but I can still see a starting point and a finishing point. I could still work out the length of the rod. 30 subtract zero is equal to 30. The length of the rod is 30. What is the length of the rod this time? Still 30, 40 subtract 10 is equal to 30. The length of the rod hasn't changed. What's changing now then? Is the length of the rod the same? Still 30, what's different then? The starting point is now 20, and the finishing point of the rod is now 50. The length of the rod is still 30. Can you have a go at telling me the equation, the subtraction equation of what we can see now? On three, say your equation aloud. One, two, three, 60 subtract 30 is equal to 30. The length of the rod is 30. Can you write this one down for me? Use the starting point, use the finishing point carefully and show me the length of the rod with difference. Ready on three hold them up. One, two, three, 70 subtract 40 equals 30. The length of the rod is 30. The difference is 30. What about now? Write this one down. Ready on three, one, two, three. Fantastic, notice what's the same and what's different. Notice the difference is the same. The length of the rod stays the same as it moves along the number line. Call this one out on three, one, two, three, 90 subtract 60 is equal to 30. And finally on this number line 100 subtract 70 is equal to 30. We can see that the length of the rod has not changed. The difference is always 30. The difference does not change but the position of the rod on the number line does. So we have different numbers for the starting and end points of the rod and we can represent those in our subtraction equations. I've got a ruler for you this time. What do you notice about the unit of measurement that this ruler is using? Millimetres. Okay, here's a line for us to measure the length of. Where does the line start? Zero. And where does it finish? Look closely, 45. So the length of the line is the difference between zero and 45. The subtraction that can help us here, 45 subtract zero is equal to 45. The length of the line is 45. The difference between zero and 45 is 45. What about this time? Now we're starting at a different number and we're finishing at a different number. So has the length of the line changed? No, the length of the line is still the same. The difference is the same, but the starting number and the finishing number have changed. So now our subtraction, pull it out with me if you're ready. One, two, three, 55 subtract 10 is equal to 45. Get your paper ready and the pen or pencil, write down for me the subtraction that would help us here to show the length of the line, to show the difference between those two numbers. Get ready to hold your paper up on three, two, one. Show me your paper and compare, 65 subtract 20 is equal to 45. The difference stays same. Another line for you. Something's different this time. What do you notice? Starts on zero, ends on 35. Ah, so the length of the line this time, what will the subtraction be? 35 subtract zero is equal to 35. The difference is 35. The length of the line is 35. What about here? What stayed the same? The difference. So the length of the line is still 35. What's changed? Starting point, finishing point. So parts of the equation are going to be different. On three, say the equation aloud with me. One, two, three, 55 subtract 20 is equal to 35. Well done. Let's have a think about this idea of seeing difference in a different context. Imagine a beach. Imagine a beach during a heatwave. At 7:00 AM, the temperature is zero degrees Celsius. By 9:00 AM, the temperature has increased. Was zero degrees Celsius, it's now 10 degrees Celsius. How can we use subtraction to represent the difference in temperatures? 10 subtract zero is equal to 10. The difference between the two temperatures is 10. 10 degrees Celsius. By 10:30 in the morning, the temperature has increased to 20 degrees Celsius. So at 9:00 it was 10 degrees Celsius, and now it's 20. How can we represent the difference between the temperature at 9:00 and the temperature at 10:30? Have a go at writing down the subtraction to represent the difference, hold it up on three. Two, one, show me. 20 subtract 10 is equal to 10. The difference between the temperatures is still the same. The temperatures increasing by the same amounts. The difference between the temperatures is 10 degrees Celsius. How about by 2:00 PM in the afternoon now? So the temperature was 20 degrees Celsius. What might it be now? If the pattern is continuing, with the same difference is continuing, what will the temperature be? 30 degrees Celsius. And the subtraction to represent the difference, on three, say it aloud with me. One, two, three, 30 subtract 20 is equal to 10. The difference is 10 degrees Celsius. The difference in temperature between those times has stayed the same, the same difference, 10 degrees Celsius. Challenge for you. I want you to keep the idea of temperatures, I want you to keep the idea of temperatures changing, but I want you to change the difference between temperatures. Can you create a little sequence a bit like the one we just had, where temperatures are changing by the same amount each time. Don't use 10, like I did. Don't use the same difference of 10, change your difference. Keep it the same with each change of temperature and use some subtraction equations to represent those changes to represent that same difference. Press pause, give it a go, and then press play again, and I'll have some questions for you about what you've done. Press pause. Are you ready? Hold up for me, whatever you've been working on to show this idea of same difference but with a different difference to 10. Hold up your paper. Fantastic, here's some questions. What did you start at? What was your starting temperature? What did the temperature change to? And what was the difference then between your starting temperature and your next temperature in your sequence? What happened next? Did you keep the same difference as you made another temperature change and what happened after that? Same difference, different starting points, different finishing points on the thermometer. What changed and what stayed the same? Say it to me. What stayed the same? The difference. What changed? The temperatures changed. The difference between the temperatures stayed the same but the temperatures changed. So here's your practise activity, to have a go at between sessions on the next page, press pause, take a photo or copy down the information, ready for you to have a go at it. At the start of the next lesson, the teacher will review the task with you. I hope you've enjoyed the lesson today, I've really enjoyed myself, see you again soon.

Introduction to same difference

Introduction to same difference

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UnitMaths / Extending calculation strategies and additive reasoning

Maths