Decimals and measures: To generate and describe linear number

Key Stage 2

Maths

Decimals and measures

﻿Hi year six. Welcome to our first lesson in the decimals and measures unit. Today we'll be looking at generating and describing linear number sequences. All you'll need is a pencil and a piece of paper. Pause the video now and get your equipment if you haven't done so already. So today you'll be generating and describing linear number sequences. We'll start with a quiz. Then we'll learn how to represent decimals in term to term rule. We'll then represent linear number sequences on a number line before completing some independent work. So first we're going to look at representing decimals. Now, if the thousandths block represents one whole, I want to know what fraction of the whole do the hundreds, tens and one block represent. So you may want to pause the video now, while you have a think about what fraction of the whole, these blocks represent. Now, the hundreds block that represents one tenth of the whole. That's because there are 10, lots of 100 in 1,000, 10 times 100 is equal to a 1,000. And if I use my knowledge at the inverse, 1,000 which divided by 10 is equal to 100. So a 10th of the whole is 100. So continuing that line of thinking, the 10 block must therefore represent one hundredth of the whole. There are a hundred lots of 10 in a 1,000. And then again, if I use my inverse knowledge, I know that 1,000 divided by 100 is equal to 10. So this 10 represents a hundredth of the whole. So finally, the one block I'm thinking how many groups of one are there in the whole, which is 1,000 So the one represents thousandth of the whole because there are 1,000 lots of one in a 1,000 or 1,000 divided by 1,000 is equal to one. So they're going to use these representations of decimals when we're representing the actual numbers in our next slide. So you can see up here that we've got these representations and I have a place which is showing me a decimal number represented with dienes block. So I can see that I've got two wholes in the one's column, which represents two ones. Then I've got 1 tenth in the tenths column so that's 1 tenth, 3 hundredths and then 1 thousandth. So these dienes represent the number 2.131. Now I want each of you to use that representation to write the number represented with these dienes blocks. So pause the video and write which decimal number is represented with the dienes blocks. So these dienes blocks represent the number 0.243. There are zero ones in the ones column, two 10ths, 4 hundredths, and three thousandths. Now, one more time without the place value. Can you write the numbers represented with these dienes blocks? So your first representation here has got no wholes, nothing in the ones column. It's got one 10th so if I've got no ones. It's 0.1 tenth, 4 hundredths, and one, two, three, four, five, six, thousandths. So this represents 0.146. In my next one again, I've got nothing in the one's column. I've got no whole numbers there. So that's zero point. I've got 3 tenths so 3 goes in the tenths column, 2 hundredths and then one, two, three, four, five, six, seven, 8 thousandths. So this represents 0.328. And finally, I've got one whole, so one in the ones column, 2 tenths, two, four, six, 7 hundredths, and two, four, six, eight, 9 thousandths. So this represents 1.279. Now those representations are important to understand how decimal numbers are made up. Now let's move on to looking at some linear number sequences. So what is linear number sequence? Well, any number pattern, which increases or decreases by the same amount each time is called a linear sequence. So if I look at this sequence here, I've got my decimal numbers, but I also have the representation with dienes so I can connect the two of them together. I can see that this sequence is increasing because I'm adding more dienes each time so the numbers are becoming greater and I know that each time what is being added is 1 hundredth, so there were two here, there's three here so I'm adding one hundredth and I'm adding 3 thousandths. So therefore what I'm adding each time is 0.013. So I was adding you can see in this column 1 hundredth, and 3 thousandths. Now, the way that we can check that is by finding the difference between two consecutive terms. So consecutive meaning ones that are next to each other. So if I want to find out what the term to term rule is, what is being added each time, I need to find the difference between these two numbers, which I can do with a column subtraction. 0.134 subtract 0.121. I begin in the thousandths column, 4 thousandths take away 1 thousandth is 3 thousandth, then the next column. And then in the tenths column, one take away one is zero or 1 tenth, sorry, take away 1 tenth is zero tenths. Zero ones take away zero ones is zero ones. So I'm subtracting, I mean sorry I'm adding 0.013 each time and I know that I'm adding because it is increasing each time. So the term to term rule here is to add 0.013. Now I want you to think about what will be the next term in the sequence. So this is the term that we've got up to the fifth term. I want you to think about what will the sixth term be? Remember, this sequence is a linear number sequence. So each time you're adding the same amount and this is the term to term rule. So pause the video and work out what the sixth number will be in the sequence So now to find the sixth number what we're looking at is the one that we're up to 0.173, and we're adding onto that what our term to term rule is. We're adding 0.013, which gives us 0.186 so that's our sixth rule, our sixth term, sorry, in the sequence. Let's have a look at another one together. So before we work on finding the term to term rule, let's think about what do we notice about the digits in these numbers? So I want you to have a think about what you notice about the thousandth digit in each term. What do you notice about the thousandth digit? Just pause the video now and have a little think about that number in each of the decimal numbers. So what we can see is that this thousandth digit alternates between even and odd. So it's even here odd, even odd, even odd, even. So we can use that to help us answer some questions. So we know that the term to term rule already, is 0.013, because we're using part of our sequence from the previous slide. Now I want you to think about what the 10th number in the sequence and in an odd or an even digit. So if you think about this as the first number, this is the second and so on. What do you think about the 10th? Pause the video now and make some notes. So we can see if we label each of these numbers. Each of these terms, we can see that the first term has an even digit in the thousandth column. The second term has an odd and so on. So if we're looking at the 10th term, we could carry on thinking, well, the seventh is even, the eighth will be odd, the ninth will be even the 10th will be odd, but we could also think about it in this. If we're looking at an even term in the sequence. So we're looking at the second, fourth, sixth, eighth, and 10th, we can see that each even term in an odd number. So the 10th number in the sequence would end in an odd digit. And we can reason that by looking at what we knew already know about the sequence. Let's have a look at another question. Do you think that 0.2 would appear in the sequence? Now, as you move more into further advanced ways of looking at number sequences, you'll workout formulas, to what, you'll use formulas, sorry to work out, whether this number would appear, but at this stage, what you need to do is to continue the sequence and see whether 0.2 would appear in it. So you have the term to term rule, continue the sequence and tell me would 0.2 appear. Pause the video now and work it out. So if I go onto my finding the eighth term, what I would do would be 0.186 plus 0.013 and I know that that will be 0.199. Now I can either go on to find the ninth, to see whether it be 0.2 or I can use my understanding of number and I know that if I added one more thousand, that would take me to 0.2. So it cannot appearing the sequence because I'm not adding one more thousandth. I'm adding 0.013, I'm adding 13 thousandths, so I know that it won't appear using my understanding of number, but I also just demonstrate that and check it by finding this was the eighth by finding the ninth term in the sequence. So by adding 0.013. 9 thousandths and 3 thousandths is 12 thousandths. 9 hundredths plus 1 hundredth is 10 hundredths, plus another one is 11 hundredths. 1 tenth plus 1 tenth is 2 tenths so my ninth number in the sequence will be 0.212. So 0.2 does not appear in the sequence. Now it's time for you to have a look at another one independently. So you have a sequence here in front of you and there's a series of questions for you to answer about this sequence, pause the video now and have a think about the questions below. So let's have a look at this one together. Here is our sequence and the sequence we can see alternates answering, our first question alternates between six and one being in the thousandths column. So if we represented them with dienes, we would have 16 wholes, 8 tenths, 0 hundredths, and 6 thousandths and so on. It'd be too big to draw out using our dienes blocks but as long as you can reason how you would represent that, then that's good enough for me. Now, the term to term rule, you can see that this was decreasing. So the term to term rule and finding the difference was to subtract 1.005 each time Now answering the next question. So the 10th digit in the sequence, sorry the 10th number in the sequence would have a one in the thousandth place. And because even terms in the sequence have a one in the thousandths place. So the first time has a six, the second term has a one, third term six, fourth term one and so on. We know that the 10th would end with a one in the thousandth column. Now here's the challenge question for you. I want you to think about whether this sequence would contain any integers. Would it contain any whole numbers? Pause the video and have a think about the challenge question. So the answer to this is that no, it would never contain any integers because it's decreasing by 1.005 with the five in the thousandth column, the thousandth digit will always be six or one. So we would not find any integers in this sequence. Now let's have a look at linear sequences on a number line. So here we have a number line and the arrow indicates the third term in a sequence. So these numbers are going to help you to figure out what the other terms are but this arrow is pointing to the third term. The term to term rule is to add 0.012, or to add 12 thousandths. So we need to find the other terms in the sequence. We need to find terms one to five. So what I'm going to do is I'm going to start off by finding the fourth and fifth terms. So I already know what the third term is. If I look at my number line, I can figure out that these lines here represent thousandths. So this is going to be 0.571, 0.572, 0.573, 0.574. So I'm going to mark that on that this is pointing to 0.574. So to find the fourth term in the sequence, what I need to do is I need to take the third term 0.574, and I need to add 0.012. That's the term to term rule. So that will be, do some quick addition, 0.586. So that's your fourth term. Then I need to put an arrow to point to the fourth term. So I've got 0.58. I know that this will be 0.585. So this one here will be 0.586, which is my fourth term. Then I'll do my fifth term by taking the fourth term 0.586 and adding the term to term rule. 0.012, which is equal to 0.598. I'm going to think about where this will end up on my number line 0.59, 0.595, 596, 597, 0.598 will be here so that will be my fifth. Now I still need my second and first terms. So if the term to term rule is to add 0.012, how am I going to go backwards in my sequence? We'll I'll use the inverse. So the term to term rule is adding. If I want to find a previous one, then I'm going to need to subtract. Subtract 0.012. So to find term two, I'm going to take term three, which is 0.574. This time I'm going to take away 0.012 and that will give me 0.562, which will go here. So that's my second. And then again, to go backwards in my sequence, I need to use the inverse. So to find the first term running out of space here, I'll take my second term, which was 0.562. And I'm subtracting 0.012. Okay, and remember that this number is a placeholder here, so I don't actually need this digit. So my answer is 0.55, which means that my first term will go here. So what you need to remember is when you're going to the right in your sequence, you're using the term to term rule. And if you're having to go backwards in the sequence, you have to use the inverse term to term rule. So let's have a go independently now. You've got an arrow indicating the second term in a sequence. Your term to term rule is to add 0.008 and I want you to give me terms one to five. So pause the video while you work it out. So we know that this arrow here is pointing to 0.433. So the second term in the sequence is 0.433 and we know that that would be somewhere over here and to get from first to second, we were adding 0.008. So to get from second back to first, we're going to subtract 0.008. And that will have given you 0.425. We know this is 0.42 here. So 0.425 will be here. So that's your term one. Now it's more straightforward. So to get to term three, we're going back in the correct direction, 0.008 is being added to term two. So 0.433 plus 0.008 will give us 0.441, which goes here 0.441. And that will be a third term. Then our fourth term, again, we're adding 0.008 will be 0.449, which will be here. And then your fifth term will be 0.457, which will be here. Okay, I think it's time now for you to complete some independent work. So pause the video now and complete the task and then click restart once you finished and we'll do the answers together. All right, so looking at question one together, you were asked not only to complete the sequence, but also to give the term to term rule. So the first job was to work out what's the term to term rule, given numbers that we already know. So you will have, used this and seen that it was decreasing so you're subtracting and then if you find the difference between these two numbers, you know that the term to time rule is minus 0.62. So to work this out the other way, you are going to add 0.62 to get from the third term to the second term, which will give you 6.65. And to go back this direction, you are adding to give you 7.27 and then to find the next two terms in the sequence, you were back in the right direction. So subtracting 0.62 from 4.17, which gives you 3.55 and then one more time to give you 2.93. So for this one, you could see that the sequence was increasing. So you knew you were adding and finding the difference between these two shows that we were adding 0.802 each time. So to find the first one we're going backwards. So we inverse minus 0.802, which gives us 3.175 and then back in the correct direction. So adding 0.802 gives us 5.581. Again, adding 7.185 and our final one is 7.98. On your second question, you were given a number line with the arrow indicating the first term in the sequence. And you were also given the term to term rule of adding 0.022. So you could see that this arrow was pointing to 0.023 and you were adding 0.022 to each time. So to find your second term, you were doing 0.023 plus 0.022 which gives us 0.045. We know that this line here points to 0.04. So this one here must point to 0.045 since each of these small lines represents a thousandth. So this is your second term. Your third term, was 0.045 plus 0.022, which gives us 0.067 and again, this one is 0.06. Oops, this is 0.065. So this one, is there a 0.067. And your final one, you're using the third term, 0.067 plus your term to term rule 0.022 is equal to 0.089. This is 0.08 so this one here, is 0.089. That's your fourth term in the sequence. Now looking at a different representation. We've got the arrows indicating the first three terms of the sequence. And you were asked to find the two terms so you can, should have labelled on what these were. So this is pointing to 0.4, this one to 0.75 and this one to 1.1 and finding the difference tells us that the term to term rule is to add 0.35. So your next two terms will be 1.1 plus 0.35, which is 1.45. And then 1.45 plus 0.35 is equal to 1.8. And then a final true or false question. So this, sorry, I should say B here, there we go. So true or false. None of the terms in the sequence have more than two decimal places. And this one is true because we were adding a multiple of 0.05 each time and we started with 0.40. The pattern will always have a zero or a five in the hundredths column. So this question here is true and it's to do with it being a multiple of this number. On to our final question, Zaara writes a linear number sequence. She begins at five and she subtracts 0.123. She says that, no matter how much we extend the sequence, there will never be another integer term. Do you agree? I need you to explain your answer. So in this question, we were starting with an integer. So the first number in the sequence was an integer. And then we were subtracting 0.123. So subtracting a decimal, but in order to get another integer, we need to subtract an integer. Each time here, though we're subtracting a decimal, but there will be a point where we're subtracting an integer. So if we got to our thousandth term, what we will ultimately have done is subtracted 0.123 a thousand times, which gives us 123. So actually the thousandth term will be an integer. It will be a negative integer. It will be minus 118, but every thousandth term will be an integer because every thousandth term we will have subtracted a whole number. Okay, you've done some really great work today. It's time your final quiz to check everything that you've learned. Click restart once you're finished. Great work today, year 6. In our next lesson will be learning to use, read, and write standard units of length, mass and volume. I'll see you then.

Decimals and measures: To generate and describe linear number

Decimals and measures: To generate and describe linear number

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