What do we mean if we describe a sequence as "descending"?

Which of the following is an example of a descending sequence?

What is the nth term of 10, 8, 6, 4, 2, ...?

What is the nth term of -4, -7, -10, -13, ...?

Find the first 5 terms of the sequence -5n + 2.

Descending Arithmetic Sequences Quiz

﻿Hi everyone and welcome to the next lesson in your sequences topic. Today's lesson is all about dot chain sequences, but before we can begin, once again, make sure you've got pen and paper ready, and that you've got rid of all of the distractions and you're in a nice quiet space if you can find one. Pause the video to make sure all of that is ready before we can begin. Right, let's start. Antoni has used this arrangement of dots to form some chains. I would like you to count the dots in the two chain, and I would also like you to count the dots in the four chain and think about, is it what you expected it to be? When you're counting those dots, don't just go one, two, three, et cetera, because eventually you're going to be doing much larger chains than this. What I would like you to do is try and group them in some way to help you predict what's going to happen because in a second you're then going to predict how many dots there are in a five or a 10 chain. So remember, we're looking at this pattern here and he's repeated it to make a two chain, a four chain and we want to think how many dots are going to be in a five or a 10 chain. If you're happy with that, pause a video here to make a start and if you want a little bit of a hint, then I'll help you now. So when you're grouping these, you're looking for patterns that you can see that are easy to group together. It might be that you want to group them in twos and count them in twos but that's going to be quite long when we get to a five or a 10 chain, or you might just see where it's repeating and where it's overlapping. I can see this pattern here comes up every time, keeps repeating itself. And then you're left with these dots at the top. And that pattern here, same thing. So you're happy to be counting in two groups of four and three. Here we've got four groups of four and a five. So pause the video here to continue counting those dots, finding out how many there are and predicting the five and 10 chain. You should have got the answer 11 and 21 for the number of dots in these patterns. And really well done if you grouped it in a way like this, or maybe there was a different way that you grouped it and you tracked what you did. What I mean by tracking is those tracking calculations we've seen before. So I see that I've got two groups of four, two multiply by four and a group of three so plus three. Here I've got four groups of four and a group of five, four multiply by four plus five. This can then be used to predict what's going to happen with a 10 chain and a five chain. So you can see the tracking calculations here. So for the five chain, instead of having two groups of four or four groups of four, I've got five groups of four add six. For a 10 chain, I had 10 groups of four add 11. So if you spot the pattern there, it is this number that changes to whatever chain it is, and this number is one larger than whatever chain it is. Let's bring this nicely onto our Connect task. You can use tracking calculations to help us find an Nth term rule or a positioned term rules, for the number of dots in an N chain. How many dots will there be with this pattern in a 20 chain and in an N chain? Some number of chain. We need to find that rule. We can do this by grouping things and tracking those calculations. So we might want to group it like this. Here, it's a similar pattern to before, we've just grouped it in a slightly different way and a little bit more of an efficient way. Here I have grouped every time the pattern is repeating, I've grouped it together. So I've got here two groups of one, two, three, four, five, two groups of five, add one leftover. Excuse my terrible writing. Here I've got three groups of five add one and four groups of five add one and you can see that written nicely here. How many dots will there be in a 20 chain and how many dots would there be in an N chain? So pause the video to have a go at that. So hopefully, we have worked out that whatever the chain is, this number here represents the number of the chain. So we'd have five lots of 20 add one and five lots of N add one, which we can simplify to 5N add one, which looks very similar to our Nth terms that we've seen before. Pause the video to complete your independent task. Here we were using a slightly different pattern, which we can see here has been repeated in different ways. Match each grouping strategy for this four chain to the tracking calculation. We can see this pattern here. They've grouped in different way and they've tracked it. There's four groups of six add one. So we know that this goes with this one. Got four groups of five, add four, add one here. So then this one goes with this one. And what I've asked you to do is using both of those grouping strategies in the first question, write an expression for the number of dots in a five chain, a 20 chain and an N chain. This is a four chains so we can recognise that here, where we've got four lots of it, we're going to have instead five lots of it here, and a group of five and a group of one. So here are your answers, five lots of five, add five, add one, or for this pattern, five lots of six, add one. Similar with the 20, same now with the N chain and an extra well done if you simplified this to 6N add one as we should with our algebraic expressions. Similarly with this one, 5N add N add one. Now move into your Explore task. Binh forms chains using this pattern of dots, so this pattern here. How could you count the dots in the chains? So again, this is now for you to group these and try and track them. What rule will tell you the number of dots in an N chain? Once you've done that, I would like you to have a little experiment of combining this pattern here in your own way. Joining the dots in different ways, rather than just in this pattern here. How many dots will there be in your N chain once you've combined those dots in a different way and made a pattern yourself? Pause the video to do that. First of all, we should have noticed that you could count these dots. You could have done it in different ways, but the easiest way to group them would be to have two groups of six add six, or you could have done it in a different way. Maybe you could have done it like this. Two groups of six, add six separate dots, four groups of six add 10, or you could have grouped it in a different way. The rule that will tell you the number of dots in an N chain was 8N add two, and you could have got that from this perhaps? You had two groups of six, and then you had another two, another two and another two, is another way of writing two lots of six. So we get another two here, two here, and a two here. That would have worked for this one where we got four groups of six, add another four, add another four, add two. So again, that was almost like 6N add N add N add two, which could be simplified to 8N add two. So well done if you grouped it. You could have grouped it in a different way. You could have done it however you wanted, but you should have got to the same conclusion. You could have done any different pattern that you wanted, this is just one that I had to go with. I rotated this around and joined it in a slightly different way. And for my one, my N chain was 9N add one, but you could have experimented with that in different ways. What I would encourage you to do is share whatever you did with your class teacher, and they can have a look at it for you, or share your work with Oak National. If you'd like to, please ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak. I would love to see your different ideas and your different patterns that you came up with, that would be fantastic. Well done so much for this lesson today. It was quite a tricky concept to get your head around. Some of you, if you don't like to learn as visually, maybe that was really tricky for you. Maybe you preferred it with just using the numbers and algebra, but for some of you, hopefully you really enjoyed that. Well done again.

What is the tracking calculation for the grouping of this 3-chain?

If the tracking calculation for this 3-chain is 8 x 3 + 1, what would the tracking calculation for a 20-chain?

If the tracking calculation for this 3-chain is 3 x 7 + 3 + 1, what would be the tracking calculation for an 100-chain?

Using the grouping of this 3-chain, what is the tracking calculation for an n-chain? Write you answer as an algebraic expression.

Dot Chain Sequences Quiz

In this lesson, we will learn how to group dot chains and generalise to find n-chains. We formulate calculations that represent dot chains.

Maths

Y8 Maths

Key Stage 3

Year 8

Miss

Dot chain sequences

Sequences

Algebra

THEME

UNIT

Number grid sequences

Tracking Calculations

Generalising arithmetic sequences through tracking calculations

The nth term rule: Position-to-term for arithmetic sequences

Descending arithmetic sequences

Growing pattern sequences

Finding the term-to-term rule

In a six column grid, the columns give us sequences that contain the multiples of ___.

Look at the number grid shown on the right. Write down the column and row that match the tracking calculation 5 x 3 - 1

Which column has the position-to-term rule 4n - 1?

Work out position-to-term rules for the sequence 3, 7, 11, 15, 19, ...

Which definition best describes an arithmetic sequence?

Which of the following statements is true?

How many squares does the next term in this sequence have?

Which sequences starts at 2 and has a term-term-term rule of multiply by 3?

The number of ______ in each grid determines the number sequence within each column.

Lesson overview:Dot chain sequences

Core Content

Sequences: