You can link together patterns of dots to form chains. Counting the dots in a ______ way can help you to find the number of dots in any length chain.

For the chain below, complete the tracking calculation that illustrates the counting strategy for the total number of dots.

Use the counting strategy from Q3 to find the total number of dots in a 10-chain.

Use the counting strategy from Q3 to find the total number of dots in an n-chain.

Counting Strategies Quiz

﻿Hi everyone. It's Ms. Jones here. Today, we are going to be building on what we've learned about counting strategies and applying them to growing tree patterns. So experimenting with what happens when we alter little bits of our chains and our patterns and seeing what happens to the tracking calculations so that we can make generalisations and count the dots quicker and count objects more quickly or more efficiently. Before we can start with that though, you need to make sure you've got a pen and some paper as well as clearing away any distractions and trying to find a nice quiet space to work. So please pause the video now to make sure you've got all of that ready so that we can begin. Okay, to start with, I would like you to have a look at these patterns here. You can make different chains by changing the repeating pattern. So this is the repeating pattern and overlapping them in a row formation. So similar to what we've seen before, you can make them by taking this pattern and repeating it lots of times and overlapping it. I would like you to tell me how many dots are in each of these six chains. So you can see that this six chain, you can use our counting strategies to find the number of dots. Look at this six chain and how it's changed from this one. You might even be able to predict the number of dots in this one by having worked up the number of dots in this one. I would then like you to create your own six chain. How many dots are there? Pause the video now to have a go at this. So you could use various different counting strategies. Here's just an example. So you group them into groups of four. So you got six groups of four, add one equals 25. Interestingly, if you use a similar counting strategy for the second chain, we've now got six groups of five add one. So we've actually just got one extra dot in each of the groups, which makes sense, because we've just added one. So not this one, we just added this dot here. So we just added one extra dot and that has been repeated. So you've now got six lots of five, add one, which equals 31. Really well done if you managed to make the link between these two. There's lots of different examples. You could have gone this one. We've just changed the position. We rotated it and changed the position of the dots. So the dot was here in this pattern, but it's been rotated and they've changed it slightly. And you could have overlapped one dot, like they've done here, or you could overlapped two dots. There's lots of ways you could have done this, so well done if you had a good experiment with different dots and the different patterns. You can explore the effects of altering the pattern used to make a chain. So just like we had slight difference in pattern here, and we've talked about how it's altered the chain, we can think about it with lots of different examples. So here Xavier has used a taller tree to create a 16 chain. So you've seen this tree quite a few times now, but he's just extended it by two dots. How could you count the dots? So think about your counting strategy for this pattern here. How could you count those dots? Pause the video to have a go at that. Here is an example where you've now got a group of six. Then we've got 16 lots of six, and still adding that one at the end equals 97. So I've got two questions here now. What happens when you vary the length of the chain? And what we mean by that is if, say you added a few more of the same pattern on. What happens if you vary the length of the chain? What's going to happen to this calculation? What also happens if you vary the number of dots added? So he's added two dots at the bottom. Let's say we added another two dots. What would happen to this calculation? If all of these had an extra couple of dots. Pause the video to have a think about that. So we can see if you vary the length of the chain, let's say to a 19 chain. This was a 16 chain. If I've got 19 chain, I've just added three more of these patterns onto three more sixes. This changes the tracking calculation to 19 lots of six rather than 16 lots of 6 add one. If I vary the number of dots added, like suggested, adding another two dots to the bottom, instead of having lots of groups of six, I now have lots of groups of eight. So my 16 chain, instead of being six multiplied by 16, I've now got eight multiplied by 16, add one. So really well done if you managed to spot what was going to change in those tracking calculations. Pause the video now to complete your independent task. The first question it's asking you to match the grouping strategy to the tracking calculation. So you can see we've got exactly the same number of dots in each of these patterns. We've just grouped them slightly differently. So if I look at this one, I can see I've got four groups, so that's what I'm looking for first of all, four groups. and then each group is four or that could also be written as two multiplied by two. So two squared. So that goes with that one. This one has been grouped very differently to what we've seen so far. I've got one add two add three add four, add three add two, add one, so it matches with this one. And this one, we've got eight groups and then each group is two. Well done if you've got those correct. For question two, chains are made using the pattern here, and that pattern has been repeated and overlapped. And here it's made easy four chain. So using a grouping strategy, find the tracking calculation first of all. And then you can use that to find the number of dots in a 100 chain. Please, hopefully nobody has written these out 100 times. It's going to take you a very long time. So there's lots of different tracking calculations you could have used. You could have used something similar to what we've seen before. Let me try and group as many dots as possible in that repeating pattern and come up with a calculation for example, or you could have just grouped groups of four. And then added whatever the number of remaining dots were and found a tracking calculation that way. This is the example that I showed you. We had four groups of one, two, three, four, five add one. So if I was to have a 100 chain, instead of having four groups, I would have 100 groups of five add one, which makes 501. Really well done if you managed to get that correct. Carla has used a different tree to create a seven chain. And hopefully you can notice actually again, we still got that original pattern. They've just changed it slightly again. She's also added two dots. So if you remember, Xavier added two dots at the bottom, but she has added two dots at the top. How could you count the dots before you actually go ahead and count the dots? I'd just like you to pause the video here and try and predict what you think is going to happen to your patterns and your tracking calculation, so just have a think now and pause the video. Right, now is your chance to see if what you thought was correct. So first of all, you need to count the dots and use your patterns and your groupings to do so. Once you've done that, I would like you to consider what happens when you vary the length of the chain. Remember that just means if you add on more of those patterns. And what happens if you vary the number of dots added? So consider, if you add more dots to the bottom, maybe even consider what happens if you add more dots to the top or somewhere else. Pause the video to have a think about that. So there were lots of different strategies we could have used. Here is one example. We've grouped it into little groups of three and another groups of three here. So we've got seven groups of three and six groups of three add two groups of two. If we were to vary the length and make it a nine chain that makes it now nine groups of three add eight groups of three add two groups of two. And so on, depending on what the size of your chain is and varying the number of dots added. So they've added another two dots to the top. And this has now made the seven chain, seven groups of three and six groups of five because this group has changed and this group has also changed to two groups of three Well done if you did this strategy, although personally, I probably would have done, what are the strategies we've seen so far? So maybe having something like this. As I said, there's lots of ways you can do this. So we could see if I did it this way. I would have one, two, three, four, five, six in each group. I might have, one, two, three, four, five, six, seven. And I would just have one leftover at the end. Six lots of seven, add one, which again makes the same number of dots. As we said before, you can group it in lots of different ways, have different tracking calculations, but we should still end up with the same number of dots, otherwise we might make a mistake. And then you can see what would happen if it changed to a nine chain. Instead of having seven groups of six, I have nine groups of six and so on. So it would just change in very similar way. Really well done if you managed to get something which got us the same number of dots as we've seen here, or you might have not used a nine chain, you might have done a different pattern here where you've added dots in a different way, but hopefully you've managed to work out that we can still use original patterns and change them and still get something similar and we could predict what's going to happen. And we can generalise, which is what's really important. So really well done for all of your hard work today. And remember to complete the quiz at the end. Please do, if you'd like to ask your parent or carer to share your work on Twitter tagging @OakNational and #LearnwithOak. We'd really love to see what you've gotten up to in these lessons because you've been absolutely brilliant. And that brings us to the end of the sequence of lessons on expressions, inequalities, and just general algebra. So really well done for all of your hard work on this topic. It's been really, really fun. Well done again.

Which of the following is a tracking calculation for the image below?

Use the tracking calculation from Q1 to find the number of dots in a 10-chain.

Use the tracking calculation from Q1 to find the number of dots in a n-chain.

Growing Tree Patterns Quiz

In this lesson, we will learn how to generalise counting strategies algebraically for different repeating patterns.

Maths

Y7 Maths

Key Stage 3

Year 7

Miss

Growing tree patterns

Expressions, equations and inequalities

Algebra

THEME

UNIT

Algebraic expressions

Collecting like terms

Distributivity and expanding

Factorising expressions

Forming and exploring equations

Exploring equality

Inequalities

Further inequalities

Perimeter expressions

Perimeter inequalities

Counting strategies

Use the distributive property to fill in the gaps: 4(10 + 5) = ____ × 10 + ____ × 5

Expand 8(n-3)

Expand -7(n+5).

Factorise 12x + 3.

Factorise −6p + 15m

Which value of y means that -3y - 1 > 2y

Fill in the gaps: 5a + 9a = ___ × a = ____

Simplify the expression -20x + 3 - 8y + x -y

Use the equation p = 4w to find the value of w when p = 16.

If 19 > 14, which symbol would go in the gap: 7 × 19 __ 7 × 14?

Which inequality best compares the expressions for the perimeters of the 2 shapes below?

Which inequality is true when a = 3?

Lesson overview:Growing tree patterns

Core Content

Expressions, equations and inequalities: