# 321. Patterns and sequences

Now what have a pair of roller skates got to do with number sequences? If you can guess before the reason, I’ll be surprised – it’ll mean there is more than one person as random as me!

As you may have guessed from my earlier post 317. Pyramid Power I’m currently doing an Algebra unit on Number Sequences. I’ve changed the way I’ve taught this topic this year to incorporate a ‘Big Picture’ view as opposed to one lesson on drawing the next picture, the next on finding the Term to Term rule and finishing with a lesson on finding the Nth term. The beauty of mathematics lies in the connections we make, not the disparate skills.

After the investigative approach of the Pyramid Numbers lesson, we did some text book work on generating number sequences (eg Start with 5, add 3) expanding to look at the physical patterns each time, so the previous rule would have looked like N groups of 3 dots plus 2 dots. As with any class (mixed ability or not) there were varying levels of progression in these lessons. To pull everyone forward I wrote structured worksheets and allowed the students to choose which they did. I described them using the following comparisons with the roller disco at our local Sports Centre:

• Sheet 1 – beginner on roller skates, need a bit of hand holding (I’ll own up to demonstrating our local instructor’s technique for teaching beginners in front of the class)
• Sheet 2 – okay on skates, just a word of encouragement every now and then
• Sheet 3 – speedskating, no fear of the next challenge
• Extension – all the skills! Some tasty questions from a tough textbook exercise

After a student completes a sheet they just move to the next – there are no duplicate questions. I printed them A5 to stick neatly in their books but you might prefer A4. Solutions are provided.

Patterns and sequences A4 one per page

Patterns and sequences A4 two per page

BTW I can tell you from personal experience that landing on your rear whilst speed skating really does hurt!

# 242. Edible Inspiration

Calling all creative thinkers!

What mathematical questions could you set from this picture?

Here are a few to start you off:

1. Sequences – do the increasing  number of chocolates in each layer form a sequence (in 2D, in 3D)? If so, what is the general term? Is it geometric or arithmetic?

2. Series – if it is an arithmetic sequence, can you find the sum of a finite number of layers? Which layer would have the 1000th chocolate?

3. Geometry – what shape must the layers be in order to form this structure? Is there a pattern to the layers? Could you stack these in a different way to form an equally stable structure?

4. Money – if a standard box holds 12 chocolates, how many boxes would a 2D or 3D version of this require? What is the cost? What if they came in a larger box? Could you save money?

5. Health – how many calories are there in the tower? How far would you have to run to burn off the calories? How many ‘average’ meals is it equivalent to? How many fastfood burgers? How sick would you feel after all that chocolate?!

# 206. Seek a number pattern

So I’m all ready to teach a lesson recapping number patterns from the basics for a lower ability group … then a visitor to the Department arrives and asks if it’s okay if they observe my lesson. They’ve been told that there is usually something ‘off the wall’ happening in my room. Thanks … I think!

Well, I’m not one to disappoint. A little fun with the starter perhaps? The sun is shining and I’ve got whiteboards and chalk …

We’ve all seen fence panel number patterns. Here is a fence:

What can you see?

We discussed the pattern linking number of posts and spacers. We then represented the fence in colour coded symbols (yes, we have chalk in more than one colour!) and annotated it.

The class were then sent off to find their own patterns. They found repeating patterns and made notes on their whiteboards. Once they were happy with their work they could chalk it out.

This group looked at number of slats on a bench with number of benches.

They represented each bench as an ‘L’ and each slat with an ‘o’.

They worked out:
No of benches x 6 = No of slats

Other groups looked at number of windows & number of classrooms and number of benches & number of picnic tables.

We then went back to our quiet number pattern work in the classroom.

This task is easily adaptable for many aspects of number, including ratio and proportion.

# 153. Sequences Starter 2

So, you’ve got term to term sequences sussed. Time to tackle Nth term!

This idea just sort of appeared in my sequences lesson.

Equipment
Giant playing cards (or numbers on two different colours of A5 card)

Set Up
1.Lay out one coloured set of cards on a table or the floor – these are the ones we needed in class. We started with all the cards in the suit.

2. Issue headbands to four pupils.

3. Pupils stand in number order.
4. Give each pupil a different. coloured card from a sequence to hold facing them.

1. Explain that each person represents a term in a sequence, given by the headband.
2. Pupil 1 turns around their card – Red 3.
Question: What is the next number?
3. Pupil 2 turns around their card – Red 5.
Question: What is the next number?
4. Pupil 3 turns around their card – Red 7.
Question: What is the next number? Why?
5. Reveal the last number – Red 9.
6. What is the pattern? Add 2 Which multiplication table has the same pattern? Twos
7. Give each pupil in the sequence the appropriate number from the two times table.
Question: How do you turn the two times table into the sequence?

8. How do you get from the headband to the sequence?
Headband x 2 + 1 = sequence

9. What about a headband with 10 on it? Or 100? Or a mystery number?
10. Try this with other sequences and develop the idea of Nth term.

Outcome
I used this as a plenary for a term to term sequences lesson with a shared class. In the following lesson my colleague, D, used this idea to develop the concept of Nth term with another class. He wanted to make something for the pupils to have in their book to remember this. This is what he came up with: Handout for sequences intro (pptx) or How to for sequences(docx). I’m currently trying out hosting my own resources, rather than using TES resources – so we’ll see how effective this is.

# 152: Sequence Starter 1

So many people have the preconcieved notion that there is only one right answer to a maths question. This is such a silly idea – they just haven’t had the right question!

Here is a simple starter for introducing term to term sequences.

Equipment
Classroom whiteboard or large sheets of paper
Imagination

Write down the next three terms in the sequence 1, 2 … and the rule used.
Eg: 1, 2, 3, 4, 5, …      Add 1
Note: Rules should be one short sentence.

Outcome
My Year 7 were frustrated that I’d given them the obvious answer and was asking for more. After a few minutes adjusting their expectations, they went for it. Some methodically wrote down rules, some abbreviated rules to symbols, some wrote rules and didn’t check them. Some didn’t write rules at all.

I randomly picked pupils to share their ideas on the board. I did the writing as I wanted to control the wording of rules and half of them can’t reach the top of the board.

Their sequences were brilliant and very creative. The stumbling block was the rules. They didn’t always work for every term of the sequence. This gave other pupils the opportunity to develop their ideas by improving or adjusting the rules to fit the sequences. We also discussed how many terms you need to make a unique sequence. By the end of the discussion we only had one sequence without a rule. I was really impressed by their numerical skills!

In the subsequent classwork, their solutions were precise and well explained.

We finished with this brain teaser:
1,2,5,10,20,50,100 …

It’s UK currency:
1p,2p,5p,10p … etc

# 16. Library Fines (Sequences)

County Library
The first day a book is overdue, you are charged 4p. Each day incurs another 4p.
What are the charges for the first week?
(4, 8, 12, 16, 20, 24, 28)
What is the Nth term?
(4N)
How much would you be charged for being 25 days late?
(100p)

Village library
The village library charges 10p for the first day and 3p for every subsequent day.
What are the charges for the first week?
(10, 13, 16, 19, 22, 25, 28)
What is the Nth term?
(3N+7)
What is the charge for 30 days?
(97p)
How many days late is one book if the fine is more than £2?
(Solve 3N+7>200)

Look back at both libraries. Under what conditions do the libraries have the cheapest fines?
(1-6 days: County Library
7 days: same
8+: Village library)

Extension
Why do the libraries have the same charge on the 7th day?
Prove it algebraically.
(Solve 3N+7=4N)

You can also extend this investigation to looking at calendar dates, with one library open 5 days a week and the other being open 6 days with fines only applying when libraries are open. How would this affect the ‘cheapness’ of fines when days are included?