# 3: Number Patterns - Mathematics

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Course Goals and Anticipated Outcomes for This Chapter:

Develop the student:

1. ability to understand number patterns and predict the pattern,
2. familiarity and facility with a wide range of number patterns and the connection to K-9 curriculum, and
3. reasoning using induction and also using finite difference Calculus.

THINKING OUT LOUD

Consider the following sequence of numbers in which only the first two terms are given: 1,3,⋯,⋯. Create four different number patterns having first two terms as 1,3, by writing out the next four terms. In each case explain the rule for your pattern. What happens if the first four terms are given as (1,3, 5,7,cdots). How many possibilities are there?

THINKING OUT LOUD

What is the perimeter of the design by joining (n) regular hexagons in a row? How can you prove your prediction?

Numbers can be organized into many different sequences. Most of these sequences have patterns which can be used to predict the next number in the pattern. Misunderstandings may occur when we list a few numbers in the sequence. For example, (3,5,7..), the next term could be either (9) (sequence of odd integers) or (11) (sequence of prime numbers). Therefore it is wise to define sequences in terms of an explicit formula for the (n)^th term.

There are many types of patterns, but we will be looking at the following:

• Arithmetic sequences
• Finite sums of arithmetic sequences
• Geometric sequences
• Finite sums of geometric sequences
• other types of sequences

All sequences, regardless of how they progress, have terms. To denote which term we wish to consider, we use (n). So, if we say that (n = 3), we are considering the third term in a sequence. The first term in a sequence is given by (a). So, if we say that (a = 23), the first term in the given sequence is 23.

So, without further ado, let's be off!

New Notation & Definitions

Terms: the numbers in a sequence

• When considering a specific term: (n = x), where x is a whole number.
• The first term in a sequence: (a)

Thumbnail: Derivation of triangular numbers from a left-justified Pascal's triangle. 9Cc BY-SA 4.0; Cmglee).

Thanks to Thomas Thangarajah for sharing his hexagonal drawing.

## Sidewalk Patterns

Cora and Cecilia each use chalk to make their own number patterns on the sidewalk. They make each of their patterns 10 boxes long and line their patterns up so they are next to each other.

Cora puts 0 in her first box and decides that she will add 3 every time to get the next number.

Cecilia puts 0 in her first box and decides that she will add 9 every time to get the next number.

1. Complete each girl's sidewalk pattern.
2. How many times greater is Cecilia’s number in the 5th box be than Cora’s number in the 5th box? What about the numbers in the 8th box? The 10th box?
3. What pattern do you notice in your answers for part b? Why do you think that pattern exists?
4. If Cora and Cecilia kept their sidewalk patterns going, what number will be in Cora's box when Cecilia's corresponding box shows 153?

## Number Patterns Worksheets Pdf Grade 3

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### Numbers patterns i.

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## 3: Number Patterns - Mathematics

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## Fibonacci Number Patterns

Here, for reference, is the Fibonacci Sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …

We already know that you get the next term in the sequence by adding the two terms before it. But let’s explore this sequence a little further.

First, let’s talk about divisors. Let me ask you this: Which of these numbers are divisible by 2?

1, 1, 2 , 3, 5, 8 , 13, 21, 34 , 55, 89, 144 , 233, 377, 610 , 987, …

Every third number, right? And 2 is the third Fibonacci number. Okay, maybe that’s a coincidence. How about the ones divisible by 3?

1, 1, 2, 3 , 5, 8, 13, 21 , 34, 55, 89, 144 , 233, 377, 610, 987 , …

Every fourth number, and 3 is the fourth Fibonacci number. Okay, that could still be a coincidence. What about by 5?

1, 1, 2, 3, 5 , 8, 13, 21, 34, 55 , 89, 144, 233, 377, 610 , 987, …

1, 1, 2, 3, 5, 8 , 13, 21, 34, 55, 89, 144 , 233, 377, 610, 987, …

Every sixth number. Now does it look like a coincidence? In fact, it can be proven that this pattern goes on forever: the nth Fibonacci number divides evenly into every nth number after it! Cool, eh?

Okay, now let’s square the Fibonacci numbers and see what happens.

The Fibonacci sequence is all about adding consecutive terms, so let’s add consecutive squares and see what we get:

We get Fibonacci numbers! In fact, we get every other number in the sequence!

So that’s adding two of the squares at a time. What happens when we add longer strings? Three or four or twenty-five?

The resulting numbers don’t look all that special at first glance. But look what happens when we factor them:

And we get more Fibonacci numbers – consecutive Fibonacci numbers, in fact. Okay, that’s too much of a coincidence. Let’s ask why this pattern occurs. We have squared numbers, so let’s draw some squares.

This is a square of side length 1. Its area is 1^2 = 1. We draw another one next to it:

Now the upper edge of the figure has length 1+1=2, so we can build a square of side length 2 on top of it:

Now the length of the rightmost edge is 1+2=3, so we can add a square of side length 3 onto the end of it.

Now the length of the bottom edge is 2+3=5:

And that makes the leftmost edge 3+5=8:

And we can do this because we’re working with Fibonacci numbers the squares fit together very conveniently. We could keep adding squares, spiraling outward for as long as we want. But we’ll stop here and ask ourselves what the area of this shape is. Well, we built it by adding a bunch of squares, and we didn’t overlap any of them or leave any gaps between them, so the total area is the sum of all of the little areas: that’s . But the resulting shape is also a rectangle, so we can find its area by multiplying its width times its length the width is , and the length is …

… and the area becomes a product of Fibonacci numbers. That’s a wonderful visual reason for the pattern we saw in the numbers earlier! If we generalize what we just did, we can use the notation that is the th Fibonacci number, and we get:

One more thing: We have a bunch of squares in the diagram we made, and we know that quarter circles fit inside squares very nicely, so let’s draw a bunch of quarter circles:

And presto! We have what’s called a Fibonacci spiral. It’s a very pretty thing. That’s not all there is to the story, though: read more at the page on Fibonacci in nature.

What’s more, we haven’t even covered all of the number patterns in the Fibonacci Sequence. In particular, there’s one that deserves a whole page to itself…

## Identifying, Continuing and Describing Increasing and Decreasing Number Patterns (First 3 Numbers Shown) (A)

Teacher s can use math worksheets as test s, practice assignment s or teaching tool s (for example in group work , for scaffolding or in a learning center ). Parent s can work with their children to give them extra practice , to help them learn a new math skill or to keep their skills fresh over school breaks . Student s can use math worksheets to master a math skill through practice, in a study group or for peer tutoring .

Use the buttons below to print, open, or download the PDF version of the Identifying, Continuing and Describing Increasing and Decreasing Number Patterns (First 3 Numbers Shown) (A) math worksheet. The size of the PDF file is 28234 bytes . Preview images of the first and second (if there is one) pages are shown. If there are more versions of this worksheet, the other versions will be available below the preview images. For more like this, use the search bar to look for some or all of these keywords: math, patterning, arithmetic, sequence, common, difference, recursive, shrinking, growing .

The Print button will initiate your browser's print dialog. The Open button will open the complete PDF file in a new tab of your browser. The Teacher button will initiate a download of the complete PDF file including the questions and answers (if there are any). If a Student button is present, it will initiate a download of only the question page(s). Additional options might be available by right-clicking on a button (or holding a tap on a touch screen). I don't see buttons!

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Spin the spinner two times. Use building bricks to build the pattern with the colors you spun.

## Patterns and structures

"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." This much quoted line is from British mathematician G. H. Hardy's famous book, A mathematician's apology, written in 1940. And any mathematician, from the ancient Greeks to those working today, would agree.

Patterns and structures are fundamental to mathematics. They allow mathematicians to spot when something interesting is going on, to identify the core of a problem and to generalise from a specific example to a more general understanding.

### Spotting patterns

Can you spot the pattern in this list of numbers?

This is one of the most famous patterns in mathematics: the Fibonacci sequence. It was spotted by Leonardo Pisano, now better known as Fibonacci, in his book Liber Abaci in the thirteenth century. One of the problems he investigated in his book was how fast rabbits could breed in ideal circumstances.

### Patterns to structures

The electron diffraction pattern of Zn-Mg-Ho which has a distinctive 5-fold symmetry, identifying the material as a quasicrystal. This discovery resulted in Dan Shechtman winning the 2011 Nobel Prize in Chemistry. (Image by Materialscientist)

Often spotting a pattern in a problem is the first step to understanding the underlying structure involved. And here lies the strength of mathematics: the same mathematical structures can appear in wildly different settings. One common mathematical structure, called a group, arises in almost every area of maths. In the 19th century people were separately studying the symmetry of shapes, trying to solve quintic equations (an equation involving a variable x, where the highest power of x is x 5 ), and investigating a deeper understanding of arithmetic. The same structure emerged in all these settings, and has since appeared everywhere from crystallography in chemistry to encoding data on CDs and hard drives.

Group theory is a very well understood area of mathematics and a subject of ongoing research. By identifying this underlying structure in each of these settings, the powerful mathematical machinery that has been developed to understand groups in one setting, can be used to better understand another setting. (You can read more in our package on group theory package.) Revealing the underlying mathematical structure is like the story of the emperor stepping out in his new, non-existent, clothes, it reveals that all these settings are actually examples of the same thing.

### Patterns of meaning

People love to spot patterns, it is something that we are intuitively good at. But this sometimes can lead us up the garden path, particularly as we will try to find patterns in anything, even in the random digits of the number π, which will contain each and every pattern of numbers that you could ever think of.

A similarly fruitless search might be in searching for a pattern within the prime numbers. Prime numbers, those whole numbers whose only factors are 1 and themselves, are the building blocks of the numbers. Any whole number can be uniquely written as a product of prime numbers. Mathematicians have been fascinated by the prime numbers for thousands of years, but they still contain many mysteries. There are infinitely many of them, but there is no discernable pattern for how they are sprinkled throughout the other numbers. They can be close together (in fact it is believed there are infinitely many pairs of prime numbers which differ by just 2, these twin primes are as close together as primes can be) or there can be very far apart (in fact there are arbitrarily long gaps between prime numbers).

Gauss's idea of counting the primes eventually lead to what is perhaps the hardest open problem in mathematics: the Riemann hypothesis, named after the 19th century mathematician Bernhard Riemann. The mathematics is complex, but essentially in trying to construct an accurate count of the prime numbers, Riemann spotted a pattern in their distribution. (You can read more about the Riemann hypothesis in The music of the primes.) A proof of the Riemann hypothesis would reveal much about the ebb and flow of primes on the number line. Just another example of the power of patterns in mathematics.

This article was inspired by content on our sister site Wild Maths, which encourages students to explore maths beyond the classroom and designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

## Early number patterns

The purpose of this unit is to develop pattern-based thinking through the exploration of a pattern that is central to our number system: odd and even numbers.

• Recognise even and odd numbers.
• Independently investigate, recognise and report on the patterns and characteristics of even numbers and of odd numbers.
• State generalisations about the addition and subtraction of even numbers and of odd numbers.
• Investigate and recognise the results of adding and subtracting combinations of odd and even numbers.
• Apply generalisations about odd and even number patterns to problem solving situations.

Our number system is made up of odd and even numbers. This is a fundamental or central pattern structure that deserves focused exploration by students.

In many junior and middle classrooms, students are provided with opportunities to recognise these two number sets, and often to count aloud using these distinct number sets, however investigation into their unique behaviours is not always given priority.

The members of each set of numbers behave in a particular way, as do the members of both sets when they work together in each of the four number operations.

The purpose of these lessons is to enable students to recognise odd and even numbers and their characteristics, to generalise their behaviours when added or subtracted, and to be able to consistently apply these generalisations with problem solving contexts.

The activities suggested in this series of lessons can form the basis of independent practice tasks. It is also assumed that throughout the school day, all class members, students and teacher alike, will look for and take opportunities to apply learning included in this unit of work.

Counting all (Stages 2 and 3)

• Packets of packaged items (eg. muesli bars, raisins, cans of drink, etc.)
• Number strips
• Coloured see-through plastic counters
• Multilink or unifix plastic cubes
• Hundreds boards
• Pencils and paper

Session 1 (Exploring even numbers)

• Recognise even numbers.
• Independently investigate, recognise and report on the patterns and characteristics of even numbers.
1. Make available to the students, number strips, and coloured see–through counters. Have students work in pairs, sharing a number strip, and counters of one colour only. (Use a single colour to better focus the student on the concept being developed.)
2. Place in front of the students, a selection of pre-packaged food or drink, which have an even number of individual content items.
Have students handle and check the number of items in each package, and then place a counter on that number on their number strip. The result will be that their number strip has several even numbers each covered with a see-through counter.
3. On the class chart or modelling book, record these numbers and have the students tell you what they notice. Elicit from the students, or tell them, that these are all even numbers. Have the students suggest reasons why commercial packaging mostly works in this way and record their ideas. (for example: ‘They’re neater that way’, ‘There’s no extra ones sticking out’, ‘The rows are equal’, ‘They’re in pairs’, etc.) Agree that these are all reasons why the identified numbers are known as even numbers.
4. Have students now place counters of the same colour on each even number on their number strip. Have one student read even numbers to twenty aloud, removing the counters as they do so. Then have the other student begin at twenty and count back in even numbers, replacing the counters as they do so. Repeat if necessary. Recognise the pattern that they make: the counters are on every second number.
1. Make available multilink or unifix plastic cubes.
Have students work together to make a cube model of each of the even numbers to at least 10, depending on the number of cubes available. Have them discuss (and record) what they notice.

2. On the class chart or modelling book, record all student ideas. For example: ‘you keep on adding 2’, ‘it’s +2 each time’, ‘they go in pairs’, ‘they match and have partners’, ‘they’re called even because there’s none left over’, ‘it’s kind of fair’ etc.). Recognise that this is a pattern that grows by +2 each time.
3. On their own paper or think-board, have each student draw a picture of the even numbers. For each number, have them record the number of pairs there are.

Have them discuss in student pairs, what they notice about the even number they have drawn, and the number of pairs, and record their ideas.
4. Have students share their ideas and once again record them on the class chart. Elicit the important language of ‘times two’, ‘double’ and ‘half’. For example: ‘The number of pairs is half the even number of cubes,’ ‘The number is 2 times or double the number of pairs’, ‘It’s like the two times tables’, ‘When you say 4 times 2 it’s like saying 4 pairs’.
5. Pose the question: “How can we find out if this is true for all even numbers?” Accept all responses and discuss.
Make available hundreds boards.
Have students place counters on even numbers greater than 20.
Record on the class chart the patterns they see and what they notice about these numbers: ‘they all end in 0, 2, 4, 6, or 8’.
Give students time to investigate the generalisations made in Step 4 above, ‘prove’ these with at least three numbers greater than 20. For example: 48 is 24 pairs, 48 is double 24, 24 is half 48, 24 x 2 = 48
6. Have students describe to their partner or to the class, an even number they have chosen. Encourage them to demonstrate their discovery, using the language of ‘times two’, ‘double’ and ‘half’.
1. Refer to the packages in Activity 1, Step 1 that have an even number of component items. List other content numbers on the class chart. For example: one pack with: 6 pottles of yoghurt, 10 ice blocks, 8 cans of drink, 2 packets of soup, 12 packets of raisins, etc.
Pose this investigation:
Sam Shoppers says: “No matter which of these packages I put into my shopping trolley, I will always have an eventotal number of items.”
Work with a partner. Use the equipment (cubes and hundreds boards) that is available to you and decide whether you agree or disagree with Sam Shopper that when you add even numbers you always get an even total number.
Explain that they should be ready to explain their position to the class and show this with materials so that others can understand your thinking.
Give students time to complete this challenge, including demonstrating their rationale using cubes.
2. Conclude this session by writing on the class chart the generalisation: When even numbers are added together the sum is always an even number.

Distribute a copy of Attachment 1 to each student. Have them individually complete the problems, then discuss their ideas with a partner.
Discuss what they notice and, on the class chart, record their agreed generalisations for subtraction and zero.
When an even number is subtracted from an even number, the result is always an even number.
Zero is an even number.
(It ‘fits into’ the pattern of even numbers and when zero is added or subtracted from an even number, the result is an even number.)

Conclude the session by having students create their own poster, poem or story about even numbers. Consider having the student choose some even numbers and give them personalities in their writing (anthropomorphism). For example: ‘Even Steven likes to. '
Challenge the students to check packaging of items at home or in the supermarket to see if all packages are made up of even numbers. Ask to find and write down the names of any products they find with an odd number of content items.

Session 2 (Exploring odd numbers)

• Recognise odd numbers.
• Independently investigate, recognise and report on the patterns and characteristics of odd numbers.

Begin with having students share their posters, poems or stories about even numbers from Session 1, Activity 5.

1. Place packages from Session 1, Activity 1 in front of the student. Ask if anyone has found packaging that has an odd number of items. Point out many packages (bags) of produce, (for example carrots, tomatoes) contain an odd number of items. Discuss possible reasons. (eg. the items are not always a uniform size and therefore it may take an odd number of them to make up the advertised weight.)
2. Record on the class chart or modeling book, student predictions of things that they think they will discover about odd numbers. Accept all suggestions, including possible misconceptions such as odd number + odd number = odd number.
3. Make available number strips and see-through plastic and counters of two colours.
Have them work individually or in pairs, covering the even numbers, saying the numbers aloud as they do so. Have them ‘fill in the spaces’, with another colour, saying the numbers aloud as they do so. Identify these as the odd numbers.
Have them discuss what they notice about the way they are placed on the number strip.
4. Make available multilink or unifix plastic cubes.
Have the students make (or draw) the odd numbers with cubes, noticing how many they add each time to make the next odd number.

5. Have students share what they notice and record these on the class chart. Elicit the important language and key ideas of : ‘There’s always one sticking out or left over,’ “Most are in pairs, then there’s an odd one without a partner.’ ‘They’re called odd numbers because they’re not all in tidy pairs,’ ‘It’s not just double or times two any more’, ‘It’s like a double of something plus one’.
6. Highlight that a feature of even numbers is that it is a plus two (+2) pattern. If it has not already been noted, ask students to describe the growing pattern of odd numbers.
Agree (and generalise) that the pattern of odd numbers grows by an even number, +2. Have students confirm this on the number strip and with their cubes.
1. Make available hundreds boards.
Have students place counters on odd numbers greater than 20.
Record on the class chart the patterns they see and what they notice about these numbers: ‘they all end in 1 3, 5, 7, or 9.
As they do, have them notice the similarities and differences in the physical patterns made with the counters on the board, for odd and even numbers. (Columns of odd numbers alternate with columns of even numbers.)
2. Ask if there are any other kinds of whole numbers other than odd and even numbers. Record responses, including odd and even negative integers if this discussion arises.

3. Have students suggest some odd numbers of produce (eg. 11 carrots, 7 tomatoes.) Write on the class chart. Have student pairs investigate addition and subtraction of these odd numbers, recording their equations as they do so. Explain that they will be reporting their finding to the class and will need to explain and show the results of their investigation using equipment or a diagram.
4. After student explanations, record on the class chart the generalisations:
When one odd number is added to another odd number the sum is an even number.
When an odd number of odd numbers are added together, the result is an odd number.
When an even number of odd numbers are added, the result is an even number.
When one odd number is subtracted from an odd number the result is an even number.

Ensure that students demonstrate each of these generalisations with materials. For example. (‘s’ is the student notation for ‘spare’ cube.)

Conclude the session by having students draw their own diagram to show the addition and subtraction of pairs of odd numbers, with an accompanying explanation.

Session 3 (Exploring combinations of odd and even numbers)

• Investigate and recognise the results of adding and subtracting combinations of odd and even numbers.
• State generalisations about the addition and subtraction of combinations of odd and even numbers.
• Apply generalisations about odd and even number patterns to problem solving situations.

On the class modeling book/chart draw an empty venn diagram with the headings odd numbers and even numbers. Have students interpret the task and suggest what they might do. Confirm that it is understood that:
They should write what they have found out about each set of numbers in the correct place. If something is true about both sets, it belongs in the segment that is the intersection of both sets (for example: when you add two of these numbers, the sum is an even number).

Have students share and discuss their venn diagrams and generalisations. Reach clear agreement.

1. Ensure that students can see a copy of the generalisations of the ‘behaviour’ of odd and even numbers when number operations are applied.
Distribute a copy of Attachment 2 to each student. Have them individually complete the problems, then discuss their ideas with a partner. Emphasise that they must use the words ‘odd’ and ‘even’ in each of their explanations.
2. As students finish have them play in pairs, Odd and Even Patience. (Attachment 3)

Conclude with a discussion of student learning about the patterns of odd and even numbers. Ask students to articulate how this information will be useful as they estimate and solve number problems in the future.

As you know, the numbers that we use are either odd or even. (Did you know that 0 is an even number?)

In class we have been exploring the patterns of odd and even numbers, including finding generalisations for what happens when they are added and subtracted. Like this:

 Even ± even = even Odd ± odd = even Even ± odd = odd Odd ± even = odd

Please take time to discuss this with your child and to enjoy playing the game, Odd and Even Patience. You’ll need a pack of playing cards. The instructions are attached.

## Grade 1 Number Pattern Worksheets

Our grade 1 number patterns numbers worksheets provide exercises in identifying and extending number patterns . All patterns are based on simple addition or subtraction. Some "2 step" patters are given in the third set of worksheets for greater challenge. Numbers up to 100 are used.

 Counting patterns 2, __, __, 8, 10, __ Extend number patterns (some with 2 rules) 2, 4, 6, __, __, __ Identify rules for number patterns Input - output charts Find the number pattern

Sample Grade 1 Number Patterns Worksheet