Module 2 Patterns and Algebra

Question 1

What is the process used in algebra? (6 stages)

Answer 1

Recognising, Describing,Continuing,Translating, Creating, Generalising

Question 2

What does algebra assist students with?

Answer 2

computation (action of mathematical calculation)
understanding patterns
conjecturing and generalising mathematical facts and relationships

Question 3

Commutative (turn around) Law

Answer 3

When we realise that when adding numbers it doesn’t matter in which order they are added.

Question 4

Growing Pattern

Answer 4

Patterns in which each section experiences consistent growth. Growing patterns are the visual link to number patterns. Growing patterns and repeating patterns lead to different types of mathematical thinking - it is important for children to understand the difference.

Experiences with growing patterns leads to the development of functional thinking.
(Serow, Callingham & Muir, 2014)

Question 5

Repeating Pattern

Answer 5

Occur in many classrooms in the early years. Students are asked to copy, create and extend repeating patterns. Children may have trouble with expressing which pattern repeats (one reason for this may be that teachers are unsure in the early years why the pattern structure is important even though they understand that patterning is important).

Every repeating pattern contains a repeating unit ie a 2, 3, 4 etc step pattern eg
123123123123
1467146714671467

Question 6

How do we develop children’s abilities/ skills in patterns

Answer 6

Recognising repeat and growing patterns (identifying repeating and growing patterns) – is it a pattern? Why?
Describing
Copying
Repeating (give first two three patterns and ask them to continue)
Extend
Finding missing elements in a pattern
Translating (Translating a pattern ie the sound pattern: clap, clap, stomp, clap, clap, stomp,

Question 7

ES1

Answer 7

Recognize, describe, continue repeating patterns and number patterns that increase or decrease.

Keywords:
Recognize, describe, extend, simple number patterns.

Question 8

S1

Answer 8

As for ES1.+ creates patterns. More emphasis on number patterns.

Keywords:
As for ES1, and copy, translate, create using increasingly more complex patterns (ie 5 , 6 step patterns)

Question 9

S2

Answer 9

Generates number patterns. Completes simple number sentences.

Keywords:
As for S1, and create, find missing elements for increasingly more complex patterns ie involving addition or subtraction starting from any number.

Question 10

S3

Answer 10

Analyzes and creates geometric and number patterns. Completes number sentences with 4 operations.

Keywords:
As for S2, and analyze and describe increasingly more complex problems using all 4 operations

Question 11

What do we expect young children to do with patterns? Provide example

Answer 11

give them a picture ie a zebra and ask them to describe the pattern they see using everyday language. Questions would include “what is this?” “What can you tell me about this?” “What is repeated?”
Describe a pattern that you can see in the classroom. As students describe the patterns, they engage with the language, the teacher can reinforce this by asking “what patterns do you see?” “What is a pattern?” ie for a picture of a sunflower, students might answer: ‘circular’, ‘spiral’, ‘whirl’, ‘fan’ type patterns.
Eventually we get to mathematical shapes, sounds, rhythmic movements etc which we describe mathematically ie patterns related to visual, hearing (sound), touch (movement) and other senses:

Question 12

Describe an activity that you would use to teach patterns and algebra in each of the four stages. What exactly is the purpose of the activity? Why do we teach it?

ES1

Answer 12

ES1 activity: Develop number sense, operations with numbers ie recognize, describe and continue repeating patterns using sounds and actions (clap, clap, stomp…). Recognise errors in a pattern, create or continue a repeating pattern using simple computer graphics.

Question 13

Describe an activity that you would use to teach patterns and algebra in each of the four stages. What exactly is the purpose of the activity? Why do we teach it?
S1

Answer 13

S1 activity: Investigate and describe number patterns formed with objects or by skip counting ie forwards and backwards by ones, twos, fives, tens from any starting point. Use objects to represent counting patterns. Represent number patterns on number lines and number charts. Recognise, copy and continue given number patterns that increase or decrease:

1, 2, 3, 4,…
20, 18, 16, 14,…

Question 14

Describe an activity that you would use to teach patterns and algebra in each of the four stages. What exactly is the purpose of the activity? Why do we teach it?
S2

Answer 14

S2 activity: Describe, continue and create number patterns resulting from addition, subtraction, multiplication and division. Identify and describe patterns when counting forwards and backwards by 3s, 4s, 6s, 7s, 8s, 9s from any starting point. Model, describe and record number patterns using diagrams, words or symbols (translate).

Question 15

Describe an activity that you would use to teach patterns and algebra in each of the four stages. What exactly is the purpose of the activity? Why do we teach it?
S3

Answer 15

S3 activity: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from any operation (addition, subtraction, multiplication, division).
½, ¼, 1/8,

Question 16

Identify an ICT tool that could be used to teach patterns:

Answer 16

Microsoft word and power point
The Cambridge University website nrich: http://nrich.maths.org/frontpage
The Mathematics Task Centre: http://www.blackdouglas.com.au/taskcentre/

Question 17

Why are patterns such an important part of mathematics?

Answer 17

“Mathematics is often described as the science of patterns. Applications of mathematics use these patterns to ‘explain’ and predict natural phenomena” (Australian Education Council (1991) as cited in Siemon et al. (2015, p. 271)

“Much of mathematics is about patterns… This transforms mathematics in the primary classroom from a focus on finding one numerical answer to arithmetic problems to providing opportunities for pattern-building, conjecturing [eg answering true/false questions], [justifying] and generalizing mathematical facts and relationships. It is these patterns that give us insights into the structure of mathematics.” (Siemon et al. (2015, p. 271).

Maths is based on the idea of relationships (patterns = relationships). No mathematical concept stands alone, there is always a link to another concept which can be utilized in order to reinforce deeper understanding of what is being taught. For example, the concept of place value is based on a factor of ‘10’. Therefore, place value is linked to the 10 times table:

10 ones (10x1) → 1 ten
10 tens (10x10) → 1 hundred
10 hundreds (10x100) →1 thousand etc

Patterns are a pervasive part of mathematics. In ES1 & S1, students recognize, describe and continue repeating patterns. Throughout primary school, students continue to increase their knowledge and awareness of patterns eg skip counting, number facts and patterns (ie commutativity: 4 + 5 = 5 + 4) and identify patterns using hundreds charts, ten frames etc. This provides a sound platform upon which children advance to ‘growing patterns’ (Siemon et al. (2015, p. 277) and algebra where they can describe patterns in words and symbols/equations.

Question 18

Functional thinking

Answer 18

Functional thinking focuses on the relationship between two (or more) varying quantities. In the early years functions often involve following rules for consistent changes, and reversing this change. The rule uniquely associates elements from one set with elements from another. For example, if the rule was ‘multiply by 2’, then this rule uniquely associates 2 with 4, 8 with 16, 20 with 40. .

Question 19

How can functions be shown?

Answer 19

Growing patterns and symbols
Tables of value
As mapping – input –}rule–}output
As graphs

Question 20

What is a work sample?

Answer 20

A work sample is any piece of student work that provides evidence (or not) of achievement and understanding. Work samples formalise the ‘thinking aloud’ process. A useful guide is that work samples should almost solely represent student-constructed ideas rather than a ‘fill in the gaps’ approach.

Question 21

What are types of work samples?

Answer 21

written or oral reports
worksheets/proformas
diagrams and graphic representations (charts, graphs, tables, models, mind maps)
students’ reflections
student self-assessment (learning logs, journals, reflections)
photos of completed work

Question 22

What are work samples used for?

Answer 22

reveal change over time (eg. before and after direct instruction);
allow students to see their own improvement and work at their own level;
become part of a feedback cycle for students, teachers and parents;
reveal any misunderstandings;
provide indications of the level of student understanding;
determine what to teach next;

provide opportunities for reflecting on, sharing and communicating learning; and
provide a shared focus for parent/teacher or three way interviews.

Question 23

1. In the early years, children engage in pattern related activities using concrete materials and pictures. They are required to recognise, describe, continue, translate and create patterns. Describe some activities that you could develop for Early Stage 1 children.

Answer 23

Use concrete manipulatives.
Students should be able to recognise, copy and continue repeating patterns using sounds and /or actions.
You could start a pattern where the students have to tap their head twice and then clap twice. At the same time have them say the pattern out loud ie, head, head, clap, clap, head, head, clap, clap. Then progress to 1, 2, 1, 2, 1, 2.
Students could use counters and cubes to create patterns. For example blue, red, blue, red.

Question 24

A child gives an answer of 17 to 12+5=?+7. What misconceptions might they have and how would you help them overcome this?

Answer 24

The child mistakenly thinks that the equals sign means ‘gives the answer’. So they have done just this and given the answer to 12 + 5. Depending on the child I might use some concrete support. Example: use some toy scales and show them 12 blue teddy bears and 5 red teddies on one side. How many teddies do we have on this side? Place 7 teddies on the other side? How many do we need to ‘balance’ the two sides? How can we write this down? If older, then explain that the equals sign means the same as. Is 12 + 5 the same as 17 + 7? Obviously not. What do we need to add to make the two sides have the same value? Try some more questions.

Question 25

3. How would you use patterns to help children learn their addition facts?

Answer 25

You can use number patterns to demonstrate commutativity 5+7=7+5.
Students can draw patterns, or build them using concrete manipulatives such as blocks or counters. How many do we have in this pile? How many did we add? How many have we got altogether?
You can also use a hundreds chart to show patterns and help students learn their addition facts, leading to skip counting which is just repeated addition.

Question 26

Children were asked to take some paddle-pop sticks and make the following “fence” with two posts. They were then asked to make a piece of fence with three posts. How would you develop this idea further into a lesson on growing patterns? Think about the questions and activities you might provide for the children (think ELPSARA). What aged children could you use this with?

Answer 26

E – experiences. What are fences used for? What do we need to build fences? When fences are built in panels, they form a repeating pattern.
L – Language. Start with fence posts and railings. Then introduce the term pattern, possibly repeated addition. Does anyone notice a pattern?
P – Pictorial representation. Have students build the fence posts. How many are needed to build three sections of fence? What about four sections?
S – Symbolic representation. Assign each section of fence a number, corresponding to the number of paddle pop sticks required to build. How many paddle pop sticks would I need to build 20 sections of fence?
A – Application. This problem shows how number patterns can be applied
R – Reflection. Do you think your answer is reasonable? What method did you use? Is there another way you could have solved this problem?
A – Assessment. Take photos of the children solving the problem. Collect work samples. Observe their engagement in the lesson

Question 27

What is Relational thinking

Answer 27

When students understand that quantities on both sides are the same. They then use this numeric relationship to solve equations rather than computing amounts. For example, to solve y + 3z = y + z + 8 they can:
Subtract y from both sides of the equation 3z = z + 8
Subtract x from both sides of the equation 2z = 8
Divide both sides by 2 z = 4
Check the answer
y + 3 x 4 = y + 4 + 8
y + 12 = y + 12
Algebraic language- use of letters to represent any number e.g. n.

Question 28

Algebraic thinking

Answer 28

it provides the tools (i.e. the language and structure) for representing and analysing quantitative relationships needed for modelling situations, solving problems and proving generalisations

Question 29

Equal=

Answer 29

both sides of equation are same (e.g. 12 – 2 = 5 + 2 + 3, or 1 + 9 = 2 + 8, 2 + 8 = 3 + 7, 3 + 7 = 4 + 6).

Question 30

Early Algebraic thinking:

Answer 30

In classrooms in the early years algebraic thinking entails:
making explicit the mathematics of pattern (particularly repeating and growing patterns), and extending these patterns to our number system
studying early functional thinking with a focus on the relationships between the operations, such as the inverse relationship between addition and subtraction, commutative law for addition, and the identities
studying the structure of the number system and operations, for example, the meaning of equals and the meaningful use of unknowns.

Question 31

Equivalence

Answer 31

Equivalence is another key concept of algebraic thinking, and must be grounded early and often. A common misconception is that equals means ‘the answer is coming’. In algebra the equal sign is used to show equivalent relationships, for example, 76 + 21 = 86 + 11. The focus is on the relationship:

Question 32

Commutative law

Answer 32

When exploring the process of adding two numbers we discover that it does not matter in which order we add them: this is an example of the commutative law

Question 33

Place value:

Answer 33

MAB blocks reveal the place value structure of our number system: the hundreds are 10 times larger than the tens, the tens are 10 times larger than the ones, and so on.

Question 34

Name the strands in mathematics where patterns occur?

Answer 34

Strand: Space
Strand: Measurement
Strand: Number
Substrand: Counting and operations
Strand: Chance & data
Strand: Patterns & algebra