WEEK SEVEN: Pre and Early Algebra

The BIG ideas covered and related concepts, skills and strategies:

- The key to working out number patterns is to look for the relationships between numbers

- Functions describe the relationship between pairs in such a way that any alteration to one number has a consequent alteration to the other. Therefore, functions may be treated as patterns, much like the growing patterns e.g., step number and number of matchsticks below. By creating tables and discussing the patterns students can explore the relationship between the sets of numbers and ultimately determine the function.

- Symmetry in a plane - Flip/reflection, slide/translation, turn/rotation

- Patterns involving geometry and numbers lead to algebra (abstract level of number reasoning)

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Concept, Skill and Strategy:

Concept: The symbolic statement of a relationship

Strategy: The big 5 strategies - Example of count on - 3 + ? = 5 (count on from 3)

Reflection: Through this week, I have formed an understanding of how patterning involving geometry and numbers leads to algebra. If children can work out the missing shape from a pattern, they are effectively doing algebra in a very early form and beginning to process the concept that missing 'parts' can be found if we look at what is happening around it. I will endeavour to always make early algebra inquiry based with the use of concrete materials, so students can explore what is happening in the problems presented to them and remove the temptation to teach via rote.

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The Language Model:

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Teaching Strategies:

Activity: Function Machine

- Exploring the true meaning of the equal sign using a visual representation, like the function machine

"If 5 went into the machine and then 10 came out - What happened in the machine"

5x2=10

5+5=10

5+4+1=10

How many can the students brainstorm?

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Misconceptions and strategies to remediate:

Misconception:

That the equal sign represents the language "the answer is" - For example, for 3 + 4 = 7 a child says, 3 + 4 and the answer is 7. NO! It means that both sides, although written differently, are equal.

Strategies to avoid or remediate misconception:

Concrete materials such as scales can be used alongside student language to explore the equal signs meaning

(Stay at Home Educator (2015). Balancing scale counting bears activities [photgraph]. Retrieved from http://stayathomeeducator.com/balancing-scale-counting-bears-activities/)

"So if we had 3 teddies on this side, how many teddies would we need on this side to make it level?" Or to extend students you can introduce fractions also "If we have 2 teddies on this side, but only have half teddies left, how many half teddies would we need to equal the scales?"

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Australian Curriculum and Scootle links:

ACARA Links to Foundation Year:

(http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1)

Scootle Resource:

Sites2See: Patterns and Algebra

(http://www.resources.det.nsw.edu.au/Resource/Access/37f3cd25-eeb2-4be3-87e0-e83f98c717b7/1)

Selected links to a range of interactive online resources for the study of patterns and algebra in Foundation to Year 6 Mathematics.

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Additional Resources:

Teaching Resource:

A good insight into how one teacher uses concrete manipulative and tools, such as scales, to help her students conceptualise early algebra.

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Chapter 15: Algebraic Thinking

- Algebra is a study of patterns and relationships

- Algebra is a way of thinking - Provides strategies for analysing representations, modelling situations, generalising ideas and justifying statements.

- Algebra is an art, characterised by order and internal consistency

- Algebra is a language that uses carefully defined terms and symbols

-Algebra is a tool

Problems:

- Routine Problems: Exercise for practising computation and for building algebraic understanding (Example - John had 10 pencils in the morning and by the afternoon he had 17 How many extra pencils did he have?)

- Non-Routine: Many non-routine puzzles such as look-for-patterns and number puzzles, lend themselves well to algebraic approach.

Patterns:

- Repeating Patterns: Has a core element that is repeated over and over again (blue, red, blue, red)

- Growing Patters: Matchsticks to make a triangle - 3 matchsticks to make a triangle - How do you grow the pattern to make a bigger triangle?

Relations:

- Properties of Numbers: Commutative, associate, distributive, identity.

- Functions : If there are 2 sets of numbers and they are related in such a way that each number in the first set is related to 1 number only in the second set - This relation is a function (Example: If you have one set of numbers that represents the number of boys, and the second set of numbers represented number of hands).

(Reys, Lindquist, Lambdin, Smith, Rogers, Falle, Frid & Bennett, 2012)