MATHEMATICAL CONTENT KNOWLEDGE -
The key concepts in early algebraic thinking include:
- Generalised arithmetic (patterns and structures)
- Functional thinking
- Equivalence, expressions, equations, and inequalities
Other concepts include:
- Understanding fundamental algebraic concepts such as the equals sign, equality, equations, properties of numbers, properties of operations, variables, unknown quantities, symbols, co-variation, and correspondence
- Applying processes oriented towards the search of similarities and differences and validation of structure and relationships, such as noticing, conjecturing, representing, generalizing, justifying and validating
- Utilizing forms of reasoning, such as abductive, inductive and deductive reasoning, which lead to the extraction of conclusions
Pattern & Structure:
- Patterns gives us insights into the structure of mathematics – teaching is about searching, describing, generalising and justifying patterns.
- Creating and describing patterns – supports the development of early number concepts.
- Structure is both defined (e.g., commutative laws) and discovered.
- Early development in one’s ability to recognise pattern and structure has a positive influence on overall mathematical achievement and enables strong foundations for algebraic thinking.
Common misconceptions:
Many students will mistake the equals sign as an operational symbol where they are required to
write an answer, rather than seeing it as a relational symbol, indicating that a relationship exists
between the numbers or expressions on each side of the equal sign.
PEDAGOGICAL CONTENT KNOWLEDGE -
When teaching students about growing patterns, the strategy to this is to copy, continue, complete
and create. This can be done using objects or drawings (such as counters or drawing shapes), to
represent repeated addition and multiplicative structures. Other representation ideas for growing
patterns and functions include physical models, tables, words, symbols and graphs.
Reference: (Van et al., 2015), (Powell, 2012)
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Activity - Growing Patterns
Year Level - 3
Classroom Context -
This is a flexible lesson that can be easily-modified to cater to the diverse needs and abilities of
students in the classroom. All students in the classroom can learn the same concept as the difficulty
levels of the activity can be adjusted according to certain skills so that all children are provided with
many opportunities to demonstrate what they know and can do. Teachers should plan early
for any adjustments required.
Content Descriptor -
Describe, continue, and create number patterns resulting from performing addition or
subtraction (ACMNA060)
Learning Objectives -
This activity is designed to help students develop further knowledge in recognising addition and
subtraction patterns, and identifying their mathematical rules. They will learn to continue these
patterns and create their own, using addition or subtraction rules of their choice.
Resources -
Key Mathematical Language:
Add, subtract, rule of, pattern, growing
Hands-on Manipulatives:
Coloured blocks and counters can be used to represent growing patterns
Other Materials:
Worksheets, coloured pens
Prior Knowledge -
Recognises simple patterns in everyday contexts
Copies simple patterns
Creates repeating patterns with numbers and shape
Instructions -
- Begin by discussing what a pattern is with the whole-class and ask questions to check prior knowledge
- Explain there are many ways to create a growing pattern, not only with numbers
- Continue to explain what a growing pattern is and how this is different (you must use addition/subtraction strategies for this)
- Demonstrate by using a drawing on the whiteboard. E.g. Draw 1 star, 1 square, 2 stars, 1 square, 3 stars, 1 square etc.
- Discuss with students about the ‘rule’ of this pattern
- Demonstrate a growing pattern with numbers. Being with beginner-level addition example E.g. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 (Discuss with students that the rule is to add the two numbers before it)
- Demonstrate a pattern with subtraction E.g. 12, 10, 15, 13, 18 (Discuss with students that the rule is to subtract 2, then add 5)
- Ask students to work in pairs to use models, drawings and manipulatives to support each other in making sense of patterns. Allow them to fill in individual worksheets
- When students finish, ask them to create 4 examples of growing patterns on the backs of their worksheets. They must create 2 numerical patterns, and 2 more using drawings
- Encourage them to use colours and to be creative
Questions to ask -
What is a basic/normal pattern?
How is a normal pattern different to a growing pattern?
How many rules can there be in 1 pattern?
Enabling Prompts -
This activity can be easily modified to suit the needs of particular students. Start by using physical
models (or drawings) such as the coloured blocks and create simple patterns with them. A good
way to support students experiencing difficulty is to say aloud what blocks you want them to connect
so they can both visually and auditory determine the rule.
Extending Prompts -
Encourage students to create their own growing patterns. For fast-learners, challenge them with
higher-level patterns, or begin including patterns with multiplication and division strategies.
Images -
Worksheet for students
Counter blocks to guide students with growing patterns
Reflection -
This week, our focus early algebraic thinking and more specifically the meaning of the equals
sign and other symbols on algebraic conceptual knowledge development. Additionally, we
looked at ensuring how all learners can be included in the learning experiences. In this particularly
activity, I focused on teaching growing patterns using addition and subtraction strategies. This
learning experiences focuses on order, sequence of events, patterns and number. Children will be
able to demonstrate their thinking while teachers check for mathematical understandings to make
decisions about how they can use scaffolding to support these children. During my time in placement,
especially with the younger grades, I have seen teachers create activities on ‘after’, ‘next’, ordering,
especially with the younger grades, I have seen teachers create activities on ‘after’, ‘next’, ordering,
matching, sorting, classifying, copying and repeating for patterning. It is beneficial for children to use
their knowledge on these above elements so they can apply these skills in games, music, and
actions. These understandings of sequences and patterns will be linked to the language of time,
which eventually will become crucial for children to understands - for example, combining this with
other familiar language to designate specific points in time such as events or transitional times.
These include lunch time, going home time, outdoor time, library day, school day, etc.
(Queensland Studies Authority, 2015). I think that in the future, I will implement fun and creative ideas
into activities on early algebraic thinking. This way, my students will have opportunities to develop
abstract ideas of number and shape that make sense to them.
into activities on early algebraic thinking. This way, my students will have opportunities to develop
abstract ideas of number and shape that make sense to them.
References -
Powell, S. R. (2012). EQUATIONS AND THE EQUAL SIGN IN ELEMENTARY MATHEMATICS
TEXTBOOKS. The Elementary School Journal, 112(4), 627–648.
Van, D. W. J. A., Karp, K. S., & Bay-Williams, J. M. (2015). Elementary and middle school
mathematics: teaching developmentally, global edition. Retrieved from
Retrieved February, 7, 2015.
