Year Six - Using Symbols to Describe the World
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About this Big Idea
Students will be able to develop understanding and fluency in mathematics through inquiry, exploring and connecting mathematical concepts, choosing and applying problem-solving skills and mathematical techniques, communication and reasoning. They will develop efficient strategies involving fractions and decimals for numerical calculation, recognise patterns, describe relationships and apply algebraic techniques and generalisation in order to understand and apply fractions and decimals in real world problem solving.
Understanding Goals:
Students will understand:
- that algebra is a language used by mathematicians to describe relationships
- that variables are used to represent unknowns and can stand for different numbers
- that the equal sign indicates that each side of an equation balance, not that one side is the answer and the other the question
- how mathematical situations might be described in a variety of ways using appropriate mathematical terminology
- and give a valid reason for supporting one reason over another
- how to select and apply appropriate problem-solving strategies
Essential Questions
- How does mathematics reveal relationships between things and how they vary?
- How do mathematicians use symbols to describe the world?
- What is the purpose of maths in the real- world?
- How does our mathematical understanding transfer to new situations?
- How do we communicate our mathematical understanding to others?
- What prior knowledge can we link to a new learning?
Background:
When we explore fractions in Year Fives "Using Fraction Equivalence" big idea students are asked to understand how a fraction represents one number defined by the relationship between its two parts. In this Big Idea students explore how symbols and variables are used in algebra to represent a relationship where at least one number is unknown but changes for a particular reason and in doing so alters the output from the balanced equation. Using algebra allows us to describe contexts where the numbers are not static and where the relationship between variables is of interest to us.
In this Big Idea, as in others, students are asked to explore multiple representations and to reason mathematically to support one solution or method over another.
Core Content from the Syllabus:
Working Mathematically
Language:
- variable, relationship, balance, equivalence, algebra, fraction, decimal, pattern, increase, decrease, missing number, number sentence, number line, symbols, representation
Connected to:
Mindset Mathematics Learning Activities
Visualise
By developing symbols to represent the relationships between Cuisenaire rods, students explore the ways that symbols can be used to efficiently and clearly communicate relationships. - See page 195
Questions for reflection:
- What ways to communicate the relationships did you develop?
- Which ways do you think are clearest? Which are the most precise?
- Why did we use symbols to communicate? How do the symbols help us?
- How do symbols help you communicate mathematical ideas?
Play
Students play with math mobiles, which show relationships of balance and equivalence, to determine the value of each shape in the mobile. Students explore how they might represent these relationships with symbols. - See page 204
In this Play activity students have the opportunity to explore the art of Alexander Calder who is perhaps best known for his hanging, mobile sculptures. His mobiles utilise carefully balanced shapes which bring movement and life into the artful forms. Calder's mobiles play with balance, light and shadow. There open space is filled and emptied as the sculpture moves with the subtle airflow around them and in this way almost interacts with the observer.
"Calder is widely regarded as the artist who made sculpture move, forging a practice in dialogue with the world in motion and the motion in things. His radical and pioneering methods of making art – understood both technically and conceptually – changed the course of modern art." NGV
Questions for reflection:
- What relationships did you notice in the mobile?
- How could you represent these relationships using symbols?
- How could these equations help you solve the puzzle?
- What is the value of each of the shapes in the mobile? How do you know?
- What were the most useful observations you made about the puzzle you tried? Why?
- How did you use symbols represent the relationships you found? How did that help you?
- How did you use the different clues, observations, or relationships together to reason about the value of the shapes?
- What was the hardest part of the puzzles you tried? Why? How did you tackle that challenge?
- What strategies were most helpful in finding the values of the shapes in the mobiles?
Investigate
Students investigate the growth of radial patterns and how to represent the relationships between the cases in the pattern and the number of tiles needed to construct it. - See page 215
Questions for reflection:
- Can you find any patterns that look similar but where the groups represented the relationships in very different ways?
- Can you find any patterns where the relationships between the tiles and cases look similar but the radial patterns look different?
- What is interesting about the collection of patterns we made?
- How did representing patterns with symbols change how you saw them?
Credit:
Boaler, Munson & Williams (2018) - Mindset Mathematics: Visualizing and investigating big ideas Grade 6
NESA - Mathematics K-10 - 2012
