Year Five - Using Numbers & Symbols Flexibly
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About this Big Idea
In this Big Idea students develop their mathematical flexibility. Students expand their understanding of the use of grouping symbols and visually represent number sentences and explain their design. Students use a number talk model to explain how number sentences can be formed in different ways to describe visual patterns and groupings.
Understanding Goals:
Students will understand:
- that mathematics requires creativity and intuition.
- that mathematicians use a blend of formal and informal representations to explore and communicate concepts.
- how the four operations and other mathematical symbols can be combined to achieve a desired result.
Background:
By the end of Stage 3, students ask questions and undertake investigations, selecting appropriate technological applications and problem-solving strategies to demonstrate fluency in mathematical techniques. They use mathematical terminology and some conventions, and they give valid reasons when comparing and selecting from possible solutions, making connections with existing knowledge and understanding. (Stage Statement)
An 'operation' is a mathematical process. The four basic operations are addition, subtraction, multiplication and division. Other operations include raising a number to a power and taking a root of a number. An 'operator' is a symbol that indicates the type of operation, eg +, –, × and ÷. Grouping symbols are used to control the order in which parts of an equation are performed.
Core Content from the Syllabus:
Working Mathematically
- determine whether a number is prime, composite or neither
- model square and triangular numbers and record each number group in numerical and diagrammatic form
- select and use efficient mental and written strategies, and digital technologies, to multiply whole numbers of up to four digits by one- and two-digit numbers
- select and use efficient mental and written strategies, and digital technologies, to divide whole numbers of up to four digits by a one-digit divisor, including where there is a remainder
- perform calculations involving grouping symbols
- apply the order of operations to perform calculations involving mixed operations and grouping symbols
Language:
- ascending order, descending order, zero, ones, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, digit, place value, expanded notation, round to, whole number, factor, highest common factor (HCF), multiple, lowest common multiple (LCM), per, operations, order of operations, grouping symbols, brackets, number sentence
Connected to:
Mindset Mathematics Learning Activities
Visualise
Grouping symbols are introduced in this visual task, where students learn to represent groups of squares with expressions. Partners work on creating multiple ways to match images to expressions and color-code the connections between them. - See page 250
Questions for reflection:
- When were the grouping symbols useful? What ways of decomposing the figures led to the need for grouping symbols? Which didn't?
- How did you tackle moving in the opposite direction— creating pictures for an expression? What was different about it? How did you have to think differently?
- What are grouping symbols useful for? How do they change the meaning of an expression?
Play
In this activity, we play with the patterns in Pascal's Triangle, representing them with expressions. We play with the starting number of Pascal's Triangle to see how it changes the row sums and their expressions. - See page 261
Questions for reflection:
- What patterns did you find that helped you find the missing values? How is Pascal's Triangle built?
- How did writing efficient expressions make the patterns in the row sums clearer?
- What do you think would happen if we started Pascal's Triangle with a larger number? What evidence is there from our work today to support your prediction?
Note for the teacher:
Pascal's Triangle is a famous two-dimensional pattern of numbers arrayed in a triangular formation. At the top of the triangle is the number 1. Each number below is the sum of the two numbers above it.
Investigate
Students investigate how they might use the digits in the number 1492 and mathematical symbols to write expressions for each value from 1 to 100. The class collects expressions as a group to determine whether or not it is possible. Students may be introduced to some new notation they can use as tools in their investigation: exponents, square roots, and factorial. - See page 268
Questions for reflection:
- Why do you think that some values were found quickly, while there are others that no one has found?
- Are there any values with more than one expression? Why do you think that is?
- Did you find any expressions that did not work? Or any places you needed to revise? What helped you catch those mistakes?
- Which answers have the most solutions? Which have only one? Are there any that have no solution? Why do you think that is?
- How close do you think we are to having found all that are possible? Why?
- What operations are most useful and flexible? Why? When are the less common operations useful? Why?
Credit:
Boaler, Munson & Williams (2018) - Mindset Mathematics: Visualizing and investigating big ideas Grade 5
NESA - Mathematics K-10 - 2012
