MAE3270 Module 2 - Lecture 2: Number sense and fluency
Chapter 1: Introduction
Multiplicative thinking involves working with a greater range of numbers
Numbers of increasing magnitude: whole numbers, decimals, fractions, and percentages
Multiplicative thinking involves recognizing and solving problems involving multiplication or division
Communication can be done through discussion, words, diagrams, symbols, expressions, and algorithms
Understanding the concept of multiplication
Encouraging students to understand that 3 multiplied by 4 means 3 lots of 4 or 4 lots of 3
Encouraging students to recognize and identify this notion through various representations
Examples of representations: tray of muffins, bags of apples, combinations of t-shirts and shorts, length of rope, bags of carrots, DVDs
Putting multiplication in a real-life context and perspective
Developing an understanding of multiplicative thinking
Helps understand the structure of the base ten number system
Understanding the place value when moving to the right or left of the decimal point
Helps with adding and subtracting fractions by understanding factors and multiples
Representations for 3 multiplied by 4
Utilizing counters
Using arrays
Multiplication and division
Multiplication is repeated addition
Arrays and combinations can help represent multiplication
Division is the inverse of multiplication
Equal sharing or equal grouping
Models for multiplication
Counters
Arrays
Number line
Chapter 2: Tackling Tables
Array for multiplication to demonstrate repeated addition
Properties of multiplication:
Multiplicative property of 0: any number multiplied by 0 is 0
Multiplicative property of 1: any number multiplied by 1 remains the same
Commutative property of multiplication: order of multiplication does not affect the product
Associative property of multiplication: factors can be grouped in any way without affecting the product
Distributive property of multiplication: operation can be distributed across an equation
Language to help students understand multiplication:
How many groups of
How many lots of
How many or how much altogether
Per or each
Product
Arrays and combinations
Hawkswan's Tackling Tables:
Strategies, activities, and games to help students understand times tables
Importance of multiplicative thinking before rote learning times tables
Activities and strategies help students learn times tables without rote memorization
Chapter 3: Blank Times Tables
If students struggle with certain multiplication tables, a trick can be used
A times table chart is shown, going up to the 9 times tables
Students quickly grasp the tens and elevens times tables
Knowledge of the tens and twos times tables, along with the distributive property, allows students to combine them to get the twelves times tables
Students can be provided with a blank times tables chart to color in certain aspects
Students can black out the times tables that multiply by 0, as the answer will be 0
The column and row for the one times tables can be blanked out, as any number multiplied by 1 is itself
The orange numbers on an angle represent the square numbers, which students learn early on
The green and white sections on either side of the square numbers are mirror images
Due to the commutative property, only the green or white section needs to be memorized
Within the green section, the two and five times tables are easy to learn
Overall, students don't need to memorize as many multiplication tables if they understand the patterns and properties
Chapter 4: Factors And Multiples
A multiplication grid, like the times tables chart, helps with understanding factors and multiples
Factors are smaller numbers that divide evenly into another number
The factors of 6 are 1, 2, 3, and 6
Factors can be paired up to multiply and give the original number
Multiples are numbers that result from counting in multiples of a certain number
Multiples of 6 include 6, 12, 18, 24, 30, etc.
The multiplication grid shows the factors of a number, such as 48
The factors of 48 are 6 and 8
Other rows and columns in the grid also show factors of 48
Further factors of 48 include 24 and 2, as well as 1 and 48
An array can be used to demonstrate factors and multiples
The number of rows and columns in the array represents the factors
The product of the two numbers is the multiple
Chapter 5: Financial Literacy and Number Fluency
Consumer and financial literacy is important for students and young Australians in the 21st century
Students need to understand how to work with money and become confident in financial matters
Understanding percentages is crucial for comprehending income tax, interest rates, and financial transactions
Fluency in addition and subtraction is necessary for representing money values and calculating change
Fluency in multiplication and division is needed for calculating purchases and solving related problems
Students should be able to calculate percentage discounts and increases accurately
Planning and amending a budget is important, distinguishing between essential and optional expenditures
Estimation and rounding are valuable skills for checking the reasonableness of results
Estimation helps students recognize whether an answer makes sense in the context of a question
Estimation saves time by providing a rough answer without going through the whole calculation process
Final Word on Number Sense and Understanding
Good number sense is essential for efficient computation
Efficient mental techniques should be encouraged
Unthinking recall of algorithms or procedures is not ideal for complex calculations
Understanding the properties and language of mathematical operations is important
Learning about money and financial maths in a relevant and relatable context is crucial
Estimation is a powerful tool for recognizing the reasonableness of results