1. Home

NA4-8: Generalise properties of multiplication and division with whole numbers.

Thanks for visiting NZMaths.
We are preparing to close this site and currently expect this to be in June 2024
but we are reviewing this timing due to the large volume of content to move and
improvements needed to make it easier to find different types of content on
Tāhūrangi. We will update this message again shortly.

For more information visit https://tahurangi.education.govt.nz/updates-to-nzmaths

Elaboration on this Achievement Objective

This means students will generalise, which means to establish properties that hold for all occurrences. This involves the ability to look at several examples, notice what changes (variables) and what does not, use appropriate mathematical terminology and symbols to describe the pattern, and apply the generalisation to other examples. At Level Four students should be able to describe and apply the properties of multiplication and division as these operations apply to whole numbers. These properties include commutativity, distributivity, associativity, inverse and identity. This includes the ability to express generalisations using words and symbols, for example 4 x 6 = 24 so 24 ÷ 6 = 4 and 24 ÷ 4 = 6 (example) leading to a x b = c so c ÷ b = a and c ÷ a = b. This is the inverse relationship of multiplication and division.

 

Teaching resources for this Achievement Objective
Resource logo

Anno’s Mysterious Multiplying Jar

Level Four
Number and Algebra
Picture book activities
This is an activity based on the picture book Anno’s Mysterious Multiplying Jar. This book may no longer be available for purchase.
  1. Students will construct model to illustrate their understanding of factorials.
  2. Students will be able to explain the power of the operation of multiplication and contrast it with addition.
Resource logo

A Prime Search

Level Four
Number and Algebra
Units of Work
In this unit we use rectangular models or arrays to explore numbers from one to fifty. We systematically identify all the factors of numbers, and are introduced to prime numbers.
  • Model the numbers from 1 to 100 as rectangular arrays.
  • Connect the possible arrays for a given number to the factors of that number.
  • Identify the factors of the numbers 1 to 100 using divisibility.
  • Identify whether natural numbers from 1 to 100 are prime, composite, or a special case (i.e. 1).
  • Calculate300
Resource logo

Counting squares

Level Three
Level Four
Level Five
Number and Algebra
This counting collections activity engages students in finding the number of squares of any size on a chessboard.
Resource logo

Digital Delights

Level Four
Number and Algebra
Figure It Out activities
This is a level 4 algebra activity from the Figure It Out series. A PDF of the student activity is included.

investigate patterns involving multiplication and division

Resource logo

Divide and Conquer

Level Four
Number and Algebra
Secondary activities
This is a collection of activities to enhance and extend division skills and understanding.

solve problems using multiplication and division facts
use divisibility rules
use place value strategies to solve division problems
use distributive property to solve equations
use rounding and compensating to solve division problems

Resource logo

Doubling and halving

Level Four
Number and Algebra
Secondary activities
These activities are designed to teach doubling and halving strategies.
solve problems using doubling and halving
identify problems that could be solved using doubling and halving
solve problems using trebling and thirding
generalise the doubling and halving strategy using algebraic notation
list factors of numbers
 
Resource logo

Factor Patterns

Level Four
Number and Algebra
Figure It Out activities
This is a level 4 number activity from the Figure It Out series. It relates to Stage 7 of the Number Framework. A PDF of the student activity is included.

find factors of numbers

generalise patterns for factors

Resource logo

Fascinated by Fibonacci

Level Four
Integrated
Units of Work
This unit looks at Fibonacci numbers and how they occur in nature. Fibonacci numbers provide a rich context in which to apply algebra at Level 4. It is recommended that students already have some experience with Level 4 algebra, prior to the introduction of this unit.
  • Use a recursive rule to generate the sequence of Fibonacci numbers.
  • Create a Fibonacci spiral using squares with Fibonacci side lengths.
  • Find a pattern of odd and even numbers in the sequence.
  • Identify and represent patterns we find for consecutive numbers in the sequence.
Resource logo

Keep Your Shirt On

Level Four
Integrated
Figure It Out activities
This is a level 4 number, algebra, and statistics activity from the Figure It Out series. It relates to Stage 7 of the Number Framework.A PDF of the student activity is included.

identify which graph best represents the data

use multiplicative strategies to calculate costs

Resource logo

Multiplication and Division Pick n Mix 1

Level Four
Number and Algebra
Units of Work
This unit presents a range of strategies for solving multiplication and division problems with multi-digit whole numbers. Students are encouraged to notice the structure of problems, and to anticipate which strategies might be best suited to solving them. This unit builds on the ideas presented in...
  • Mentally solve whole number multiplication and division problems using:
    1. proportional adjustment
    2. place value partitioning
    3. rounding and compensation
    4. factorisation.
  • Use appropriate recording techniques.
  • Predict the usefulness of strategies for given problems.
  • Evaluate the effectiveness of selected strategies300
Resource logo

Mystery Hexagons

Level Four
Number and Algebra
Figure It Out activities
This is a level 4 number and algebra activity from the Figure It Out series. It relates to Stage 7 of the Number Framework. A PDF of the student activity is included.

find a pattern using mutliplication

explore the commutative property

Resource logo

Number Juggling

Level Four
Level Five
Number and Algebra
Figure It Out activities
This is a level 4 and 5 algebra strand activity from the Figure It Out series. A PDF of the student activity is included.

write an equation to generalise an equation

Resource logo

Number Returns

Level Four
Number and Algebra
Figure It Out activities
This is a level 4 algebra activity from the Figure It Out series. A PDF of the student activity is included.

investigate patterns when multiplying numbers

Resource logo

Pascal's Patterns

Level Four
Number and Algebra
Figure It Out activities
This is a level 4 algebra activity from the Figure It Out series. A PDF of the student activity is included.

investigate patterns in Pascal's Triangle

Resource logo

Series-ous Escape

Level Four
Number and Algebra
Secondary activities
These exercises, activities and games are designed for students to use independently or in small groups to practise number properties. Some involve investigation (Mikes Investigation Sheet link) and may become longer and more involved tasks with subsequent recording/reporting.

generalise a number pattern

Resource logo

So Many Relatives

Level Four
Number and Algebra
Rich learning activities
The purpose of this activity is to engage students in solving a problem involving investigating with whole numbers using graphical means and/or generalising.
Resource logo

The number devil: A mathematical adventure

Level Four
Number and Algebra
Picture book activities
This is a series of activities based on the novel The number devil: A mathematical adventure
  • Students will explore several key concepts from the world of mathematics through narrative and through the construction of models.
  • Students will be able to express their observations about patterns and groups of numbers with an accurate vocabulary.
Resource logo

True or False?

Level Four
Number and Algebra
Figure It Out activities
This is a level 4 algebra link activity from the Figure It Out series. A PDF of the student activity is included.

investigate patterns of adding and multiplying consecutive numbers

Resource logo

What's going on? Properties of multiplication and division

Level Four
Number and Algebra
Units of Work
This unit develops students’ recognition of pattern (consistency) in equations involving multiplication and division with whole numbers.
  • Describe and represent the commutative property of multiplication
  • Describe and represent the distributive property of multiplication by attending to place value
  • Recognise that multiplication and division are inverse operations, and interpret division as either equal sharing or measuring.
  • Find300
Resource logo

Writing in Words and Symbols

Level Four
Number and Algebra
Secondary activities
These exercises and activities are for students to use independently of the teacher to develop and practice algebraic reasoning. They follow from teaching sessions that are described below. The initial teaching episode is an attached lesson, while other teaching sessions are outlined in the...

use symbolic notation to record simple word phrases
explain what symbol notation says should be done to the numbers
follow conventions for the use of brackets
interpret conventions for recording multiplications and divisions