Algebra

Learning Sequence

Curriculum Achievement Objectives

Specific Learning Outcomes
The students will be able to:

Algebra Units of Work

Level One


Exemplar Link

Link to Learning Objects

Level 1 Patterns and relationships AO1

Level 1 Number Strategies AO1

Number Knowledge AO1

  • recognise that counting tells how many objects are in the set irrespective of how they are arranged or the order in which they are counted.
  • solve problems involving one more or less to a given set using their knowledge of the forward and backward number sequences
  • skip count in 2s

Counting on Counting

Level 1 Patterns and relationships AO1

Level 1 Number Strategies AO1

  • see what a number pattern is
  • be able to guess and check the next number in a pattern
  • skip count in 2s, 5s, and 10s

Ten in a Bed

Level 1 Patterns and relationships AO2

  • describe patterns
  • continue a pattern
  • create patterns

Pattern Makers

Level 1 Patterns and relationships AO2

  • record patterns on grid paper
  • make predictions about ‘missing’ sections of a pattern
  • use words to describe linear patterns

Snakes and Scarves

Level 1 Patterns and relationships AO2

  • "read" a repeating pattern and predict what may come next
  • create a repeating pattern with two elements
  • create a repeating pattern with three elements

Mary, Mary quite Contrary

Level 1 Patterns and relationships AO1

Level 1 Number Strategies AO1

  • continue a sequential pattern
  • systematically count to establish rules for sequential patterns
  • skip count in 2s, 5s and 3s

The Three Pigs

Level 1 Patterns and relationships AO1

Level 1 Number Strategies AO1

  • continue a skip-counting pattern
  • describe skip-counting patterns
  • use graphs to illustrate skip-counting patterns

Beetle Wheels

Level 1 Number strategies AO1

Level 1 Equations and expressions AO1

  • draw representations to show a simple addition equations
  • write an equation/number sentence to match their diagram

Ways to Add

Level Two


Exemplar Link

Level 2 Equations and expressions AO1

Level 2 Patterns and relationships AO2

  • continue a simple pattern
  • generalise the pattern

Pede Patterns

Level 2 Equations and expressions AO1

  • use the mathematical symbols of =, <, >

Cuisenaire Mats

Level 2 Patterns and relationships AO1

Level 2 Measurement AO2:

  • partition numbers less than 10
  • know and use "teen" facts
  • solve addition problems by making a ten, or making a decade
  • solve addition problems involving measurements

Partitions

Level 2 Patterns and relationships AO2

Level 2 Equations and expressions AO1:

  • continue a sequential pattern
  • develop bar charts to show relationships

Staircases

Level 2 Patterns and relationships AO2

  • draw the next shape in a pattern sequence
  • see how the pattern continues from one shape to the next
  • draw up a table of values

Letter Patterns

Level 2 Patterns and relationships AO2

  • identify patterns in number sequences
  • systematically “count” to establish rules for sequential patterns
  • use rules to make predictions

Supermarket Displays

Level Three


Exemplar Link

Link to Learning Object

Level 3 Patterns and relationships AO1

Level 3 Equations and expressions AO1

  • consolidate understanding of simple properties of addition, subtraction, multiplication and division
  • discover and use some more complex properties of addition, subtraction, multiplication and division

Properties of Operations

Level 3 Patterns and relationships AO2

  • predict the next term of a spatial pattern
  • find a rule to give the number of matchsticks (tiles) in a given member of the pattern
  • find the member of the pattern that has a given number of matchsticks (tiles)

Matchstick Patterns

Level 3 Patterns and relationships AO2

  • show number patterns using the hundred’s board and other grid arrangements for whole numbers
  • find the rule for a pattern of numbers shown on a hundred’s board or for input/output pairs from a calculator;
  • relate sequential spatial patterns to how they appear as a number sequence on a hundreds board.

Hundreds of Patterns

Level 3 Patterns and relationships AO2

  • continue a pattern
  • find the recurrence rule of a pattern
  • look at relations between two patterns
  • have some idea of what a general rule is

Building Patterns Constantly

Level 3 Patterns and relationships AO2

  • use a "cups and Cubes" model to describe relationships

Cups and Cubes

Level Four


Exemplar Link

Link to Learning Object

Level 4 Equations and expressions AO1

  • write and calculate arithmetic expressions precisely using the order of operations.
  • realise the importance of the order of operations on a calculator.

Four Fours

Level 4 Equations and expressions AO1

  • predict further members in patterns of equations using relationships within the equations
  • develop function rules to describe relationships
  • find specific values for variables from given relationships

Balancing Acts

Level 4 Equations and expressions AO1

Level 4 Patterns and relationships AO2

  • devise a rule for ensuring that sets of numbers can be arranged into 3-by-3 magic squares
  • represent 3-by-3 magic squares algebraically
  • devise rules for determining the Magic Number for magic squares
  • represent magic squares using parametric equations
  • solve equations that have been formed from magic squares.

Magic Squares

Level 4 Patterns and relationships AO1

Level 5 Number strategies and knowledge AO2

  • use powers of two in problem situations
  • find number patterns in practical situations
  • experiment to find patterns

Two's Company

Level 4 Measurement AO3

Level 4 Patterns and relationships AO2

  • explore the relationship between rows and columns in finding the areas of rectangles
  • calculate the area of rectangles, parallelograms and triangles

You Can Count on Squares

Level 4 Patterns and relationships AO2

  • develop, justify and use rules to solve problems that involve number strips
  • identify and clearly articulate patterns, and make generalisations based on these .

Matilda's Waltz

Level 4 Patterns and relationships AO2

  • find a rule to describe any member of a number sequence and express it in words .

The Truth about Triangles and Squares

Level 4 Patterns and relationships AO2

  • find the number of crosses in Tukutuku panels by using areas of squares and rectangles
  • find the number of crosses in repeating Tukutuku panels by using linear formulae.

Tukutuku Panels

Level 4 Patterns and relationships AO2

  • solve problems using linear relationships shown on tables and graphs.

Drive

Level Five


Exemplar Link

Exemplar Link

Link to Learning Object

Level 5 Equations and expressions AO1

  • devise rules based on numerical patterns to solve triangular arithmagons
  • explain the condition for the solution of any square arithmagon
  • form and use linear equations to solve triangular arithmagons
  • develop proofs of rules and conditions for the solution of arithmagons

Arithmagons

Level 5 Equations and expressions AO1

  • investigate situations involving ratios
  • understand that there are many ways to solve ratio problems
  • solve simple equations of the form ax = b
  • see the relevance of algebra to ratio problems.

Beanies

Level 5 Patterns and relationships AO1

  • infer the precise limit as a fraction from the limit inferred as a decimal number
  • set up a spreadsheet to explore the convergence of certain sequences

What happens on Average?

Level 5 Patterns and relationships AO2

  • solve linear equations
  • describe a linear relationship between two variables in words and as an equation
  • make a table of one variable against another
  • use a graph to find the value of y, given x, and x, given y

Holistic Algebra

Level 5 Patterns and relationships AO2

Level 5 Transformation AO2

  • devise an algebraic rule to identify tilted squares that can fit on geoboards of different sizes
  • devise an algebraic rule to identify the size of the smallest square geoboard on which tilted squares can fit
  • devise and use an algebraic rule for Pythagoras’ theorem
  • devise algebraic rules to find Pythagorean triples

Tilted Sqaures and Triangles

Level 5 Patterns and relationships AO2

  • understand the concept of Fibonacci numbers and how they are generated
  • find factors of a number
  • make conjectures and attempt to prove them
  • find generalisations

Fibonacci I

Level 5 Patterns and relationships AO2

  • find pairs of whole number co-ordinates and use them to draw graphs in problem contexts
  • link the graphs to formulae of the kind ax±by=c
  • find the nth whole number pairs in a context that solve ax-by=c

Linear Graphs and Patterns

Level Six


Level 6 Equations and expressions AO1

  • find the recurrence relation for simple sequences
  • construct tables of values for a pattern
  • find the value of the general term of a sequence algebraically
  • find the value of the general term of a sequence geometrically

The Why and How of General Terms

Level 6 Equations and expressions AO1

  • find the smaller Fibonacci numbers by using the recurrence relation
  • find the initial members of other sequences that can be found using a recurrence relation like the Fibonacci one
  • find the general term of these recurrence relations using quadratic equations
  • understand the concept of a limit of a sequence
  • find various limits relating to sequences

Fibonacci II

Level 6 Patterns and relationships AO2

  • find patterns in the lengths of the sides of standard paper formats
  • use patterns relating to the lengths of the sides of standard paper formats
  • see that fractions can be ‘continued’ in order to calculate basic surds

All Shapes and Sizes