Algebra units of work

Learning sequence

Curriculum achievement objectives

Specific learning outcomes
The students will be able to:

Algebra units of work

Level 1

Link to learning objects

• 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

• 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

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

The three pigs

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

Beetle wheels

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

Ways to add

Level 2

Link to learning objects

• continue a simple pattern
• generalise the pattern

Pede patterns

Level 2 Equations and expressions AO1

• use the mathematical symbols of =, <, >

Cuisenaire mats

• 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

• 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 3

Link to learning objects

• 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 4

Exemplar link

Link to learning objects

• 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

• 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 aquares

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

Two's company

• 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

• 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

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

The truth about triangles and squares

• 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

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

Drive

Level 5

Exemplar link

Link to learning objects

• 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

• 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

• 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?

• 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

• 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 squares and triangles

• 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

• 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 6 

• 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

• 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

• 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