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NA5-9: Relate tables, graphs, and equations to linear and simple quadratic relationships found in number and spatial patterns.

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Elaboration on this Achievement Objective

This means students will recognise the features of tabular, graph and equation representations of linear and quadratic relationships. This includes connecting constant first or second order difference in tables with linear and quadratic relations respectively, with the graph (linear and parabolic) and standard equation forms (y = mx + c and y = a x 2 + bx + c) for such relations. For example, given the spatial pattern below students can use a table, graph or equation to represent the relation and solve problems.

spatial patterns.
 

This includes finding both a recursive and direct (functional) rules and using them to find further terms using a spreadsheet or calculator, for example:

spreadsheet.
 

Students should also use these tabular and graphic representations for other relationships, such as simple exponential and step relations, but it is acceptable for them to use recursive rules for these more difficult relations.

 

Teaching resources for this Achievement Objective
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Alien Critters

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

use a table to find a pattern

write a quadratic equations to describe the relationship

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A neutral solution

Level Five
Number and Algebra
This unit provides an opportunity for students to explore an integrated mathematics and science context: chemical reactions involving acids and bases. Within this, students develop their use of rates and ratios through exploration of linear relationships using tables and graphs. Students also...
  • use rates, ratios, and reasoning with linear proportions to solve a problem.
  • make connections between representations of linear data in tables and graphs.
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A Parabolic Investigation

Level Five
Number and Algebra
Rich learning activities
The purpose of this activity is to engage students in an investigation of parabolas in a practical context.
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Areas of Interest

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

find a rule to describe a geometric pattern

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Can stack

Level Five
Number and Algebra
Problem solving activities
This problem solving activity has an algebra focus.
  • Find the rule for summing consecutive numbers.
  • Identify the pattern of triangular numbers.
  • Devise and use problem solving strategies to explore situations mathematically (be systematic, use algebra).
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Circuit Problems

Level Five
Number and Algebra
The purpose of this unit is to engage students in applying their understanding of algebra to solve problems within the science - physical world context of electrical circuits. Subsequently, students develop their skills and knowledge on the following mathematics learning progression: Using Symbols...
  • use algebra to generalise the results of a practical investigation.
  • apply knowledge of circuits, power, voltage, currents, and resistance to problem solving.
  • use graphical techniques to identify and describe a linear relationship found in a practical investigation. 
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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.
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Cuisenaire Rod Fractions: Level 5

Level Five
Number and Algebra
Units of Work
This unit explores how the properties, that hold for multiplication and division of whole numbers, also apply to fractions (commutative, associative, distributive, and inverse). The operations of multiplication and division can be developed using a length model (Cuisenaire rods). While the size of...
  • Name the fraction for a given Cuisenaire rod with reference to one (whole) and with reference to another rod.
  • Fluently multiply two or more fractions.
  • Divide a fraction by another fraction, including when the divisor is greater than the dividend.
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Diagonals

Level Five
Number and Algebra
Rich learning activities
The purpose of this activity is to engage students in recognising non-linear patterns, in this case a quadratic relationship. Students might use tables, graphs and equations to represent the relationship between sides of a polygon and the number of diagonals for the polygon.
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Domino Stacks

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

usea  table to continue a pattern

use a rule to describe a prediction

write a rule as a quadratic equation

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Fences and Posts

Level Five
Integrated
Units of Work
The focus of this measurement unit is around finding linear patterns. This is explored through Pick’s Rule which applies to finding areas of polygons where all the vertices are lattice points (in this application, the nails on geoboards).
  • Find areas of shapes.
  • Find simple two-variable linear patterns relating to areas.
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From One to Another

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

use a graph to find a pattern for a non linear relationship

 

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Holistic Algebra

Level Five
Number and Algebra
Units of Work
In this large range of activities some basic concepts in algebra are developed. The intention of this unit is not to teach skills in isolation, but rather to use each activity for developing all the concepts, with opportunities for discussion and consolidation.
  • 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.
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Jam Jars

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

interpret a relationship shown by a graph

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Linear Graphs and Patterns

Level Five
Number and Algebra
Units of Work
Techniques for drawing linear graphs are introduced via whole number problem situations where the focus is on the solutions rather than the technique. Only linear equations of the form ax±by=c are graphed by finding two or more points on the line. No attempt is made to link these equations by...
  • 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
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Mary's Garden

Level Five
Number and Algebra
Units of Work
In this unit students represent a quadratic relationship using spatial patterns, tables, graphs and equations. In doing this, they recognise the key features of a parabola, find unknowns from the graph of a parabola, and find unknowns from simple quadratic equations. Procedural fluency and...
  • Make a table of one variable against another to represent a quadratic relationship.
  • Represent a quadratic relationship between two variables in words and as an equation.
  • Represent a quadratic relationship as a parabola on the Cartesian Plane.
  • Recognise the key features of a parabola, including the300
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Mats, Patterns, and Rules

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

use a table to find a rule for a geometric pattern

write the rule for a relationship as a linear equation

use rules that describe quadratic relationships

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Multiplicative sequences

Level Five
Number and Algebra
Secondary activities
These exercises and activities are for students to use independently of the teacher to practice number properties.
  • Continue multiplicative sequences.
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Overlapping powers

Level Five
Number and Algebra
Rich learning activities
The purpose of this activity is to engage students in using algebraic equations, graphs and/or tables to solve a problem.
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Poppy meets Pythagoras

Level Five
Integrated
Problem solving activities
This problem solving activity has a geometry focus.
  • Find connections between numbers in a table.
  • Use Pythagoras’ theorem in a general algebraic form.
  • Measure accurately from a scale drawing.
  • devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram).
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Power play

Level Five
Number and Algebra
Rich learning activities
The purpose of this activity is to engage students in evaluating powers of whole numbers and generalise the repeating sequence in the ones digit of those powers.
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Quadratics in context

Level Five
Number and Algebra
Units of Work
In this unit procedural fluency and conceptual understanding are developed together. Students will represent quadratic relationships, particularly using graphs and equations, and will use quadratic functions to model phenomena such as paths of projectiles, stopping distances and profits. In turn...
  • Make a table of one variable against another to represent a quadratic relationship.
  • Represent a quadratic relationship between two variables in words and as an equation.
  • Represent a quadratic relationship as a parabola on the Cartesian Plane.
  • Recognise the key features of a parabola, including the300
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Renting a Car

Level Five
Number and Algebra
Rich learning activities
The purpose of this activity is to engage students in using their knowledge of linear relationships to solve a problem.
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Roller Coasting

Level Five
Number and Algebra
Figure It Out activities
This is a level 5 algebra activity from the Figure It Out theme series.

make a graph from a data table

interpret a relationship shown on a graph

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Rows of numbers

Level Five
Number and Algebra
Problem solving activities
This problem solving activity has an algebra focus.
  • Generate patterns from a structured situation.
  • Find a rule for the general term (extension problem).
  • Devise and use problem solving strategies to explore situations mathematically (be systematic, guess and check, make a table, look for a pattern).