Patterns and relationships

This unit brings together a number of valuable mathematical ideas using the practical context of international paper sizes. In this context, the number √2 appears naturally as do arithmetic and geometric means. The international paper formats provide a way for students to practice some basic algebra by finding the general terms of patterns and then using these general expressions to find other specific terms. More algebra is present in the continued fractions of the last session. Students are also encouraged to practice their measuring skills with the aim of furthering their knowledge about the paper formats that are the subject of this unit.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 and 6 number activity from the Figure It Out theme series.

These exercises and activities are for students to use independently of the teacher to practice number properties

We have available a large range of activities that can be used for developing some basic concepts in algebra. Students often complain that they cannot see the point of learning algebra so all of the teaching of skills here is placed within contexts. The intention of this unit is not to teach skills in isolation, but rather to use each activity for developing all the concepts. It is not suggested that this unit should replace the skills practice traditionally used in the teaching of algebra. It does, however, provide the opportunities for discussion and development of concepts. As each learning outcome is explored there will probably be need for consolidation through more traditional exercises. With the first activities one will probably not wish to explore all aspects with the whole class, but the possibility is there for extending individuals. It would also be appropriate to use the first activity for diagnostic assessment.

The initial focus of this unit involves students in gathering data from their investigation of squares that can be made on different sized geoboards, (an array of dots arranged on a square grid).  Students need to be systematic in their work, and to record their results in ways that are likely to help them notice patterns and relationships.

In this unit we look at Fibonacci numbers and how they occur in nature. We also look at what factors Fibonacci numbers can have.

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 algebra to the gradient intercept form y=mx+c.

These exercises and activities are for students to use independently of the teacher to practice number properties.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 algebra strand activity from the Figure It Out series.

This is a level 5 algebra activity from the Figure It Out theme series.

In this unit students are introduced to convergence of sequences based on averaging previous terms in the sequence. The power of the “fill-down” function in spreadsheets is exploited. By using various starting numbers, and inferring the limit, students are encouraged to make generalisations of formulae for the convergence of sequences using arbitrary starting numbers.

These exercises and activities are for students to use independently of the teacher to practice number properties

These exercises and activities are for students to use independently of the teacher to practice number properties

These exercises and activities are for students to use independently of the teacher to practice number and measurement skills.

This unit allows the class to work on two different pattern themes. It lays the foundations for Algebra at Level 5 by requiring them to articulate the patterns.

In this unit we are paying close attention to the general rule of a pattern for the first time.  We look at odd and even numbers as well as multiples of small numbers.  We also look at squares and cubes of numbers and see that it is possible to make more than one pattern from the same three initial terms.

This unit explores some of the algebra behind triangular and square numbers.

In this unit some Tukutuku patterns are introduced. Rotations of these patterns produce simple shapes whose area formulae are well known. From these formulae algebraic formulae for sequences can be deduced.  While the unit is written to the Level 4 Achievement Objectives, the work is quite advanced and may be most suitable for students entering Level 5.

In this Unit, students develop and use algebraic representations of 3-by-3 magic squares in which the sum of the three numbers in each row, column, and diagonal, add to a constant number known as the Magic Number. The unit provides opportunities for students to use their problem solving skills, to begin to consolidate their skills with algebraic manipulation and to solve an array of problems using their algebraic tools.

This unit uses one of the digital learning objects, Drive: easy problems, to support students as they investigate linear relationships. The context of driving illustrates the relationship between time, distance and speed. The students use the information from one situation to establish the relationship between the variables and apply it to new numbers. Information is given in a table and on a graph. The students’ answers are shown on the graph. It is suitable for students working at stage 7 of the number framework. It includes a sequence of problems and questions that can be used by the teacher when working with a group of students on the learning object, and ideas for independent student work.

These exercises, activities and games are designed for students to use independently or in small groups to practise number properties. Some involve investigation (see Related Resources) and may become longer and more involved with subsequent recording/reporting.

This is a level 3 and 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series. It also has an achievement objective from the Level 4 Number strategies and knowledge substrand.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand activity from the Figure It Out series.

This is a level 4 algebra strand link activity from the Figure It Out series.

This is a level 4 algebra strand link activity from the Figure It Out series.

This is a level 4 algebra activity from the Figure It Out series.

This is a level 4 number activity from the Figure It Out series.

These are level 4 number, algebra and statistics problems from the Figure It Out series.

These are level 3 number and statistics problems and level 4 algebra problems from the Figure It Out series.

These are level 3 number problems and level 4 number and algebra problems from the Figure It Out series.

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.

This is a level 4 number and algebra activity from the Figure It Out series. It relates to Stage 7 of the Number Framework.

This is a level 4 number and algebra activity from the Figure It Out series. It relates to Stage 7 of the Number Framework.

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.

This is a level 3 algebra activity from the Figure It Out theme series.

This is a level 4 algebra activity from the Figure It Out theme series.

This is a level (3+ to 4+) mathematics in science contexts activity from the Figure It Out series.

This is a level (3+ to 4+) mathematics in science contexts activity from the Figure It Out series.
 

This is a level 4 algebra activity from the Figure It Out series.

This is an activity based on the picture book The rabbit problem

This is an activity based on the short illustrated novel Fibonacci’s Cows

This is a level 3-4 activity from the Figure It Out series.

This unit presents students with several problems that develop the idea of binary (base two) arithmetic. In the course of solving the problems, students are required to look for patterns and relationships and describe them using symbols and mathematical language.

In this unit we look at a range of strategies for solving multiplication and division problems with whole numbers with a view to students anticipating from the structure of a problem which strategies might be best suited. This unit builds on the ideas presented in the Multiplication Smorgasbord session in Book 6: Teaching Multiplication and Divsion.

These activities are designed to teach doubling and halving strategies.

This is a collection of activities to enhance and extend division skills and understanding.

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.

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 ‘background’ section.

This is a level 4 and 5 algebra strand activity from the Figure It Out series.

This is a level 4 number and algebra activity from the Figure It Out series. It relates to Stage 7 of the Number Framework.

This is a level 4 algebra link activity from the Figure It Out series.

This is a level 4 algebra link activity from the Figure It Out series.

This is a level 4 algebra link activity from the Figure It Out series.

This is a level 4 algebra link activity from the Figure It Out series.

This is a level 4 number activity from the Figure It Out series. It relates to Stage 7 of the Number Framework.

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.

This is an activity based on the picture book Anno’s Mysterious Multiplying Jar.

This is a series of activities based on the novel The number devil: A mathematical adventure

The unit investigates patterns made using matchsticks and tiles, so it involves the children in active learning. The relation between the number of the term of a pattern and the number of matchsticks (tiles) that that term has, is explored with a view to finding a general rule. This rule can be expressed in words or any other way that the class is comfortable with.

The unit investigates how the link between recursive rules and pattern on the hundred’s board and other types of arrangements for whole numbers. Students are encouraged to find rules for relationships as shown through patterns on number grids and calculator outputs.

In this unit we introduce some number patterns using buildings constructed with squares. The aim of the unit is for the students to be fluent in constructing tables of patterns and in finding the recurrence rule for a simple pattern where the increase from term to term is a constant.  We also talk about the general rule but don’t expect all students to be able to fully grasp and use this concept.

This unit aims to give students the opportunity to explore recurrence rules for patterns where the increase from one term to the next is not constant.  Encourage them to find the recurrence rule.  Actually some students will find it easier to find the general rule here as in some ways it is less complicated than the recurrence rule. 

In this unit students explore the use of cups and counters as a model to analyse the effects of operations rather than focusing on specific numbers.

These exercises and activities are for students to use independently of the teacher to practice number properties.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand activity from the Figure It Out series.

This is a level 3 algebra strand link activity from the Figure It Out series.

This is a level 3 algebra strand link activity from the Figure It Out series.

This is a level 3 algebra strand link activity from the Figure It Out series.

This is a level 3 algebra strand link activity from the Figure It Out series.

This is a level 3 algebra strand link activity from the Figure It Out series.

This is a level 3 algebra strand link activity from the Figure It Out series.

This is a level 3 algebra strand link activity from the Figure It Out series.

This is a level 3 algebra strand link activity from the Figure It Out series.

This is a level 3 algebra strand link activity from the Figure It Out series.

This is a level 3 algebra link activity from the Figure It Out series.

This is a level 3 algebra activity from the Figure It Out series.

This is a level 3 algebra activity from the Figure It Out series. It relates to Stage 6 of the Number Framework.

This is a level 3 algebra activity from the Figure It Out series.

These are level 3 number, algebra, and statistics problems from the Figure It Out series.

These are level 3 number, algebra, measurement problems from the Figure It Out series.

These are level 3 number and algebra problems from the Figure It Out series.

These are level 3 number, algebra, and geometry problems from the Figure It Out series.

These are level 3 number and algebra and level 5 number problems from the Figure It Out series.

These are level 3 number and algebra problems from the Figure It Out series.

These are level 3 algebra, geometry and number problems from the Figure It Out series.

These are level 4 number, and measurement problems and level 3 algebra problems from the Figure It Out series.

use common multiples to solve problems (Problem 1)

use addition and problem solving strategies to solve problems (Problem 2)

continue a sequential pattern to find a rule and predict a future member (Problem 3)

This is a level 3 number and algebra activity from the Figure It Out theme series.

This is a level 3 algebra and level 4 number activity from the Figure It Out theme series.

This is a level 3 algebra activity from the Figure It Out theme series.

This is a level (2+ to 3+) mathematics in science contexts activity from the Figure It Out series.

This is a level 3 activity from the Figure It Out series.

This is a level 3 activity from the Figure It Out series.

 This is an activity based on the picture book Spaghetti and Meatballs for All! A Mathematical Story

 Students work out their own generalizations of the properties of number operations through their exploration of addition, subtraction, multiplication and division.

This is a level 3 and 4 algebra strand activity from the Figure It Out series.

This is a level 3 algebra activity from the Figure It Out series.

This is a level 3 algebra link activity from the Figure It Out series.

This is a level 3 number activity from the Figure It Out series. It relates to Stage 6 of the Number Framework.

These are level 3 algebra and number problems from the Figure It Out series.

These are level 3 algebra, geometry and measurement problems from the Figure It Out series.

This is a level 3 number, algebra, statistics activity from the Figure It Out series. It relates to Stage 6 of the Number Framework.

This unit is about generating number patterns for certain ‘insects’ from the mythical planet of Elsinore. Each ‘Pede’ is made up of square parts and has a number of feet. The patterns all link the number of feet and the number of squares. The students are encouraged to explore the patterns for themselves and try to make the connections between feet and squares. The patterns range from counting by 2s and 3s to being the number of feet plus three.

This week we explore the patterns in different types of staircases.

In this unit we look at the number patterns we get from letters and numbers. We keep track of the numbers involved by drawing up a table of values. It’s important here to look for the pattern and see how the number of tiles changes from letter to letter.

In this unit the students look at the number patterns created when tins are stacked in different arrangements. The aim of the unit is for students to keep track of the numbers involved by drawing up a table of values.  They are then encouraged to look for patterns in the numbers.

This is a level 2 algebra strand activity from the Figure It Out series.

This is a level 2 algebra strand activity from the Figure It Out series.

This is a level 2 algebra strand activity from the Figure It Out series.

This is a level 2 algebra strand activity from the Figure It Out series.

This is a level 2 algebra strand activity from the Figure It Out series.

This is a level 2 algebra strand activity from the Figure It Out series.

This is a level 2 algebra strand activity from the Figure It Out series.

This is a level 2 algebra strand activity from the Figure It Out series.

This is a level 2 algebra strand activity from the Figure It Out series.

This is a level 2 algebra strand activity from the Figure It Out series.

This is a level 2 algebra strand activity from the Figure It Out series.

These are level 2 number and algebra problems from the Figure It Out series.

This is a level 2 number and algebra activity from the Figure It Out theme series.

This is a level 2 algebra activity from the Figure It Out theme series.

This is a level 2 algebra activity from the Figure It Out theme series.

This is an activity based on the picture book The Biggest Number in the Universe.