Stage 4 - Patterns and Algebra

Strategy

Students can:

  • Identify and build a geometric pattern
  • Record the results in a table of values
  • Describe the pattern in words and algebraic symbols
  • Extend the number pattern using a table of values and determine its rule
  • Use a rule generated from a pattern to calculate the corresponding value for a larger number

Activities to support the strategy

Students will first need to understand that this type of problem is about understanding linear relationships. Whilst an understanding of perimeter and geometry is necessary to interpret the problem, the key concepts involve identifying the underlying patterns between each figure and developing a generalised rule.

To solve this problem we need to begin from the first case of a triangle and build the shape as we go.

1st shape - perimeter = 2 = 1 cm

2nd shape - perimeter = 2 = 3 cm

3rd shape - perimeter = 2 = 3 = 4 cm

The pattern shown above shows that the perimeter when a 10-sided figure is added is

Perimeter = 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 54 cm

Activity 1

Students need to be shown how to build a geometric pattern, record the results in a table of values, describe the pattern in words and algebraic symbols, and represent the relationship on a number grid and to calculate the corresponding value for a larger number. Shown below is an example from the syllabus (MA4-11NA).

Written statements describing patterns can be replaced with equations written in algebraic symbols, e.g. 'You multiply the number of pentagons by four and add one to get the number of matches' could be replaced with 'm=4p+1'

Activity 2

A number of examples should be provided with a variety of patterns and an increasing level of complexity. Some examples are shown below.

1 x triangle, 2 x triangle, 3 x triangle

No of triangles (t)12345Algebraic rule
No of sides (s)3691215S = 3t

1 x square, 2 x square, 3 x squares

 
No of squares12345Algebraic rule
Number of lines47101316I = 3s + 1

1 dot, 2 dots, 9 dots

 
Side length (s)12345Algebraic rule
No of dot (s)1491625d = s2

Activity 3

Students should also complete activities exploring mathematical reasoning. The ‘Bad Apples’ resource develops this concept of identifying and generalising a geometric pattern. Other explorations using mathematics reasoning can be found on the Top draw teachers site.

References

Australian curriculum reference: ACMMG193

Plot linear relationships on the Cartesian plane, with and without the use of digital technologies

NSW syllabus reference: MA4-11NA

Creates and displays number patterns; graphs and analyses linear relationships; and performs transformations on the Cartesian plane

MA4-1WM: Communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

MA4-3WM: Recognises and explains mathematical relationships using reasoning

Student resources

This is a website that covers five mathematical topics: fractions, mental computation, patterns, statistical literacy and ‘reasoning’. It also explores good teaching practice linked to classroom activities including how to address common misunderstandings. There is also a section for assessment, as well as worksheets with solutions, video transcripts, templates, slide presentations and teaching notes.

Build bridges by adding pentagonal sections (each made up of four beams plus a shared beam). Examine a table and graph of the total number of beams used in bridges of different sizes. Predict the number of beams needed to build a wider span. Describe the number pattern. This learning object is the last in a series of five objects that progressively increase in difficulty.

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