Stage 4 - Patterns and Algebra
- 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
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'
A number of examples should be provided with a variety of patterns and an increasing level of complexity. Some examples are shown below.
|Number of triangles (t)||1||2||3||4||5||Algebraic rule|
|Number of sides (s)||3||6||9||12||15||S = 3t|
|Number of squares||1||2||3||4||5||Algebraic rule|
|Number of lines||4||7||10||13||16||I = 3s + 1|
|Side length (s)||1||2||3||4||5||Algebraic rule|
|No of dot (s)||1||4||9||16||25||d = s2|
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.
ACMMG193: Plot linear relationships on the Cartesian plane, with and without the use of digital technologies
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.
- Top drawer teachers - this website 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 - 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.