Lesson Plan

SUBJECT/UNIT Triangles
YEAR/SET 10X1 / DATE 23 May 2007 / PERIOD 0950 – 1040 (50mins)
No.of Pupils: 28 / No. of G & T / No. of SEN School Action:
Staff Initials: F Rix / School Action Plus: Statement:

CONTEXT

Range of prior attainment point scores, Level or grade of lesson content, Content of previous lesson

Assume high level of ability but lesson plan may need to be adapted accordingly

LEARNING OBJECTIVES/TEACHING ACTIVITY

During this lesson pupils will develop their skills, knowledge & understanding by:
Investigating pattern and structure within and between Pascal’s and Sierpinski’s triangles with links to fractals.

LESSON STRUCTURE

STARTER

/ TIMING
Show the equilateral triangle. Give me 3 properties of it (regular polygon, 3 lines of symmetry, rotational symmetry order 3).
What about the broccoli? Break off a floret and smaller part thereof. Explain similarity in structure between each part. The small parts, when magnified, are replicas of the whole thing.
This is an example of a fractal – rough or fragmented geometric shape that can be subdivided in parts, each of which is approximately a smaller copy of the whole. They are often considered to be infinitely complex e.g clouds,snowflakes.
Show other examples of fractals.
Explain that today will be seeing the link between the structure of fractals and 2 closely linked triangles. / 5 – 10 mins

MAIN ACTIVITY

/ TIMING
1.  Issue sheet of blank Pascal’s triangles. Explain/ask if have heard of it/him: French mathematician Blaise Pascal in 1653 described a triangular arrangement of numbers corresponding to probability theories involving tossing coins. Show how it starts (OHP) and how created. Give pupils 3-4 mins to complete the first few rows.
2.  What patterns or particular numbers do we see here? Give pupils 2-3 mins to find some:
v  All 1’s down sides
v  Positive integers
v  Triangle number sequence
v  Tetrahedral (pyramid) numbers
v  Square numbers (when add consecutive triangle numbers)
v  Fibonacci sequence (when total shallow diagonals)
v  Powers of 2 when total each row etc.
3.  Issue isometric paper. Transfer first 10 rows of numbers onto it. With a pencil, or colour, start to shade in the odd numbers only. What happens? Pattern emerges of triangles.
4.  Issue sheet with outline of equilateral triangle. Sides measure 20cm. Mark midpoints and join them up. Continue to mark the midpoints of the outer triangles (leaving centre one blank) and join them up. What do you notice? Pattern of smaller equilateral triangles. If you shade some in you will see the same pattern as in Pascal’s triangle. This is known as Sierpinski’s Triangle and is named after a Polish mathematician who discovered it in 1915. This is an example of one of the simplest fractals that can be constructed.
5.  Show how pattern can be generated by computer from http://britton.disted.camosun.bc.ca/blaise/blaise.html if internet available. / 5 mins
25 - 30 mins

PLENARY

Summarise lesson and findings – the link between the broccoli, an equilateral triangle, a French and a Polish mathematician. Important to be able to see pattern and structure in maths as many patterns in numbers occur in different situations.
For any interested show the chaos game and give internet sites to explore for more fractals on http://www.ocf.berkeley.edu/~wwu/fractals/fractal_gallery.html / TIMING
5 mins
GIFTED & TALENTED/EXTENSION TASKS
Chaos game / SEN DETAILS/SUPPLEMENTARY TASKS/SUPPORT STAFF ROLE
Assist as necessary with each activity
ICT / NUM
4, 5 / CIT
3, 4, 6 / LIT
6, 7, 9
RESOURCES
Broccoli, laminated equilateral triangle
Fractal examples
OHP
OHTs and copied Pascal’s triangle, equilateral triangle sheets
Isometric paper
Spare Sierpinski blank triangles if required
Colouring pencils/pens
Rulers
Pencils /
RISK ASSESSMENT
Trip hazard with OHP flex
ASSESSMENT FOR LEARNING (Peer / self / tutor)
Tutor from pupil responses
Self assessment from completion of task
FUTURE LEARNING / NEXT STEPS
Generation of Sierpinski’s triangle with ICT?
Further examination of Pascal’s triangle?
CROSS CURRICULAR KEY
ICT / NUMERACY / CITIZENSHIP / LITERACY
1 / Word processing / 1 / Fractions / 1 / School Environment / 1 / Spelling
2 / Internet / 2 / Basic Calculations / 2 / Local Community / 2 / Paragraphing
3 / Desktop Publishing / 3 / Percentages / 3 / Organisation Skills / 3 / Reading/Research
4 / Data Logging Spreadsheets / 4 / Numbers and Counting / 4 / Relationships and Social Skills / 4 / Planning/Revising text
5 / Accessing CD ROM’s / 5 / Measurement and Scale / 5 / Extra Curricular Activities / 5 / Note Taking
6 / Digital Imaging / 6 / Interpreting tables and graphs / 6 / Using Initiative and Management / 6 / Oral communication and Presentation
7 / E-Mail / 7 / Drawing tables and graphs / 7 / Charities and Fund Raising / 7 / Comprehension
8 / Satellite Messages / 8 / Hypothesis Testing / 8 / School ethos and the pupils’ role / 8 / Description
9 / Video / 9 / Ratio and Proportion / 9 / Analysis/Evolution
10 / Graphs / 10 / Co-ordinates / 10 / Report Writing
11 / Averages