Spaces 2 Sat, 15 th Jan 2011

1. Spaces 2Sat, 15th Jan 2011

2. 9:30 - 11:00 Algebra Consolidation 11:30 - 13:00 Painted cube Investigation 14:00 - 15:30 Solving optimization problems (minimizing or maximizing some quantities) – Computer room 4, 543

3. 9:30 - 11:00 Algebra Consolidation

4. Simplifying expressions How would you explain to a Year 7 group (new to algebra) how to simplify algebraic the following expression? • 4g+3b-g+2b How about this? • 2g+4b-5g+b How about expanding this? • 2(a+3) or (m+2)(p+3)

5. Classic errors in Algebra • 2a+3b treated as 2 apples + 3 bananas ...pupils in the past suggested 2a+3b=5f • Thinking letters stands for whole numbers only, rather than standing for any number whatsoever; • Thinking that x and y, being different letters, must stand for different numbers; • Confusing 2metres, 2m and 2 x m, where m is a number; • The sum of any two even numbers is represented as 2n +2n, thus thinking that two occurrences of the same letter can stand for different numbers; • Confusing ab is 41, when substituting a=4,b=1 • not accepting or recognising l x b as an answer to the area of a rectangle, because they expect a single symbol or even a number as an answer. • Last but not least 3 x x instead of 3 x x (not recognised in the literature) Mason et al (2005), Developing Thinking in Algebra, p.149

6. 11:30 – 13:00 Painted Cube Investigation

7. Starter Activity Imaginary Cubes • Imagine a box that will fit a 3x3x3 cube. • Now imagine the box is twice as high, three times as long and twice as wide. Work out how many small cubes will fit into the new box.

8. Painted Cube Investigation Imagine dipping a 3 x 3 x 3 cube into a pot of yellow paint so the whole outer surface is covered, and then breaking the cube up into its small (unit) cubes. http://nrich.maths.org/2322 • How many of the small cubes will have yellow paint on their faces? • Will they all look the same? Now imagine doing the same with other cubes made up from small red cubes. • What can you say about the number of small cubes with yellow paint on?

9. Poster presentation 3/4 groups • Resources unit cubes, paper, coloured pens, poster paper • Group work ensure and encourage each other so that everyone contributes to the task and understands before the group moves on clearly assign roles (who writes, who presents, who does the practical, etc) • Poster display One person from each group presents the work, while the rest of the class act as critical friends requiring clear mathematical explanations and justifications

10. Painted Cube InvestigationPossible extension The number of smaller cubes of a n x n x n cube with one face painted yellow is equal to the number of smaller cubes completely red. • What is the value of n?

11. 14:00 – 15:30 Solving optimization problems

12. The sheep pen Farmer Jones has to create a new grazing pen for his sheep. He has 32 metres of fencing and all the necessary posts. However, he also has a stone wall that he can use for one side of his pen, so he only has to worry about the other three sidesof the rectangular pen. He is building a rectangular pen. • What is the maximum area in square metres that the sheep can have and what will be the dimensions of the pen? See BECTA advice for planning for such a lesson

13. The sheep pen Farmer Jones has to create a new grazing pen for his sheep. He has 32 metres of fencing and all the necessary posts. However, he also has a stone wall that he can use for one side of his pen, so he only has to worry about the other three sidesof the rectangular pen. He is building a rectangular pen. • How should he lay out this pen in order to give the sheep the maximum area to graze? • What is the maximum area in square metres that the sheep can have and what will be the dimensions of the pen? See BECTA advice for planning for such a lesson

14. The sheep pen – The lesson This lesson is about problem solving. It is an investigation into the relationship between area and perimeter. The lesson uses a spreadsheet program to perform the calculations and the graphing facility to illustrate the results. What will the children learn? • To recognise that shapes with the same perimeter may have different areas • To develop a strategy for solving the problem • To enter formulae into a spreadsheet • To use a spreadsheet to draw a graph. What prior experience do the children need? • The ability to work out the area and perimeter of a rectangular shape • Entering numbers and formulae into a spreadsheet. Differentiating the activity • For less able children it might be necessary to start with a length of 32 cm string to help them create some fields. They can then work out the area by multiplying length by breadth or by counting squares.

15. The sheep pen – The lessonQuestions to ask • What do we notice about the area? • At what point has it reached its maximum value? • Why does the area start to get smaller again after this point? • Do you notice anything particular about the shape of the pen that provides the largest area?

16. MaxBox Take a piece of A4 paper, cut squares from its corners, and fold it to make an open box. Experiment with other boxes, cutting out different sized squares. Which box has the largest volume? Reading:Ainley, J. and Pratt, D. Unpacking MaxBox, Micromath Autumn 1994.

17. MaxBox Think about how you would introduce this task to your pupils.

18. Keeping warm Why do people keep babies well wrapped up? Fact: In general, things cool as quick as the ratio between the surface area to their volume increases.

19. Keeping warm Why do people keep babies well wrapped up? Fact: In general, things cool as quick as the ratio between the surface area to their volume increases. Chose different 3D geometrical shapes which approximate the shape of the human body.  In each case, investigate which shape loses heat quickly.

20. Dialogue NotebookSat 15th Jan 2011 1. Attempt as many questions from the Algebra handouts as possible 2. MaxCone Use a spreadsheet to find the largest volume of a cone that may be made from a circle of card with radius 10 cm. Reading: Murray, J. Max cone, Micromath, Autumn 1993.