1 Warm up Connect these nine dots with only four straight lines without lifting your pencil from the paper.
2 Sometimes we need to think outside the box! Warm up Solution
3 Warm up Insert the Numbers 1 8 into the boxes provided. Consecutive numbers cannot be beside, adjacent, or diagonal to each other. Example:
4 Warm up Insert the Numbers 1 8 into the boxes provided. Consecutive numbers cannot be beside, adjacent, or diagonal to each other. Example:
5 Warm up Solution Insert the Numbers 1 8 into the boxes provided. Consecutive numbers cannot be beside, adjacent, or diagonal to each other.
6 Problem Solving Addressing misconceptions during teaching does actually improve achievement and long-term retention of mathematical skills and concepts. Drawing attention to a misconception before giving the examples was less effective than letting the pupils fall into the trap and then having the discussion. (Askew and Wiliam 1995)
7 What's the difference? Problem Solving Solving Problems
8 text Solving Problems Solving Problems Everyday and essential activity in Mathematics Classes Features of existing classroom tasks Has one solution Provide limited scope for discovery or discussion Problem has limited objectives Solving Problems Show limited graduation Rely on formulaic & Procedural approach Develop the student skill set
9 Solving Problems
10 Problem Solving Apollo 13: Fit a square peg into a round hole CLICK TO PLAY VIDEO Link:
11 Problem Solving
12 Mathematical Problem Solving People are generally better persuaded by the reasons which they have themselves discovered than by those which have come in to the mind of others. Blaise Pascal Link: Is problem solving the same for us all? What is a problem for one student may not be a problem for another!!
13 Four Stage Problem Solving Process Stage 1 Understand the problem Stage 4 Look back or Review Stage 2 Devise a plan Stage 3 Carry out the Plan (Pólya, 1945)
14 Understanding the Problem Is the problem well defined? Do you understand all the words used in stating the problem? What are you asked to find or show? Can you restate the problem in your own words? Can you think of a picture or a diagram that might help you understand the problem? Is there enough information to enable you to find a solution? Do you need to ask a question to get the answer?
15 Solve an equation Look for a pattern Draw a picture Solve a simpler problem Use a model t e x t Use direct reasoning Consider special cases Devise a Plan Work backwards Use a formula Eliminate possibilities Use your head Make a list Guess & Check Be creative
17 Monitor progress to make sure that things go according to plan. Check each step Try a new plan if this plan fails Carry out the Plan Record errors
18 Review your Plan Can you check the result? Try to understand why you succeeded/failed? Reflect and look back at what you have done What worked and what didn t work.
19 Proof Proof and Numbers Proof can be used to motivate or revise areas of the curriculum. Look to develop and apply proof in areas other than geometry. Use of a counter example Prove that the statement a four sided figure with all sides equal in length must be a square is untrue.
20 Activity 1 [Exploring Numbers] Four bags contain a large number of 1 s, 3 s, 5 s and 7 s. Pick any 10 numbers from the bag so that their sum equals 37. Justify your solution.
21 Activity 1 [Exploring Numbers] Four bags contain a large number of 1 s, 3 s, 5 s and 7 s. Pick any 10 numbers from the bag so that their sum equals 37. Justify your solution.
22 Activity 1 [Exploring Numbers] SOLUTION. Pick any 10 numbers from the bag so that their sum equals 37. Justify your solution.
23 Exploring Numbers Curricular Links: 1. Patterns in number 2. Expanding algebraic expressions Purpose: Enhance the students appreciation of number and to motivate the expansion of algebraic expressions. *
24 Activity 2 [Exploring Numbers] Let s look at the properties of even and odd numbers... Write down as many properties of even and odd numbers that you can! Activity
25 Try discover. Activity 2 [Exploring Numbers] (a) A rule to represent every even number. (b) A rule to represent every odd number. (c) The outcome when two even numbers are added. (d) The outcome when two odd numbers are added. (e) The outcome when two even numbers are multiplied. (f) The outcome when two odd numbers are multiplied. Prove: The outcomes for (c) to (f) above.
26 (a) (b) (c) (d) Rule for even numbers is: 2n Rule for odd numbers is: 2n + 1 Addition Adding two even numbers is: 2n + 2k = 2(n + k) an even number. Adding two odd numbers is: 2k n + 1 = 2k + 2n + 2 = 2 k + n + 1 an even number.
27 Multiplication (e) Multiplying even numbers; 2k 2n = 4nk = 2(2nk) an even number. (f) Multiplying two odd numbers; (2k + 1)(2n + 1) 4nk + 2n + 2k + 1 = 2(2nk + n + k) + 1 an odd number *
28 Activity 3 Prove that a square with side an odd number in length, must have an odd area. * Activity
29 Activity 3 Prove that a square with side an odd number in length, must have an odd area. 2n+1 2n+1 Area (2n 1)(2n 1) 2 4n 4n 1 2 2(2n 2 n) 1 * Activity
30 Prove : Proof : 2 2 Proof by Contradiction - LCHL 2 is irrational Assume the contrary: 2 is rational i.e. there exists integers p and q with no common factors such that: p q p q p p p If p 2k 2 is even is even p 2 k for some k p 2q 2 q becomes (2 k) 2q 4k 2q 2k q 2 2 p 2k p q 2m q Square both sides Multiply both sides by q 2 it s a multiple of 2 even 2 = even Then similarly q 2 m from some m has a factor of 2 in common. This contradicts the original assumption. 2 is irrational Q.E.D.
31 Proof Links to the syllabus 1. Variables and constants in expressions and equations 2. Whole Numbers/Integers 3. Integer Values
32 Consecutive Numbers Try to discover a rule to represent the outcome when two consecutive numbers are multiplied: First even and second odd: 2n 2n + 1 = 4n 2 + 2n which is even. First odd and second even?
33 Activity Find all integer solutions of the equation: x 2 + y 2 + x + y = 1997 x x y y + 1 = 1997 Here we have the product of two consecutive integers. Since the product of two consecutive integers is even (or zero), we have the sum of two even integers. This can never be equal to the odd number, 1997, and thus the solution set is empty.
34 12 days of Christmas
35 Activity 4 [Triangular Numbers] This pattern continues indefinitely. Triangle number Can you find a rule to define the number of circles in any given triangle?
36 Activity 4 [Triangular Numbers] This pattern continues indefinitely. Triangle number For any triangular number eight times the number of circles when increased by 1 yields a perfect square. Can you prove this?
37 Activity 4 [Triangular Numbers] (b) What seems to happen if any two consecutive triangle numbers are added? Can you prove this to be true? + or + 1, (1 + 2), ( ), ( ), Adding 2 consecutive triangle numbers yields a perfect square.
38 Triangular Numbers Triangle Pattern General pattern is given by : n 2 + n 2
39 Solving a Problem Expand the following: x( x 2) ( x 1) 2 ( x 1)( x 1) 2 ( x 2) x( x 4) ( a b)( a b) 2 ( ) ( 2 ) p q p q p Problem Solving Write down 3 consecutive numbers. Square the middle number. Multiply the other two numbers together. What do you notice? e.g. 81, 82, = = Try other groups of consecutive numbers. What happens if you use decimals? e.g. 51.5, 52.5, 53.5 What happens if the numbers are not consecutive, but go up in two s : e.g. 412, 414, 416 Generalise your result
40 Syllabus Synthesis and problem-solving skills explore patterns and formulate conjectures explain findings justify conclusions communicate mathematics verbally and in written form apply their knowledge and skills to solve problems in familiar and unfamiliar contexts analyse information presented verbally and translate it into mathematical form devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions.
41 Numbers and Indices How many digits has the number, (2 ) (5 ) (2 5 ) (10 ) So, there are 82 digits Indices
42 LCOL (2011) Numbers and Indices The number is a prime number. Using your calculator, or otherwise, express its value, correct to two significant figures, in the form a 10 n, where 1 a < 10 and n N. How many digits are in ? LCFL (2012) Let a = Express a as a product of it s prime factors. If b = Express ab as a product of prime factors.
43 Proof and the Curriculum Curricular Links: 1. Patterns in number 2. Expanding algebraic expressions Purpose: Enhance the students appreciation of number and to motivate the expansion of algebraic expressions.
44 Linking our Thinking
45 You must have a reason for asking questions. Students think about how they thought about it. The student voice is LEAST clearly heard in maths than in any other subject. Use at start of lesson to motivate it. Some don t have an obvious solution. The children you regard as best at maths aren t always the best. Rigor is necessary.
46 Activity 5 [Linking Triangles] (a) (b) (c) (d) How many different triangles have a perimeter of 12 units? What kinds of triangles are they? Explain how you determined this. Theorem 8 2 sides of a triangle are together greater than the third. Explain what you have discovered.
47 Curricular Links Activity 6 [Estimation of π] 1. Geometry 2. Area 3. Algebraic manipulation 4. Inequalities Purpose To consolidate and link certain concepts in Geometry and to see how these links can be used to estimate π. *
48 Activity 6 [Estimation of π] Find the ratio of the areas of the circles below. Can you get a better approximation for π?
49 Area of Outer Circle> Area of Triangle> Area of Inner Circle o r 1 r tan30 x r 3 x 3 x Area of Triangle in the original diagram x( height) x( R r) As R 2 r, area of the triangle r 3(3 r) 3 3r Area of outer circle Area of Triangle Area of inner Circle R 3 3r r (2 r) 3 3r r 4 r 3 3r r
50 Area of outer circle > Area of Square > Area of Inner Circle R (2 r) r or 4 2 or <4 2< <4 Discussion: R = 2r, why?
51 Area of outer circle> Area of Regular Pentagon > Area of inner circle 1 R 5 R sin72 r o r 1 r 5 sin72 r o o cos36 2 cos r r r or o 2 Discussion: where do the figures come from?
52 Early engagement with formulas for regions with curvilinear boundaries
53 Syllabus Synthesis and problem-solving skills explore patterns and formulate conjectures explain findings justify conclusions communicate mathematics verbally and in written form apply their knowledge and skills to solve problems in familiar and unfamiliar contexts analyse information presented verbally and translate it into mathematical form devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions.
54 Problem Solving Incorporating Problem Solving Introduce questions that Promote discussion Promote discovery Relevant to the topic being covered Link to different areas of the curriculum Have reduced levels of scaffolding Problem Solving Lead toward the notion of proof
55 Problem Solving Collaborative Problem solving Purpose Engage students Motivate curricular content Place the student at the centre of the learning process Encourage collaboration and discussion Facilitate research, hypothesis development and testing Classroom organisation Minimal Instruction Group work Unseen Problems Collaboration Discussion Internet Access Presentation of Solutions
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