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Mathematics

Mathematics. Matariki ahunga nui Ka p ē r ā hoki te taurangi Matariki provider of plentiful food So too does algebra. Kua whiti te rà, he rangi hou anò. The sun has risen, a new day is born. What Building confidence with algebraic skills through developing understanding Why

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Mathematics

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  1. Mathematics

  2. Matariki ahunga nui Ka pērā hoki te taurangi Matariki provider of plentiful food So too does algebra. Kua whiti te rà, he rangi hou anò. The sun has risen, a new day is born.

  3. What Building confidence with algebraic skills through developing understanding Why Algebra can be a block to mathematics learning when misunderstood How Through the use of multiple representations and contexts

  4. Some Food for Thought! What do we mean by 2x? Does it mean x + x, 2 × x, (x + x) or doesn’t it matter? • Work through the following: • What does 2x mean? 2. What is the value of 2x when x = 3 ?. 3. What is the value of 12 × 2 × 3 ? 4. What is the value of 12 ÷ 2 × 3 ? 5. What is the value of 12 ÷ 2 × x when x = 3 ? 6. What is the value of 12 ÷ 2x when x = 3 ? • Did you get these answers? • 2 × x or similar expression2. 63. 72 • 18 (convention tells us to work from left to right)5. 18 • 6. Most people give the answer 2, different to that of question 5. What does it tell us about 2x and 2 × x? Nothing is quite as simple as it first appears, perhaps. Perhaps the moral of the story here is to use brackets everywhere where there might possibly be ambiguity

  5. nzmaths / algebra A patterned approach vs a generalised arithmetic approach. Is this a 6,4,4,4,4,4,. . . or a 2,4,4,4,4,4,4,4,4, . . . pattern To get <6, 10, 14, 18, 22, . . .> What’s the rule? Total = 6 + 4(n - 1) or Total = 4n + 2

  6. Ka ngaere mai nga ngaru ki te one. The rollers came tumbling up the beach. The Ngaru Nui represent the waves of theNgatokimatawhaorua. The zig zag part are the waves. The rectangle part is the waka Taniko Niho Taniwha http://whakaahua.maori.org.nz/tukutuku.htm

  7. Algebra Strand Algebra gives us a vehicle to reason why things are. algebra applets http://www.active-maths.co.uk/algebra/investigations/polygon1.html

  8. A Message from Lewis Carroll  The red Queen says to Alice, “What’s one and one and one and one and one and one and one and one and one and one?” “I don’t know”, said Alice, “I lost count.” “She can’t do addition”, said the Queen. Are there similarities for Number and Algebra? Are there similarities for mathematical Processes – Logic and Reasoning and Algebra?

  9. First we state a law that all parents will agree with: Teenagers = Time × Money so Teenagers = Money × Money = Money2 because Time is Money Because money is the root of all evil Money = (root of all evil)1/2 So Money2 = root of all evil and hence: Teenagers = root of all evil

  10. Looking for patterns. Draw / display examples to give you some ideas Plotting points on a graph to display your examples Making links Graphically Numerically What type of graph is it? Generating a numerical pattern / sequence Straight Line Linear? Algebraically -generalising Other? What is the rule?

  11. "Choose any two digit number, add together both digits and then subtract the total from your original number. When you have the final number look it up on the chart and find the relevant symbol. " mysticalball

  12. Make the magic square so that all of the rows, columns and major diagonals add to 1 2 3 4 34 5 6 7 8 9 10 11 12 16 2 3 13 13 14 15 16 5 11 10 8 9 7 6 12 4 14 15 1

  13. What’s coming up next? 1 2 3 4 5 6 7 8 9 1 1 2 3 5 8 13 21 34 0 1 2 3 4 7 6 15 8

  14. At this stage in the exploring patterns progression, students are able to copy given elements in a pattern, work out the next element in the pattern, and show it in some way. Copy a pattern and create the next element At this stage in the exploring patterns progression, students are able to use systematic counting to continue a pattern. This allows them to work out the next element in the pattern more efficiently and accurately. Use a systematic approach to continue a pattern and find number values

  15. Algebra: Level 3 Achievement Objectives Exploring patterns and relationships Exploring equations and expressions • solve problems of the type:

  16. Algebra: Level 4 Achievement Objectives Exploring patterns and relationships • find and justify a word formula which represents a given practical situation; • solve simple linear equations such as: Exploring equations and expressions

  17. At this stage in the exploring patterns progression, students are able to recognise relationships between successive elements in a pattern and may be able to use a table to list values. Predict values using relationships between successive elements At this stage in the exploring patterns progression, students are able to explain a rule to predict the value of any given element in a pattern. They no longer need to rely on knowing the previous element to work out any given element. Predict values using rules

  18. Algebra: Level 5 Achievement Objectives Exploring patterns and relationships Exploring equations and expressions

  19. At this stage in the exploring patterns progression, students are able to state and use an algebraic expression for a relationship. They are able to use symbols and the variable "n" to express their rule. Find an algebraic expression for a relationship Students are now able to use equations for a pattern to solve a range of problems related to that pattern. For example, they might use equations to solve the problem: "If 53 red tiles are used, how many blue tiles are used?" Solve linear equations related to patterns

  20. Copy a pattern and create the next element Predict relationship values by continuing the pattern with systematic counting Predict relationship values using recursive methods e.g. table of values, numeric expression Predict relationship values using direct rules e.g.   ? x 3 + 1 Express a relationship using algebraic symbols with structural understanding e.g. m = 6f + 2 or m = 8 + 6(f – 1)

  21. http://arb.nzcer.org.nz/nzcer3/keywordm.htm Fill in the grid so that every row,every column, and every 3x3 boxcontains the digits 1 through 9. 6 f 1 a a d 4 e 5 8 h c 3 5 e 6 f b 2 1 a 8 h d 4 g 7 f 6 6 f 3 c 7 g 9 i 1 a 4 d e 5 2 b g 7 b 2 f 6 i 9 d 4 e 5 h 8 g 7

  22. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 2 2 4 4 6 6 8 8 10 1 3 12 14 5 7 16 18 9 3 3 6 6 9 9 3 12 15 6 9 18 21 3 6 24 27 9 4 4 8 8 12 3 16 7 20 2 24 6 28 1 32 5 36 9 5 5 10 1 6 15 2 20 7 25 30 3 8 35 4 40 45 9 6 6 12 3 18 9 24 6 30 3 36 9 6 42 48 3 9 54 7 7 14 5 3 21 1 28 35 8 42 6 4 49 56 2 63 9 8 8 16 7 6 24 5 32 40 4 48 3 2 56 1 64 9 72 9 9 18 9 9 27 9 36 9 45 9 54 63 9 9 72 81 9 ·Vedic numbers and patterns – completing the multiplication table and converting to vedic then connect up all of the number 1’s with lines OR 5’s etc to link with the GEOMETRY and NUMBER strand.

  23. Algebra: Number • Algebra allows analysis and descriptions of number patterns. 2, 4, 6, 8, …….. 99, 96, 93, 90, …… 1, 4, 9, 16, …..

  24. Algebra: Measurement • Algebra allows analysis and descriptions of relationships between things we can measure. • e.g. height and weight, length of foot and hand span, how we feel over time, savings over time, • Algebra also provides formulae for common measurement problems. e.g. area = base x height

  25. Algebra: Geometry • Algebra allows investigation and descriptions of relationships between geometrical properties e.g. the number of sides of a polygon and the number of diagonal lines or the number of sides of a polygon and the number of degrees in its angles.

  26. Algebra: Statistics • Algebra allows analysis and descriptions of data and graphs that show trends. • Category data (e.g. favourite breakfast cereal, eye colour, languages we speak) is often unable to be described in terms of algebraic relationships.

  27. Algebra: Mathematical Processes • Mathematical Processes - the A.O.’s link closely with the Algebra A.O.’s. e.g. use words and symbols to describe and continue patterns, record and talk about the results of mathematical exploration, interpret information and results in context

  28. http://www.nzmaths.co.nz/algebra/Background.htm A good platform for students to move forward to algebra is provided by a sound knowledge of the properties of number and of the four basic operations. • understanding equality; • understanding operations; • using a wide range of numbers; • describing patterns.

  29. What is Jane’s pattern? Jane started adding up odd numbers. She got a surprise when she realised what pattern she was making. • What was the pattern?  • Can you show why Jane’s pattern works using counters or some other method? http://www.nzmaths.co.nz/PS/L5/Algebra/SquareandTriNos.htm

  30. Magic squares = L a b c = L = L d e f g h i = L = L = L = L = L (a + e + i) + (g + e + c) + (d + e + f) = 3L Rearranging a + d + g + 3e + c + f + i = 3L L + 3e + L = 3L 3e = L

  31. e + d – 2a e + a e – d + a e + d e e – d e – a e – d + 2a e + d - a Choose a value for the variables e, a and d e.g. e = 15, a = 1 and d = 5

  32. 11 18 16 20 15 10 14 12 19

  33. Patterning 1 = 2 + 3 – 4 2 = 3 + 4 – 5 3 = 4 + 5 – 6 4 = 5 + 6 – 7 …….. n = (n + 1) + ( n + 2) - ( n + 3) generalising or = + 1 + ( + 2) - ( + 3) Using lego blocks to get

  34. generate patterns from a structured situation, find a rule for the general term, and express it in words and symbols; (paper folding, choose a number, Kava bowl, number chants) • generate a pattern from a rule; • sketch and interpret graphs which represent everyday situations; (a race, reading, flagpole…) • graph linear rules and interpret the slope and intercepts on an integer co-ordinate system. (exposition if time)

  35. evaluate linear expressions by substitution; • solve linear equations; (jelly beans, rod representations) • combine like terms in algebraic expressions; • simplify algebraic fractions; (exposition if time) • factorise and expand algebraic expressions; • use equations to represent practical situations. (tea leaves, lightning, power prices, cooking a roast…)

  36. Fold number Sections Creases 0 1 0 1 2 1 2 4 3 3 8 7 4 16 15 5 32 31 6 64 63 7 128 127 8 256 255 9 512 511 …… …… ……. n 2n 2n - 1 Folding a strip of paper:  What about if the paper was folded into 1/3‘s

  37. choose a number N N + 4 2N + 8 2N + 4 N + 2 2 add 4 multiply by 2 subtract 4 divide by 2 subtract the number you started with and the answer is …

  38. Kava Bowl (Samoa) Pattern(two-variable relationships)

  39. 1 1, 2, 1 1, 2, 3, 2, 1 1, 2, 3, 4, 3, 2, 1 1, 2, 3, 4, 5, 4, 3, 2, 1

  40. What patterns could we analyse from this number chant?

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