Michaels corporsate master

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National Numeracy Hui

• Year ten topics
• Pythagoras and right angled trig
• Algebra
• A numeracy approach

Michael Drake Victoria University of Wellington
College of Education
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Something to think about

• What do you need to know if you are to learn
  Pythagoras Theorem and right angled trigonometry
  with understanding?
• Discuss in groups of 3 4

3

• So what numeracy stage do you need to be at to
  deal with

Pythagoras?
Right angled trig?
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Take a look at this one
Which line is longest? (a) the top (b) the
bottom (c) they are both the same
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Predict how you think students will answer

• Results
• Lots of students think they are the same length

First year Second year Third year
51 50 46
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• By the way these were English secondary school
  students, so

year 1 students were 11 12 year olds year 2
students were 12 13 year olds year 3 students
were 13 14 year olds Hart, K. (1978).
Mistakes in mathematics. In Mathematics
teaching. Number 85, December 1978, 38-41
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The broken ruler problem

• This problem can take several forms one is to
  measure an item with a broken ruler, the other is
  to measure the length of an object where the
  object is not aligned with zero.

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• Here is the Chelsea diagnostic version
• How long is the line in centimetres?

The answer 7 occurred thus
First year Second year Third year
46 30 23
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• 7 can be worked out through
• starting at one when measuring
• failing to consider the distance between the
  start and the finish point (just looking at the
  highest number reached)

What are the implications of what you have just
seen for the teaching of Pythagoras and right
angled trig?
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Implications

• There are a lot of things we take for granted
  when we teach. The basics need to be covered for
  a lot of students, and the basics may not be what
  we think they are!

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Developing similarity

• From a numeracy approach, new ideas are based on
  developed understanding. As similarity underpins
  right angled trig, it seems sensible to spend
  time developing this first

• The second activity (II similar) extends the
  understanding of similarity to that of
  calculating a fraction (the scale factor of
  enlargement) from lengths of comparable sides

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Algebra

• What is algebra?

Think
Discuss in pairs

• If you ask students (average year 11 types), what
  would they say?

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How would you average year 11 student solve this
problem?

• Sian has 2 packs of sweets, each with the same
  number of sweets. She eats 6 sweets and has 14
  left. How many sweets are in a pack?

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When is a problem a number problem and when it
is an algebra problem?
6 ? 10
57 ? 83
57 x 83
3.64 ? 4½
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• So when does algebra become something that is
  important for students to know and understand?

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• So how do we use letters in
• mathematics?

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How do we use letters in mathematics?

• 1) A letter can be used to name something
• In the formula for the area of a rectangle,
  the base of the rectangle is often named b
• 2) A letter can be used to stand for a specific
  unknown number that needs to be found
• In a triangle, x is often used for the angle
  students need to find
• 3) A letter is really a number
• evaluate a b if a 2 and b 3
• 4) A letter can be used as a variable that can
  take a variety of possible values
• In the sequence, (n, 2n - 1), n takes on the
• values of the natural numbers sequentially

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Developing an understanding of symbols

• A number of pieces of research indicate students
  have difficulty with understanding the equals sign

Falkner, Levi Carpenter (1999). Cited in
carpenter, Franke Levi (2003). Thinking
mathematically integrating arithmetic and
algebra in elementary school. Portsmouth, NH
Heinemann 8 4 ? 5
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Results
Grade 7 12 17 12 17
1 2 5 58 13 8
3 4 9 49 25 10
5 6 2 76 21 2

• Note that years 5 6 are worse than the years 3
  4

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Scenario

• You have discovered that some students in your
  class have wrong conceptions of the equals sign

• How do you fix it?

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• Children may cling tenaciously to the conceptions
  they have formed about how the equals sign should
  be used, and simply explaining the correct use of
  the symbol is not sufficient to convince most
  children to abandon their prior conceptions and
  adopt the accepted use of the equals sign

Carpenter Franke Levi (2003), p. 12
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Dealing with misconceptions

• If a student has a misconception it must be
  challenged

• Try introducing problems for discussion. What
  different conceptions exist, and need to be
  resolved?

• This forces students to articulate beliefs that
  are often left unstated and implicit. Students
  must justify their principles in a way that
  convinces others

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Try this part of the puzzle

• How many matches for 5 squares?

• How many matches for 100 squares?

• How many matches for n squares?

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• Did you notice that Hamish changed his solution
  method when dealing with n sides?

• What methods from the flip chart could he
  describe (or write) without knowing the
  conventions of algebraic notation?

• So what algebra does a student need to know
  before they can record these statements correctly?

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Sorting out algebra in years 9 and 10
Year 9 emphasis Year 10 emphasis
Letters for naming Abbreviations Development of simple formulae Units Letters for naming Substitution into formulae Algebraic substitution
Letters as specific unknowns Box equations (letter equations for brighter students) or word problems that can be solved mentally Letters as specific unknowns More formal methods based on knowledge of the equals sign as a balance
Letters as a variable Patterning Generalisation from number Letters as a variable Graphical functions Manipulation
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Year 9

• Introduction to generalising thinking beyond
  the getting the answer
• Learning to express mathematical ideas with
  symbols learning the language of mathematics
• Learning the conventions of symbol sentences
• Learning the different meanings of letters

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Year 10

• Exploring algebra itself for what we can learn.
• For example we have now met letters (algebra)
  in a variety of situations lets study them in
  more detail to see what happens when we /-//?
  them like we did for fractions, decimals,
  integers

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Starter Warmdown

• (1) Draw a picture to show that an odd number
  plus an odd number always gives you an even
  number

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25 25
(2)
Show that 25 36 36 25

• Will this always work?

• How do you know?

• If you think yes can you write a sentence that
  shows it will always work, regardless of the
  number you start with, and the number you add.

• If you think no how can you prove it doesnt
  always work?

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• (3) What is 4 ? 4?

• What about 16 ? 16?

• 250 ? 250?

• Does this always work?

• Explain why this works using a drawing or some
  equipment from the box

• Can you write this as a rule in words?

• Write a sentence with symbols to show your rule