Transcript Michaels corporsate master

National Numeracy Hui
Year ten topics:
• Pythagoras and right angled trig
• Algebra
A numeracy approach
Michael Drake
Victoria University of Wellington
College of Education
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
So what numeracy stage do you need
to be at to deal with:
Pythagoras?
Right angled trig?
Take a look at this one…
Which line is longest?
(a) the top
(b) the bottom
(c) they are both the same
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%
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
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.
Here is the Chelsea diagnostic version:
• How long is the line in centimetres?
cm
1
2
3
4
5
6
7
8
The answer 7 occurred thus:
First year
Second year
Third year
46%
30%
23%
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?
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!
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
Algebra
• What is algebra?
Think…
Discuss in pairs…
• If you ask students (average year
11 types), what would they say?
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?
When is a problem a number problem –
and when it is an algebra problem?
6 +  = 10
57 +  = 83
57 + x = 83
3.64 +  = 4½
• So when does algebra become something
that is important for students to know
and understand?
So how do we use letters in
mathematics?
1)
2)
3)
4)
5)
How do we use letters in
mathematics?
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
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
A letter is really a number
evaluate a + b if a = 2 and b = 3
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
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
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
Scenario
You have discovered that some students
in your class have wrong conceptions of
the equals sign
• How do you fix it?
• 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
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
Try this part of the puzzle…
• How many matches for 5 squares?
• How many matches for 100 squares?
• How many matches for n squares?
• 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?
Sorting out algebra in years 9
and 10
Year 9 emphasis
Year 10 emphasis
Letters for naming
Letters for naming
•Abbreviations
•Substitution into formulae
•Development of simple formulae •Algebraic substitution
•Units
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
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
Year 10
• Exploring algebra itself for what we can
learn.
For example: we have now met letters
(algebra) in a variety of situations – let’s
study them in more detail to see what
happens when we +/-/×/ them – like we
did for fractions, decimals, integers …
Starter Warmdown
(1)
Draw a picture to show that an odd
number plus an odd number always
gives you an even number
(2)
•
•
•
•
25 = 25
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 doesn’t always work?
(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