Transcript Van rekenen naar algebra

Positive Algebra
From arithmetic to algebra
Jaap den Hertog
Freudenthal Instituut
Universiteit Utrecht
[email protected]
“I
used to be good at arithemetic, but
now I don’t understand anything
anymore.”
Counting in primary school grows into
advanced and more sophisticated counting
You cannot maintain what you never learned
When do you use your calculator?
Continuous learning trajectories
To introduce negative numbers and to use
them
Knowledge about fractions as a preparation
to working with algebraic expressions
Rules, patterns, structures
27 – 38 = ….?
A pattern
5 × -3 = -15
4 × -3 = -12
3 × -3 = -9
2 × -3 = -6
1 × -3 = -3
0 × -3 = 0
-1 × -3 = 3
-2 × -3 = 6
always 3 more
What is the power of algebra?
Reasoning and generalizing: is it always?
Are you sure? Is it certain?
Not only knowledge of (f.e. number system)
but also knowledge about
Development of thinking models
A continous learning trajectory
Developing a fraction language
Reasoned divide
Perform operations within the context
To relate ‘Part of’ to multiplication
Towards the development of routine
procedures
Fractions on the number line
And what is next …?
Two thirds of 4500
2/3 times 4500
2
3 × 4500
A learning process and struggles
 π/4; 1/4π; π ÷ 4; they are all the same, but different
 Add up the same number with the nominator and
the denonminator
 You divide a number and the result is larger. Why?
 Add up the nominators and the denominators. Is the
new fraction bigger or smaller than the sum of the
fractions?
 Is there a smallest fraction greater than zero?
 How is the number system extended?
1
--5
4
--5
2
--5
3
--5
A square of 1 bij1. Write the area of each piece as
a fraction and add up.
When is formal arithmetic with
letter fractions introduced?
For which students is it important?
In which grade do we start?
What are the preparations for the students?
Which formula is equivalent with…
2
3
y 
x x 1
x  3 x 1


2
5
x3 5


2
x 1
5( x  3)
2( x  1)
Are there more examples?
Is there a formula?
1
1
4  1  4 1
3
3
Simplify fractions
1 a
2a
Reasoning with formulas
Adjust / prepare formulas yourself
Discus the effect of changes in variables and /
or numbers
Recommended maximum heart rate
For years, the following formula was used:
Maximum heart rate = 220 – age
Who has a higher maximum heart rate,
someone in your class or one of the teachers?
Recommended heart rate
Recently the formula has been changed
Maximum heart rate = 208 - (0.7 x age)
What are the consequences of using this
formula: is your heart rate higher or lower
than the recommended rate?
Summary
 Continuous learning trajectories from Primary school
and Secondary school
 Introducing negative numbers in primary school, but
the formal operations in secondary school
 Fractions are not “ready” after the primary school
 Fractions in secondary school
 Do not avoid fractions in secondary education, but
also include letters
 Learning processes in developing and adapting
formulas