The four roots of algebra are:
- Generalised arithmetic
- Expressing generality
- Possibilities and Constraints
- Rearranging and Manipulating
Some tasks to uncover misconceptions:
Invite a class to write down an arithmetic expression such as 13=3*4+1 and then rearrange it without actually performing the operations. Calculators will be most helpful for discussions about whether a rearrangement is correct.
and 3 can be isolated,
²Engage the class or group in discussion about how to write down the rearrangement. You can expect to expose all sorts of weaknesses and confusions about the four operations and their rules!
When a class has caught onto the idea, get pairs of pupils to produce arithmetic all expressions and then challenge other pairs to rearrange them with a particular number isolated on the left-hand side. often it is the case that a number which appears more than once cannot be isolated. Can the class come to any conclusions about when a number can be isolated?
Here are 3 consecutive numbers: 10,11,12
Multiply the first and third numbers: 10 *12=
Now square the middle number: 11*11=
What is the difference between the numbers?
Does this always work for consecutive whole numbers?
(A formal algebraic solution to this problem is:
Let x, x+1,x+2 represent three consecutive numbers.
I want to prove that the difference between the square of the Middle number and the product of the first and third, is one. In symbols, I want (x+1)²-x(x+2)=1
Express algebraically the statement that “There are three feet in a yard.”
One probable response is 3f=1y (f=feet and y=yards). This conveys the sense of the relationship between feet and yards, but it isn’t clear exactly what the letters f and y stand for.
The annotation would be a shorthand rather than an algebraic statement. As the f and y are being seen as units (feet and yards) rather than as generalised numbers.
When the letters are used in a more algebraic sense the appropriate formulation becomes:
3y = 1f (three times the number of yards…)
many children never properly grasp how a letter can be used to represent a generalised number. They tend to think of 2a + 3b as 2 apples plus 3 bananas, using the letters as units of measure rather than things which can take on different values ( twice the number of apples…)
Question a) If a + b = 43, a + b + 2 =
Question b) If e + f = 8, e + f + g =