Robert Shinn, Ph.D.
Reasons to Learn Algebra:
1. Symbolic manipulation
2. Graphical presentation
3. Analysis of (more) complex
situations
4. Multiple correct approaches
to a solution
5. Higher math and science
courses
All math is a tool – get a
good answer quickly!
Suzuki Method: Have the student learn the whole process with easy material: bring in the more difficult and special cases after skill and confidence are in place.
General Strategies
1. Games
2. Real-life examples
3. One point each day
4. Individualized explanations
5. Independent analysis and
practice
6. No calculators!
Ø There will be an exception
for graphics
Always hone number sense!
Four goals for each skill or
process:
1. Translation
Ø Converting situations into
mathematical symbols
2. Manipulation
Ø Performing logical and
mathematical operations
3. Interpretation
Ø Making real-world sense of
the answer(s)
4. Extension
Ø Building upon those
previously learned and providing a platform for future learning
If
it can not do these, why bother teaching it?
General Mathematical Skills
1.
Basic mathematics
operators (+, -, x, ¸)
2.
Factoring and factor
trees; prime numbers
3.
Fractions, operations
and simplification
4.
The number line!
5.
Basic equalities (& inequalities) graphed on it
6.
Operations with negative
numbers
7.
Exponents, in fractions
and simplifications
8.
Memorize perfect squares
(and square roots)
9.
Square root
manipulations
Ø Irrational numbers between
perfect squares
Ø Factor out perfect squares
from radical
10.
Bar graphs and other charts
11.
Measurements and conversions
12.
Recipes!
13.
Scientific notation and decimal manipulations
14.
Order of operations (PEMDAS)
15.
Basic geometric shapes and formulas
When
to begin algebra?
Ø After basic skills are
mastered
Ø After the brain can handle
symbolic logic and analysis (about 15)
10 Algebra Goals
1. Introduction of the variable
A letter (usually x)
replaces the simple blank
____ + 4 = 9
x + 4 = 9
Work with
bar graphs to build up this concept of a letter representing a number.
2. Setting up equations with variables
This has two distinct parts:
Ø An expression with an
operation being done to a variable (unknown)
Ø An equation of this to another
value
3. Solving simple equations & graphing
Ø What is done to one side is
done to both
Ø Object is to get variable by
itself on one side
Ø Order of operations reversed
to undo what has been done to the variable
Ø One operation at a time; one
operation per line
Ø Variable should equal a
numeric value
Ø Put a dot on this value on
the number line
Ø Trouble with two-step
equations indicates a lack of understanding of the process.
Example
problems:
One
step:
A) x + 3 = 7 B) x
– 2 = 3
C) 3x = 18 D) x
/ -1 = -7
Two
step:
E) 2x + 1 = 9 F) x
/ 2 – 3 = 1
Explain
cross-multiplication as a special-case shortcut here.
WARNING! If the child does not understand why x
equals different values for different problems, STOP! The child has not matured
enough to handle symbolic logic of this sort.
Review
other skills and wait.
4.
Two variable equations
Concept: one value changes in response to changes in
another
Ø Be sure to start with
examples! (If a cookie recipe will make
20 cookies with one egg, and I have 3 eggs, how many cookies can I make? With 4
eggs? With 5 eggs? With zero eggs? With negative one egg?)
Ø Usually best to stick to x
and y at this point
Ø Setting up equations is the
same as earlier, EXCEPT the “equals a value” becomes “equals y”
Ø DO NOT INTRODUCE EQUATIONS
WHERE y IS NOT ALONE ON ONE SIDE UNTIL THESE AND GRAPHING SKILLS ARE
MASTERED!
5.
Solving two variable equations
Concept: These have an entire battery
of answers.
Ø They are much more like
geometric formulas.
Ø Start with problems that
give the independent variable.
Ø Younger learners will need
more help selecting values.
Example: y = 2x – 1
|
x |
y |
|
-1 |
|
|
0 |
|
|
1 |
|
|
2 |
|
A
table such as this keeps work tidy, introduces order, eliminates confusion, and
sets up for the concept of coordinate pairs for graphs.
6.
Graphing on Cartesian coordinates
Ø These are like two number
lines at right angles.
Ø Convert table values to
coordinate pairs in parentheses
Ø Emphasize x-axis is the old
number line
Ø Emphasize y-axis is elevation
in relation to it
Ø First over, second up or
down to plot each point
Ø Connect the dots for a
straight line
Ø Play Battleship
Ø Other games with
connect-the-dots as coordinate pairs
Ø Make a floor plan on graph
paper, measure distances and find areas by calculating and counting.
7.
Slopes and intercepts
Ø All linear equations will
have both
Ø And fit the formula y = mx +
b
Ø Slope = rise / run
Ø Build some ramps and
calculate the slopes
Ø Figure slopes from linear
graphs
Ø Construct equations from
graphs
Ø Give examples of usefulness
of intercepts (fixed costs in a business…)
If
you are going to cover systems of equations, now is the time to do so, after
all the above has been comprehended.
There are games similar to Battleship where lines are “torpedoes” and
the players must calculate to see if they have been hit (intersected); their
position is either a point or line.
Do
not forget to include that parallel lines have identical slopes; otherwise they
will intersect!
Slopes
of perpendicular lines are negative reciprocals.
8.
Quadratic equations
Do not cover this one until the preceding goals have been mastered. There are too many “exceptions.”
Ø This graph is curved, not
straight
Ø Now the “correct” answers
are the x-axis intercepts
Ø Many more points need to be
plotted
Ø Preselected values may not
work
Ø A graphing calculator can
often enhance understanding at this point
Ø Orders of equations come
into play
Ø Increased level of variable
manipulation
Ø Several different
techniques:
§ Factoring
§ Completing the square
§ Quadratic formula
Each
technique should be pursued to show how the manipulations work and what utility
it can have; these are not ends but means to grasping an understanding of the
process.
As
such, when the student understands the process sufficiently, any one technique
may be put aside. The point is for the
student to choose the technique he or she can best use in a given situation to
quickly determine the answer.
Understanding
how the processes work is the most important thing.
Best
examples:
Ø Newtonian physics
Ø Parabolic mirrors
Ø Arches (and old bridges)
Ø Building extensions
Please
do not use FOIL; it is not extensible.
9.
Inequalities
Handle this one very much as the equations previously studied.
Ø Start with one-variable
inequalities
Ø For one-variable
inequalities, keep x on the left
Ø Change sign if you change
sides! (But do not do this to the younger set for a while.)
Ø “Arrow” points in the
solution direction
Ø Use real-world examples
(recipes!)
Ø For two-variable
inequalities, keep y on the left
The
“big deal” with these is that multiplying or dividing by a negative number
“flips” the inequality sign around. If
the student tries to change the sign with ANY multiplication or division
operation, review the number line and negatives operations. If this does not clear matters up, you may
have to wait on more cognitive development.
10.
Graphing inequalities
Ø Keep correct variable on the
left side
Ø In one variable, the “arrow
points the way”
Ø Introduce the idea of using
test points
Ø (Introduce the idea of
logical truth tables)
Ø Use this table for graphing
conventions:
|
|
>, < |
³, £, = |
|
One
variable |
Open
circle |
Closed
dot |
|
Two
variables |
Dashed
line |
Solid
line |