IV. Factoring
After teaching all the various techniques of factoring it
is extremely beneficial to spend a day in class creating a flow chart of
factoring techniques. Sub titles would include
A) Look for the common factor, factor it out. If the lead
coefficient is negative , factor out the negative
B) If the expression has two terms : is it difference of
squares? Sum or difference of cubes? Are you done?
C) If the expression has 3 terms: is it a perfect square
trinomial? Is the lead coefficient one? If necessary use the AC or grouping
method
E) If the expression has 4 or more terms factor by
grouping. Remember that you can group in a variety of ways.
F) An interesting way to check whether a trinomial of
degree two is factorable is to calculate b squared – 4ac. If the result if a
perfect square the trinomial is factorable. If it isn’t, it isn’t. Talk about
the “Why” with your class.
V. Work Problems:
When students see how one solution method can be applied
to a variety of problems, they are getting the “Big Picture”. When this happens
in your classroom it can be very exciting. Distance, rate, time; work and simple
interest problems all can be solved using a similar technique.
A) John mows the lawn at Wrigley field in 4 hours. Mary
can do the job in 5 hours. Working together how long will it take to mow the
lawn. Assume that their rates remain constant. (Talk about this assumption with
the class. Is it valid?) John’s rate of work is ΒΌ Th of the job per hour. Mary’s
rate of work is 1/5th of the job per hour. Using D= rt + rt set up the equation
1 (1lawn) = (1/4) t + (1/5) t.
Solve the equation. t =20/9 hour or 2 hours and about 13 minutes.
B) Now have fun with the problem. Have Mary come late or
leave early. Have John work for 45 minutes and then leave because he had a fight
with Mary.
C) Make John and Mary house painters. Suppose that they
are painting 40 identical living rooms. Set the rates for John and Mary. Now
determine how long it would take to paint 40 living rooms
D) Extend the problem into rates of rowing or biking or
walking. Find total interest earned by 3 different accounts with different rates
of interest and different principals. Connect all the problems together. Help
students see the “Big Picture”.
VI. Explain concepts in a concrete way – especially to
Math 9 and Math 1 classes.
A) Suppose that you want to show place value and borrowing
for the example
300 – 12. Talk about the problem in terms of money. (This promotes immediate
interest!) You have 3 hundred dollar bills and must make change. Talk the class
through the solution. You have made borrowing real.
B) We can multiply as a shortcut for repeated addition.
2+2+2 = 6 is 3(2) =6. Then a+a+a = three a’s = 3a
C) Five minus two equals three: 5-2=3. Five y’s minus 2
y’s is 3 y’s. 5y-2y=3y.
D) 4x + 5x+9y = 4 bananas + 5 bananas + 9 apples= 9
bananas + 9 apples.
E) 3(x+4) = (x+4) + (x+4) +(x+4) = (x+x+x) + (4+4+4) =
3x+12
F) Students who fail math courses do not understand
concepts, they memorize everything. They cannot transition between the concrete
and the abstract. They lack study skills and have math anxiety.
VII Simplifying rational expressions
A) If a variable or constant lacks a visible exponent, put
one (1)!
B) If a variable has a negative exponent , move it to where
it is not. (Numerator moves to denominator and vice versa). Apply this to
fractions that have negative exponents. It is beneficial to show students why
you multiply by the reciprocal when you divide by a fraction . (Many students do
the process without thinking at all about what they are doing) Use this example
as you do compound fraction problems (I call those “4 story problems with
interesting views”)
C) When simplifying a radical, factor the radical into two
separate parts. The first radical should be the” Perfect power pile ” (or ppp)
and the second radical should be the ”Left over “(or lo). Explain that x to the
nth power is a perfect square if n is divisible by 2, a perfect cube if n is
divisible by three, a perfect 4th power if n is divisible by 4 etc. Then I
create a list on the board of perfect squares (up to20 squared), perfect cubes
and 4th and 5th powers. I tell the students that they can write down the
perfect powers on an index card. I suggest that they have this card out as they
simplify radicals. I stress that
the only divisors that are important when you simplify radicals are perfect
power divisors. I suggest that in simplifying a fairly large number they should
first cut the number in half. I explain that any divisor would have to divide
the number into at least two parts. Then go backwards from that half-way point,
checking for exact divisors of the radicand. I let my Math 1 and 3 students use
the index card on their exam.
D) When rationalizing the denominator for a problem such
as 1 divided by the 4th root of x, you can use the “Fill Er Up” technique. Draw
a rectangular house that has 4 floors. Point out that the first floor is full of
x. Ask the class how many more x are needed to “Fill Er Up”. Expand the concept
to a problem such as 5 divided by the cube root of y squared and the 5th root
of z cubed. Draw a tower for your y’s. It will have 3 floors, 2 of which have
y’s .Draw a tower for your z’s. It will have 5 floors, three of which have z’s.
Now “Fill Er Up”.
VIII Absolute value equations and inequalities
A) As you introduce absolute value draw pictures on a
number line. For example the absolute value of x = 5 is asking for the location
of any and all points that are 5 units from 0 on the number line. The absolute
value of the quantity (x-6) = 8 is asking for the location of any and all points
that are 8 units from 6 on the number line. Make the concept real by letting
your visual learners see the solution.
B) Kathy Fink shared a fun way to describe the absolute
value of x is more than some number (say 10). She told her students to think of
x as the bad husband and 0 as the home of the good wife. (Feel free to
interchange husband and wife – or use your favorite movie star and paparazzi)
The absolute inequality is equivalent to a restraining order telling x (the
husband) to stay more than 10 miles away from home and wife. Another analogy is
to suppose that a bomb is dropped on a particular location (call it zero).
Depending upon the strength of the bomb everything within a certain radius (say
1 mile) would be destroyed. You could say that you would be safe if the absolute
value of x (your location) is greater than one.
C) To make the concept of less than or equal to real use
the “Stay Close To Home” analogy. Consider a mother telling her child that she
must say on the street (the number line) and go no more than 6 houses away in
either direction. This problem is equivalent to the absolute value of x is less
than or equal to 6.
D) You can teach absolute value inequalities in a variety
of ways. One method that I have found successful is to group together the
solutions of absolute value equations and inequalities. Begin your solution by
always solving the problem as if it were an absolute value equation. Stress to
students that they should expect 2 answers in most situations. (Take a moment to
talk about the exceptions to this rule). Then, if the question is an inequality,
the solutions that you just found are the boundaries of your solution set. Talk
about whether the boundaries themselves solve the inequality. Plot these
boundaries on a number line. Test a value larger than the largest boundary on
the number line to see whether the inequality is true. Continue testing areas
between boundaries from right to left on the number line. Write your solution in
interval notation. This technique is another example of creating a unifying
concept which helps students understand the “Big Picture”.
