Mental Math

Students can sometimes procedurally solve arithmetic problems but have to use different skills when asked to compute 85 x 99 without a pencil.
The idea of 99 x 85 being equivalent to 100 x 85 - 1 x 85 is a different non-mechanical way of thinking for many students. The mechanics of Mathematics are important -- however, when used in conjunction with mental Math and clever but easier procedures, students are empowered and become more confident and self-reliant.

Is a function increasing or decreasing?
The advent of the graphing calculator has helped students and teachers understand and explain this material much more easily and clearly. The best example is to use a right side up parabola and ask the student verbally. When shown the Power Rule of Calculus at the same time, the student can then support his/her answer (or change it accordingly!!). The next task is show the contrast between the parabolic and the cubic so that the student can see that y = x^3 is ALWAYS increasing!!
Then we can revisit the idea of a line and how it has a constant rate of change -- all of these concepts can be well demonstrated by using the xy table on the graphing calculator.

When working with a tenth grader on imaginary numbers, we used the graphing calculator to study the contrast between y = x^2 - 4 (that has 2 zeros -- at 2 and -2) and y = x^2 + 4 that hovers up above the x-axis and has no real roots only imaginary. When students are able to visualize this difference and then tie it together with the quadratic formula (seeing that the discriminant b^2 - 4ac will result in a negative number for x^2 + 4 -- and that there is no sqrt of a negative number), they can then understand and process the concepts and procedures that are key to success in working with imaginary numbers.

Imaginary numbers are best studied and remembered by sequentially (and boringly) rewriting each component of the equation. For example,
sqrt(-4) x sqrt(-36)
sqrt 4 sqrt (-1) sqrt 36 sqrt(-1)
2 i 6 i
12 i^2
Then from up above drop in a -1 for i^2
12 (-1) =

An interesting Letter to the Editor regarding Math homework in the New York Times today.
Here is the text:

To the Editor:
I wonder if the educators cited in the article have ever tried to teach the New York City-mandated math curriculum to students who enter high school at a sixth-grade level. Perhaps gifted students can “chill” during vacations.
But the 98 percent of students who don’t attend Stuyvesant and the other specialized high schools need to catch up — and homework is one of the few tools we teachers control that can help us provide individualized instruction.
We either raise the level of homework for these kids or lower our expectations of them — the choice is obvious.
Mitch Kurz
New York, April 4, 2007
The writer is a math teacher at the Bronx Center for Science and Math.

The Calculus in a freshman core terminal class!!

The idea of slope extended beyond the straight line

Contrast a line with a parabola graphically and with a table on the TI-84 and equation wise (algebraically) so that students can see that the rate of change for a nonlinear function changes in a nonconstant manner)

Show the slope of line is the "m"

Write the slope formula using dy/dx or as delta y / delta x (using the Greek letter delta triangle thingie)

Sixth grade homework can sometimes look like this: 5/12 divided by 5/9. I wish fractions problems would use easier numbers so that the student can feel if their answer makes sense. I like to use 12 divided by 1/2 or perhaps the more difficult problem used by Liping Ma and/or Deborah Ball : 1 3/4 divided by 1/2.

Students can build intuition by asking themselves if the answer would be more than 1 or less than 1 by using 3/12 or 12/3 and seeing which of these examples their problem resembles.


These fraction problems are usually also related to attention to detail and reading. One of the other problems was you have 12 yards of fabric and will use 2/3 of a yard for each item -- how many items can you make? The student will often perform the multiplication of12 x 2/3 without asking themselves the question: Can I make more than 12 items or less? Once they have answered that questions, then they have a more clear sense of what to do. 1/2 vs. 2 can be used here.

1/2 x 1/2 is a great example as students often add instead of multiply and do not feel that multiplication should make their answer smaller!!

For division of fractions, why do we invert and multiply? If the student just memorizes the procedure, they will not know which procedure to use. It might (definite maybe) be ok if students were able to memorize these concepts and procedures but they get all jumbled!! Students will better comprehend the material and have the ability to more accurately retrieve it if they rely upon examples rather than memorizing procedures.