1. Winning at Math Series Set I Successful Homework
12/2/2018
Winning at Math Series Set I Successful Homework
LEARNING RESOURCE SERVICES
FLORIDA GULF COAST
UNIVERSITY
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

2. Sources
Some of the topics herein are adapted from the text, “Winning at Math” by Paul D. Nolting, Ph.D., Academic Success Press, Inc., 1997.
The author of this presentation, Robert Brownell, Ph.D., Applied Mathematics, University of Virginia, has over 30 years combined experience teaching university level mathematics and training engineers in industrial simulator design and operation.
“Hands on training”, a common term in industry, is a basic method used to teach mathematics and engineering concepts in both industry and academia.
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

3. Dreading a math exam. Lost in a math course
Dreading a math exam ? Lost in a math course ? Feeling that it’s hopeless to catch up ?
You might be surprised at how many people feel the same way about math exams and math homework ...
Did you know that most people are not trained to take math exams or even to study mathematics?…
…and that it’s hard for most people to prepare for a math exam or do math homework ?
This study guide can help you prepare for an exam or do math homework more effectively, plus it can help you in other courses as well.
It does take persistence on your part, and it can produce good results quickly if you learn and practice the steps in this study guide.
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

4. Sounds good…, too good maybe ?
Most people who are successful at learning mathematics have developed ways to reduce the amount of rote memorization.
They get new facts into long-term memory efficiently.
When faced with a challenging math problem, they start from basic concepts and extend their knowledge to the larger problem.
You can learn to do math problems the same way and :
Improve your understanding of the basic concepts.
Commit concepts to long-term memory without rote memorization.
There is no secret here - the concept is not new. Stated simply:
Use simple, basic steps, one at a time, in your solution.
Write down each step in words and symbols as you work, using as many pathways in the brain as possible.
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

5. Use as many pathways in the brain as possible as you learn -------------------
This means you use as many senses as possible while you work….
By writing each step in words
By writing each step in symbols
By seeing symbols and words together for each step
By speaking the words as you write them
That pretty much uses the hands, the eyes, the ears, eye-hand coordination, and speech all at once!
Learning tests have shown that using all of the senses can increase learning efficiency from less than 25% to more than 75% --- plus it helps keep you alert.
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

6. PLUS ---
Writing steps gives you a written record of each type of problem, which you can use at any time for exam review, or for a reference from which to work other problems.
If you get stuck at a step in a problem, your written work will show just where the problem is. A teacher, tutor, or friend can then give fast, efficient help.
You might call this method “HANDS ON LEARNING” because you use your hands, but there is more to it ...
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

7. “Hands On Learning” - Learn by doing it yourself, with your own hands
“Hands on Training” has long been the choice in industry, because it is very effective.
The trainee walks through the procedure himself...”hands on”
Each basic step of the training problem is written down, checked off a list, consciously anticipated, and physically done.
No steps are skipped. None are done “in the head”
Memory is not trusted. The training procedure shows all basic steps and results.
Visualize training for landing a 747 airplane. Would you skip any steps?
“Hands on Training or Learning” engages the trainee
Uses the physical senses and the communication pathways of the brain all at the same time. That is why it is so effective.
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

8. “Hands on Learning” used in Preparation for Math Exams and Math Homework
The following are examples of math problem solutions which
use this method.
Remember, take small basic steps; write down each step completely.
DO NOT try to memorize, but DO re-read each step to understand it before you go on to the next step.
The step-by-step method is what counts, not your IQ !
This may seem tedious, but it ----
reinforces learning significantly
saves time in the long-run
increases your accuracy
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

9. Example 1 Mixed number to an improper fraction
Problem: Convert the mixed number 3 to an improper fraction
Step 1: Multiply the whole number by the denominator of the fractional portion
3 • 4 = 12
Step 2: Add the numerator of the fractional portion to the result
= 13
Step 3: Replace the numerator of the fractional portion of the original mixed number with the new result, and remove the whole number 13 4
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

10. Example 2 Linear relationships in a word problem
The relationship between Centigrade and Fahrenheit is a linear relationship.
The Centigrade temperature scale was designed deliberately so that boiling would occur at 100 degrees Centigrade and freezing would occur at 0 degrees Centigrade.
On the Fahrenheit temperature scale, boiling occurs at 212 degrees Fahrenheit and freezing occurs at 32 degrees Fahrenheit .
Problem: Determine the relation between Fahrenheit, F, and Centigrade, C.
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

11. Example 2 page 2
Your solution notes do not have to be just like these. Simply write down your reasoning clearly for each step of your solution.
Step 1: Since C and F are linearly related, their relationship must satisfy
the slope-intercept equation for a straight line y = mx + b
the two-point equation for a straight line y - y1 = (x - x1)
The problem gives two values for freezing and two values for boiling. These values determine the two points (x1, y1) and (x2, y2).
Rename the variables. Let x = C and y = F. Then
(freezing) x1 = C1 = (boiling) x2 = C2 = 100
y1 = F1 = y2 = F2 = 212
(y2 - y1 )
____
(x2 - x1 )
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

12. Example 2 page 3 ____ ____ _____
The two-point equation can be used, since the values are known:
y - y1 = (x - x1) becomes F - F1 = (C - C1)
Step 2: Replace the variables with their corresponding values
F - 32 = (C - 0)
Step 3: Simplify the equation
F - 32 = C (subtract inside parentheses)
F - 32 = C (fraction reduces by a common factor of 20)
(y2 - y1 )
____
(F2 - F1 )
____
(x2 - x1 )
(C2 - C1 )
( )
_____
( )
180
100 9 5
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

13. Example 2 page 4
Step 4: Solve for either F or C. The original problem did not specify. It is easier, in this problem, to solve for F.
F = C (add 32 to both sides)
With the solution written down in words and in symbols, you will have a permanent reference for future problems. You will be able to do a quick run-through as many times as you want before an exam. 9 5
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

14. Example 3 Solve a system of equations
Your solution notes do not have to be just like these. Simply write down your reasoning clearly for each step of your solution.
Problem: Solve the system of equations x + y = 18 (Eq 1)
= 2 (Eq 2)
Step 1: Multiply both sides of (Eq 2) by x to clear the fraction and solve for y in terms of x
y = 2x
Step 2: Substitute this value for y into (Eq 1)
x + 2x = 18 { y x
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

15. Example 3 page 2 Step 3: Add the x terms together x + 2x = 18 3x = 18
Step 4: Divide both sides of the equation by 3 to solve for x
x = 6
Step 5: Substitute the value of 6 for x in (Eq 1)
x + y = 18
6 + y = 18
Step 6: Subtract 6 from both sides to solve for the corresponding value of y
y = The solution is the ordered pair (6, 12)
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

16. Example 4 Solve a quadratic
12/2/2018
Example 4 Solve a quadratic
Your solution notes do not have to be just like these. Simply write down your reasoning clearly for each step of your solution.
Problem: Solve for y as a function of x in the relation = 1
Step 1: Isolate the term with y by subtracting the term with x from both sides of the equation = 1 -
Step 2: Clear the y-term fraction by multiplying both sides of the equation by 9
y2 = 9( ) __ x2 __ y2 4 9 __ y2 __ x2 9 4
This equation represents and ellipse. The first move, getting the y terms on one side of the equation and the x terms on the other side is chosen because there is only one term in y and it is a quadratic term.
Since it is possible to solve a second order equation using the quadratic formula, it is possible to solve all relationships which are quadratic in y.
(second and first powers of y ). _ x2 4
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

17. Example 4 page 2      _ _ _ _ _
Step 3: Take the square root of both sides of the equation
y = ( )
Step 4: Separate the two answers
y = ( ) and y = ( )
Step 5: Simplify under the radical
y = 3 ( ) and y = ( ) _ x2  4 _ _  x2  x2 4 4 _ _  x2  x2 4 4
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.

18. “Hands on Learning” emphasizes :
Enhancing your thinking with your physical senses as you study.
Using a methodical procedure to focus your thinking on the mathematical method of each step.
Using small basic steps to encourage steady success toward the solution and to increase learning of basic math skills.
Documenting each step clearly for understanding and for future reference.
Reducing rote memorization while increasing long-term retention of basic math skills.
Adapted from "Winnning at Math", Paul D. Nolting, Ph. D.