The exactness of an answer weighs on all math students. Namely, the significance of a right or wrong answer on a math exam is critical. Not only does the student's overall grade take a hit from a poor score, but also their future expectations takes a hit, as well, eroding future confidence for better grades.

Studies show that poor scores today can erode a student's confidence in earning higher scores in the future. When students score poorly, they build an all-too-common monologue, "math is too hard, "I'm not good in math," and even worse, "I should avoid math in my future at all costs."

Consider a student who makes the smallest of mistakes when solving a math problem. Most recently, I was working with a student who fully understood how to solve a particular math problem -- her numeric answer fell within one one-hundredth of the correct value. "WRONG," came back the response. What a waste of my student's time. Her immediate response was simply, "I hate math."

Any other class in secondary education, including English, history, science, biology, and Spanish, award almost perfect answers with a response, "Correct." But that is not the case in math. If the test specifications call for answers within one one-hundredth of a value, then by golly your answer is WRONG until you find the more precise value expected.

A strategic question is a question that asks about a higher-level concept regarding a problems. Questions like what problems are similar to the one asked? How are these problems the same? How are they different.

Arranging concepts (sorting) is equally useful. Solving math problems often entails following a sequence of steps. Knowing those steps and when they apply is a strong example of a strategic question.

In general, asking these higher-level questions requires a student respond to a question in the form of sets, ordered and non-ordered.

Hypothesis. Questions that have a student take action on ordered and non-ordered sets is a substantially better learning experience for the student than having this student suffer the exactness of questions that commonly align to true-false and multiple-choice style questions.

Hypothesis. Taking actions on set-style questions helps students learn strategies that promote their confidence in understanding. In particular, strategies are patterns. Recognize the pattern? Know what to look for in that pattern? Perfect, now you can solve the problem.

When factoring 2nd-degree equations, a student groups equations by their leading coefficient. After the student groups their equations, they apply the appropriate method (guess-and-check, the AC Method, or the Quadratic Formula) to solve their problem.

There is no doubt that using the quadratic formula to find a solution to a 2nd-degree equation works every time. However, using the quadratic formula is both time-consuming and error-prone. Most of the time, using guess-and-check -- if appropriate -- is quicker and less error-prone.

When finding the absolute minimum and absolute maximum values of a continuous function $f(x)$ over a closed interval $[a,b],$ there are several steps required.

1. Take the first derivative of $f(x).$ Find $f'(x) = \ddx f(x).$

2. Make a table TABLE with two columns, labeled $x$ and $f(x),$ respectively.

3. As your first two entries in TABLE, write in the values of the two end-points $a$ and $b$ under the first column.

4. Solve for all values of $x$ where either $f'(x) = 0$ or $f'(x)$ is undefined.

5. As your next entries in TABLE, write in the values of each $x$ from Step 3 under the first column.

6. For each value of $x$ in TABLE, calculate the value of $f(x).$ Write these values into their proper row under column for each value of $x$ from Step 3. Write these values to their corresponding row in TABLE, forming a list of 2-tuples $\lang x, \ f(x) \rang$ for each $x$ from Step 3.

7. Finally, write TABLE, include the pairs of values $<a, f(a)$ and $<b,f(b)$ for the two end-points of the closed interval.

8. Choose the absolute minimum value of $f(x)$ as $n$ for all $x \in [a,b]$ where $n$ is the least value in the table.

Normally, when students find the minimum and maximum values of a function, they simply find the first derivative of their function, set this derivative to zero, and then solve for values of $x$ where $f(x)=0.$

However, when it comes to finding the absolute minimum and absolute maximum value of a function, the student must be more methodical in their approach and solution. When asked about absolute minimum and maximum values on the AP Exam, the graders expect a table of values, as described here.