Mental Shortcuts - Intermediate

Mathematics is much more than the tedious, boring memorization of formulas. A mathematician plays around with the equation, recognizes facts and principles seen previously, and builds up on what he/she already knows in order to tackle the great unknown. By simply playing, we are becoming a mathematician.

You are never too young (or old) to start learning the fundamentals of being a mathematician. Even if all you have is basic knowledge of the order of operations, you can still brainstorm different ideas and suggestions for dealing with potentially nasty calculations. Play around with the numbers, move them about, and tweak them slightly. By identifying patterns, we are able to create new methods to solve problems that have never been seen before!

Here are some tips to get you started:

1. Apply the distributive property in creative ways.
2. Recognize common identities that you have seen in the past, and use them.
3. When you learn a new formula, think about why it works and explain it to someone else.
4. When you spot a pattern, figure out why it exists.

Using the perfect square identity, evaluate \( 81^2. \) \[\]

Hint: \( (a+b)^2 = (a+b) \times ( a +b) \).

In this case, we're already given the hint of an identity to use.

Letting \( a = 80\) and \(b = 1 \), we get

\[ 81^2 = (80+1)\times (80+1) = 80^2 + 80 \times 1 + 1 \times 80 + 1 ^ 2 = 6400 + 80 + 80 + 1 = 6561. \ _\square \]

Back to Quiz: Mental Shortcuts - Basic

In Basic Mathematics, you will learn about:

• Recognizing Patterns: This is the bread and butter of being a mathematician. A quick recognition of patterns allows you to form hypothesis and ideas, paving a route to attacking the problem.
• Order of Operations: Basic arithmetic skills that shouldn't be forgotten. Doing this quickly will speed up your thought processes on harder problems.
• Fractions, Decimals and Percentages: Relating these concepts with each other allows you to gain a deeper understanding of each of them. Drawing linkages between "different" areas is at the heart of mathematics.
• Angles: An introduction to geometry, that helps you recognize patterns in shapes.