To multiply a number by four, double it and then double it again! Remember to disregard any decimal points or zeros when starting the calculation.

Example 1: 32 x 4
Step 1: Double the 32: 32 x 2 = 64
Step 2: Double the 64: 64 x 2 = 128 (the answer)

Example 2: 18 x 4
Step 1: Double the 18: 18 x 2 = 36
Step 2: Double the 36: 36 x 2 = 72 (the answer)

Example 3: 2.4 x 40
Step 1: Disregard the decimal point and zero and think “24 x 4″
Step 2: Double the 24: 24 x 2 = 48
Step 3: Double the 48: 48 x 2 = 96
Step 4: Apply ToR (Test of Reasonableness): The original problem was a little more than 2 times 40, which would be more than 80. The final answer should therefore not be 9.6 or 960, but 96.

Example 4: 1900 x 0.4
Step 1: Disregard the decimal point and zero and think “19 x 4″
Step 2: Double the 19: 19 x 2 = 38
Step 3: Double the 38: 38 x 2 = 76
Step 4: Apply ToR (Test of Reasonableness): The original problem was a little more than 1900 times 0.4, which is close to 2000 times about a half (.5 would be a half). 2000 times 0.5 would be 1000, so the final answer should be close to that. Our intermediate answer was 76, so we add enough zeros to bring the answer close to 1000. 76 isn’t big enough; 7600 is too big. 760 is close to 1000. The final answer is therefore 760.

Step 2: Do your multiplication or division
24 x 15
= (4 x 6) x (3 x 5) (this is a new trick in its own right, but  handy)
= 4 x 5 x 3 x 6
= 20 x 18 – apply Trick No. 1 to get 2 x 18 = 36, and add a zero to get 360, although we don’t really need this, as we’ll see in a minute.

Step 3: Apply a Test of Reasonableness (ToR)
The original problem was 2.4 x 1.5. That’s about 2 and a half times 1 and a half: More than 2, but less than 4.
This tells us where to put the decimal point:

3.6 is our final answer (not 360 or 36 or 0.36…)

Example 1: 1.2 x 1.2
Step 1: Convert to 12 x 12
Step 2: Multiply to get 144
Step 3: ToR for decimal place; answer: 1.44
(The original problem was close to 1 x 1)

Example 2: 48 ÷ 2.4
Step 1: Convert to 48 ÷ 24
Step 2: Divide to get 2
Step 3: ToR for decimal place; answer: 20
(The original problem was close to 50 ÷ 2)

Example 3: 930 ÷ 3.1
Step 1: Conver to 93 ÷ 3 (remember dropping zeros?)
Step 2: Divide to get 31
Step 3: ToR for decimal place; answer: 310
(The original problem was close to 900 ÷3)

The concept I’ve latched on to recently is that of the three critical arms of learning: Facts, Procedures and Understanding. Without any one of these, learning will be at best incomplete, and at worst, non-existent.

Case studies are given which show that children who learn their addition and multiplication facts by rote will often still make “obvious” conceptual errors when working out a problem, because they lack the understanding of why 3+5=8. They’ll be able to recite that sum (or tell you “eight” when you ask “What’s three plus five?”) but when presented with 13 + 35, they won’t necessarily apply the fact, or if they do, they may apply it in the wrong way. In fact, children who understand what addition is but don’t know their facts were shown to have been MORE successful later on, because they were able to work out the answer to a problem even if they didn’t know the answer to a smaller, “easier” one, such as 3 + 8.

Similarly, children (and adults, I should add) may know a procedure for multiplying two-digit numbers (carrying, and all that), but may apply the steps in the wrong order if they don’t understand why the procedure works.

Even understanding isn’t enough by itself; if you don’t have the facts and procedures in the first place, you may be able to work out simpler problems, but more complex ones will be too difficult. It may be that the facts, the memorisation of certain key details that seem to come up often, will happen as a result of pracitcing a concept (understanding), and that procedures may even reveal themselves to the learner over time. But then that just suggests that these two elements are important after all, at least in terms of efficient calculation.

The answer, then, is to ensure children learn with all three principles in mind.

* The copy I have of this book has clearly been marketed to the US, but there is strong evidence that Mr Butterworth’s original audience, or at the very least his own experience, is from the UK, specifically England.

To square a number, say “N”, add a small number “a” to it to make it a round number.

Then find (N+a) x (N-a) +a².

Here are some examples:

77²
= (77 + 3) x (77 – 3) + 3²  – [here our "a" is 3, because it rounds 77 to 80]
= (80) x (74) + 3² – [think "8 x 74" and then add a zero]
= 5920 + 9 – [okay, 8 x 74 isn't exactly easy, but it's easier than 77²!]
= 5929

32²
= (32 + 8) x (32 – 8) + 8²  – [here our "a" is 8, because it rounds 32 to 40]
= (40) x (24) + 8² – [think "4 x 24" and then add a zero]
= 960 + 64
= 1024

We could have done 32² this way:

32²
= (32 – 2) x (32 +2) + 2²  – [here our "a" is 2, because it rounds 32 to 30]
= (30) x (34) + 2² – [think "3 x 34" and then add a zero]
= 1020 + 4
= 1024

One more:
59²
= (59 + 1) x (59 – 1) + 1²  – [here our "a" is 1, because it rounds 59 to 60]
= (60) x (58) + 1² – [think "6 x 58" and then add a zero]
= 3480 + 1
= 3481

There are other ways to square numbers, but this is my current favourite, since I’ve just “discovered” it. Do you know a different trick for squaring a number? Leave a comment!

Think:
good = pos
bad = neg

A good thing happening to a good person is good.
(pos × pos = pos)

A good thing happening to a bad person is bad.
(pos × neg = neg)

A bad thing happening to a good person is bad.
(neg × pos = neg)

A bad thing happening to a bad person is good.
(neg× neg = pos)

Thanks again to OnlineMathLearning.com for another great idea!

How do we come to know this number? If the proportion between two lines is such that cutting the longer one to make it the same length as the shorter one and the bit you have left over has the same proportion to the new longer bit than the old shorter bit to the old longer bit, that proportion is the number (√5 + 1)/2.

In other words, if you take a line and cut it in just the right place, you will get two shorter lines.

Now cut the longer line so that you get one line the same as the shorter line from your first cut, and another, even shorter line. THESE two lines have exactly the same proportions as the lines you got from your first cut, if you cut in just the right place. That proportion is “The Divine Proportion”.

If you make a rectangle out of these lines, you can cut the rectangle into a square and a rectangle with the same proportions as the original. If you do it again with the inside rectangle, and then again and again… and then draw a spiral through the squares, you get a curve that shows up a lot in nature. Sunflower seeds align themselves along such a curve; leaves on the branches of some trees have a similar pattern; an unfolding fern frond looks like this as well. Perhaps the most well-known example of this curve is the nautilus shell.

Here are some links to other people’s musings on the Golden Ratio:

Click to read article

Click to read article

Click to read article

Click to read article

Example: 4800 ÷ 120
Remove all zeros and perform 48 ÷ 12 = 4
Count the number of zeros on the left and right of the origianl problem: two on the left, one on the right
Cancel one zero from each side of the original problem and add one zero to the result.
Answer: 40

Example: 4800 ÷ 1200
Remove all zeros and perform 48 ÷ 12 = 4
Count the number of zeros on the left and right of the origianl problem: two on the left, two on the right
Cancel two zeros from each side of the original problem and add two zeros to the result.
Answer: 400

Example: 48,000 ÷  120,000 – THIS IS A TRICKY ONE!
Remove all zeros and perform 48 ÷ 12 = 4
Count the number of zeros on the left and right of the origianl problem: three on the left, four on the right
Cancel three zeros from each side of the original problem and add one zero to the result.
Answer: 40?

*BUT: 120,000 is BIGGER than 48,000, so this answer doesn’t make sense. Instead of adding and removing zeros, we can talk about places. In this case, instead of adding one zero (moving one place to the right) we have to move one place to the left. Answer: 0.4. We’ll discuss this type of problem again in a later trick.

It looks like this:
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

It’s easy to write it out; doesn’t take long at all. But if you had a googol pennies you couldn’t count them all in your lifetime. In fact, you couldn’t count them in a million lifetimes!

Here’s another number: a Googolplex, which is a 1 followed by a googol zeros. That’s so big it can’t even be written out in a million lifetimes!

*”1997: Larry and Sergey decide that [their] search engine needs a new name. After some brainstorming, they go with Google — a play on the word “googol,” a mathematical term for the number represented by the numeral 1 followed by 100 zeros. The use of the term reflects their mission to organize a seemingly infinite amount of information on the web.”

Moral: Never put Descartes before the horse.

*Taken from The Penguin Book of Curious and Interesting Mathematics, by David Wells