5 Easy as 1-2-3

/ ivasallay / 2 Comments
  • 5 is a prime number.
  • Prime factorization: 5 is prime.
  • The exponent of prime number 5 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 5 has exactly 2 factors.
  • Factors of 5: 1, 5
  • Factor pairs: 5 = 1 x 5
  • 5 has no square factors that allow its square root to be simplified. √5 ≈ 2.236.

How do we know that 5 is a prime number? If 5 were not a prime number, then it would be divisible by at least one prime number less than or equal to √5 ≈ 2.2. Since 5 cannot be divided evenly by 2, we know that 5 is a prime number.

5 is the only number that is the sum of ALL the prime numbers less than itself.

2² + 1² = 5 and 3² + 4² = 5² so 5 is the smallest Pythagorean triple hypotenuse.

When 5 is a clue in a FIND THE FACTORS puzzle, use 1 for one of the factors and 5 for the other.

Being able to identify factors of a whole number is a very important skill in mathematics.  It is a skill that is commonly used in many areas of mathematics ranging from reducing fractions to solving differential equations.  The Find the Factors puzzles can help make that skill second nature.

2013-11-11

Click 10 Factors 2013-11-11 for more puzzles.

To solve the puzzles, we are only interested in the limited set of factors that are represented in the following table:

Puzzle Clues Chart

What about all the other factors of these numbers?  And what about all the other whole numbers not on the chart?  How do you find ALL of the factors of a given whole number?  For example, suppose you were asked to find all of the factors of 435.  Some people might notice right away that it is divisible by 5 because its last digit is 5.  While that is true, beginning with 5 is not the best place to start because there is an advantage in considering all possible factors in an organized way.  When you are asked to find ALL of the factors of any number, starting at 1 will make finding all of the factors as easy as 1-2-3. So what are the factors of 435?  Using a calculator, I notice that the square root of 435 is about 20.85.  That means I can find absolutely all of the factors of 435 by considering as  divisors just the whole numbers from 1 to 20!  Each factor will have a partner that is greater than 20 but will be found at the same time with these few short calculations. To demonstrate my thinking process, I will put each possible factor from 1 to 20 in a chart and write my thoughts as I consider each one.

Thinking part 1

Thinking part 2

As you may notice, once a possible factor is eliminated, it is not necessary to do any actual division by ANY of the multiples of that number. (4, 6, 8, 10, 12, 14, 16, 18, and 20 are all multiples of 2, which was not a factor, so I didn’t actually divide 435 by any of those multiples.)

As I carefully consider each possible factor, I only WRITE DOWN a number if it is an actual factor.  Therefore, with only a little bit of effort I would list ALL of the factors of 435 in one tidy list: 1 x 435, 3 x 145, 5 x 87, 15 x 29.

See, it was as easy as 1-2-3!  Now let’s find all of the factors of 144.

factors of 144

Even though 144 is less than 435, it has more factors. One of its factors is paired with itself because the square root of 144  is 12.  That fact is also the signal that we can stop looking for more factors, and we can list all the factors of 144 on the following chart:

144 table

There are 8 multiplication facts that produce 144, but 12 x 12 = 144 is the only fact we consider when solving a Find the Factors 1-12 puzzle with 144 as one of the clues. In every other case one of the pair of numbers in the multiplication fact will be greater than 12 and not eligible to be written in the factor row or factor column. However in solving mathematical problems, any of the factors of a whole number could be the star of the show. Knowing how to find those factors is indeed an important skill and is as easy as 1-2-3.

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4 Rhyme and Rhythm

/ ivasallay / 1 Comment

4  is the first composite number, and it is 2 squared. 4 = 1 x 4 or 2 x 2. Factors of 4: 1, 2, 4. Prime factorization: 4 = 2 x 2, which can also be written 4 = 2².

Since √4 = 2, a whole number, 4 is a perfect square.

When 4 is a clue in the FIND THE FACTORS puzzles, the factors might be 1 and 4 or they might be 2 and 2.

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Seventeen years ago, when my daughter was learning the multiplication facts, I came across a rhyme that taught one fact:

5, 6, 7, 8……….56 is 7 times 8.

Coincidentally,

1, 2, 3, 4……….I know twelve is 3 times 4.

3 x 4 = 12 isn’t as difficult to remember as 7 x 8 = 56. Still I find it fun to notice the relationship between counting to eight and these two multiplication facts. I enjoyed the rhymes mentioned in my 10/31/2013 post, and I have found yet another site with rhymes for learning the multiplication facts. Two rhymes similar to these 12 and 56 counting rhymes were even included!  The site is : http://www.teacherweb.com/NY/Quogue/MrsLevy/MULTIPLICATION-RHYMES.pdf

Some children have no problem memorizing number facts, but for some children, a rhyme makes learning the facts more fun and much easier.  Even though I already know all of the basic multiplication facts, I am going to memorize these rhymes simply because I personally enjoy them.  I also know I will have at least one opportunity every week to share them with someone trying to memorize the facts: Already when a student asks me, “what’s 7 x 8?” I always answer in rhyme.

Besides the two rhymes listed above, my favorites are:

Times One: Mirror, mirror look and see, it’s the other number, not me.
Times Zero:  Zero is always the hero
Six times six / Magic tricks / Abracadabra / thirty-six
A tree on skates fell on the floor / Three times eight is twenty-four.
A 4 by 4 is a big machine, Iʼm going to get one when Iʼm 16.

This week I even wrote one myself:  Twelve times twelve / Is a dozen dozen / A gross one forty-four / Just ask my cousin.

When a very young child loves a rhyme, he or she will want to hear it over and over again. Mother Goose rhymes have been enriching lives for years. Being able to fully comprehend what the rhyme is about isn’t necessary at first so even preschoolers can be introduced to these rhymes.

Robin Liner writes a blog (crazygoodreaders.wordpress.com) that discusses reading and dyslexia.  On October 5, 2013 , she wrote Rhythm and Rhyme: A Phonological Power Tool.  She wrote, “Rhymes provide subconscious clues.” That means someone is more likely to get an answer right when that answer rhymes with the question. What a fun and powerful way to learn!

Much of what she wrote not only applies to learning to read but also to learning math, science, history, ……. anything.

Twice a week I put 6 new puzzles in an excel file that is attached to this blog. The puzzles can be solved using logic and knowledge of the multiplication table. Here is one of the puzzles I created this week.

2013-11-07

Click on this link, 12 Factors 2013-11-07,  for more puzzles and the previous week’s solutions. How do you solve the puzzles? Place the numbers 1 – 12 in both the top row and the first column so that those numbers are the factors of the given clues.

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Update: I like the idea of using rhymes to learn multiplication facts so much that I compiled my own list:

Multiplication Rhymes

3 Puzzles for a November Day

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3 is the smallest odd prime number.

3 is the only number that equals the sum of ALL the counting numbers less than itself.

  • 3 is a prime number.
  • Prime factorization: 3 is prime.
  • The exponent of prime number 3 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 3 has exactly 2 factors.
  • Factors of 3: 1, 3
  • Factor pairs: 3 = 1 x 3
  • 3 has no square factors that allow its square root to be simplified. √3 ≈ 1.732.

How do we know that 3 is a prime number? If 3 were not a prime number, then it would be divisible by at least one prime number less than or equal to √3 ≈ 1.7. Since there are no prime numbers less than or equal to 1.7, we know 3 is a prime number.

When 3 is a clue in the FIND THE FACTOR puzzles, the factors are always 1 and 3.

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Click on this link, 10 factors 2013-11-04, to get printable copies of the puzzles below and last Monday’s answers. To solve the puzzles, place the numbers 1 – 10 in both the top row and the first column so that those numbers are the factors of the given clues. Obviously level 6 is more difficult to solve than level 1.

2013-11-04.12013-11-04.2 2013-11-04.32013-11-04.42013-11-04.52013-11-04.6

2 Gross Multiplication Facts

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Factors 1-2 Blog 10-30-13

This level TWO multiplication table is only a little more difficult than the table in my previous post.  This puzzle uses 12 factors while the previous one used just 10. To find all these factors you have to be familiar with 144 multiplication facts, a requirement that sounds much more intimidating than it really is. (For example, 3 x 5 and 5 x 3 are counted as 2 different facts.) 144 is a fascinating number.  It is a dozen dozen, otherwise known as a gross. Therefore, completing this puzzle will help you review the gross multiplication facts!   If you haven’t memorized all of the facts, there is a gross way to learn some of the gross multiplication facts.

Edwin A. Anderson Elementary School uses manipulatives and rhymes to teach the multiplication facts.  Their students learn two rhymes a week for sixteen week and are masters of the gross multiplication facts when they are done. Five of the rhymes are actually a little gross:

  • 3×3, nine cuts on my knee
  • 3×8, 24 horseflies on my plate
  • 4×7, 28 spiders are webbin’
  • 6×8, fishing bait, count slimy worms all 48
  • 7×7, roaches on a vine, yucky and scary all 49

It was a very gross multiplication rhyme about the number 64 that inspired me to collect some favorite rhymes and even write a few rhymes myself. I collected them in one list I titled Multiplication Rhymes:

Multiplication Rhymes

That gross, but inspiring, rhyme about the number 64 and some other gross multiplication rhymes are listed below:

  • 18 The two Frankenstein monsters ate teens. 2×9=18
  • 18 These three stinky trolls chewed six garlic cloves whenever they ate beans. 3×6=18
  • 36 That shelf has three cans of ginger ale if you get thirsty and sick. 12×3=36
  • 63 Kevin whined by a prickly tree. He whined about learning 7×9=63. Now Kevin whines ’cause the prickles didn’t let him be!
  • 64 I ate and I ate and got sick on the floor, 8×8 is 64.
  • 96 The boy ate everything on the shelf. Then he felt naughty and sick. 8×12 is 96.

 

Knowing all 144 (gross) multiplication facts will help you complete my puzzles.

As you Find the Factors and fill in those gross multiplication facts, there are two things I’d like you to remember:

1. Don’t write anything on the inside of the puzzle until you write down all the factors on the outside of the puzzle.

2. The only numbers allowed in the factor row (top row) and the factor column (1st column) are the numbers from 1 to 12, and all 12 numbers must appear in both places.

Now you can click on 12 Factors 2013-10-31, to find the 6 puzzles appearing in this post.

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Now let’s think about the number 2:

2 is the smallest prime number and the only prime that is an even number.

2 is also the largest number that is equal to its number of factors.

In particular situations, there are several prefixes and words that mean two:

  • 2 is the second counting number.
  • A bicycle is a two-wheeler. A binomial is a polynomial with two terms.
  • Two musicians form a duet. Take a look at duet’s synonyms and translations in other languages.
  • The playing card with a 2 on it is called the deuce.
  • A pair can be two shoes, two eyes, or lots of other possibilities.
  • Two people in a relationship are called a couple.

Can you think of any more?

Base 2 uses only the digits 1 and 0 to represent each counting number. Computers use base 2, or binary, as it is also called.

Wikipedia informs us of a Calculus topic about the number 2:

  • If you take the sum from 0 to ∞ of b⁻ᵏ, it will converge to b, ONLY when the base number, b, is 2. Thus,
  •  ⅟₁ + ⅟₂ + ⅟₄ + ⅟₈ + ⅟₁₆ + ⅟₃₂ + ⅟₆₄ + ⅟₁₂₈ + … = 2

Here is 2’s factoring information:

  • 2 is a prime number.
  • Prime factorization: 2 is prime.
  • The exponent of prime number 2 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 2 has exactly 2 factors.
  • Factors of 2: 1, 2
  • Factor pairs: 2 = 1 x 2
  • 2 has no square factors that allow its square root to be simplified. √2 ≈ 1.414.

How do we know that 2 is a prime number? If 2 were not a prime number, then it would be divisible by at least one prime number less than or equal to √2 ≈ 1.4. Since there are no prime numbers less than or equal to 1.4, we know 2 is a prime number.

When 2 is a clue in the FIND THE FACTOR puzzles, the factors are always 1 and 2.

1 Perfect Square

/ ivasallay / 4 Comments

1 has 1 factor. 2 has 2 factors…that is the end of that pattern because no number greater than 2 equals its number of factors.

  • 1 is not a prime number, and 1 is not a composite number. 1 is in a category all by itself. It is classified as a unit.
  • 1 has no Prime factorization.
  • p⁰ = 1, where p is any prime number, so 1 is a factor of every prime number and every composite number.
  • 1 is also the only number to have exactly 1 factor.
  • Factors of 1: 1
  • Factor pairs: 1 = 1 x 1
  • √1 = 1. Since its square root is a whole number, 1 is a perfect square.

1(n) = n and n ÷ 1 = n for every number n.

Also 1⁰ = 1, 1¹ = 1, 1² = 1, 1³ = 1, 1⁴ = 1, 1⁵ = 1, 1⁶ = 1, 1⁷ = 1, 1⁸ = 1, 1⁹ = 1. In fact, 1 raised to any power equals 1. Even 1⁻⁹⁸⁷⁶⁵⁴³²¹⁰ = 1.

Not only that, but any number (EXCEPT 0) raised to the zeroth power is equal to 1.

One of my college professors wrote something like the following on the board to show why 0º is NOT defined:

When 1 is a clue in the FIND THE FACTORS puzzle, write 1 in both the corresponding factor row and the corresponding factor column.

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Like most people, you probably know how to fill in a multiplication table even if it looks like this:

standard random table

The numbers that are given on a table can be called clues. The table above has 20 clues. What is the least number of clues that a table could have and still only have one way to fill it out?

343-1

Although the table above has just nine clues, there is still only one way to complete it. Nine is the fewest number of clues that will still yield a unique solution.  All of those clues would have to be perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. One of the clues will be missing, but it isn’t difficult to figure out where the missing clue should go.  Always find the factors on the outside of the puzzle BEFORE writing down the products on the inside of the puzzle.

The puzzle above is rated difficulty level ONE because you only need to know 10 multiplication facts to find all the factors. If this puzzle is too easy for you, you can try a more difficult puzzle. Levels FOUR, FIVE, or SIX will be much more challenging, even for adults.

This link, 10 Factors 2013-10-28, will bring up an excel file with the puzzles that are on this post.  After you enable editing, you can print the puzzles or type the factor answers directly onto the excel file.

An answer key will be posted one week after a puzzle is published.

If you don’t want to open the excel file, the rest of the puzzles will be printed below. If you cut and paste them on a document, you can make them any size you want.

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2013-10-28.32013-10-28.4

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If you want to check your work, the answers are given in a tab of the excel file that was published a week later: 10 factors 2013-11-04.