58 and OXOXOXOXOXOXOX

58 is a composite number. 58 = 1 x 58 or 2 x 29. Factors of 58: 1, 2, 29, 58. Prime Factorization: 58 = 2 x 29.

58 is never a clue in the FIND THE FACTORS puzzles.

2014-05 Level 4

Hugs and Kisses, OXOXOXOXOXOXOX, these were some of my stepmother’s favorite things. This week in anticipation of Valentine’s Day next week, I’ve included both an X puzzle and this O puzzle. Valentine’s Day was also my dad’s and my stepmother’s wedding anniversary, and she always made sure it was a special day for everyone who lived in our house. I’m sure that you also have many happy Valentine’s Day memories.

To solve this O puzzle, consider the clues given in the puzzle. Then write the numbers 1-12 in the top row and again in the first column so that the puzzle becomes a multiplication table.

This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles.

Here are the factors for last week’s level 4 Puzzle:

2014-04 Level 4 Answer

Here is one way those factors could be found:

2014-04 L-4 steps

57 and Cupid’s arrow is on its way!

57 is a composite number. 57 = 1 x 57 or 3 x 19. Factors of 57: 1, 3, 19, 57. Prime factorization: 57 = 3 x 19.

57 is never a clue in the FIND THE FACTORS puzzles.

How do you feel about factoring?

If you will let Cupid hit you with this arrow, you might just fall in love with factors.

Factoring is very important in many levels of mathematics, so enjoying it can be a very good thing.

2014-05 Level 3

I love factoring. I hope you will let Cupid’s arrow hit you so you will love factoring, too.

This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles.

The factors for last week’s level 3 Puzzle:

2014-04 Level 3 Answer

Here’s how last week’s puzzle was solved:

2014-04 L-3 steps

56 and Level 2

56 is a composite number. 56 = 1 x 56, 2 x 28, 4 x 14, or 7 x 8. Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56. Prime factorization: 56 = 2 x 2 x 2 x 7, which can also be written 56 = 2³ x 7.

Sometimes 56 is a clue in the FIND THE FACTORS puzzles. Even though it has other factors, the only multiplication fact we use is 7 x 8 = 56.

2014-05 Level 2

This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles.

The factors for last week’s level 2 Southeast Arrow Puzzle:

2014-04 Level 2 Answer

Here are three different factor trees for the number 56:

 

55 and Square Roots for Third Grade

55 is a composite number. 55 = 1 x 55 or 5 x 11. Factors of 55: 1, 5, 11, 55. Prime factorization: 55 = 5 x 11.

Sometimes 55 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, the puzzle only uses 5 x 11 = 55.

——————————————————————————————-

What is the square root of a number? If you had a square made with 9 squares, how many squares would be on each side?

3 x 3 squareEven a third grader can count 3 squares on each side. That is one way to explain why √9 = 3 to someone so young. Here’s another explanation:

  1. 1 x 1 = 1, so √1 = 1
  2. 2 x 2 = 4, so √4 = 2
  3. 3 x 3 = 9, so √9 = 3
  4. 4 x 4 = 16, so √16 = 4
  5. 5 x 5 = 25, so √25 = 5
  6. 6 x 6 = 36, so √36 = 6
  7. 7 x 7 = 49, so √49 = 7
  8. 8 x 8 = 64, so √64 = 8
  9. 9 x 9 = 81, so √81 = 9
  10. 10 x 10 = 100, so √100 = 10

These are the kinds of square roots that third grade students may need to know. When they get older they will learn about other square roots like √2 ≈ ±1.414, but third graders don’t need to worry about those yet.

On January 24 someone hurriedly typed “aquare roots for third grade” into a search engine, and a post from this blog came up. The person took ONE quick look and closed the post and the blog just as hurriedly. I’m not exactly sure what aquare or square roots for third grade means, but I felt bad that the blog disappointed this unknown person. I thought, “That person MAY have been searching for something that actually might be on the blog SOMEWHERE…..Perhaps a puzzle with perfect square clues would have been a welcome sight.” Hey, I can dream, can’t I?

To solve today’s puzzle (It actually has perfect square clues!), you will have to take the square roots of the clues, and place those square roots in the top row as well as the first column. After you write down all those square roots, figure out what numbers go in the empty cells so that the entire puzzle works as a multiplication table.

2014-05 Level 1 These square roots may or may not be the square roots for third grade that person was looking for. Nevertheless, I have personally seen several bright THIRD grade students solve similar perfect square puzzles that I’ve made. This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles. Here are the factors for last week’s level 1 puzzle: 2014-04 Level 1 Answer

54 and How Many Squares Are in This Puzzle?

How Many Squares?

How many squares are in this puzzle? Finding that answer is too tedious for me to pursue. There is 1 square that is bigger than all the rest, 169 of the smallest size squares, and some different number between 1 and 169 for each size square in between. Also, I think the clues seem to form three little squares, and one of the clues, 16, happens to be a perfect square. It’s a trick question, and most people don’t like trick questions!

Solving the actual puzzle will actually be less work and more fun. The actual puzzle challenges you to write the numbers 1 through 12 in the top row and again in the first column so that the answers you write and the clues inside the puzzle work together as a multiplication table. Use logic to find the unique solution to the puzzle. 2014-04 Level 6 This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles.

Factors of 54:

54 is a composite number. 54 = 1 x 54, 2 x 27, 3 x 18, or 6 x 9. Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54. Prime factorization: 54 = 2 x 3 x 3 x 3, which can also be written 2 x 3³.

Sometimes 54 is a clue in the FIND THE FACTORS puzzles. Even though it has other factors, the only multiplication fact the puzzle uses is 6 x 9 = 54.

Sum-Difference Puzzle:

The number 6 has two factor pairs. One of those factor pairs adds up to 5, and the other one subtracts to 5. Can you put those factors in the correct places in the first puzzle?

The number 54 has four factor pairs. One of those factor pairs adds up to 15, and another one subtracts to 15. If you can identify those factor pairs, then you can solve the second puzzle.

The second puzzle is really just the first puzzle in disguise. Why would I say that?

Solution to Last Week’s Level 6 Puzzle:

2014-03 Level 6 Answer

Here is one way those factors can be found using logic.

2014-03 L-6 steps

53 and Animated Gif FIND THE FACTORS Level 5

I’m a mother and a grandmother. My most recent grandson, Oliver, was born three days ago on Tuesday.

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Steve Wilhite, the creator of gif graphics format, said, “Choosy mothers choose GIF.” I think that can apply to grandmothers, too.

I made my very first animated gif! You can see the factors from last week’s level 5 puzzle appear one by one right before your eyes! I set the gif at the slowest possible setting, but it still goes fairly fast. The puzzle is solved from start to finish in about 15 seconds.

FIND THE FACTORS 403-5

Here is this week’s level 5 puzzle. To solve it, write the numbers from 1 to 12 in the top row and again in the first column so that the numbers you write are the factors of the given clues. There is only one solution.

2014-04 Level 5

This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles.

Here are the factors from last week’s level 5 puzzle:

2014-03 Level 5 Answer

How were those factors found? Look at the animated gif above or click here to see the puzzle being solved, or you can look at the chart below for a slightly different way to solve it.

2014-03 L5 steps

  • 53 is a prime number.
  • Prime factorization: 53 is prime.
  • The exponent of prime number 53 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 53 has exactly 2 factors.
  • Factors of 53: 1, 53
  • Factor pairs: 53 = 1 x 53
  • 53 has no square factors that allow its square root to be simplified. √53 ≈ 7.2801

How do we know that 53 is a prime number? If 53 were not a prime number, then it would be divisible by at least one prime number less than or equal to √53 ≈ 7.3. Since 53 cannot be divided evenly by 2, 3, 5, or 7, we know that 53 is a prime number.

50 and Multiples

Multiples and factors are related.

For example, 50 is a multiple of all these numbers: 1, 2, 5, 10, 25, and 50.

And 1, 2, 5, 10, 25, and 50 are all factors of 50.

50 is a composite number. 50 = 1 x 50, 2 x 25, or 5 x 10. Factors of 50: 1, 2, 5, 10, 25, 50. Prime factorization: 50 = 2 x 5 x 5, which can also be written 2 x 5².

The first few multiples of 50 are 50, 100, 150, 200, 250, 300, and so on…

If the difference between factors and multiples is confusing, this poster should help. Thanks to Resourceaholic for mentioning this great resource in one of her posts.

Sometimes 50 is a clue in the FIND THE FACTORS puzzles. Even though it has other factors, we only use 50 = 5 x 10 to fill in the table.

Each of the clues inside this puzzle are MULTIPLES of a number from 1 to 12. Can you write every number from 1 to 12 in the top row as well as in the first column so that the clues are multiples of the numbers that you write?

2014-04 Level 4

This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles.

Here are the factors to last week’s level 4 puzzle:

2014-03 Level 4 Answer

The chart below shows one possible way to arrive logically at the solution.

2014-03 L4 steps

49 and V is for Victory Puzzle

49 is a composite number, and it is 7 squared. 49 = 1 x 49 or 7 x 7. Factors of 49: 1, 7, 49. Prime factorization: 49 = 7 x 7 which can also be written 49 = 7².

Since √49 = 7, a whole number, 49 is a perfect square. 

Whenever 49 is a clue in the FIND THE FACTOR puzzles, write 7 in both the corresponding factor column and factor row.

Level 3 is the first level of the higher level FIND THE FACTORS puzzles. For many people going from level 2 to level 4 would be too overwhelming. Level 3 is a bridge between those two levels and allows a person to move on from the lower levels and victoriously solve a little more difficult puzzle. V is for that victory.

Level 3 puzzles are designed to be solved starting from a row or column at the top of the puzzle with 2 clues. First find the common factor of those two clues that will allow you to write only numbers from 1 to 10 in the first column of the puzzle. Then factor row by row to the bottom of the puzzle making sure each number from 1 to 12 is written only once in both the factor row and the factor column. You will notice a rhythm for the answers as you work.

2014-04 Level 3

 

This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles. May you be victorious in your efforts!

Here are the factors to last week’s level 3 puzzle:

2014-03 Level 3 Answer

 

This puzzle was solved starting with the column with 2 clues near the top of the puzzle and then factoring row by row until we reached the bottom of the puzzle.

46 and South East Arrow Puzzle

46 is a composite number. 46 = 1 x 46 or 2 x 23. Factors of 46: 1, 2, 23, 46. Prime factorization: 46 = 2 x 23.

46 is never a clue in the FIND THE FACTORS puzzles.

Wikipedia has a chart of 116 different directional arrow symbols. Each one of them has a name. Today’s puzzle represents the South East Arrow.

2014-04 Level 2

To solve the puzzle write the numbers 1 -12 in the top row and again in the first column so that the numbers you write and the clues in the puzzle work together to make a multiplication table.

This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles.

Here are the factors of last week’s hook-shaped level 2 puzzle:

2014-03 Level 2 Answer

45 and A Change in Order

45 is a composite number. 45 = 1 x 45, 3 x 15, or 5 x 9. Factors of 45: 1, 3, 5, 9, 15, 45. Prime factorization: 45 = 3 x 3 x 5, which can also be written 45 = 3² x 5.

When 45 is a clue in the FIND THE FACTORS puzzles, always use 5 and 9 as the factors.

2014-04 Level 1

When I introduce the FIND THE FACTORS puzzles to elementary school students, I only print copies of level 1 and level 2 puzzles. After students have completed those puzzles, I will demonstrate how a level 3 puzzle is completed.

A few students have downloaded the puzzles at home. One very bright sixth grade student was able to complete a level 6 puzzle with the help of his dad.

Kids can be very determined. They may die several times fighting dragons, but they will eventually find the way to save the princess. They can be just as determined to find the way to write the numbers from 1 to 12 in the top row and the first column to make the puzzle work as a multiplication table, especially when I compare it to winning a computer or video game.

Many students tend to guess and check and will get frustrated unless they are shown how to solve the puzzles logically. Frustration can be the enemy of determination.

This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles. If you click on the excel file you may notice that I have changed the order of the puzzles on the main page.

puzzle order

If the puzzles are printed on both sides of a sheet of paper, the paper can be cut down the middle. In this way I can print puzzles for an entire classroom using only half as much paper.

I instruct students to treat the sheet of puzzles like a computer game and see what level they can achieve. They know upfront that I will be extremely surprised if any of them find all the factors to all six puzzles. That statement motivates some students while it reduces stress in others. Not all students like solving the puzzles, but it is a joy to observe the ones who do.

front & back sides

Here is solution for last week’s level 1 puzzle:

2014-03 Level 1 Answer