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.

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52 and Maths in the snow

Here are some incredible and beautiful sketches in the snow. These pictures must have been taken from an airplane or helicopter. Be sure to click on the picture so you can see many other sketches including a gorgeous fractal. I’m sure you want to know how this fabulous art was made.

52 is a composite number. 52 = 1 x 52, 2 x 26, or 4 x 13. Factors of 52: 1, 2, 4, 13, 26, 52. Prime factorization: 52 = 2² x 13.

52 is never a clue in the FIND THE FACTORS 1- 10 or 1- 12 puzzles.

51 and Magic forcing grid – Maths Magic

Mathematics is full of magic, sometimes more than we even expected. There appears to be a connection between this math trick and 4 x 4 magic squares. I will be interested to see if anyone can form a 4 x 4 magic square from the information provided by The Science Magician in this post. There are 880 ways to make a 4 x 4 magic square, so it seems like it would still be a lot of work. This same process works on a 3 x 3 square and results in numbers that add up to 15 in every case.

51 is a composite number. 51 = 1 x 51 or 3 x 17. Factors of 51: 1, 3, 17, 51. Prime factorization: 51 = 3 x 17.

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

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.

48 and What’s wrong with this?

Here is a four piece puzzle arranged two different ways. There are 24 blue squares and 24 yellow squares. 24 + 24 = 48. Read on to see why the factors of 48 are significant to this puzzle.

Those triangles are some fabulous mathematical slight of hand.  Something is wrong with the puzzle, but what?

Something is fishy with these two triangle puzzles, too:

And with these two triangles made with six puzzle pieces!

If I had time, I could make an infinite number of these types of puzzles. Why do we get a solid triangle sometimes, but one with a hole in it the other times?

Believe it or not, the answer to the triangle puzzles has something to do with factoring.

Here’s that first puzzle again, but smaller:

The first triangle doesn’t have a hole it while the second one does because

Who would have thought that comparing fractions could turn into a type of magic trick?!!

Both triangles have 48 blue or yellow squares, but the second triangle wants you to think it should have 49.

(Psst….6/7 is also the slope of the green triangle and 7/8 is the slope of the red triangle. Since the two triangles don’t have identical slopes, the big “triangles” formed in the puzzles aren’t really triangles at all.)

I made all the puzzles in this post by comparing fractions. In each case, I simply used two consecutive composite numbers and then factored them and came up with the fractions I wanted to use. The numerators and the denominators of the fractions became the sides of the triangles.

Now, what do you think of this video of a never-ending chocolate bar?

Let me tell you a little bit about the number 48:

48 is a composite number. 48 = 1 x 48, 2 x 24, 3 x 16, 4 x 12, or 6 x 8. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Prime factorization: 48 = 2 x 2 x 2 x 2 x 3, which can also be written 48 = 2⁴ x 3.

Sometimes 48 is a clue in the FIND THE FACTORS puzzles. Even though it has many other factors, we use only 6 x 8 for the FIND THE FACTORS 1-10 puzzles and 6 x 8 or 4 x 12 for the FIND THE FACTORS 1-12 puzzles.

 

factor tree

 

47 and Understanding the Equal Sign with Candy

Understanding the Equal Sign Using Peanut Butter Cups. What a wonderful lesson! Teachers or parents can easily follow Jen’s lesson plan. I’m certain children will always remember what an equal sign means forever after.

  • 47 is a prime number.
  • Prime factorization: 47 is prime.
  • The exponent of prime number 47 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 47 has exactly 2 factors.
  • Factors of 47: 1, 47
  • Factor pairs: 47 = 1 x 47
  • 47 has no square factors that allow its square root to be simplified. √47 ≈ 6.85565

How do we know that 47 is a prime number? If 47 were not a prime number, then it would be divisible by at least one prime number less than or equal to √47 ≈ 6.9. Since 47 cannot be divided evenly by 2, 3, or 5, we know that 47 is a prime number.

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

 

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

                                     

44 and Games again…

Mathematicsgames.wordpress.com that I’ve written about previously is part of the Mathematics, Learning, and Technology blog. This blog is a great resource for teachers, parents, and lovers of mathematics. I really appreciate Find the Factors being included in the post Games, again!

44 is a composite number. 44 = 1 x 44, 2 x 22, or 4 x 11. Factors of 44: 1, 2, 4, 11, 22, 44. Prime factorization: 44 = 2 x 2 x 11, which can also be written 2² x 11.

When 44 is a clue in the FIND THE FACTORS 1 -12 puzzles, always use 4 and 11 as the factors.