991 Carry Your Own Weather

This snowflake puzzle isn’t for beginners, but making snowflakes goes very well with the idea of carrying your own weather.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

I like when I can sneak a little bit of mathematics into a completely unrelated lesson. This lesson is about being proactive and is called “Carry Your Own Weather”.

Carry Your Own Weather (Be Proactive) discussion:

  1. If you could choose the weather, what kind of weather would you choose? (Sunny weather and snowy weather seem to be chosen the most.)
  2. If the weather outside was ALWAYS sunny, would you like that? Would that be a good thing? Would you appreciate the sunny days if every day was sunny? (Variety is good. Without rain and snow, how could food grow?)
  3. Does your mood depend on the weather?
  4. How do you feel when it’s sunny outside?
  5. How do you feel when it’s gloomy outside?
  6. The author of the 7 Habits of Highly Effective People, Stephen R. Covey, talked about carrying your own weather. What do you think that might mean? (Decide for yourself how you will feel. Don’t let the outside weather or other people make that decision for you.)
  7. Let’s listen as Stephen R. Covey talks about Carrying Your Own Weather:

7Habits – CARRY YOUR OWN WEATHER VIDEO.flv.

Have you ever made a snowflake before? Did you know that you can decide how the snowflake will look before you make a single cut? Choosing how the snowflake will look ahead of time is like deciding what kind of weather you will carry with you. You will not leave it up to chance. You will begin with the end in mind. You will decide ahead of time how your snowflake will look. In real life, a snowflake has 6 sides, but you can choose to make your snowflake have four sides, six sides or eight sides. Several layers of a napkin are much easier to cut than the same number of layers of regular paper so you will use white paper napkins to make your snowflakes. Afterwards, you can glue your snowflake onto a sheet of dark blue construction paper. Then you can use a white crayon to sign your name and decorate the dark blue construction paper around your snowflake.

First, you need to fold your napkin. Folding into eighths is the easiest. Just find the corner where all the folds in the napkin already meet and fold that corner again, thus making a 45° angle. Fold that corner in half again and you’ll get the 22.5° angle that you see at the bottom of the napkin in the picture below. The other napkin is folded into sixths and then into twelfths. If you don’t cut off the uneven edges at the top of those napkins, your snowflake will only have four sides, but if you do cut off the uneven edges, you will get a six-sided or an eight-sided snowflake. (Depending on that cut, you might also get a twelve-sided or sixteen-sided snowflake. They’ll look great, too!)

Making a perfect six-sided snowflake is a little more difficult than an eight-sided one. I recommend reading this post from Paula Beardell Krieg for complete instructions on six-sided snowflake cutting: ‘Tis the Season to Make Paper Snowflakes (She is the one who told me about using easy-to-cut paper napkins for the snowflakes, too.)

I found these three triangles helpful in making snowflakes with perfect 60° angles.

 

Place the center of the folded napkin at the bottom center of triangles. You can easily see through the napkin to see where the folds need to go.

When each side of the napkin has been folded up, it will look have a 60° angle at the bottom. The red line shows where to cut the top off the napkin to get a perfect hexagon folded into an equilateral triangle.

Fold the napkin in half again so that bottom angle becomes a 30° angle before making your decorative cuts. These next instructions tell how to make those cuts to get the exact snowflakes that you want. These tips were made for six-sided snowflakes, but you can also apply the tips to eight-sided snowflakes:

Paper Snowflake Cutting Tips

Please be aware that these snowflakes are delicate. They can rip easily. I suggest you mount them on sturdy paper as soon as possible.

After you finish making your snowflakes, I’m sure you will enjoy a story that I like very much, even though you have heard it many times before. The main character felt sorry for himself because he was bullied and nobody thought he was good at anything. When he decided to find a way to be helpful, he started to carry his own weather. He became proactive. Doing so not only lifted him but lifted everyone around him, too. Can you guess the name of the story? (Rudolph the Red-Nosed Reindeer)

Rudolph the Red-Nosed Reindeer story and song from youtube

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This is my 991st post, so I’ll tell you a little bit about the number 991:

991 is a prime number that can be written as the sum of consecutive prime numbers two different ways:
127 + 131 + 137 + 139 + 149 + 151 + 157 = 991; that’s seven consecutive primes.
191 + 193 + 197 + 199 + 211 = 991; that’s five consecutive primes.

991 is a palindrome two different ways:
33133 in BASE 4 because 3(4⁴) + 3(4³) + 1(4²) + 3(4¹) + 3(4⁰) = 991
131 in BASE 30 because 1(30²) + 3(30) + 1(1) = 991

  • 991 is a prime number.
  • Prime factorization: 991 is prime.
  • The exponent of prime number 991 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 991 has exactly 2 factors.
  • Factors of 991: 1, 991
  • Factor pairs: 991 = 1 × 991
  • 991 has no square factors that allow its square root to be simplified. √991 ≈ 31.48015

How do we know that 991 is a prime number? If 991 were not a prime number, then it would be divisible by at least one prime number less than or equal to √991 ≈ 31.5. Since 991 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 991 is a prime number.

 

884 Put First Things First

Tangrams have two large pieces, three medium size pieces, and two small pieces.

Most Tangram puzzles are easier to solve if you figure out where to put the two big triangles first. Making a daily plan is easier if you figure out where to schedule the important items like homework and chores first. One of the Seven Habits of Highly Effective People is Put First Things First.

I wrote a 30-40 minute lesson plan to teach habit number 3 of the seven habits with the seven Tangram shapes. The lesson has now been taught to a third of a local elementary school. It was taught to students from first grade to sixth, and all of them really liked the lesson. The teachers who taught it enjoyed it very much as well. The rest of the school will be taught the same lesson later.

I made a pdf copy of the lesson plan here: Put First Things First. You can use it if you would like to teach that principle to your children or your students.  Part of the lesson is reading the adorable book, A Small Brown Dog with a Wet Pink Nose, by Stephanie Stuve-Bodeen. Our county library system had more than enough copies for us to use. It is also available on Amazon.com.

 

Image result for a small brown dog with a wet pink nose

The Tangram square above was copied on light brown paper so each student in the class could make their own small brown dog.

Students could make both dogs or just one of them. Ability levels vary in surprising ways. There were a few first graders who could put the Tangram puzzle together without any help while a few of the older kids struggled. It was okay if a student had difficulty putting the puzzle together. In fact, I made that potential difficulty an important part of the lesson plan. After playing with the puzzle pieces, some students chose to glue the pieces onto the puzzle. Some of them used crayons to add details to their dogs.

The book and the puzzle were the funnest parts of the lesson, but the lesson began with a serious discussion. We used a document camera to show the four time quadrants to the classes.

(I used the term “Pressing” instead of the more common term “Urgent.” Elementary students probably don’t know what either of those words mean, but they do know what “Pressure” means. They will feel a lot of pressure if they wait until the last minute to do something important. They might feel peer pressure to follow the crowd. Pressing and pressure have the same root word.)

So go ahead, click on the pdf file, Put First Things First, and teach planning and responsibility using the seven Tangram shapes.

Now I’ll write a little bit about the number 884:

884 is the hypotenuse of four Pythagorean triples.

  • 84-880-884 which is 4 times (21-220-221)
  • 340-816-884 which is (5-12-13) times 68
  • 416-780-884 which is (8-15-17) times 52
  • 560-684-884 which is 4 times (140-171-221)

If we had more than ten fingers, 884 might be written in some of these interesting ways:

  • 2D2 in BASE 18 (D is 13 base 10), because 2(18²) + 13(18) + 2(1) = 884
  • 202 in BASE 21, because 2(21²) + 2(1) = 2(441 + 1) = 2(442) = 884
  • QQ in BASE 33 (Q is 26 base 10), because 26(33) + 26(1) = 26(34) = 884
  • Q0 in BASE 34 because 26(34) = 884

What are the factors of 884?

  • 884 is a composite number.
  • Prime factorization: 884 = 2 × 2 × 3 × 73, which can be written 884 = 2² × 13 × 17
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 884 has exactly 12 factors.
  • Factors of 884: 1, 2, 4, 13, 17, 26, 34, 52, 68, 221, 442, 884
  • Factor pairs: 884 = 1 × 884, 2 × 442, 4 × 221, 13 × 68, 17 × 52, or 26 × 34,
  • Taking the factor pair with the largest square number factor, we get √884 = (√4)(√221) = 2√221 ≈ 29.732137

12 The Doorbell Rang

12 is a composite number. 12 = 1 x 12, 2 x 6, or 3 x 4. Factors of 12: 1, 2, 3, 4, 6, 12. Prime factorization: 12 = 2 x 2 x 3, which can also be written 12 = 2² x 3.

When 12 is a clue in the FIND THE FACTORS 1 – 12 puzzles, any pair of its factors could be the correct choice. In the 1 – 10 puzzles, only 2 x 6 or 3 x 4 will be the correct choice.

The Doorbell Rang by Pat Hutchins is about cookies and sharing. It takes less than five minutes for an adult to read every delightful word aloud to a child.  It is also a good book for beginning readers because it is filled with reliable repetition, and it is also sprinkled with a few interesting multi-syllabic words. Some words that do NOT appear in the text are mathematics, multiplication, division, or factoring. Still, the book very cleverly helps children recognize all the factors of 12. Chiix Moses wrote in a review, “Something I firmly believe is that learning is best when it doesn’t feel like learning, and that is precisely what this book accomplishes.” This book almost effortlessly teaches students to think win-win, so it is also an excellent choice for reinforcing the Seven Habits.

Here is part of an email that my blogging friend, Paula Krieg, sent after reading this post, “I’ve been looking at some Islamic Geometry, learning to draw some of those rosettes, and was struck by how the 12-fold pattern seemed particularly rich. I may be wrong about this, but it got me thinking about 12. 12 makes a dozen. 12 months to a year. 12 inches to a foot. 12 days of Christmas, 12 numbers on a clock,  12 apostles. My cupcake pans make 12 cupcakes, and I guess Grama’s cookie pan makes 12 cookies in The Doorbell Rang book.”

I should also mention that some people think we should switch from base 10 to base 12 because 12 is divisible by 50% of the numbers less than or equal to it while 10 is only divisible by 40% of the numbers less than or equal to it.

The puzzle below will require knowledge of the factors of 12 as well as thirteen other numbers. It is a level five puzzle, meant to be completed by adults or very bright children.

 2013-11-21

Click 12 Factors 2013-11-21 for more puzzles.

 

7 Spaghetti and Meatballs for All!

  • 7 is a prime number.
  • Prime factorization: 7 is prime.
  • The exponent of prime number 7 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 7 has exactly 2 factors.
  • Factors of 7: 1, 7
  • Factor pairs: 7 = 1 x 7
  • 7 has no square factors that allow its square root to be simplified. √7 ≈ 2.64575.

How do we know that 7 is a prime number? If 7 were not a prime number, then it would be divisible by at least one prime number less than or equal to √7 ≈ 2.6. Since 7 cannot be divided evenly by 2, we know that 7 is a prime number.

When 7 is a clue in the FIND THE FACTORS puzzles, one factor will be 7 and the other will be 1.

Spaghetti and Meatballs for All! by Marilyn Burns is a delightful story, the kind that children enjoy hearing over and over again.

I work at a Leader in Me school, where we promote the Seven Habits. I used this book when I taught about habit 4, think win-win. When we think win-win, we do not allow someone to “step on us’ to give them a win. Mrs. Comfort’s relatives stepped on her over and over again, and they didn’t even realize it. Finally, she cried, “I give up!” and planted herself on a chair. She definitely felt like she was losing. The class listened to the story intently trying to identify places where the Seven Habits were used or could have been used. We had a great discussion afterward. Also since the book did not use the words, “area” or “perimeter” at all, the class hardly realized that the story was also about those concepts. When we followed the suggestions at the back of the book, the class was able to learn about perimeter and area as we had a great discussion about those topics as well. 

2013-11-18