884 Put First Things First

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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

884 is in this cool pattern:

883 and Level 5

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833 is the sum of the eleven prime numbers from 59 to 103.

833 is also the sum of these three consecutive prime numbers: 283 + 293 + 307 = 833.

883 is palindrome 373 in BASE 16 because 3(16²) + 7(16) + 3(1) = 883.

Print the puzzles or type the solution on this excel file: 10-factors-875-885

  • 883 is a prime number.
  • Prime factorization: 883 is prime.
  • The exponent of prime number 883 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 883 has exactly 2 factors.
  • Factors of 883: 1, 883
  • Factor pairs: 883 = 1 × 883
  • 883 has no square factors that allow its square root to be simplified. √883 ≈ 29.7153159

How do we know that 883 is a prime number? If 883 were not a prime number, then it would be divisible by at least one prime number less than or equal to √883 ≈ 29.7. Since 883 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 883 is a prime number.

882 Factor Trees for the First Day of Autumn

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September 22 was the first day of autumn. Leaves are already beginning to fall from the trees.

To rake up the leaves for 882, you might first notice that it’s even. The logical thing to do would be to first divide 882 by two. . . But perhaps you might notice that 8 + 8 + 2 = 18, a number divisible by nine, so you might just as logically begin by dividing 882 by 9. The first step you take determines how the factor tree looks.

882 has many possible factor trees but these two are probably the most common.

You can rake the leaves up this way or you can rake them up that way, but when you rake up the leaves from 882’s factor trees, you always get the same prime factors: 2, 3, 3, 7, and 7.

Here’s a little more about the number 882:

882 has eighteen factors. The greatest number less than 882 with eighteen factors is 828. Now get this: 288 also has eighteen factors. That means that every possible combination of 8-8-2 has exactly eighteen factors!

882 has some interesting representations in some other bases:

  • 616 BASE 12, because 6(12²) + 1(12)¹ + 6(12º) = 882
  • 242 BASE 20, because 2(20²) + 4(20)¹ + 2(20º) = 882
  • 200 BASE 21, because 2(21²) = 882

882 is also the sum of consecutive primes: 439 + 443 = 882

  • 882 is a composite number.
  • Prime factorization: 882 = 2 × 3 × 3 × 7 × 7, which can be written 882 = 2 × 3² × 7²
  • The exponents in the prime factorization are 1, 2 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1)(2 + 1) = 2 × 3 × 3 = 18. Therefore 882 has exactly 18 factors.
  • Factors of 882: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 441, 882
  • Factor pairs: 882 = 1 × 882, 2 × 441, 3 × 294, 6 × 147, 7 × 126, 9 × 98, 14 × 63, 18 × 49 or 21 × 42
  • Taking the factor pair with the largest square number factor, we get √882 = (√441)(√2) = 21√2 ≈ 29.6984848.

881 What Level Should This Puzzle Be?

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Before I started this blog I shared a sheet of six puzzles with a coworker. The most difficult puzzle on the sheet looked similar to this one.

Print the puzzles or type the solution on this excel file: 10-factors-875-885

He skipped ALL the easier puzzles and went straight for the most difficult one. Even though I advised him to use logic to solve the puzzle, he used guess and check and solved the puzzle within a couple of minutes. He then bragged that he could also solve a difficult Sudoku puzzle in about five minutes. He told me that my puzzles could never be a challenge to him and he wasn’t interested in ever doing another one. Ouch.

After that experience, when I began publishing my puzzles on my blog, I made sure the most difficult puzzle on the sheet were more difficult than his puzzle was.

Still I like this level of puzzle. I’m just not sure where it should be categorized.

It’s more difficult than level 4 because 20 and 16 have more than one possible common factor. However, 20 and 16 are the only set of multiple clues in any row or column, so it’s easier than a level 6. It doesn’t exactly qualify as a level 5 so I’m not assigning it that level.

Logic is still very important in finding the solution, although I suppose some lucky guess-and-checker might find it without logic. I think most people would only get frustrated if they just guessed and checked.

So give this puzzle a try. I’m calling it level ????, and its difficulty level is somewhere between a level 4 and a level 6.

Here are a few facts about the number 881:

25² + 16² = 881, so 881 is the hypotenuse of a Pythagorean triple which happens to be a primitive:

  • 369-800-881, which is calculated from 25² – 16², 2(25)(16), 25² + 16²

881 is the sum of the nine prime numbers from 79 to 113.

  • 881 is a prime number.
  • Prime factorization: 881 is prime.
  • The exponent of prime number 881 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 881 has exactly 2 factors.
  • Factors of 881: 1, 881
  • Factor pairs: 881 = 1 × 881
  • 881 has no square factors that allow its square root to be simplified. √881 ≈ 29.681644

How do we know that 881 is a prime number? If 881 were not a prime number, then it would be divisible by at least one prime number less than or equal to √881 ≈ 29.7. Since 881 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 881 is a prime number.

Here’s another way we know that 881 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 25² + 16² = 881 with 25 and 16 having no common prime factors, 881 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √881 ≈ 29.7. Since 881 is not divisible by 5, 13, 17, or 29, we know that 881 is a prime number.

 

880 Flip the Diagonals!

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There are 880 different 4 × 4 Magic Squares. (In addition, each of those can be rotated and/or reflected to make 880 × 8 Magic Squares.) I have found a few that start with some simple ways to order the numbers 1 to 16 in the boxes and then flip the diagonals to create a perfect Magic Square.

For this first one I began by putting the numbers 1-4 in one box with the numbers 13-16 in the cater-corner box. Then 5-8 and 9-12 occupy the other spaces. Then I flipped the diagonals and got a Magic Square!

If we flip the locations of the numbers 5-8 and the numbers 9-12, it still works:

This third Magic Square begins with the most common way to order the numbers from 1 to 16:

With just a little tweaking of that most common way to order the numbers, I was able to make the following three Magic Squares. Begin by switching the first two rows with each other and the last two rows with each other. Notice in every beginning square the diagonals already equal 34.

We can also switch the first two rows with the last two rows, but that just gives us a reflection of the one just above it. (The colors don’t match, but it is still a reflection!)

Try switching the first two columns with the last two columns. You’ll get another different one.

There are MANY more ways to do a 4 × 4 Magic Square.

Last week’s excel file, 12 factors 864-874, included some Magic Square templates including the 4 × 4 one. You can use that file to find more Magic Squares. Go ahead give it a try!

Here’s a little more about the number 880:

880 is the hypotenuse of a Pythagorean triple:

  • 528-704-880, which is (3-4-5) times 176.

880 is the sum of the twelve prime numbers from 47 to 101.

  • 880 is a composite number.
  • Prime factorization: 880 = 2 × 2 × 2 × 2 × 5 × 11, which can be written 880 = 2⁴ × 5 × 11
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 880 has exactly 20 factors.
  • Factors of 880: 1, 2, 4, 5, 8, 10, 11, 16, 20, 22, 40, 44, 55, 80, 88, 110, 176, 220, 440, 880
  • Factor pairs: 880 = 1 × 880, 2 × 440, 4 × 220, 5 × 176, 8 × 110, 10 × 88, 11 × 80, 16 × 55, 20 × 44 or 22 × 40
  • Taking the factor pair with the largest square number factor, we get √880 = (√16)(√55) = 4√55 ≈ 29.66479.

879 and Level 4

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879 consists of three consecutive numbers, 7, 8, 9, so it is divisible by 3.

879 is the hypotenuse of Pythagorean triple 204-855-879 which is 3 times (68-285-293)

Print the puzzles or type the solution on this excel file: 10-factors-875-885

  • 879 is a composite number.
  • Prime factorization: 879 = 3 × 293
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 879 has exactly 4 factors.
  • Factors of 879: 1, 3, 293, 879
  • Factor pairs: 879 = 1 × 879 or 3 × 293
  • 879 has no square factors that allow its square root to be simplified. √879 ≈ 29.64793

878 and Level 3

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218 + 219 + 220 + 221 = 878; that’s the sum of four consecutive numbers.

438 + 440 = 878; that’s the sum of two consecutive even numbers.

878 is a palindrome in base 10 but not in any of the other bases from 2 to 36.

Print the puzzles or type the solution on this excel file: 10-factors-875-885

  • 878 is a composite number.
  • Prime factorization: 878 = 2 × 439
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 878 has exactly 4 factors.
  • Factors of 878: 1, 2, 439, 878
  • Factor pairs: 878 = 1 × 878 or 2 × 439
  • 878 has no square factors that allow its square root to be simplified. √878 ≈ 29.63106

877 and Level 2

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29² + 6² = 877

That makes 877 the hypotenuse of a Primitive Pythagorean triple:

  • 348-805-877 calculated from 2(29)(6), 29² – 6², 29² + 6²

Print the puzzles or type the solution on this excel file: 10-factors-875-885

  • 877 is a prime number.
  • Prime factorization: 877 is prime.
  • The exponent of prime number 877 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 877 has exactly 2 factors.
  • Factors of 877: 1, 877
  • Factor pairs: 877 = 1 × 877
  • 877 has no square factors that allow its square root to be simplified. √877 ≈ 29.6141858

How do we know that 877 is a prime number? If 877 were not a prime number, then it would be divisible by at least one prime number less than or equal to √877 ≈ 29.6. Since 877 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 877 is a prime number.

Here’s another way we know that 877 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 29² + 6² = 877 with 29 and 6 having no common prime factors, 877 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √877 ≈ 29.6. Since 877 is not divisible by 5, 13, 17, or 29, we know that 877 is a prime number.

 

876 and Level 1

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876 consists of three consecutive numbers 6, 7, 8, so 876 has to be divisible by 3. We can also conclude the following:

  • Since it’s even and divisible by 3, we know that 876 is also divisible by 6.
  • Since it is divisible by 3 and it’s last two digits are divisible by 4, we know that 876 is also divisible by 12.

Print the puzzles or type the solution on this excel file: 10-factors-875-885

876 is a palindrome in four other bases:

  • 727 BASE 11, because 7(121) + 2(11) + 7(1) = 876
  • 525 BASE 13, because 5(13²) + 2(13¹) + 5(13º) = 876
  • 282 BASE 19, because 2(19²) + 8(19¹) + 2(19º) = 876
  • 1A1 BASE 25 (A is 10 base 10), because 1(25²) + 10(25¹) + 1(25º) = 876

876 is also the hypotenuse of Pythagorean triple, 576-660-876 which is 12 times (48-55-73).

  • 876 is a composite number.
  • Prime factorization: 876 = 2 × 2 × 3 × 73, which can be written 876 = 2² × 3 × 73
  • 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 876 has exactly 12 factors.
  • Factors of 876: 1, 2, 3, 4, 6, 12, 73, 146, 219, 292, 438, 876
  • Factor pairs: 876 = 1 × 876, 2 × 438, 3 × 292, 4 × 219, 6 × 146, or 12 × 73,
  • Taking the factor pair with the largest square number factor, we get √876 = (√4)(√219) = 2√219 ≈ 29.597297

875 Multiplication and Division Facts

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This simple illustration of $8.75 can be used to show a variety of multiplication facts. We can easily see that this much money can be divided by 5, 7, 25, and even 125.

Likewise, if we know that the amount of money is $8.75, we can write that as 8¾ and do some other easy division problems if we divide the money into groups:

  • 8¾ ÷ 1¼ = 7 because 1¼ is 5 quarters.
  • 8¾ ÷ 1¾ = 5 because 1¾ is 7 quarters.

Now, it might not be obvious that there really is $8.75 in money unless we divide the money into groups of four quarters.

Since $8.75 is the same as 8¾, we can try this division problem, too:

How much is 8¾ ÷ 1½?

What other fractional division problems can you do with these 35 quarters?

Here’s a little more about the number 875:

875 is the hypotenuse of three Pythagorean triples:

  • 525-700-875 which is (3-4-5) times 175.
  • 245-840-875 which is (7-24-25) times 35.
  • 308-819-875 which is 7 times (44-117-125)

Factors of 875 were shown in red in those triples. Here are all of 875’s factors:

  • 875 is a composite number.
  • Prime factorization: 875 = 5 × 5 × 5 × 7, which can be written 875 = 5³ × 7
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 875 has exactly 8 factors.
  • Factors of 875: 1, 5, 7, 25, 35, 125, 175, 875
  • Factor pairs: 875 = 1 × 875, 5 × 175, 7 × 125, or 25 × 35
  • Taking the factor pair with the largest square number factor, we get √875 = (√25)(√35) = 5√35 ≈ 29.5803989

875 is in this cool pattern: