Mathematics

What Kind of Shape Is 946 In?

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First of all, 946 is the sum of the numbers from 1 to 43, so it is the 43rd triangular number.

Every other triangular number is also a hexagonal number. Since 946 is the 43rd triangular number, and 43 is an odd number, 946 is also the 22nd hexagonal number. 946 is the 22nd hexagonal number because 22(2(22) – 1) = 22(43) = 946.

But that’s not all. 946 is different than any previous hexagonal number. 946 is the smallest hexagonal number that is also a hexagonal pyramidal number. It is, in fact, the 11th hexagonal pyramidal number. That means if you stack the hexagons in the graphic below in order from largest to smallest, you would get a hexagonal pyramid made with 946 tiny squares. That’s pretty cool, I think.

 

467 + 479 = 946 so 946 is the sum of two consecutive prime numbers.
946 is also the sum of the twenty prime numbers from 11 to 89.

946 is palindrome 181 in BASE 27 because
1(27²) + 8(27¹) + 1(27⁰) = 729 + 216 + 1 = 946

  • 946 is a composite number.
  • Prime factorization: 946 = 2 × 11 × 43
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 946 has exactly 8 factors.
  • Factors of 946: 1, 2, 11, 22, 43, 86, 473, 946
  • Factor pairs: 946 = 1 × 946, 2 × 473, 11 × 86, or 22 × 43
  • 946 has no square factors that allow its square root to be simplified. √946 ≈ 30.75711

There’s Something Odd about the Number 945

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945 = 1 × 3 × 5 × 7 × 9

The sum of the proper divisors of a number determines if the number is abundant, deficient, or perfect. If the sum is greater than the number, the number is abundant. If the sum is less than the number, the number is deficient. If the sum is equal to the number, the number is perfect.

What is a proper divisor? All the factors of a number except itself. Proper divisors are ALMOST the same thing as proper factors. (The number 1 is always a proper divisor, but NEVER a proper factor.)

The first 25 abundant numbers are 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, and 108. Notice that all those numbers are even.

OEIS informs us that 945 is the 232nd abundant number. The first 231 abundant numbers are all even numbers.

Wow, 945 is the smallest ODD abundant number. OEIS also lists the first 31 odd abundant numbers. Every one of the first 31 is divisible by 3 and ends with a 5, but if you scroll down the page you’ll see some that aren’t divisible by 3 or aren’t divisible by 5.

Since 1 × 3 × 5 × 7 × 9 = 945 is the smallest number on the list, you may be wondering about some other numbers:
1 × 3 × 5 × 7 × 9 × 11 = 10,395 made the list.
1 × 3 × 5 × 7 × 9 × 11 × 13 = 135,135 which is too big to be one of the first 31 odd abundant numbers. I was curious if it is also an abundant number, so I found its proper divisors and added them up:

945 is also the hypotenuse of a Pythagorean triple:
567-756-945 which is (3-4-5) times 189

945 looks interesting in a few other bases:
1661 in BASE 8 because 1(8³) + 6(8²) + 6(8¹) + 1(8⁰) = 945
RR in BASE 34 (R is 27 base 10), because 27(34¹) + 27(34⁰) = 27(35) = 945
R0 in BASE 35 because 27(35) + 0(1) = 945

  • 945 is a composite number.
  • Prime factorization: 945 = 3 × 3 × 3 × 5 × 7, which can be written 945 = 3³ × 5 × 7
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 945 has exactly 16 factors.
  • Factors of 945: 1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945
  • Factor pairs: 945 = 1 × 945, 3 × 315, 5 × 189, 7 × 135, 9 × 105, 15 × 63, 21 × 45, or 27 × 35
  • Taking the factor pair with the largest square number factor, we get √945 = (√9)(√105) = 3√105 ≈ 30.74085

Is There Anything Else Special about the Palindrome 939?

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Yes, 939 is a palindrome in base 10, but also all of its factors (1, 3, 313, and 939) are palindromes. It is also palindrome
32223 in BASE 4 because 3(4⁴) + 2(4³) + 2(4²) + 2(4¹) + 3(4⁰) = 939

Okay, that’s nice. Is there anything else special about 939?

The first ten decimal places of the cube root of 939 contain ALL ten digits 0 to 9. That’s unusual, and a reason why 939 is a special number. I made this gif to highlight its uniqueness.
Cube Root 939

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***********

Thank you OEIS.org for informing us of that amazing fact about 939’s cube root.

939 is also the hypotenuse of a Pythagorean triple:
75-936-939 which is 3 times (25-312-313)

  • 939 is a composite number.
  • Prime factorization: 939 = 3 × 313
  • 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 939 has exactly 4 factors.
  • Factors of 939: 1, 3, 313, 939
  • Factor pairs: 939 = 1 × 939 or 3 × 313
  • 939 has no square factors that allow its square root to be simplified. √939 ≈ 30.64310689

 

Why Is Prime Number 937 the 13th Star Number?

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Even though 937 is a prime number, 937 tiny rectangles can be arranged into this beautiful star. Why?

937 is the 13th star number because 6(13)(13 – 1) + 1 = 937.

It is also the 13th star number because it is 12 times the 12th triangular number plus one: Look at this pattern:

The first star number is 12 times the 0th triangular number plus 1. Thus, 12(0) + 1 = 1 (1 yellow rectangle in the center)
The second star number is 12 times the 1st triangular number plus 1. Thus, 12(1) + 1 = 13 (12 green + 1 yellow rectangle in the center)
The third star number is 12 times the 2nd triangular number plus 1. Thus, 12(3) + 1 = 37 (24 blue + 12 green + 1 yellow rectangle in the center)
and so on. . .until
The thirteen star number is 12 times the 12th triangular number plus 1. Thus, 12(78) + 1 = 937 (144 yellow + 132 orange + 120 red + 108 purple + 96 blue + 84 green + 72 yellow + 60 orange + 48 red + 36 purple + 24 blue + 12 green + 1 yellow rectangle in the center)

I made the star so that it consists of one tiny rectangle in the center surrounded by 6 triangles with 78 (the 12th triangular number) rectangles each with another 6 triangles of the same size to form the 6 points of the star.

I very much enjoyed making this star. If you look closely you will see thirteen concentric stars in it following the pattern yellow, green, blue, purple, red, and orange repeated. I added star outlines to make the three smallest stars easier to see.

I think the graphic says a lot about the number 937 all by itself. I hope you enjoy looking at it.

Here’s a little more about the number 937:

24² + 19² = 937, so 937 is the hypotenuse of a Pythagorean triple:
215-912-937 which can be calculated from 24² – 19², 2(24)(19), 24² + 19²

937 is also a palindrome in three other bases:
1021201 in BASE 3 because 1(3⁶) + 0(3⁵) + 2(3⁴) + 1(3³) + 2(3²) + 0(3¹) + 1(3⁰) = 937
1F1 in BASE 24 (F is 15 in base 10) because 1(24²) + 15(24¹) + 1(24⁰) = 937
1A1 in BASE 26 (A is 10 in base 10) because 1(26²) + 10(26¹) + 1(26⁰) = 937

  • 937 is a prime number.
  • Prime factorization: 937 is prime.
  • The exponent of prime number 937 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 937 has exactly 2 factors.
  • Factors of 937: 1, 937
  • Factor pairs: 937 = 1 × 937
  • 937 has no square factors that allow its square root to be simplified. √937 ≈ 30.6104557

How do we know that 937 is a prime number? If 937 were not a prime number, then it would be divisible by at least one prime number less than or equal to √937 ≈ 30.6. Since 937 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 937 is a prime number.

Here’s another way we know that 937 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 24² + 19² = 937 with 24 and 19 having no common prime factors, 937 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √937 ≈ 30.6. Since 937 is not divisible by 5, 13, 17, or 29, we know that 937 is a prime number.

933 Negative and Positive Reflections

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Magic mirror on the wall.
Am I teaching one and all?

Teachers reflect. They often ask themselves how their lesson went, what went well, and how they could improve.

Many years ago I taught an algebra class. The textbook suggested I use algebra tiles to teach a lesson on adding negative and positive integers. I had never heard of algebra tiles before. The school didn’t have any, and there wasn’t time to order some online. Later that September day, I looked at some Halloween candy in a store. When I saw a package of Pumpkin Mix m&m’s, I knew I had found the perfect algebra tiles. All the m&m’s in the package were brown or orange and had pumpkin faces with an “m” for the pumpkin’s nose, but this is how I saw them:

The algebra students learned about adding and subtracting positive and negative integers without any problems, and they LOVED it.

Pumpkin Mix m&m’s have been replaced with other varieties. The colors don’t matter. You could have the sides with the “m” be positive and the side without the “m” be negative.

That summer I enrolled in a Teaching Secondary Mathematics class at the university. I needed to do some volunteer work in a school, reflect on the experience, and write a paper about it. I share that slightly edited paper with you today:

I worked with Mark’s classes. Shon and Serena volunteered there as well. Mark’s students are adults many of which are learning English as a second language as they prepare for the GED test.  Every student I observed was motivated to learn.  One of the students struggled with basic addition facts.  Another understood the concepts but wrote the symbols for algebraic sentences in a different order than we use.  Most of the students are learning Pre-Algebra concepts and getting individualized instruction from the computer program “Classworks.”  If students don’t pass a pretest, they can read a brief lesson on the computer, use some virtual manipulatives to learn the concept, and demonstrate what they have learned.  Several students worked on a lesson that required them to solve for x by balancing equations.  Some of the students seemed confused.  I thought it might be helpful if they had physical manipulatives that required them to do the balancing rather than the computer.  Mark allowed me to plan and prepare a lesson for the twelve students in his first-period class.

I typed and printed a worksheet that consisted of four equations and two large rectangles.  I purchased twelve 1.5 oz packages of Reece’s Pieces to use as Algebra Tiles.  The empty packages represented the variable “x,” each orange candy represented “+ 1” and each brown (or yellow) candy represented “-1.”  Shon and Serena also helped the students understand how to use the manipulatives.  I thought the lesson would only last about five minutes, but it lasted the remainder of the class period.  The students did well with the activity, but it would not have gone so well if my fellow students were not there assisting mostly because of English language issues.

Mark asked me to teach the lesson again to his third-period class.  He even bought more candy so I wouldn’t have to.  His third-period class had six or seven students in attendance.  Because Mark bought M&M’s which come in many different colors, I labeled the diagram I drew on the board with +’s and –‘s instead of O’s and B’s when I explained how their mats should look as we did each step.   Doing that made my explanation to third period clearer than my explanation was to the first period.

When we were almost finished, Mark asked me to write more problems so the students could continue practicing balancing equations.  Instead, I asked a student to write a problem for the class.  She quickly wrote one on paper and then on the whiteboard.  After most of the class members had solved her problem, I had her explain the steps to the class.  She did a terrific job and we all clapped.  I asked another student to write a problem.  She shared a problem, and we cheered for her after she explained the steps.  Eventually, every class member wrote a problem for the class to solve, and we cheered after they explained the steps to solve it with their newly acquired English skills.  Mark also wrote a problem, one that I had thought to be too simple to put on the board:
x – 2 = -2.  It turned out not to be too trivial.  Some students needed to manipulate what happens in that case as well.  Shon and Serena assisted some of the students, but clearly before the class period was over most of the students did not need much help.

When class was finished, Mark met with the three of us.  He told us he really liked the activity and that any time you mix candy and learning together, it’s going to be a hit.  He said when I introduce an activity, I need to slow down.  I need to make sure everyone understands what they are supposed to do.  He thanked me and started his next class.  Shon and Serena both enjoyed helping students with the activity.  Serena said slowing down when giving directions seems to be one of the most common suggestions she hears given to pre-service teachers.  She said it might be helpful to have an equal sign between the two rectangles on the mat so students would know that the two sides are supposed to be equal.  Shon mentioned that when I explained what to do, I didn’t stress that we were solving for x so students might not understand what they need to do when they have similar problems to solve but no candy to use as a manipulative.  All of these are good suggestions that will improve my presentation to help students learn better.

As this paper illustrated, reflection when teaching is very important.

Now for a little about the number 933:

All of its digits are divisible by 3, so 933 is divisible by 3.

  • 933 is a composite number.
  • Prime factorization: 933 = 3 × 311
  • 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 933 has exactly 4 factors.
  • Factors of 933: 1, 3, 311, 933
  • Factor pairs: 933 = 1 × 933 or 3 × 311
  • 933 has no square factors that allow its square root to be simplified. √933 ≈ 30.545049

What Kind of Shape is 925 In?

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925 is the 22nd Centered Square number because 22² + 21² = 925. I made this graphic to show this fact through the use of color. Look at the center of the centered square. Can you see how the single yellow square and the four small green squares in the center correspond to the same colored squares in the smaller squares? The pattern continues from the inside of the centered square to the outside.

925 is the sum of two squares these THREE ways:

 

  • 22² + 21² = 925
  • 27² + 14² = 925
  • 30² + 5² = 925

925 is the hypotenuse of SEVEN Pythagorean triples:

  • 43-924-925
  • 259-888-925
  • 285-880-925
  • 300-875-925
  • 520-765-925
  • 533-756-925
  • 555-740-925

925 is the 25th pentagonal number because 3((25²) – 25)/2 = 925. The shape in the graphic below of 925 tiny squares may look more like a house, but it is still very much a pentagon.

925 looks interesting in a few other bases:

4141 in BASE 6
1K1 BASE 22 (K is 20 in base 10)
151 BASE 28
PP in BASE 36 (P is 25 in base 10)

  • 925 is a composite number.
  • Prime factorization: 925 = 5 × 5 × 37, which can be written 925 = 2² × 37
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 925 has exactly 6 factors.
  • Factors of 925: 1, 5, 25, 37, 185, 925
  • Factor pairs: 925 = 1 × 925, 5 × 185, or 25 × 37
  • Taking the factor pair with the largest square number factor, we get √925 = (√25)(√37) = 5√37 ≈ 30.41381

 

Why Prime Number 919 is the 18th Centered Hexagonal Number

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919 is a prime number, but if you had 919 little squares, they could be formed into this fabulous shape:

This hexagon is made from 18 concentric hexagons using the pattern yellow, green, blue, purple, red, and orange repeated. You can easily count that there are 3 × 6 hexagons. (Yes, that’s counting the yellow square in the center as a hexagon because 1 is the first centered hexagonal number.) Here’s why prime number 919 is a centered hexagonal number:

919 = 1 + 6 + 12 + 18 + 24 + 30 + 36 + 42 + 48 + 54 + 60 + 66 + 72 + 78 + 84 + 90 + 96 + 102, the number of squares contained in those concentric hexagons listed in order from smallest to largest . Thus,
919 = 1 + 6(1 + 2 + 3 + 4 +  5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17) = 1 + 6(153)

919 is a centered hexagonal number because 919 – 1 is 918. What was special about 918? Well, consecutive numbers, 17 and 18, are two of its factors. That made the 17th triangular number a factor of 918. Because 918 is 6 times a triangular number (153), the next number, 919, is a centered hexagonal number.

919 is the 18th centered hexagonal number because 817 (the 17th centered hexagonal number) plus 6(17) = 919.

919 is the 18th centered hexagonal number because 630 (the 18th hexagonal number) plus 17² = 919:

919 is also the 18th centered hexagonal number because 18³ – 17³ = 919. Even though the difference of two cubes can always be factored, 919 is still a prime number because
18³ – 17³ = (18 – 17)(18² + (17)(18) + 17²) = (1)(919)

919 is not only the 18th centered hexagonal number, but it is a palindrome in base 10 and two other bases:

414 in BASE 15

171 in BASE 27

919 uses its same digits, 199, in BASE 26

  • 919 is a prime number.
  • Prime factorization: 919 is prime.
  • The exponent of prime number 919 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 919 has exactly 2 factors.
  • Factors of 919: 1, 919
  • Factor pairs: 919 = 1 × 919
  • 919 has no square factors that allow its square root to be simplified. √919 ≈ 30.31501

How do we know that 919 is a prime number? If 919 were not a prime number, then it would be divisible by at least one prime number less than or equal to √919 ≈ 30.3. Since 919 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 919 is a prime number.

 

917 How to Solve for X with Candy

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How many m&m’s are there in one fun size Halloween pack of m&m’s?

I don’t know. You don’t know. Nobody knows. That’s why these little fun size packs make the perfect UNKNOWN. For this activity, there may even be a negative number of m&m’s in a pack because I’m only using blue and orange m&m’s, and I’m letting each blue m&m equal negative one and each orange m&m equal positive one.

I’m also letting the front side of the fun size package equal negative x, and the back side equal positive x. In algebra we often call our unknown x.

(The colors chosen don’t matter, as long as there are only two of them, and you are consistent with that color being positive or negative. The front of the package could just as easily be +1 and the back -1. Consistency is important. Choose the values you want to use and stick with them. You can also use ALL the m&m’s in a few single packs and have the side of the candy with the m be positive and the side without the m be negative. You can use the empty wrappers as x and -x.)

We can figure out how many m&m’s are in the pack by balancing an equation. The number of m&m’s in a pack is x. We will solve for x by using the very best algebra tiles in the world, m&m’s!

Besides fun size m&m’s (or Skittles or Reece’s Pieces) we need a paper balance for our equations:

Click Equation Balance for a printable pdf of the paper balance.

Now let’s solve x – 3 = 5 by using the m&m’s to find x. This is how the equation balance should look to begin:

We want to get the wrapper by itself, so what do we do? To keep the equation balanced, we add three (positive) orange m&m’s (one at a time) to both sides of the paper balance:

Three (negative) blue m&m’s plus three (positive) orange m&m’s are equal to zero, so we can remove them.

Mmm. I just ate zero m&m’s, and they tasted so good! That leaves us with x = 8, so we have found x, and the equation has been solved! (If you have more equations to solve, you might want to wait to eat the m&m’s until you’re just about finished.)

Now let’s try finding x when the equation is a little more complicated, -2x = x + 12. This is how the balance should look at the beginning:

We want to get all the x’s on one side so we subtract x from both sides of the equation by adding a (negative) front-facing wrapper to both sides of the balance.

Since x – x = 0, we can remove the front-facing and back-facing wrappers from the right side of the equation:

We can arrange the 12 candies into 3 rows of 4.

Now we can divide both sides of the equation by 3.

All that’s left to do is change the signs of EVERYTHING on both sides of the equation:

Thus x = -4. We solved for x correctly because we kept the equation balanced every step of the way.

Now let me tell you a little bit about the number 917:

917 is the sum of five consecutive prime numbers:
173 + 179 + 181 + 191 + 193 = 917

Rearrange its digits and 917 becomes 197 in BASE 26.

  • 917 is a composite number.
  • Prime factorization: 917 = 7 × 131
  • 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 917 has exactly 4 factors.
  • Factors of 917: 1, 7, 131, 917
  • Factor pairs: 917 = 1 × 917 or 7 × 131
  • 917 has no square factors that allow its square root to be simplified. √917 ≈ 30.282007859

900 Pick Your Pony. Who’ll Win This Amount of Factors Horse Race?

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I really like this rhyme that I saw for the first time this week (even though it’s all over the net):

Hey diddle diddle, the median’s the middle,
You add then divide for the mean.
The mode is the one that appears the most,
And the range is the difference between.

All of the numbers from 801 to 900 have at least 2 factors, but no more than 32 factors. 32 – 2 = 30, so 30 is the range of the amount of factors.

There are 100 numbers from 801 to 900. If you list the amount of factors for each number, then arrange those amounts from smallest to largest, the amounts that will appear in the 50th and 51st spots will both be 6. That means that 6 is the median amount of factors. If we had different amounts in the 50th and 51st spots, we would average the two amounts together to get the median.

If you add up the amounts of factors that the numbers from 801 to 900 have, you will get 794. If you divide 794 by 100, the number of entries, then you will know that 7.94 is the mean amount of factors.

What about the mode? Which amount of factors appears the most? That’s why we are having a Horse Race, to see if more numbers have 2 factors, 3 factors, 4 factors, or a different amount of factors. So pick your pony. We’ll see which amount wins, and we’ll find out what the mode is at the same time.

The contenders are these amounts: 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 27, 32.

I should tell you that only perfect squares can have an odd amount of factors, so you probably don’t want to pick an odd amount.

Here are some interesting facts about the numbers from 801 to 900 that might help you decide which pony to pick.

  • We had the smallest two consecutive numbers with exactly 12 factors: (819, 820)
  • We had the fourth prime decade: (821, 823, 827, 829). All four of those numbers are prime numbers and have exactly two factors.
  • We had five consecutive numbers whose square roots can be reduced: (844, 845, 846, 847, 848). Three of those numbers had 6 factors, one had 10, and one had 12.
  • We also had 840, the smallest number with exactly 32 factors
  • 900 is the smallest number with exactly 27 factors. Coincidentally, the amount that is the mode will appear 27 times.

As the following table shows, there are 42 integers from 801 to 900 that have square roots that can be simplified. 42 is more than any previous set of 100 numbers has given us. Even still we are still holding close to just under 40% of integers having square roots that can be simplified.

Okay. If you’ve picked your pony, NOW you can watch the Horse Race:

900 Horse Race
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Hmm…

The race was exciting for a second or two.

As you can see from the Horse Race the mode is 4. How did your pony do?

Here’s a little more about the number 900:

900 is the sum of the fourteen prime numbers from 37 to 97.

24² + 18² = 900

900 is the hypotenuse of two Pythagorean triples:

  • 252-864-900, which is 24² – 18², 2(24)(18), 24² + 18². It is also (7-24-25) times 36.
  • 540-720-900, which is (3-4-5) times 180.

900 is the sum of the interior angles of a heptagon (seven-sided polygon).

  • 900 is a composite number and a perfect square.
  • Prime factorization: 900 = 2 × 2 × 3 × 3 × 5 × 5, which can be written 900 = 2² × 3² × 5²
  • The exponents in the prime factorization are 2, 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 3 = 27. Therefore 900 has exactly 27 factors.
  • Factors of 900: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, 900
  • Factor pairs: 900 = 1 × 900, 2 × 450, 3 × 300, 4 × 225, 5 × 180, 6 × 150, 9 × 100, 10 × 90, 12 × 75, 15 × 60, 18 × 50, 20 × 45, 25 × 36, or 30 × 30
  • Taking the factor pair with the largest square number factor, we get √900 = (√30)(√30) = 30.

 

896÷8=112. Math Teachers, It’s Carnival Time!

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  1. Welcome to the 112th Math Education Blog Carnival! There are many fabulous rides at this carnival, and hopefully you won’t get motion sickness on any of them!

At this first booth, we have the mystifying number 112. What is amazing about the number 112?

Well, not only is 112² = 2² + 4² + 6² + 7² + 8² + 9² + 11² + 15² + 16² + 17² + 18² + 19² + 24² + 25² + 27² + 29² + 33² + 35² + 37² + 42² + 50², but 112 is also the side length of the SMALLEST square that is composed entirely of smaller distinct sized squares with integer sides. Sources: Squaring a Square and OEIS.org. There are 21 different squares in this square with side length 112. Click on the image below and it will magically become bigger. You can then print it, cut it into pieces to make a puzzle, and take that home as your first carnival prize today.

I don’t know if it is significant, but 112 is 7 × 4², and 112’s square was divided into 3 × 7 different squares . 175 is also such a side length, and 175 is 7 × 5², and 175’s square was divided into 3 × 8 different squares.

 

Addition, Subtraction, Multiplication, Division

Bedtime Math has a quick math activity for kids every evening. Here’s one about adding, subtracting, and/or multiplying at the Giraffe Hotel.

Algorithms

Rodi Steinig leads a math circle of students ages 11-17 in a course titled “Our Algorithmic Culture,”. It may surprise you that algorithms are not just for math; they are for real, real life, too. See for yourself as several activities are described in Introducing Algorithms, a post about the first of 8 sessions on the subject.

Math Art

RobertLovesPi’s blog regularly features a rotating solid geometric shape or a beautiful tessellation such as this one that can be enjoyed by young or old alike.

David Mitchell of Latticelabyrinths explains how he and his friend, Jacob, made a beautiful structure for the September 1917 Wirksworth Art and Architecture Trail using a large peg board, pegs and 1302 red or blue precisely-cut wooden equilateral triangles. Amazing! I wish I could have seen it in person.

What is a Rotogon? Katie Steckles of aperiodical.com blogged about this beautiful computer generated, constantly transforming piece of art that could mesmerize young and old alike.

Bell Ringers

What is the Same? What is Different?  has a wide variety of thoughtful activities that can get your students’ brain juices flowing.

Try Math Visuals for other great bell ringers.

Life Through a Mathematician’s Eyes sees a great deal of beauty in Pascal’s Triangle, but it isn’t the significant part of the curriculum she wishes it could be. She got around that though. In her post, Pascal’s Triangle, she shared some great articles that she turned into bell ringers for her students to contemplate when they arrive to class. She also made assignments that can be completed in class or at home.

Bulletin Boards

Paul Murray creates bulletin boards near the lunch room with questions on them.  Some of the questions are math problems. For the last ten years or so, students have stopped by his bulletin boards, read them, and pondered the questions. Read how he does it in A Math Bulletin Board that Actually Gets Read!

Carnival of Mathematics

Earlier this month Just Maths published the more advanced 149th Carnival of Mathematics. It has several great links in it that could pique high schoolers’ interest even if those students aren’t able to understand all the mathematics yet. Next month Alexander at CoDiMa will host the 150th Carnival of Mathematics.

Math Competitions

Some people like entering math competitions. If you have a student that likes them, look at this post from Resourceaholic. It has questions that can be a fun challenge whether you like competitions or not.

Games

Alan Parr often plays his envelop game with students who are learning many different mathematical concepts. His students all enjoy it. The game described in A Wow! Conversation with Amy was easy to put together, let Amy display some brilliant reasoning and provided its creator a very memorable experience, perhaps his most memorable this school year. .

Geometry

Paula Beardell Krieg of Bookzoompa’s wrote a post about Symmetry for 4 – 5 year olds that I adore. Even kids that young can make some gorgeous geometric art.

Mike’s Math Page does so many great math videos with his sons that its difficult to pick only one or two. I went with a couple of posts about geometry: Playing with Some Mathy Art ideas this morning which will appeal to kids of all ages and Lessons from a great geometry homework problem for older kids.

Before teaching congruent triangle proofs, Mrs. E Teaches Math recommends a paper folding, cutting, and arranging activity that helps students visualize overlapping triangles.

Mathematical Humor: 

A little mathematical humor can help students access prior knowledge, or it can make a new concept memorable. Joseph Nebus reads lots of comics to find ones with some mathematics in them. This August 17, 2017 edition of Reading the Comics has a few that could help students remember what irrational means, what sum means, or what the < sign means.

Math Literature Books

Math Book Magic introduces us to a new book, Stack the Cats by Susie Ghahremani. The book is just right for kindergarten and earlier elementary grades. It can help you talk to kids about counting, adding, even decomposing numbers in a very fun way. Kids may not even be aware they are learning math as they find different ways to stack toy cats and other objects. Stack the Cats joins the #Mathbookmagic

Ben Orlin of Math with Bad Drawings delights us with Literature’s Greatest Opening Lines, as Written By Mathematicians.

Denise Gaskins has written Word Problems from Literature. It has a more serious but still very fun approach to exploring mathematics and problem solving through good literature.

Math Magic

Alex Bellos of Science Book a Day introduces us to Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks. Every trick in the book introduces a different mathematical idea, and lots of magician secrets are revealed. Sounds fun!

Managing the Mathematics Classroom

Mrs. G. of Give Me a Sine blogged that this school year she had the best first week of school she’s ever had. She mixed some of her tried and true favorite activities with some new ones that she learned this summer from NYC Math Lab and Sara Vanderwerf. Mrs. G gives specifics detailing what she did. Her students’ response described in the next to the last paragraph is enviable.

Years ago I made a seating chart for a Pre-Algebra class. I didn’t know the students yet and arranged my seating chart in alphabetical order by first name. It was a disaster. I had placed the tallest student in the class at a desk in front of the shortest student in the class. Once one legitimate complaint was voiced, other complaints followed. If I could have read Mrs. E Teaches Math post How to Create a Seating Chart before that school year, I would have had one less frustration.

Math Music

Coleen Young has updated the Mathematical Songs on her website to include MinuteMath’s quadratic formula sung to a One Direction song. She wrote that the song makes her students smile and sing along.

Number Sense

What’s your favorite number? John Golden of Math Hombre asked that question to people and got some wonderful responses. Really cool mathematics is attached to several numbers including the Top Ten Favorite Numbers.

Problem Solving

True problem solving is more about “Why?” than it is about an exact “What?” Denise Gaskins of Let’s Play Math has written several questions to teach students How to Succeed in Math: Answer-Getting vs. Problem-Solving

Puzzles

Long ago mathematicians were often philosophers and philosophers were often mathematicians. Simona Prilogan is both. Every day she posts a mathematical meme, a puzzle such as this one, and a philosophical mathematical thought. Search her site for all kinds of goodies to appeal mostly to middle school children and up.

Alok Goyal’s Puzzles Page shared a puzzle titled 10 Friends. Upper elementary students will be able to understand what the puzzle is asking but would probably need a lot of guidance to solve it.

Rupesh Gesota of Math Coach shared an interesting 6 rectangle puzzle and revealed several different methods students used to solve the puzzle.

My blog typically features a factoring puzzle such as this one that I fancied up for back to school:

This month I did something different: I wrote an elementary-school-age time management lesson plan with an object lesson that uses Tangram puzzle pieces. It was a big hit with the teachers and the students.

Resources

Three J’s Learning wrote math recommendations for 3 year-olds! It includes a list of measuring devices a 3 year-old would love to use and learn from.

Singapore Maths Tuition has assembled a list of the Best Online Resources to Improve Your Math Skills along with their pros and cons.

Resource Room Dot Net Blog wrote about an experience using Illustrative Mathematics (a resource for 6-8 grades) and how it is worth the effort to give feedback.

Square Roots

Last month’s 111 Math Teachers at Play Carnival was hosted at High Heels and No. 2 Pencils by Jacqueline Richardson. This month the Amazing Jacqueline doesn’t just guess your age, she can guess the square root of your age within a few decimal places! I have never seen anything like this before. She uses tiles and grid paper to model square roots of non-perfect squares and is amazingly accurate. Teach your students this way, and they will be amazed. They will understand square roots so well that they will be amazing as well.

Teaching Practices

What are the zeros of this polynomial? Julie Morgan of Fraction Fanatic lets her students give their solutions in a colorful way. This practice has become her new favorite.  Lots of discussion happens. Young children could also use this fun method to give answers.

Ed Southall of Solvemymaths wrote a sobering post titled Sorry we keep lying to you… about lies our math teachers told us and we continue to perpetuate. Read it. Share it. Maybe the lies will stop.

Sara Vanderwerf explains how her Stand and Talks engage students much more than Think/Pair/Share does and REALLY gets the whole class talking about math and contributing to a whole class discussion.

Statitistics

Business blogger Lanisha Butterfield wrote a fascinating article titled Arithmophobia. A major portion of the article was an interview with statistician Jennifer Rogers who did well in math as a kid but HATED it until she was introduced to A-level maths and statistics in school. High School teachers and students should be especially interested in this article.

Sue VanHattum of Math Mama Writes detailed her first day teaching algebra, statistics, and calculus this year. When she discussed the class syllabus, she inserted some fun mathematics here and there. She summed up that first day and shared every math teacher’s universal dream, “I think the class went well. If they really feel good about it, they’ll end up thinking I’m their best teacher ever.”

To everyone who plays at this Carnival, I hope your students think that way about you!

Thank you to everyone who blogged about teaching mathematics this month, and especially thank you to those who submitted a post to this carnival.

I’d like to encourage everyone who blogs about math to submit a post to next month’s carnival which will be hosted at Three J’s Learning.

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Since this is my 896th post, I’ll also write a little bit about that number which happens to be 8 times 112. A factor tree for 112 is contained in this factor tree for 896.

896 is SS in BASE 31 (S is 28 base 10), because 28(31) + 28(1) = 28(31 + 1) = 28(32) = 896.

896 is also S0 in BASE 32 because 28(32) + 0(1) = 896.

896 is the sum of six consecutive prime numbers: 137 + 139 + 149 + 151 + 157 + 163 = 896.

Here is the factoring information for 896 with the ten factors of 112 in red.

  • 896 is a composite number.
  • Prime factorization: 896 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7, which can be written 896 = 2⁽⁴⁺³⁾ × 7 or 896 = 2⁷ × 7
  • The exponents in the prime factorization are 7 and 1. Adding one to each and multiplying we get (7 + 1)(1 + 1) = 8 × 2 = 16. Therefore 896 has exactly 16 factors.
  • Factors of 896: 1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 128, 224, 448, 896
  • Factor pairs: 896 = 1 × 896, 2 × 448, 4 × 224, 7 × 128, 8 × 112, 14 × 64, 16 × 56, or 28 × 32
  • Taking the factor pair with the largest square number factor, we get √896 = (√64)(√14) = 8√14 ≈ 29.933259