1408 Powers of 2 in the Multiplication Table

number, puzzle, factors, factor pairs, prime factorization,

I have a 10 × 10 multiplication table poster in my classroom to help students who haven’t memorized the times’ table yet. We have to spend our time going over more advanced topics. One student struggled with the idea of raising two to a power. I went to the poster and boxed in all the powers of two on it. While I boxed them in, I recited, “2⁰ = 1, 2¹ = 2, 2² = 2×2 = 4, 2³ = 2×2×2 = 8, 2⁴ = 2×2×2×2= 16, 2⁵ = 2×2×2×2×2= 32, 2⁶ = 2×2×2×2×2×2=64.”

I liked the pattern those powers of two made on the poster so I made this 32×32 multiplication chart on my computer and continued the pattern.

I expect the chart has many things for you to notice and wonder about. You could also do it with powers of 3, or another number, but you would need to use a much bigger multiplication table to show as many powers.

Now I’ll tell you a little bit about the number 1408.

1408 is not a power of 2, but it is 11 times a power of 2, specifically, it is 11 × 2⁷.

  • 1408 is a composite number.
  • Prime factorization: 1408 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 11, which can be written 1408 = 2⁷ × 11
  • 1408 has at least one exponent greater than 1 in its prime factorization so √1408 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1408 = (√64)(√22) = 8√22
  • The exponents in the prime factorization are 7 and 1. Adding one to each exponent and multiplying we get (7 + 1)(1 + 1) = 8 × 2 = 16. Therefore 1408 has exactly 16 factors.
  • The factors of 1408 are outlined with their factor pair partners in the graphic below.

Here is a festive multilayered factor cake for 1408:

So delicious! And here is a nicely balanced factor tree showing all of its prime factors:

 

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1407 Please Stop Making Excuses for My Dear Aunt Sally

Please Excuse My Dear Aunt Sally. You’ve heard math teachers say that phrase many times. Supposedly, Aunt Sally is supposed to help you remember Parenthesis, Exponents, Multiplication/Division, Addition/Subtraction as the correct order to do operations when simplifying math problems.

I say, please stop making excuses for my dear Aunt Sally!

My Dear Aunt Sally. People think they know her, but too often they really don’t. Lots of people have tried to please her. Sometimes they succeed, but just as often they fail. She seems to relish the fact that so many people misunderstand her.

I clearly remember my first year teaching a classful of seventh graders at a new school. I was trying to develop a good relationship with my students and be the best teacher I could be. One of the first lessons I was supposed to teach them was order-of-operations.

I wish I knew about the better mnemonic PEMA back then, but I didn’t. Instead, I brought my dear Aunt Sally to class with me: I introduced her to my students and tried to make it clear that multiplication and division were equals so they must be done in order from left to right whichever one comes first. The same is true of addition and subtraction.

“That’s not what we learned last year!” students responded. Their teacher last year brought Aunt Sally to class, too. but she gave them the impression that all multiplication was supposed to be done before any division, and the same for addition and subtraction. Yeah, Aunt Sally went to class their previous year and didn’t say a word when their teacher gave them misinformation. Now that I was telling them the truth about her, she didn’t speak up and tell them I was right either. Instead, she allowed me to lose credibility with my students that day as I insisted on sticking with the truth. If I had retold the lie, the students would have believed me more. I also discovered that for some problems in the textbook, you would get it right either way.

I seriously couldn’t believe that their teacher from the last year would have given them the wrong information. Surely the students misunderstood what had been taught. However, since that day, I have heard more than one teacher incorrectly tell students to do all the multiplication, division, addition, and subtraction in that order from left to right. Those teachers put the students’ next teachers in a catch-22:

That is why I prefer to keep “my dear Aunt Sally” away from kids. She always shows up at the beginning of the school year when students and teachers are trying to start off on the right foot.  She torments students and immediately causes them to feel bad about themselves or mathematics. She makes them question the teaching of their current teacher or their past teachers. She gets a kick out of making children and even adults feel like there’s no way to understand math:

Why do we allow Aunt Sally to abuse children like this? I want to shout, “please, stop making excuses for my dear, Aunt Sally!”

Let me tell you the story of when I decided not to introduce this abusive aunt to children every again.  It was 2016. I was substituting in a 5th-grade class. I wrote an expression I saw on twitter on the board and told the students it was my favorite order-of-operations problem. Here’s what I wrote:

10 + 9 + 8 × 7 × 6 × 5 – 4 + 321 = 

I, along with my dear Aunt Sally,  encouraged the students to figure it out. The students knew that 8 × 7 was 56. I watched them struggle to multiply 56 by 6 and then by 5. When I mentioned that they could multiply the 6 and the 5 first to get 56 × 30 to make the problem easier, they argued that doing that wasn’t allowed. They said that the order-of-operations demanded that the multiplication be done in ORDER from left to right.

They thought that order-of-operation makes multiplication no longer commutative?!!  How do you counteract that misinformation? After that day, not only do I not invite my dear Aunt Sally to meet my students, but I also avoid the phrase “order-of-operations”!

Order-of-operations is just an ALGORITHM! It doesn’t trump the commutative property, and it doesn’t even have to be used to solve these kinds of problems!

Jo Boaler’s tweet especially applies to this kind of problem and this algorithm.

Besides, are these kinds of problems still necessary since typing on a computer no longer has the same limitations as typing on a typewriter? I hope you think about that! If you insist on using an algorithm, I suggest you use PEMA instead.

Since this is my 1407th post, I’d like to tell you a little bit about that number:

  • 1407 is a composite number.
  • Prime factorization: 1407 = 3 × 7 × 67
  • 1407 has no exponents greater than 1 in its prime factorization, so √1407 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1407 has exactly 8 factors.
  • The factors of 1407 are outlined with their factor pair partners in the graphic below.

1407 looks interesting when it is written in some other bases:
It’s 111333 in BASE 4,
21112 in BASE 5, and
727 in BASE 14.

1406 Has a Very Cool 4th Root

To find the 4th root of 1406, all you need to do is take its square root twice. The square root of 1406 is 37.4966665185. . .

Take the square root of that and you get a decimal starting with 6.12345…

That’s pretty cool. I’m glad Stetson.edu let me know about it!

Here’s a little more about the number 1406:

  • 1406 is a composite number.
  • Prime factorization: 1406 = 2 × 19 × 37
  • 1406 has no exponents greater than 1 in its prime factorization, so √1406 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1406 has exactly 8 factors.
  • The factors of 1406 are outlined with their factor pair partners in the graphic below.

1406 is the sum of the first 37 EVEN numbers because 37 × 38=1406.

1406 is the hypotenuse of a Pythagorean triple:
456-1330-1406 which is (12-35-37) times 38

1405 is the Sum of Squares

I knew that 1405 was the sum of two consecutive squares, but Stetson.edu let me know that it was the sum of even more consecutive squares, ELEVEN to be exact!

Because it is the sum of the 26th and the 27th squares, 1405 is also the 27th centered square number. Here are 1405 tiny squares illustrating that fact:

Here’s more about the number 1405:

  • 1405 is a composite number.
  • Prime factorization: 1405 = 5 × 281
  • 1405 has no exponents greater than 1 in its prime factorization, so √1405 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1405 has exactly 4 factors.
  • The factors of 1405 are outlined with their factor pair partners in the graphic below.

I’ve mentioned one of these before, but 1405 is the sum of TWO squares in TWO ways:
27² + 26² = 1405
37² + 6² = 1405

1405 is also the hypotenuse of FOUR Pythagorean triples:
53-1404-1405 calculated from 27² – 26², 2(27)(26), 27² + 26²
444-1333-1405 calculated from 2(37)(6), 37² – 6², 37² + 6²
800-1155-1405 which is 5 times (160-231-281)
843-1124-1405 which is (3-4-5) times 281

1404 Texas Tessellation

I recently visited family members in Texas. My daughter-in-law is awesome at both mathematics and quilting. My photo does not do her work justice, but Texas is tessellated in this quilt! She also carefully chose the fabrics she pieced together. Do they remind you of anything for which Texas is famous?

Someone else designed the pattern, but piecing these pieces together was not the easiest sewing project.

I wondered if anyone else had thought to tessellate Texas and found a couple of examples on twitter. As this first one asked, should we call this Texellation?

Now I’ll tell you something about the number 1404:

  • 1404 is a composite number.
  • Prime factorization: 1404 = 2 × 2 × 3 × 3 × 3 × 13, which can be written 1404 = 2² × 3³ × 13
  • 1404 has at least one exponent greater than 1 in its prime factorization so √1404 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1404 = (√36)(√39) = 6√39
  • The exponents in the prime factorization are 2, 3, and 1. Adding one to each exponent and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 × 4 × 2 = 24. Therefore 1404 has exactly 24 factors.
  • The factors of 1404 are outlined with their factor pair partners in the graphic below.

1404 is the hypotenuse of a Pythagorean triple:
540-1296-1404 which is (5-12-13) times 108

Since 1404 has so many factors, it also has MANY different factor trees. Here are four of them mixed in with some Texas tessellations!

1403 Multiplication Table Challenge

Just because you’re not in elementary school anymore doesn’t mean that the multiplication table can’t be a challenge. This one certainly is. Can you write the numbers 1 to 10 in the four factor areas so that this multiplication table works with the given clues? Don’t get discouraged; it will probably take you at least 15 minutes just to put those factors in the right places.

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll share some information about the puzzle number, 1403:

  • 1403 is a composite number.
  • Prime factorization: 1403 = 23 × 61
  • 1403 has no exponents greater than 1 in its prime factorization, so √1403 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1403 has exactly 4 factors.
  • The factors of 1403 are outlined with their factor pair partners in the graphic below.

1403 is the hypotenuse of a Pythagorean triple:
253-1380-1403 which is 23 times (11-60-61)

 

 

1402 Mystery Level

Mystery level puzzles may be very difficult or relatively easy. How much trouble will this one be? You’ll have to try it to see!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll tell you a little bit about the number 1402:

  • 1402 is a composite number.
  • Prime factorization: 1402 = 2 × 701
  • 1402 has no exponents greater than 1 in its prime factorization, so √1402 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1402 has exactly 4 factors.
  • The factors of 1402 are outlined with their factor pair partners in the graphic below.

1402 is the hypotenuse of a Pythagorean triple:
31² + 21² = 1402

1402 is the hypotenuse of a Pythagorean triple:
520-1302-1402 which is 2 times (260-651-701)
and can also be calculated from 2(31)(21), 31² – 21², 31² + 21²

1401 Roasting Over an Open Fire

I went camping last week. My family roasted hotdogs. Some people refer to them as mystery meat. Others roasted marshmallows. I was surprised to learn that almost all brands of marshmallows have blue dye in them.  I’m told that without that blue dye the marshmallows will lose their whiteness as they sit on store shelves. Why they have to be that white is a mystery to me.

Here’s a mystery level puzzle for you to solve. It looks a lot like the utensil that was used to roast the hotdogs and marshmallows.

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll tell you something about the number 1401:

  • 1401 is a composite number.
  • Prime factorization: 1401 = 3 × 467
  • 1401 has no exponents greater than 1 in its prime factorization, so √1401 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1401 has exactly 4 factors.
  • The factors of 1401 are outlined with their factor pair partners in the graphic below.

1401 is the difference of two squares in two different ways. Can you figure out what those ways are?

1398 and Level 5

You might find this puzzle to be a little tricky, but if you always use logic before you write any of the factors, you should succeed!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here is some information about the number 1398:

  • 1398 is a composite number.
  • Prime factorization: 1398 = 2 × 3 × 233
  • 1398 has no exponents greater than 1 in its prime factorization, so √1398 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1398 has exactly 8 factors.
  • The factors of 1398 are outlined with their factor pair partners in the graphic below.

1398 is the hypotenuse of a Pythagorean triple:
630-1248-1398 which is 6 times (105-208-233)

1397 and Level 4

I bet you know enough multiplication facts to get this puzzle started. Once you’ve started it, you might as well finish it. You will feel so clever when you do!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll write a little bit about the puzzle number, 1397:

  • 1397 is a composite number.
  • Prime factorization: 1397 = 11 × 127
  • 1397 has no exponents greater than 1 in its prime factorization, so √1397 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1397 has exactly 4 factors.
  • The factors of 1397 are outlined with their factor pair partners in the graphic below.

1397 is the difference of two squares two different ways:
699² – 698² = 1397
69² – 58² = 1397