In elementary school, we learned about improper fractions. Should we call them that? Is it even possible for any kind of number to be IMPROPER? They are simply fractions greater than one. I’ve recently heard the term fraction form used, and ever since I’ve made a point of saying that fractions greater than one are in fraction form.
On Twitter, I’ve found a few people who also don’t like using the word improper to describe any fraction.
This first tweet has a link explaining why it is improper to use the term improper fraction:
Whether it is an improper fraction or mixed number, terminology in maths matters just as much as it does in English, writes Kevin O’Brien https://t.co/B9ZuQThUqf
I always wanted to analyze the "behavior" of any fraction that was called improper! Should this fraction receive some sort of penalty for their deeds? Seriously, knowing their equivalence and when one form may be more appropriate use-wise, is the issue: whether 5/4; 1 1/4 or 1.25
In my 3rd grade class we had a conversation about the term "improper" and how it doesn't fit the fraction. The kids all agreed that fractions can have many different representations and there's nothing "improper" about that. They were super cute.
Does the term ‘improper fraction’ lead to misunderstanding?Does it suggest that a /real/ fraction is less than 1?My goal is to use the term ‘rename’ rather than ‘convert’. We aren’t changing anything but the way it looks. #TVDSBmathpic.twitter.com/jlKKx8uN7l
Beautiful shamrocks with their three heart-shaped leaves are not difficult to find. Finding the factors in this shamrock-shaped puzzle might be a different story. Sure, it might start off to be easy, but after a while, you might find it a wee bit more difficult, unless, of course, the luck of the Irish is with you!
Now I’ll share some information about the number 1365:
1365 is a composite number.
Prime factorization: 1365 = 3 × 5 × 7 × 13
The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1365 has exactly 16 factors.
1365 has no square factors that allow its square root to be simplified. √1365 ≈ 36.94591
1365 is the hypotenuse of FOUR Pythagorean triples:
336-1323-1365 which is 21 times (16-63-65)
525-1260-1365 which is (5-12-13) times 105
693-1176-1365 which is 21 times (33-56-65)
819-1092-1365 which is (3-4-5) times 273
1365 looks interesting in some other bases:
It’s 10101010101 in BASE 2,
111111 in BASE 4,
2525 in BASE 8, and
555 in BASE 16
I’m feeling pretty lucky that I noticed all those fabulous number facts! If you haven’t been so lucky finding the factors of the puzzle, the same puzzle but with more clues might help:
By simply changing two clues of that recently published puzzle that I rejected, I was able to create a love-ly puzzle that can be solved entirely by logic. Can you figure out where to put the numbers from 1 to 12 in each of the four outlined areas that divide the puzzle into four equal sections? If you can, my heart might just skip a beat!
Now I’ll tell you a few things about the number 1350:
is a composite number.
= 2 × 3 × 3 × 3 × 5 × 5, which can be written 1350 = 2 × 3³ × 5²
The exponents in
the prime factorization are 1, 3 and 2. Adding one to each and multiplying we
get (1 + 1)(3 + 1)(2 + 1) = 2 × 4 × 3 = 24. Therefore 1350
has exactly 24 factors.
I was in the mood to make a Find the Factors Challenge Puzzle that used the numbers from 1 to 12 as the factors. I’ve never made such a large puzzle before, but after I made it, I rejected it. All the puzzles I make must meet certain standards: they must have a unique solution, and that solution must be obtainable by using logic. Although the “puzzle” below has a unique solution, and you can fill in a few of the cells using logic, you would have to use guess and check to finish it. Besides that, you wouldn’t be able to know if you guessed right until almost the entire puzzle was completed. Thus, it doesn’t meet my standards.
Even though the puzzle was rejected, there were still some things about it that I really liked. In my next post, I’ll publish a slightly different puzzle that uses some of the same necessary logic that I appreciated but doesn’t rely on guess and check at all. This is NOT the first time I have tweaked a puzzle that didn’t initially meet my standards to make it acceptable. I just thought I would share the process this time. If you try to solve it, you will be able to see the problem with the puzzle yourself.
Now I’ll share some information about the number 1349:
1349 is the sum of 13 consecutive primes, and it is also the sum of three consecutive primes: 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 1349 443 + 449 + 457 = 1349
2019 is the hypotenuse of a Pythagorean triple: 1155-1656-2019 so 1155² + 1656² = 2019²
2¹⁰ + 2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2¹ + 2⁰ = 2019
2019 is a palindrome in a couple of bases: It’s 5B5 in BASE 19 (B is 11 base 10) because 5(19²) + 11(19) + 5(1) = 2019, and 3C3 in BASE 24 (C is 12 base 10) because 3(24²) + 12(24) + 3(1) = 2019
Every year has factors that often catch people by surprise. Today I would like to give you my predictions for the factors of 2019: 2019 will have four positive factors: 1, 3, 673, and 2019 However, 2019 will also have four negative factors: -1, -3, -673, and -2019
Which factors, positive or negative, will be your focus in the coming year?
Finally, I’ll share some mathematics-related 2019 and New Year tweets that I’ve seen on twitter. Some of these tweets have links that contain even more facts about the number 2019.