A Multiplication Based Logic Puzzle

Posts tagged ‘numbers’

740 and Level 2

  • 740 is a composite number.
  • Prime factorization: 740 = 2 x 2 x 5 x 37, which can be written 740 = (2^2) x 5 x 37
  • 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 x 2 x 2 = 12. Therefore 740 has exactly 12 factors.
  • Factors of 740: 1, 2, 4, 5, 10, 20, 37, 74, 148, 185, 370, 740
  • Factor pairs: 740 = 1 x 740, 2 x 370, 4 x 185, 5 x 148, 10 x 74, or 20 x 37
  • Taking the factor pair with the largest square number factor, we get √740 = (√4)(√185) = 2√185 ≈ 27.202941.

740-factor-pairs

This level 2 puzzle isn’t too difficult to solve:

740 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

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Here’s more about 740:

740 is the sum of consecutive numbers several ways:

  • 146 + 147 + 148 + 149 + 150 = 740; that’s 5 consecutive numbers.
  • 89 + 90 + 91 + 92 + 93 + 94 + 95 + 96 = 740; that’s 8 consecutive numbers.
  • 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 = 740; that’s 37 consecutive numbers.

Because 5 and 37 are two of its factors, 740 is the hypotenuse of four Pythagorean triples:

  • 240-700-740
  • 228-704-740
  • 416-612-740
  • 444-592-740

740 is also a palindrome in two bases, one of which is double the other:

  • 252 BASE 18; note that 2(18²) + 5(18) + 2(1) = 740.
  • KK BASE 36 (K = 20 base 10); note that 20(36) + 20(1)

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740 Factors

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275 Are First Graders Too Young to Learn About Odd and Even Numbers?

  • 275 is a composite number.
  • Prime factorization: 275 = 5 x 5 x 11, which can be written (5^2) x 11
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 275 has 6 factors.
  • Factors of 275: 1, 5, 11, 25, 55, 275
  • Factor pairs: 275 = 1 x 275, 5 x 55, or 11 x 25
  • Taking the factor pair with the largest square number factor, we get √275 = (√11)(√25) = 5√11 ≈ 16.583.
number line evens 2

Designed and photographed by Paula Beardell Krieg; http://bookzoompa.wordpress.com/

I recently read a post at mathfour.com that discussed the “basic” concept of odd and even numbers and children’s ability to understand the difference. The article made me very curious so I talked briefly to 45 first grade students about even and odd numbers. What did I find out?

 

  • Almost all of them had been introduced to the concept in kindergarten and knew that 1, 3, 5, 7, 9 are odd numbers while 2, 4, 6, 8, 10 are even.
  • A few accelerated learning students were able to explain to me that the ones digit of a number determines if the number is even or odd,
  • But most of these first graders did not understood that fact because about a third of the students thought that 32 is odd!
  • One little girl explained to me how odd and even numbers alternate. She said, “If 99 is even, then 100 will be odd.” She remembered that concept but didn’t understand it well enough to apply it to the example she gave!

Even though odd and even numbers may be a difficult concept to learn, teach the concept and use it anyway. In fact, talk about it to preschoolers while you put on their socks, shoes, or mittens. One,_Two,_Buckle_My_Shoe.

Children learn to recite numbers in order before they learn how to count, and that helps them learn how to count and later how to add or subtract 1 from a number. I have tutored bewildered looking students who weren’t sure what to do with 8 + 1 = until I told them that 8 + 1 = means “what number comes right after 8 when you count?” Likewise, 8 – 1 = means “what number comes right before 8 when you count?” After hearing those questions, these students immediately knew the answer, and they didn’t count to find it.

Children who can quickly recite the odd numbers to 11 and the even numbers to 10 will have an easier time adding or subtracting two from a number. When they see 3 + 2 =, they can remember that 3 is odd and then ask themselves what odd number comes after 3. Likewise when they see 8 – 2, they can remember that 8 is even and recall that 6 is the even number right before 8.

The way I remember it, I was in second grade when I first was told that an even number plus an even number is even, an odd number plus an odd number is even, while an even number plus and odd number is odd. Any student learning to add or subtract would benefit from that tip.

Adding 3 to an odd number gives an even number, in fact it’s the second even number after the original number. Adding 3 to an even number gives an odd number which is the second odd number after the original number. Subtracting 3 has the same rule, but substitute the word “before” for the word “after.”

Adding 4 to an odd number gives the second odd number after it while adding 4 to an even number gives the second even number after it. Subtracting 4 has a similar rule.

Adding 3 or 4 will mean additionally memorizing that 12 and 14 are even and 13 is odd, but that will be all a first grader needs to know about odd and even numbers. Later these two categories of numbers will be useful throughout their lives for many, many reasons.

What are some ways to help children to memorize odd and even numbers? Paula Beardell Krieg has designed the most captivating number line in the world.

It can easily go from looking like this:

Designed and photographed by Paula Beardell Krieg; http://bookzoompa.wordpress.com/

to looking like this:

evens 4

Designed and photographed by Paula Beardell Krieg; http://bookzoompa.wordpress.com/

The transformation can be done by a child or an adult. This number line that is made with envelopes is pretty enough to hang on a classroom wall, but it can fold up like a book, or be played with and changed so that real learning can take place. Paula Beardell Krieg shows several uses of it in her post, the-flux-capacity-of-an-artful-number-line, and promises to give directions on how to make one soon!

Try these rhymes: 0, 2, 4, 6, 8; Being EVEN is just great! 1, 3, 5, 7, 9; Being ODD is just fine!

Smartfirstgraders.com has several activities and rhymes to help students memorize the odd and even numbers.

  • Finally, if you clap when you say ODD, you will clap one time. 1 is an odd number.
  • If you clap when you say EVEN, you will clap two times, 2 is even.
  • And as mathfour.com pointed out with more detail then I’m showing here: ODD has 3 letters, and 3 is odd.
  • Also EVEN has 4 letters to help us remember that 4 is even.

 

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