A Multiplication Based Logic Puzzle

Posts tagged ‘even’

680 What Would Happen If Ten-Frames Looked Like This?

680 is a number made using only even digits. (There’s much more about 680 at the end of the post.)

Numbers ending in 0, 2, 4, 6, or 8 are even. Numbers ending in 1, 3, 5, 7 or 9 are odd. Those two simple concepts are not always easy for young children to understand.

Sometimes we teach rhymes to children to help them know the difference:

  • 0, 2, 4, 6, 8; being EVEN is just great.
  • 1, 3, 5, 7, 9; being ODD is just fine.

Still students in early grades struggle with the concepts of odd and even.

Another seemingly simple concept is what pairs of numbers add up to ten. That concept also isn’t as easy for children to understand as adults might think.

Donna Boucher is an elementary school math interventionist with many years experience. Besides many other topics, she is an expert on teaching adding and subtracting to first and second graders. Here are a couple of her tweets with links to her site:




Free ten-frame flash cards are available on her site to help students learn addition and subtraction facts. What a powerful way for students to learn! She also has Halloween/Thanksgiving ten-frames for sale at Teachers Pay Teachers.

As I read her post about how to use the ten-frame flash cards I wondered what would happen if we followed her instructions EXACTLY, but the ten-frames looked like this:

7 mitten ten-frame
Children would still learn how to add and subtract, but would they also instinctively learn the difference between odd and even numbers?

Would they figure out for themselves that adding two even numbers or adding two odd numbers ALWAYS makes an even number? Or that adding an odd number and an even number together ALWAYS makes an odd number? Or would changing the ten-frames not make any difference at all? Will the mitten ten-frames only make a difference if the parent/teacher/tutor talks about the odd and even numbers?

I don’t know the answer to those questions, but I think the idea is worth trying. I’ve made Mitten Ten-Frames for all the numbers from 0 to 10. The “empty” frames have outlines of mittens to help children know if a left or a right mitten belongs there. The mitten ten-frames don’t have a second border to guide in cutting them out, so the flashcards might not look as good as Donna Boucher’s, but they should still work as flashcards. Follow Donna Boucher’s instructions exactly. If you use the mitten ten-frames, please add a comment to let me know whether or not they make any difference helping students learn the properties of odd and even numbers.


Here’s more about the number 680:

1² + 3² + 5² + 7² + 9² + 11² + 13² + 15² = 680.

Because 5, 17, and 85 are some of its factors, 680 is the hypotenuse of four Pythagorean triples. Can you find the greatest common factor of each triple?

  • 104-672-680
  • 288-616-680
  • 320-600-680
  • 408-544-680

680 the 15th tetrahedral number. Stetson.edu tells us that it is also the smallest tetrahedral number that can be made by adding two other tetrahedral numbers together, specifically the sum of the 10th and the 14th tetrahedral numbers equals this 15th tetrahedral number as shown below:

  • (10)(11)(12)/6 = 220
  • (14)(15)(16)/6 = 560
  • 220 + 560 = 680
  • (15)(16)(17)/6 = 680

Finally, here is the factoring information for 680:

  • 680 is a composite number.
  • Prime factorization: 680 = 2 x 2 x 2 x 5 x 17, which can be written 680 = (2^3) x 5 x 17
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 680 has exactly 16 factors.
  • Factors of 680: 1, 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, 340, 680
  • Factor pairs: 680 = 1 x 680, 2 x 340, 4 x 170, 5 x 136, 8 x 85, 10 x 68, 17 x 40, or 20 x 34
  • Taking the factor pair with the largest square number factor, we get √680 = (√4)(√170) = 2√170 ≈ 26.0768096.



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

Are six year olds too young to learn about odd and even numbers?

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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