Wednesday, April 25, 2012

Ten Black Dots

I am an avid reader, and I love books that integrate math and literature. Occasionally, my blog will feature a book that links the two.  I will summarize the book, give its overall mathematical theme, as well as list various activities you can use in your classroom.
Two dots can make the eyes of a fox
Or the eyes of keys that open locks.
Today's book is Ten Black Dots by Donald Crews (Greenwillow Books, 1986).  This picture book for grades PreK-2 deals with numbers and operations. 

The book asks the question, What can you do with ten black dots?  This question is answered throughout the book by using  illustrations of everyday objects beginning with one dot and continuing up to ten. Simple rhymes accompany the pictures.

Materials Needed: 
  • Unifix cubes or Snap Cubes (multi-link cubes) as seen on the right
  • Black circles cut from construction paper or black circle stickers
  • Crayons
  • Pencils
  • Story paper
  • Calculators -simple ones like you purchase for $1.00 at Walmart


1)  Read the book a number of times to your class.  Let the students count the dots in each picture. On about the third reading, have the children use the snap cubes to build towers that equal the number of dots in each picture.

2)  Have the children think of different ways to make combinations, such as: How could we arrange four black dots?  (e.g. 1 and 3, 4 and 0, 2 and 2)  Have the children use black dots or snap cubes to make various combinations for each numeral from 2-10.

3)  This is a perfect time to work on rhyming words since the book is written in whimsical verse. Make lists of words so that the students will have a Word Wall of Rhyming Words for activity #4.
  • How many words can we make that rhyme with:  sun?  fox?  face?  grow?  coat?  old?  rake?  rain?  rank?  tree?
  • Except for the first letter, rhyming words do not have to be spelled the same.  Give some examples (fox - locks or see - me)
4)  Have the children make their own Black Dot books  (Black circle stickers work the best although you can use black circles cut from construction paper.  I'm not a big fan of glue!)  Each child makes one page at a time.  Don't try to do this all in one day.  Use story paper so that the children can illustrate how they used the dots as well as write a rhyme about what they made.  Collate each book, having each child create a cover.

5)  Have the children figure out how many black dots are needed to make each book. (The answer is 55.)  This is a good time to introduce calculators and how to add numbers using the calculator.

If you can't find Ten Black Dots in your library, it is still available on Amazon.

I found a wonderful blog posting where Holly, who is an art teacher, used this book for an art project.  Check it out at Links, Dots and Doodles.

Thursday, April 19, 2012

Bathroom Math

Which roll is today's product?
Consumer’s Report (May, 2012, pp. 44-47) featured an article about the number games of toilet paper.  (Sounds like math to me!) Since I thought the article was interesting, I mentioned it to my husband who, being a science teacher, had to investigate.  His motto: Never take anyone’s word for it.

So he marched to our bathroom and discovered that our toilet paper was smaller than the holder which had been there since 1975. (Yes, our house is old - like us).  There was a little more than 1/2 inch showing on each side of the roll.  To further investigate, my husband went to the trusty Internet.  There he discovered the following facts.

1)  Toilet paper was first manufactured in 1857.  Before this, corncobs and many other "soft" items were used for this purpose.

Hey Elmer!
Look what's on sale
at Sears!
2)  In the early American west, pages torn from newspapers or magazines were often used as toilet paper. The Sears catalogue was commonly used for this purpose and even the Farmer's Almanac had a hole in it so it could be hung on a hook in the outhouse.

3) In 1935, Northern Tissue advertised "splinter free" toilet paper. (Yes, splinter free!)  Early production procedures frequently left splinters embedded in the paper. And you thought cheap toilet paper was rough!

4)  Toilet paper was originally manufactured in the shape of a square, 4.5" by 4.5" which was about the average size of a man's hand.  The square made it handy to fold over a few times, but still be considered acceptable for sanitary use.  Basically, this size was established because it worked, sort of like the 90 foot pitcher's mound or the ten foot basketball rim.

5)  In the last ten years, the size of toilet paper has been reduced because manufacturers are tyring to cut costs by trimming the sheet size.  (Try placing one "square" in your hand now, and you will see what I mean.)

6)  Most toilet paper producers have decreased the width of a roll from 4.5 inches to 4.2 inches (or something close to that).

7)  Not only have many manufacturers diminished the size of the square (which is now a rectangle), but they have also placed fewer "squares" on a roll.

8)  Unfortunately, it is not just the width of the roll that has been altered.  The size of the cardboard tube in the middle now has a larger diameter, and that is not something you can easily compare in the store!

9)  Typical sizes of popular brands which I had available to measure:
    • Kleenex Cottenelle - Standard: 4.5" x 4.0"
    • Angle Soft - Standard:  4.5" x 4.0"
    • Quilted Northern:  4.5" x 4.0"

What's really comical (or depressing) is that even though toilet paper is smaller, it still costs the same as the old size.  It is just like so many other products we purchase.  No longer can we buy three pounds of coffee or a one pound can of beans.  (I noticed the beans because I used them for students to feel how heavy 16 ounces was. The can now weighs 14 ounces!)  Then there is the 1/2 gallon of ice cream which decreased overnight to 1.75 quarts and half gallon containers of Tropicana Orange Juice which suddenly became 59 ounces instead of 64!  But toilet paper?  I never thought they would play the number games with toilet paper.  Is nothing sacred in the world of mathematics?

Thursday, April 12, 2012

Dump and Divide

When working with fractions, my remedial math college students are never quite sure which number to divide by.  This same thing often occurred when I taught middle school and high school.  So the question I had to answer was, "How can I help my students remember what number goes where?"

First, the student must understand and know the vocabulary for the three parts of a division problem. As illustrated above, each part is correctly named and identified.

Side Note:  The symbol separating the dividend from the divisor in a long division problem is a straight vertical bar with an attached vinculum (you might have to look this word up) extending to the left, but it seems to have no established name of its own. Therefore, it can simply be called the "long division symbol" or  the division bracket.  I wish it were named something fancier, but sometimes plain and straight forward is the best!
Now let's look at a fraction that the student is asked to rewrite as a decimal.  The fraction on your right is two-fifths and is read from top to bottom as two divided by five.  That's easy enough, but when my students enter this into their calculators, many will put in the 5 first, and then press the division sign, followed by the 2. Of course, they get the wrong answer. Now let's look at the dump and divide method.

First, dump the 2 into the calculator.  Then press the division sign; then divide by 5.  The answer is 0.4.

I am aware that many of students are not allowed to use calculators; so, let's look at how this method would work using the division bracket. We will use the same fraction of 2/5 and the same phrase, dump and divide.

First, take the numerator and dump it inside the division bracket.  (Note: Use N side instead of inside so that numerator and N side both start with "N".)  Now place the 5 outside of the long division bracket and divide. The answer is still .4.

Dump and Divide will also work when a division problem is written horizontally as a number sentence such as:  15 ÷ 3.  First, reading left to right, dump 15 into the division bracket. Now place the 3 on the outside. Ask, "How many groups of three are in 15?"  The answer is 5.
Try using Dump and Divide with your students, and then let me know how it works.  You can e-mail by clicking on the page entitled Contact Me or just leave a comment.

Since many students do not know their multiplication tables, reducing fractions is almost an impossible task. The divisibility rules, if learned and understood, can be an excellent math tool.  The resource, Using Digital Root to Reduce Fractions, contains four easy to understand divisibility rules as well as the digital root rules for 3, 6, and 9.  A clarification of what digital root is and how to find it is explained. Also contained in the resource is a dividing check off list for the student. Download the preview to view the first divisibility rule plus three samples from the student check off list.