A Perfect Ten

Don't you love tests where you ask a question which you believe everyone will get correct, and then find out it just isn't so?  I gave my PreAlgebra college students a pretest to see what they knew and didn't know.  One of the first questions was:  Why is our number system called Base Ten?  This is an extremely important concept as it reveals what they know about place value.  Below are some of the answers I received.

1)  It is called Base Ten because we have ten fingers.  (Yikes! If that is so, should we include our toes as well?)

2)  It is called Base Ten because I think you multiply by ten when you move past the decimal sign.  (Well, sort of.  You do multiply by ten when you move to the left of the decimal sign, going from the ones place, to the tens place, to the hundreds place, etc.)

3)  I think it is called Base Ten because it's something we use everyday.  (Really????)

Enough!  It is called Base Ten because we use ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to write all of the other numbers.  Each digit can have one of ten values: any number from 0 through 9. When the value reaches 9, just before 10, it starts over at zero again.  (Notice the pattern below.)

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, etc.

In addition, each place is worth ten times more than the last. Ten is worth ten times more than 1, and 1,000 is ten times more than 100. The pattern continues infinitely both ways on a number line.

The decimal point allows for the place value to continue in a consistent pattern with numbers smaller than one. As we move to the right of the decimal point, each place is divided by ten to get to the next place value.  One hundredth is one tenth divided by ten, and one thousandth is one hundredth divided by ten.  The pattern goes on infinitely. 

          100's, 10's, 1's . 0.1, 0.01, 0.001, 0.0001, 0.00001, etc.

Since all mathematics is based on patterns, this shouldn’t be a new revelation. Perhaps on the posttest, my students will omit the fingers and instead rely on patterns to answer the questions!

For more information about teaching place value, refer to the September 7^th posting entitled: There’s A Place for Us.

Beware of Math Cannibals!!!

I have just started to teach Basic Algebra Concepts to my college mathphobics.  This is where the rubber meets the road" as they say.  The biggest hurdle for my students is understanding positive and negative numbers.  Multiplying and dividing seem to be no problem (see my June 13th posting entitled: Accentuate the Positive; Eliminate the Negative), but addition and subtraction are another story.  To state that subtracting a positive number is the same as adding a negative number is considered hieroglyphics to many.  Since many of my students are visual/kinesthetic learners, I needed a strategy that would connect the abstract to the concrete. 

I took film canisters (yes, another Trash to Treasure idea!) and filled them with two sided beans. One side of the bean is red (negative), and the other side is white (positive). Suppose the students have the problem -5 + 2.  They would get out five red beans and two white ones as illustrated on the left. Then the fun begins because suddenly the beans become "cannibalistic".  The red ones begin to "eat" the white ones and vice versa. (In reality, the students are matching each red bean with a white one and moving them aside; see illustration on the right.)  After each bean has been “eaten” by the opposing color, three red beans remain.  As a result, the answer to the problem of -5 + 2 is -3.

Don't stew; study!

 If the problem were -2 - 6, the students would lay out two red beans and six red beans.  Since all the beans are the same color and no bean desires to "eat" anyone on their team, the student simply counts all of the red beans.  So  -2 - 6 = - 8.

What happens with a problem such as 5 + ^-3?  At the beginning, I have the students get out five white beans and three red ones; then match them resulting in the answer of 2.  Unfortunately, in our Algebra book, the double signs vanish by about the third page of the chapter; so, the students must recognize what to do. 

 

The first option is to insert a + sign such as in the problem – 4 – 2 = -4 - ⁺2.  This allows them to see that, in reality, they are subtracting a positive number. 

However, what do they do with -4 - ^-2?   I instruct them to circle the two signs, and use the multiplication rule for a negative times a negative to change the double minus signs into a plus sign as seen in the illustration on the left. They can then proceed to use their beans to solve the problem.  This may seem unusual, but it makes sense to my mathphobics.

You might ask, "How long do the students use the beans?"  It’s interesting, but all of my students put them away, just at different times.  A few only need them for the first assignment whereas others need them for many.  I once had a special education student who was mainstreamed into my regular PreAlgebra class.  He was the last one to rely on the beans, but he did eventually put them away.  The important thing was he had a picture in his head that he could use over and over again.  Incidentally, he passed the class with a “C”, completing all of the same work the other students did.

Need a game instead of a worksheet to practice adding and subtracting positive and negative numbers?  Try Bug Mania or Roll and Calculate.  Just click on the name of the game.

Very "Ghoul" Math

Welcome to My Web!

Here is a question for all you mathematicians:  What is a mathematician's favorite dessert?  Give up?  It's Pumpkin Pi!!   (Ha!  Ha!)   Here is another Halloween riddle. Why didn’t the skeleton dance at the party?  He had no body to dance with!  

Okay, so what do these riddles have to do with teaching math?  I have been trying to come up with ways for my students to recognize fractional parts in lowest terms.  As you know from this blog, I have used Pattern Sticks, the Divisibility Rules, and finding Digital Root.  These are all strategies my students like and use, but to be a good mathematician requires practice - something most of my students dread doing.  I can find many "drill and kill" activities, but they tend to do just that, drill those who don't need it and kill those who already know how to do it. 

So to "drill and thrill" I created some fractional word puzzles for specific times of the year.  The newest addition is entitled Halloween Fraction Riddles and contains eight riddles and answers which the students must discover by correctly identifying fractional parts of words.  For instance, my first clue might be: the first 2/3's of WILLOW.  The word WILLOW contains six letters.  It takes two letters to make 1/3; therefore, the first 2/3's would be the word WILL.  This causes the students to group the letters (in this case 4/6), and then to reduce the fraction to lowest terms.  The letters are a visual aid for those students who are still having difficulty, and I observe many actually drawing lines between the letters to create groups of two.  At first, I thought my students would breeze through the activity, but to my surprise, it proved to be challenging as well as somewhat tricky - just perfect for a Trick or Treat holiday.  

Very Ghoul!



Maybe this is an activity you would like to try with your intermediate or middle school students.  Just click on this link:  Halloween Fraction Riddles

Trash to Treasure

http://flapjackeducationalresources.blogspot.com/

While searching teacher blogs, I came across Flap Jack Educational Resources.  Sra. Corra, also known as Mrs. Green, creates clever and useful things for her classroom from what might be considered trash.  I think she might be better named "The Trash to Treasure Lady".  She also has quality products at her Teachers Pay Teachers store:  FlapJack Products. 

Her blog caused me to think about: "What sort of extraordinary things could I create from ordinary things which might otherwise be thrown away?"  Here is one of my Trash to Treasure ideas.

Go to any Quick Trip or a store similar to that and ask if you could have some plastic cup lids, two for each child.  (Stores are usually happy to help out teachers.)  I like the sturdy 4" red ones.  Instead of placing a straw in the designated spot, place a brad to connect two of the lids.  These should be touching each other top to top or flat side to flat side.  After the lids are together, place a few stickers on the outside of the lids.  What do you have?  A card holder!  Just slide the game cards in between the two lids, and they will actually stay there!  These are great for little hands which have difficulty holding several cards, or for older hands which aren't functioning like they use to, or for disabled or crippled hands.  My granddaughters love them because they can now play Go Fish without dropping and showing everyone all of their cards.

Do you have a Trash to Treasure idea?  Share it with us by leaving a comment.

Free Resource

Check out a free resource entitled Trash to Treasure.  It is an eight page handoutthat features clever ideas, fun and engaging mini-lessons in addition to cute and easy to construct crafts made from recycled or common, everyday items.  In this resource, discover how to take old, discarded materials and make them into new, useful, inexpensive products or tools for your classroom.

Just click on the link below the resource cover.