Various Tools to Help Students Graph Equations

I work in the math lab at the community college where I also teach. The math lab is staffed by only math instructors and offers free math tutoring to any of our students. We try to have many resources available for our students. When it comes to graphing, we have found that the computer can be very unfriendly. The graphs are often hard to see because they are small, and so finding points is next to impossible. We keep in stock some items that help our students.

First, we have graph paper that is always available. We keep an assortment of different kinds for our students:

1. 1/4" grid paper
2. Four co-ordinate graphs per page
3. Full co-ordinate graph paper
4. Six small co-ordinate graphs per page

On the right, you see Markwan holding the example of #2.

We also have two-sided white boards. One side is blank while the opposite side contains a coordinate graph outline. Our students make good use of these. They like the fact that they can do the linear or quadratic equation on one side and then construct the graph on the other.  (They don't have to draw the X-Y axes and tick marks for each problem or get out the ruler for accuracy.) Since the white boards are erasable, they can be used over and over again. On the left, Sam is "modeling" the white board. (Both young men wanted to be on my blog and were anxious for me to use their first names.)

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BUT my favorite item we have on hand are graphing sticky notes.  I often use them in my math classes because students can take notes while drawing examples of graphs and then stick the example right into their math book.  These post-it-notes are called MiniPLOTs®.  They are a unique brand of Post-It Notes designed for math students, teachers and tutors. MiniPLOTs® are 3x3" paper pads with 50 coordinate grid, polar coordinates, or 3D solid shapes printed on each sheet. They are the perfect size for homework and tests. In addition, the company makes them for algebra, geometry, trigonometry, statistics and K-6 math (provides an innovative method of teaching students the basic multiplication and division factors in about six weeks).

These work great when I am grading math homework. When a student has graphed an equation wrong, I simply take a graphing sticky note, correctly graph the equation and stick it beside their incorrect answer. It's important that students see the correct answer so that the wrong one doesn't remain stuck in their heads!

The Math Lab also supplies a reference sheet entitled Graphs of Some Common Functions. It gives an example of the equation being graphed (i.e. f(x) = x ), a visual of the what the graph should look like, the domain, range, and symmetry origin. The students are free to use the laminated ones in the Math Lab, but can also take home a paper one to place in their math notebooks.

On the reverse side of this reference sheet are examples of: Graph of f(x) = a^x and Graph of f(x) = log_a(x). Besides a visual of the graph, it includes domain, range, decreasing on and horizontal or vertical asymptote.

Most students are visual learners and can see lines and curves and project how they behave intuitively. Their brains can easily understand, understand and recognize pictorially better than just remembering abstract equations.  It is therefore important for students to construct and draw graphs so they can picture them in their minds. Hopefully some of these graphing tools will make constructing those graphs easier.

The Changing Size of Toilet Paper - A Math Dilemma!

Which Roll is Today's Product?

Consumer’s Report 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 1989. (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 catalog 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 trying 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"
• Angel 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 and sometimes thinner and more transparent, 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. They 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 game with toilet paper.  Is nothing sacred in the world of mathematics?

A Multiplication "Trick" To Know When You Are Multiplying by 11

Knowing how to do math does NOT require magic; although, sometimes working a problem can appear to be done magically.  This week I want to talk about multiplying by eleven. Before I demonstrate the "trick", I have to get on my soap box for just a moment. In my humble opinion, all students should know their times tables through 12 even though the Common Core Standard for third grade says through 10 x 10. Remember, Common Core is the minimum or base line of what is to be learned. In Algebra, I insist that my students know the doubles through 25 x 25 and the square roots of those answers up to 625. It saves so much time when we are working with polynomials.

Now to our our amazing mathematical "trick". Let's look at the problem below which is 231 x 11.

 

First we write the problem vertically. Next, we bring down the number in the ones place which in this case is a one. Now we add the digits in the ones and tens place which is 3 + 1 and get the sum of four which is brought down into the answer.


Moving over to the hundreds place, we add that digit with the digit in the tens place 2 + 3 and get an answer of five which we bring down. Finally, we bring down the digit in the hundreds place which is a two. The answer to 231 x 11 is 2,541.

Now try 452 x 11 in your head. Did you get 4,972? Let's try one more. This time multiply 614 by 11. I'm waiting...... Is your answer 6,754?

Now it is time to make this process a little more difficult. What happens if we have to regroup or carry in one of these multiplication problems?

We will multiply 784 by 11. Notice that we start as we did before by just bringing down the number in the ones place. Next, we add 8 + 4 and get a sum of 12. We write down the 2 but carry or regroup the one. We now add 7 + 8 which is 15 and then add in the 1 we are carrying. That makes 16. We bring down the 6 but carry the 1 over. We have a 7 in the hundreds place, but must add in the one we are carrying to get a sum of 8. Thus our answer is 8,624.

Let's see if you can do these without paper or pencil. 965 x 11 768 x 11 859 x 11 After working the problems in your head, write down your answers and check them with a calculator. Try making up some four and five digit problems because this is a non-threatening way to have your students practice their multiplication facts. Have fun!

Hands-On Math Using FREE Milk Lid Jug Lids

$3.00

Start saving milk jug lids because there are countless hands-on math activities you can do in your classroom using this free manipulative. Here are just four of those ideas.

1) Sort the lids by various attributes such as:

• Color
• Snap-on or Twist-on 
• Label or No Label
• Kind of edge (smooth or rough)

2) Let the students grab one handful of lids.

• Ask the students to count the lids.
• See if the students can write that number.

3) Make a pattern using two different colors of lids.

 

• Identify the pattern using letters of the alphabet or numbers. The pattern above would be an A, A, B pattern or a 1, 1, 2 pattern.

• Now ask the students to use more than two colors to make a pattern

• Once more, have the students identify the pattern using alphabet letters or numbers.

4) Decide on a money value for each color of lid. (Example: Red lids are worth a nickel, blue lids are worth a dime, and white lids are worth a penny.) Put all of the lids into a bag and have the students draw out four lids. Have the students add up the total value of these four lids.

• Use play money (coins) to have the students show the value of the lids. 

• Have the students practice writing money as either a part of a dollar or as cents.

• Another idea is to have the students find all the combinations of lids that would equal a nickel or a dime or a quarter.

The resource, Milk Lid Math, contains over 15 hand-on ideas with numerous activities listed under each idea. The activities may be used with a whole group, small groups or as center activities.