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Tools for Helping 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, 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) = ax and Graph of f(x) = loga(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.


A Go Figure Debut From A Kansan Who is New

Today, my Go Figure Debut is for someone from my hometown, Wichita, Kansas. (I personally have never met Dianna, but I would like to. In fact, we live very close to one another.) Dianna has been in education since 2004. She spent nine years in the classroom teaching different grade levels - PreK through 6th grade. She now works as a success coach (university supervisor) and instructor for Wichita State University (I, too, worked there for five years.).

Dianna likes working with students who are studying to be teachers. She helps them come up with lesson ideas for grades PreK through 6th grade, and she delights in watching these students phase into teachers. Since she teaches online, her home office is her classroom. Right now she is teaching her four year old preschooler; so, she has turned her office into a classroom for her with a calendar, letter of the week, weather, and fun learning activities. She and Evangelene enjoy playing file folder games together.

Dianna has been married for 13 years to her husband Greg. They have two children; Joshua who is nine, and Evangelene who is four. They have two dogs, a Collie named Blue Bell, a silver Labrador named TC and a leopard gecko named Spike. Dianna loves reading, especially romance novels. Her favorite author is Jane Austen, and her favorite TV show is Friends. Spending time with her family, travelling, swimming, fishing, going for walks, etc. is her favorite thing to do.

Teachers R Us
Dianna’s store is called Teachers R Us. Currently her store contains a total of 107 resources with five of those being free. Her resources are suitable for grades PreK-2. 

One of her free items is titled, Kindergarten Popcorn High Frequency Words Activity Freebie. It comes with:
Free Resource
  • 9 High Frequency Words on pieces of popcorn
  • 9 High Frequency Words on popcorn containers  
  • Popcorn High Frequency Words Writing Page 
  • Take Home Word Cards 
This product teaches sight words, nine words from Literacy First. It is a fun way for your students to interact with high frequency words.

Her featured paid item is a $15.00 bundle (you save $2.00) called Long e (-y), ey, and y Literacy Activities Bundled with Assessment. This product teaches Long e (-y). It has puzzle matches, picture and word sorts, word scramble, and worksheets. This product includes: 

Only $15.00
  • 6 worksheets with answer keys
  • Picture Sort Worksheet with answer key 
  • Word and Picture sort 
If you are looking for morning work, center work, or homework, then this product is for you.


Dianna says she loves teaching and believes God has given her the best job she could possibly ask for. If you check out her store, you will observe that she has top ratings (4.0) from her buyers with many of them saying that they “love” her resources! I expect you will, too.

There's A Place For Us! Teaching Place Value

When my college students (remedial math students) finish the first chapter in Fractions, Decimals, and Percents, we focus on place value. Over the years, I have come to the realization how vital it is to provide a careful development of the basic grouping and positional ideas involved in place value. An understanding of these ideas is important to the future success of gaining insight into the relative size of large numbers and in computing.  A firm understanding of this concept is needed before a student can be introduced to more than one digit addition, subtraction, multiplication, and division problems. It is important to stay with the concept until the students have mastery. Often when students have difficulty with computation, the source of the problem can be traced back to a poor understanding of place value.

It was not surprising when I discovered that many of my students had never used base ten blocks to visually see the pattern of cube, tower, flat, cube, tower, flat.  When I built the thousands tower using ten one hundred cubes, they were amazed at how tall it was.  Comparing the tens tower to the thousands tower demonstrated how numbers grew exponentially.  Another pattern emerged when we moved to the left; each previous number was being multiplied by 10 to get to the next number.  We also discussed how the names of the places were also based on the pattern of:  name, tens, hundreds, name (thousands), ten thousands, hundred thousands, etc. 

I asked the question, "Why is our number system called base ten?"  I got the usual response, "Because we have ten fingers?"  Few were aware that our system uses only ten digits (0-9) to make every number in the base ten system.

We proceeded to look at decimals and discovered that as we moved to the right of the decimal point, each number was being divided by 10 to get to the next number. We looked at the ones cube and tried to imagine it being divided into ten pieces, then 100, then 1,000. The class decided we would need a powerful microscope to view the tiny pieces.  Again, we saw a pattern in the names of each place:  tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, etc.


I then got out the Decimal Show Me Boards.  (See illustration on the left.)  These are very simple to make. Take a whole piece of cardstock (8.5" x 11") and cut off .5 inches. Now cut the cardstock into fourths (2.75 inches).  Fold each fourth from top to bottom. Measure and mark the cardstock every two inches to create four equal pieces. Label the sections from left to right - tenths, hundredths, thousandths, ten thousandths. Numbers (see free handout below) will fit into the slots which are the unfolded part of the cardstock. (You can type up the names of the places which then can be cut out and glued onto the place value board).

Here are some examples of how I use the boards.  I might write the decimal number in words.  Then the students make the decimal using their show me boards by putting the correct numbers into the right place.  Pairs of students may create two different decimals, and then compare them deciding which one is greater.  Several students may make unlike decimals, and then order the decimals from least to greatest.  What I really like is when I say, "Show me", I can readily see who is having difficulty which allows me to spend some one-on-one time with that student.

If you aren't ready to do decimals, Show Me Boards can also be made for the ones, tens, hundreds and thousands place.  Include as many places as you are teaching. I've made them up to the hundred thousands place by using legal sized paper. As you can see in the photo above, my two granddaughters love using them, and it is a good way for them to work on place value.

I have attached a link to a number handout which is FREE. Just run it off onto cardstock, laminate, cut apart, and place the numbers into small zip lock bags (one sheet per child). Try using different colors of cardstock, so if a number is lost, it is easier to find the bag from which the number is missing.

Free Resource

Under the resource cover on your right is the link to a free page of numbers which anyone is welcomed to download and use.



Only $3.00
A good way to practice nay math skill is with a game. Your students might enjoy the place value game entitled: Big Number.  Seven game boards are included in this eleven page resource packet. The game boards vary in difficulty beginning with only two places, the ones and the tens.  Game Board #5 goes to the hundred thousands place and requires the learner to decide where to place six different numbers.  All the games have been developed to practice place value using problem solving strategies, reasoning, and intelligent practice.

Let's Go Fly A Kite - Using the Correct Geometry Term for Diamond!


This was a comment I received from a fourth grade teacher, "Would you believe on the state 4th grade math test this year, they would not accept "diamond" as an acceptable answer for a rhombus, but they did accept "kite"!!!!!  Can you believe this? Since when is kite a shape name? Crazy."

First of all, there are NO diamonds in mathematics, but believe it or not, a kite is a geometric shape! The figure on the right is a kite. In fact, since it has four sides, it is classified as a quadrilateral. It has two pairs of adjacent sides that are congruent (the same length). The dashes on the sides of the diagram show which side is equal to which side. The sides with one dash are equal to each other, and the sides with two dashes are equal to each other.

A kite has just one pair of equal angles. These congruent angles are a light orange on the illustration at the left. A kite also has one line of symmetry which is represented by the dotted line. (A line of symmetry is an imaginary line that divides a shape in half so that both sides are exactly the same. In other words, when you fold it in half, the sides match.) It is like a reflection in a mirror.

The diagonals of the kite are perpendicular because they meet and form four right angles. In other words, one of the diagonals bisects or cuts the other diagonal exactly in half. This is shown on the diagram on the right. The diagonals are green, and one of the right angles is represented by the small square where the diagonals intersect.

There you have it! Don't you think a geometric kite is very similar to the kites we use to fly as children? Well, maybe you didn't fly kites as a kid, but I do remember reading about Ben Franklin flying one! Anyway, as usual, the wind is blowing strong here in Kansas, so I think I will go fly that kite!

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Only $1.95
This set of two polygon crossword puzzles features 16 geometric shapes with an emphasis on quadrilaterals and triangles. The words showcased in both puzzles are: congruent, equilateral, isosceles, parallelogram, pentagon, polygon, quadrilateral, rectangle, rhombus, right, scalene, square, trapezoid and triangle.  The purpose of these puzzles is to have students practice, review, recognize and use correct geometric vocabulary. Answer keys are included.