Ed Tech 4 Math

Last week, I had the pleasure of observing an 8^th grade classroom.  The planned activity was the next part of an ongoing study of data collection and representation.

On this particular day of the lesson, the students were given the task of conducting a survey in order to collect and display data in a circle graph.    Students were provided with a handout and were asked to do the following:

• Write a question
• Determine 6 choices
• Survey the class and complete a frequency table
• Convert data to fractions, decimals, and percents
• Divide the circle graph into quarters, then fifths
• Divide up circle graph into percents
• Create another type of graph to represent the data

After a clear explanation of the task, the class of 15 students stood up and began to collect data.  For about 20 minutes, the students worked together and discussed the data they needed to collect.  Then, the students sat down and began work on the construction of their circle graphs.

Overall, the students were engaged and were having a good success rate at completing the task.  The teacher made the task meaningful to the students in the sense that they had ownership in the creation of the survey topic and who they asked to collect the data.

Enhanced with technology

Adaptive Curriculum, the award winning, online learning environment, offers a similar Activity Object called “Circle Graphs.”  In this multi-part activity, students are first asked to plan their 24-hour day by choosing from a variety of events (typical of a middle school student) and determine the amount of time to be spent on each event.  Then, students divide a 24-hour clock according to the hours selected for the events.

The 24-hour clock is then recreated as a circle graph. To do this, students need to determine the angle measurements needed for each section of the circle graph by finding the fraction of the 24-hour day each event needs and then multiplying by 360⁰ to get the actual angle measurement.  With using sliders on the circle, students can easily draw the correct angle measurements needed for each section of the graph.

In Section 2 of this Activity Object, students design their own data set and categories and then practice creating a circle graph with the new set of data. 

“Circle Graphs” would be a great complementary activity for the 8^th grade students to work on; in fact, I would recommend that it take the place of the handout and paper-and-pencil task that the teacher provided.  The Activity Object provides meaning to learning and applying math to the middle school student; it is more engaging as it allows for ease of the creation of the circle graph with intelligent explanations and feedback on the construction as it is needed.  After completing “Circle Graphs”, students will have gained practice and knowledge on how to construct circle graphs without unnecessary time spent on attempting to draw a circle graph with accuracy.

You can view more of Adaptive Curriculum’s Activity Objects at www.adaptivecurriculum.com. 

As with many mathematical operations, most of us have been taught the rules.  And most probably wouldn’t be able to explain what we did to get the answer.

Multiplying integers is an example.

An explanation for 2 x -4 = -8

One way to explain this rule is to look at the pattern that results from a series of multiplying.

2 x 3 = 6

2 x 2 = 4

2 x 1 = 2

2 x 0 = 0

What do you notice?  The first factor remains constant.  The second factor decreases by 1.  And the product decreases by 2. 

If we continued the patterns,

2 x -1 = -2

2 x -2 = -4

2 x -3 = -6

As we analyze the patterns in the multiplication series, one can conclude that when you multiply a postivie integers by a negative integer, the product is a negative integer. Therefore, 2 x -4 = -8.

An alternate explanation enhanced by technology

Making an array is often a means of explaining multiplication.  But it is mostly only used with positive whole number and in the elementary grades.  The National Library of Virtual Manipulatives offers Rectangle Multiplication of Integers [1] an online applet for creating arrays with negative integer factors. 

Take a look at 2 x -4 = -8 

With the use of a coordinate grid and the ease of moving the slider along the axes, students can create arrays with both positive and negative integers.

In this explanation, the first factor represents a value on the y-axis and the second factor represents a value on the x-axis.  The array is created in Quadrant II and is colored red.

Likewise, here’s -2 x -4 = +8  

This array is created in Quadrant III and is colored blue.

With continued exploration of integer multiplication using arrays, students will more likely be able to make meaning of the operation.  Taking notice of where the array is created on the coordinate grid and the positive or negative factors used to create the array, students will be able to understand why 2 x -4 = -8, but  -2 x -4 = +8.

Representation to conceptualization

Often times, the rules of mathematical operations don’t mean much to our students.  It’s not until they can physically manipulate an object or create a visual representation that students make meaning of the operation.

NCTM  states:

“Representations should be treated as essential elements in supporting students’ understanding of mathematical concepts and relationships … representation associated with electronic technology create a need for even greater instructional attention to representation.”[2]

References:

[1] Rectangle Multiplication of Integers.  National Library of Virtual Manipulatives.

[2]  The National Council of Teachers of Mathematics.  Principals and Standards for School Mathematics.  2000.  Page 66.

“It was much different in our times…” is a cliché that we often hear from our elders. But now, it is my turn to say it. When I was in high school, which corresponds to the late 1980’s, we did not have the luxury of exploiting technology in our math classes. In order to sketch the graph of a function, we had to perform a series of tedious steps which would yield a graph that we were never a hundred percent sure of, and unfortunately, we would have to count on the sketch presented in the text book. I can also remember going over a number of books and tens of graphing exercises prior to a really scary precalculus exam.

Once when I was in high school, I came upon a question similar to this one:

How many real solutions does the equation given by e^x = x^3 + 4 have?

At first sight, it really did not ring a bell and I desperately looked up the solution: Sketch the graphs of y = e^x and y = x^3 + 4 on the same set of coordinate axes and observe the number of intersection points. This was a complete awakening for me (and would later constitute a major corner stone in my math teaching career: I wrote a book, based upon how to exploit the capabilities of the graphing calculator, which was recognized by many as well as the US Department of Education).

Now, if I make the following claim, what would you think?

It is possible to solve any type of equation or inequality with a graphing calculator, whether it is algebraic, trigonometric, exponential, logarithmic, polynomial, or transcendental, in a similar way without having to perform tedious steps.

With a graphing calculator, yes, it is possible. However, it would be such a waste not to exploit other useful functions of these brilliant handheld computers that are given the modest name of “graphing calculators”. With our mathematical knowledge and what is already there within those handheld giants, the sky is the limit to what can be performed, mathematically speaking!

For instance, let us consider the following question:

What is the equation of the parabola that passes through the points (2, 0), (4, 6), (-3, 20)?

Here is one approach to the solution: Let the parabola be y = ax^{^}2+ bx + c. Plug in the points and you get the three equations given by 4a + 2b + c = 0, 16a + 4b + c = 6 and 9a – 3b + c = 20. Now you have a linear system of three unknowns and three equations that you can use to solve for a, b and c to find the correct solution: a = 1, b = – 3 and c = 2.

Now, how can you use technology to help you? Can you use it to solve the linear system for a, b and c? Yes, indeed you can. But let me propose a rather “radical” approach. Does your graphing calculator perform quadratic regression? I bet it does. I suggest that you use quadratic regression to find the equation of the parabola; since you have three points and no more, quadratic regression will exactly give you the parabola that passes through all three of these points and in one easy step. I believe this approach is much better than what I used to be doing when I was in high school since it saves lots of time avoiding the danger of making a mistake while solving the system of three equations for the three unknowns.

Consequently, in the level of civilization attained today, we do have the luxury of using technology. All it takes is to use this option smartly by combining the capabilities of technology with our knowledge of mathematics. Thus, we can not only increase the quality of our teaching, but also speed up the teaching and learning process. Here is a humble opinion of mine: There will be times when the difference between a fair math teacher and a good one will depend on how competently that teacher employs technology in the classroom. Believe me, my dear friends, those times have already arrived!

Should the United States go metric?  What role do math teachers have to play?

My husband recently decided to renew our subscription to TIME magazine.  While I was enjoying a Sunday morning cup of coffee, I picked up the latest edition.  As I was reading about the expansion of O’Hare airport (p. 12), I noticed an unusual sentence:  “…gobbling up land in a 300-acre (120 hectare) ‘acquisition of area’….”

Here are some other metric friendly sentences that also caught my attention:

“The Kisling’s grow more than 3,000 acres (1,200 hectares) of hard red winter wheat…” (p.89) 

“…the main ingredient in instant noodles [is] produced nearly 10,000 miles (16,000 km) away in a factory…” (p. 89)

“…the inventor of a 27-oz (.8 L) stainless bottle …” (p. 92)

What is so unusual about these sentences? 

I don’t know about you, but I don’t have many memories of learning about the metric system in school.  The majority of those memories were reading and memorizing countless tables of conversions in high school and college science. 

Until very recently, articles from newspapers and magazines (and even textbooks!), only gave measurements in US Customary units.  When I read the articles in TIME magazine this morning I was surprised to see the variety of measurements noted in metric!  When did this start happening?

Current Reality

The debate about using the metric system versus the U.S. Customary system is not a new one.

The United States is now the only industrialized country in the world that does not use the metric system as its predominant system of measurement (Liberia and Myanmar are other countries who have not officially converted to the use of the metric system). 

You can go to the store and buy a 2 liter of soda as compared to a gallon of milk.

In the final report of the study “A Metric America: A Decision Whose Time Has Come,” the National Institute of Standards and Technology concluded that “the U.S. would eventually join the rest of the world in the use of the metric system of measurement”. [2]

The National Council of Teachers of Mathematics position on the use of the US Customary and the metric systems is:

“To equip students to deal with diverse situations in science and other subject areas, and to prepare them for life in a global society, schools should provide students with rich experiences in working with both the metric and the customary systems of measurement while developing their ability to solve problems in either system.” [3]

A Sign of the Time (no pun intended!)

If what  NIST says is true, and from what I observed today in TIME magazine, perhaps the United States is catching up to the rest of the world.

Ironically enough, the feature article of TIME for September 22, 2008 is “21 Ways to Fix Up America.”  

References:

[1] TIME.  September 22, 2008. 

[2] The United States and the Metric System (LC 1136). National Institute of Standards and Technology.  http://ts.nist.gov/WeightsAndMeasures/Metric/lc1136a.cfm  

[3] http://nctm.org/about/content.aspx?id=6346

Technology can help you teach math more efficiently. Too often students engage in paper and pencil tasks when they could be actively engaged with math concepts.

Recently, I observed a middle school math lesson. Students were sitting quietly, following the directions of the teacher. They were drawing a coordinate grid on a piece of graph paper. It took 15 laborious minutes for the completion of this one coordinate grid. Here are the steps they took:

• At first, students were confused on the direction their paper should be in…..horizontally or vertically (the teacher did give directions about this). 
• Then students were meticulously counting the number of squares on the graph paper to find the
  exact middle (both horizontally and vertically) in order to draw the x- and y-axes.
• The teacher needed to work with many students in order to correct the placement of the axes…
  the counting took place again.
• Once the grids were finally set up, students began plotting ordered pairs.
• The last step of this “activity” was to connect the ordered pairs to find out what picture resulted.

The purpose of this activity was to practice locating points on a coordinate grid (as a review to get ready for work with scatter plots and transformation of shapes). But, why spend 15 minutes drawing one coordinate grid and then using paper and pencil to locate the ordered pairs? What’s the better alternative?

Many teachers might say, “Well, just print out the graph paper with the coordinate grid already drawn.”

Yes, this will save time, but when you really think about it, does this solution allow for maximizing instructional time with meaningful activities and learning?

Take a look at these options:

December Lites using the TI-73 Graphing Calculator

In this activity, students will graph ordered pairs in order to create a picture of a candle. I particularly like the activity because of it connection to the timeless connect-the-dots activities so beloved of younger children! But, here, technology, through the use of a TI-73 graphing calculator, enhances the experience. Once created, students are then challenged to recreate the picture by changing the ‘rule’ of the ordered pairs.

Teaching Coordinate Graphing with Microsoft Excel
This lesson plan idea provides teachers a way to use technology to create meaningful activities that connect to the real world. Step-by-step directions are given on how to import an outline of a state (or country) map into an Excel spreadsheet that serves as a coordinate grid. Student can be asked to identify certain map features or draw in features using coordinate locations.

And for something cute…

Catch the Fly

Flies move along a coordinate grid and settle down in one location.  Students identify the fly’s location and when correct, a frog releases his tongue to capture the fly.

Locate the Aliens

Students have 90 seconds to locate as many aliens as possible on Planet Algebra.