Ed Tech 4 Math

At the beginning of the school year, one of my fourth grade teachers, asked me if I would come to her class and introduce the Product Game to her students as a way of having some fun while practicing multiplication facts (we’ve done this for the past 3 years). 

Each time I visited or walked through Miss Rogowski’s classroom this past quarter, we did a little update with the students on the progress they were making with working on their multiplication facts and built up the excitement of me coming to class to “play a fun game” with them and the ice-cream party they would have at the end of the year as a celebration for their mastery of multiplication facts.   It doesn’t take much to excite fourth graders!  It got to the point that when I walked in, some students would mention right away where they were…one student in particular continually reminded me that he had already memorized all his 12s! 

Yesterday marked the end of the first grading quarter.  So, Miss Rogowski and I decided that it would be a perfect day to play the Product Game.

The object of the game

The Product Game is fairly easy to play.  Get four squares in a row (horizontally, vertically, or diagonally) by using the factors 1 – 9 to create products located on the game board.

NCTM’s Illuminations website [1] features the online applet that the Connected Mathematics Project created to use the Product Game online.  There are no bells and whistles to this applet, but it’s worth a mention and recommendation to all math teachers because the game is fun – I can attest to that! 

I first came across the Product Game when I taught sixth grade math using the Connected Mathematics Project [2] textbook series.  The game was part of the Investigations in Prime Time: Factors and Multiplies (1996), the book focusing on Number Sense (GCF, LCM, prime numbers, etc.).  I always had success with students in playing the game and couldn’t see just using it in sixth grade.  I’d recommend the game to any group of students who need a way to practice their multiplication facts!

Adaptations to the game

The Connected Mathematics Project revisits the Product Game in the seventh grade series in Accentuate the Negative (1996).  This time, positive and negative factors are used to play the game.

Teachers can easily enough change the format of this game to fit the needs of their students. You can recreate the game to fit the needs of any students:

• Change the factors to be used to 4 – 12 (don’t forget to change the products in the game board!)
• Make the game board smaller and decrease the number of factors for students who are struggling.
• Go back to the good old days of playing bingo and use some of the variations of that game to win:  four corners or blackout.
• Play in trios

Please feel free to post any other ideas on how to adapt the Product Game to fit the needs of students!

References:

[1] National Council of Teachers of Mathematics.  Illuminations. The Product Game.

[2] Prime Time: Factors and Multiples. Connected Mathematics Project. G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips. Dale Seymour Publications (1996), pp. 17‑25.

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.

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.