Problem Solve with Presidential Data

Problem Solve with Presidential Data

Sue Pawula

Summary

Presidential data is the source of a math data lesson integrated into the content area of a Social Studies classroom, linking students to the real world.  Students were expected to (CCSSI) understand a set of data, use measures of center and variability for numerical data, and find probabilities of compound events.  In other words, they gathered data based on presidential records and did bell work involving creating measures of central tendency such as mean, median, and mode.  Working in cooperative groups where they each held specific jobs, students were asked to study eight or nine presidents and determine their age at inauguration.  In order to calculate that they needed to subtract the president’s birth year and date from the inauguration year and date factoring in the month and day in which the presidents were born. 

The next step was creation of a stem-and-leaf plot where students were in charge of putting sticky notes with their particular presidents name and age.  The “age” sticky note leaves were placed into a graph, which depicted a range of tens representing from 40 to 70 years.  If there were identical leaves, they were place horizontally to allow for a better mathematical picture of the data collected.  These leaves had to be moved around as new leaves were placed on the graph to insure accuracy of the data.  Students then discussed the data and drew a collaborative graph.  This allowed for students to confer and reason mathematically concerning placement of the leaves on the graph.

Data was then analyzed for the median, mode, and mean.  The mode was the easiest measure to identify by just looking at the stem-and-leaf plot graph.  Students counted sticky notes to find the median but later, through discussion decided that it would be easiest to just total the number of data entries and divide by two to find out which one would be the median.  Then students were allowed to use calculators to find the mean.  However, they discovered the inaccuracy that can occur from trying to add all the numbers at once and agreed it would be better if they solved this by only adding a few at a time.  One of the teachers made a suggestion that they enter the data into Microsoft Excel in order to keep track of the numbers.  At the end of class, students filled in exit slips where most indicated that they understood the day’s lesson. 

This lesson was continued through a tie into literature the next day when the teacher read a story and students collaboratively created graphic organizers to illustrate the distribution of presidents’ ages. Student groups created different representations of the data with one group making an ascension bar graph while another group created a circle graph. Plans were made for future lesson incorporating more technology and student creation of histograms to show data.

Reflection

This article demonstrates that teachers do have the ability to create cross-curricular lessons through which different subject areas can be integrated to increase student motivation and understanding.  By allowing learning to move forward through student groupings and discussion, teachers can demonstrate best practices as was done in this instance.  Taking real world information and making it come to life for the students and allowing them to create graphs from the data they created gives them a real grasp and connection to how to record data and what it means.  When these students need to look at other graphs recording data on a different topic, they will have a much better understanding of what they are observing and how to interpret it.

Patterson, L. G., & Patterson, K. L. (2014). Problem solve with presidential data. Mathematics Teaching in the Middle School, 19(7), 406-411.

Linking Lego and Algebra

Linking Lego and Algebra

Sue Pawula

Summary

            Using legos as a medium for student connection to real world commerce, investments of time and intellect while looking for solutions, use of information from multiple sources while looking at a variety unique perspectives, collaboration development, and solution location, as well as interpretation of the solutions in context to the problem, is a model that will help students connect the classroom to real life situations. Students were required to think concretely and abstractly while discussing, exploring, and investigating variables related to the different combinations of solutions available.  They were encouraged to track the relationships of their variables by using a table representation.  Student discussion and analysis moved the project forward as they were asked to look at the relationship of the quantities of legos needed to make maximum profit if they were to produce sets of legos that could allow consumers to build “ducks” and “dogs” as a product line.  Students also were asked to speculate on how an additional lego piece could affect the profit margin.  The term “marginal value” was introduced in respect to adding another piece and reflection was encouraged on how that would affect the final profit dependent on the cost of the additional piece.

Using computers, students entered data in spreadsheets on a different lego scenario and connected its elements to the context of the problem.  They were introduced to the term “objective function” and told that their objective was to generate the most profit from their lego product line.  They also found that spreadsheet cells contain formulas that are actually algebraic representations and shown their correlation to an equation.  This was and extension of the table to actual algebraic representation. 

Next students were asked to discuss each other’s reasoning and look at multiple combination scenarios to evaluate.  They were surprised that they could work out solutions on their computers and that it enhanced their exploration.  The technology allowed the students to be introduced to algebraic formulas that are built into spreadsheets and to see how these could be used in real-life connections.

Finally, the framework for this exercise provided an authentic task and a modeling cycle for students to build real work connections, identify variables and relate variable in real problems, and develop varied solutions through reasoning alone.

Reflection

The student connections developed and described in this article were very complex.  It is amazing that taking such a basic and simple toy like a lego and giving students only a few scenarios with variable to explore could result in such a plethora of idea generation.  Students said it was challenging and made them think and consider other perspectives, which is what we want to hear from our students.  This truly allowed them a to participate in a project that could be segued into real world situations.  The help provided by technology, surprised students and at the same time allowed them to see that spreadsheets are set up like algebraic equations.  I think students really bought into the entire experience.  I think the only caveat that classroom teachers might have would be related to the amount of time it took to conduct the lessons and the students buy in that is necessary to accomplish all that was done for this article.

Ozgun-Koca, S.. A., Edwards, T. G., & Chelst, K. R. (2015).  Linking lego and algebra, Mathematics Teaching in the Middle School, 20(7), 400-405.

A Balancing Act: Making Sense of Algebra

A Balancing Act: Making Sense of Algebra

Sue Pawula

Summary

            This article demonstrates ways that students can use balance scales, bar diagrams, and shared mathematical reasoning to develop deeper understanding of equality and variables in relationship to equation creation.  Students need to develop their skill using numbers in computational procedures and operations through making sense of problems, reasoning abstractly, and modeling their thinking using graphs, tables, diagrams, and more.  They need to explore the structure of the numbers and create dialog to justify their thinking as well as analyze others reasoning in a respectful way.  Understanding equality, by making sense of problems, justifying their reasoning, and considering others tendered solutions, allow students to gain a stronger hold on how to use and interpret the equal sign when they see it.  By showing their reasoning using a variety of different visual models, they also aid others in their class through access to diverse solutions that might be the particular lenses that these other individuals can see and learn more effectively through.  Compensation strategies were also discussed to help students build mental computation fluency as well as reasoning abstractly and quantitatively.  Final focus of the article was student knowledge development on interrelationships among variables as well as the effects of operations on the variables.  Overall skills were focused on helping students become better mathematicians in equality and build deeper understanding of variables and their relationships in equations.

Reflection

            This was a very intriguing article to read since many of the practices discussed in it are also practice in the early elementary classrooms.  The idea of using scales with objects on them to show students what it means to be equal is a very powerful visual.  Also, the use of the bar graph to depict the differences in the number values is another visual that is meaningful and easier to draw as the numbers become bigger.  Justifying answers and being able to explain and defend how they came up with their answer can help many other students, as well as developing the ability to listen to someone else’s reasoning of how they developed their answer and view it through another mental construction, is so valuable in providing students with a variety of strategies to use when solving real life problems.  Finally, letting students “play” with the variable can be like letting them play at putting a puzzle together.  That can be such a difficult skill for them to make the associations necessary to understanding.  Building deeper understanding of equality and variables will give all students a strong base on which to build their Algebraic foundation skills.

Gavin, K. G., & Sheffield, L. J. (2015). A balancing act: Making sense of algebra. Mathematics Teaching in the Middle School, 20 (8), 460-466.

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Sue