Common Core Criminology

Embracing the critical thinking and applied problem solving entailed in the common core, I used a vacant classroom on our campus to create a crime scene for my students.  The students used their understanding of relationships in right triangles to not only solve a murder mystery, but to also meet the eight Standards for Mathematical Practice.

1.  Make sense of problems and persevere in solving them.  As the culminating activity in our trigonometry unit, the only suggestion I offered students was to use measurements from the crime scene to find right triangle relationships.  Many of the students were frustrated initially by the open-ended nature of this activity, however, they eventually found freedom in the fact that there was no "right answer."  
  

2.  Reason abstractly and quantitatively.  This activity challenged students to find mathematical relationships in real world situations.  The students needed to move fluidly between seeing a measurement as the width of the blood droplet to seeing this same distance as the hypotenuse of a right triangle.    

3.  Construct viable arguments and critique the reasoning of others.  By far, the most valuable element of this activity was the opportunity for students to formulate arguments and justify their reasoning using mathematics.  This activity did not involve a clean, carefully constructed scenario with a correct "solution."  Students differed in which blood droplets they chose to measure.  Some students began their measurements from the center of the blood spatter, while others measured from the first droplet.  Students used inches, feet, centimeters, meters, and even paces to measure the scene.  Together, these variables combined to create enormous variation in student conclusions.  Some students pointed the finger at a 5'3" female shooter, while others painted a picture a 6'9" gunman.  The key is that each group was able to justify their conclusion using trigonometry (and the debates got fairly heated!).  For example, the student work samples below highlight the diversity of student conclusions:  

          Mistress Murder   vs.  Crime Scene  vs.  The Bust  vs.  The Crime

4.   Model with mathematics.  Students were enthralled by the real world applications of trigonometry.  Using publications in forensic science (Jackson & Jackson, 2008), students discovered that the sine function can reveal blood spatter angles.  The students were then challenged to find their own trigonometric relationships within the crime scene.    

5.  Use appropriate tools strategically.  In order to gather evidence at the crime scene, students were challenged to select appropriate tools, juggling between rulers, meter sticks, and tape measures.  Many students attempted to use protractors, only to discover the tool is better suited to the two dimensional world.  The students also relied on scientific calculators to evaluate trigonometric ratios.    

6.  Attend to precision.  This activity highlighted the importance of precision.  Many students carefully measured distances and painstakingly labeled their crime scene map, but neglected to include units of measurement.  Partners returned to the classroom to complete their calculations only to discover that "6" might mean feet or inches or centimeters or number of droplets . . .  Most students returned to the crime scene multiple times to double check the accuracy of their measurements and to clarify their units.  

7.  Look for and make use of structure.  The central theme the students discovered through this activity is relationships in right triangles.  Prior to the activity, the students demonstrated a mechanical approach to solving trig problems.  By exploring real world scenarios, the students showed a deeper understanding that given any two measurements within a right triangle, it is possible to use ratios to find the remaining measurements.  


8.  Look for and express regularity in repeated reasoning.  Through this activity the students demonstrated increased facility with solving algebraic equations.  Initially students worked tediously through trigonometric equations such as tan 41= 5/x by finding tan 41, then writing this decimal as a fraction over 1, then cross multiplying, then dividing both sides by the decimal, and so on.  In time, the students created shortcuts for themselves, based on the location of the variable (in the numerator or the denominator) and began to manipulate trig ratios with ease.

TI-Nspire Summer Workshop

After participating in the 2012 TI-Nspire Summer Workshop, I am eager to incorporate more technology in my math and science classes.

It seems unlikely that I will be able to afford a class set of the new TI-Nspire CX handheld calculators anytime soon.  In the meantime, I am excited to begin using the teacher software that accompanied my new toy.  The Math Nspired website offers a wealth of  activities, for use on either the handheld calculator or the computer.


For example, one lesson provides students with the an image of/the formula for the area of a rectangle.  Students are then challenged to manipulate this image to derive the formulas for area of a triangle, parallelogram, and trapezoid.  Together, the visuals and the inquiry opportunities make for a powerful learning experience!


While I wait to win the lottery, I will recommend that our site technology committee purchase one class set of devices for teachers to check out of the library.  The software and web-based activities would also make for engaging whole class discussions when paired with an interactive whiteboard, allowing students to interact with and manipulate lines, figures, graphs, equations . . .    

Forensic Science (and Math!)

Using some of my summer downtime, I have been creating new lessons based on the idea of forensic science.  Over the past few years, a common problem I have run into with my teaching is convincing my students that the topics we discuss in class have applications in the real world.  I understand their doubts - at first glance it can be challenging to imagine using similarity ratios, unit conversions, or density calculations in our day-to-day lives.

Cleaning out the cabinets in my classroom, I discovered Crime Scene Investigations:  Real Life Science Labs by Pam Walker and Elaine Wood.  Using this resource as my inspiration, I created a series of lessons for integration into my Scientific Method Unit for my Earth Science and Life Science classes.  For example, next year, my students will use fiber analysis to explore using a compound microscope and to learn how to make detailed, scientific observations.  Later, they will use broken vehicle class collected from a "crime scene" to determine mass, use displacement to find volume, and calculate density.  I am confident these hands-on activities will boost engagement while also highlighting the relevance of science in the real world.

    

More exciting for me has been applying this idea of forensic science to my geometry classes!  Fortunately, geometry is a more hands-on topic than some other areas in math, but it can still seem context-reduced.  Building on my work with the science curriculum, I also created/adapted a set of application activities to supplement the pen-and-paper lessons in my geometry repertoire.  For example, working from a Missing Persons activity by Walker and Wood, I created a lesson where my geometry students will take what they have learned about similarity ratios and apply this understanding to forensic anthropology.  The students will be given a recovered arm bone and a missing persons roster and then use the proportional relationship between humerus length and height to identify the victim.  My personal favorite (though implementation will be quite an undertaking) is a lesson I generated based on blood spatter analysis.  Given a staged crime scene, students will use trigonometric ratios to re-create the locations of perpetrators and victims.  

This will be my first time using these activities based on forensic science and I will report back over the course of the year to reflect on challenges and successes . . .