Co-Creating Solutions. A CTL-sponsored blog to foster and encourage professional conversation about teaching and learning.

I’m always trying to think about how I can bring activities together thatachieve the goals of getting students to use math, things like: measure,

gather data, use the data to design something or predict an outcome, and have some kind of application that might be engaging to them. As I was walking my dog this morning, I started thinking about (don’t ask me why, I just did) having students draw a human figure. It’s a skill that is hard for me, but it’s not difficult for artists, because they have explored and memorized the expected proportions for “average” people. Check out Sherri’s blue sausage people if you don’t believe me. When I talk with Catherine or Sherri about drawing a person’s face they start with identifying the basic shape of the face, and then find the mid-line and half-way point.

Interestingly, the half-way point is approximately where the eyes go. In fact here are some guidelines for drawing a human face:

• Eyes: The eyes are always halfway down, between the top of the head and the bottom of the chin. They are also an eye-length apart. This means that, however long you decide to make the eyes, there will be that much space between the eyes as well (yes, break the horizontal distance into thirds; who knew artists did so much rational number thinking when they drew).
• Nose: The rest of the face underneath the eyes is divided into thirds (breaking the bottom half vertically distance into thirds or sixths). At the one-third line will be the bottom of the nose.
• Mouth: The next third, or two-thirds of the way from the eyes to the chin, should be the mouth. The mouth’s edges should be in line with the middle of each eye. To check this, put your pencil on the middle of one of the eyes. If the lower part of the pencil touches the outer corner of the mouth, it is aligned correctly.
• Ears: The top of the ears line up above the eyes, on the eyebrows (It doesn’t say in this set of guidelines, but I wonder what the ratio of the top of the eyebrow is with the overall face? Hmmmm…).  The bottom of the ears line up with the bottom of the nose.

Here is a sketch of a face with places to record measurements for all kinds of different distances. (Note the image doesn’t necessarily reflect the estimates above so that students need to use their own data to identify those ‘averages’.)

With these two ideas (drawing bodies/faces & measurement) in mind I think it is interesting to have students gather all kinds of measurement data (see my previous post on Measurement in the POS for KY), identify the means of the different measurements and use the measurements to create ‘average people’ or the ‘average face’.

This activity is a simple concept, not overly exotic, can be pulled off in relatively short amount of time, and opens up other opportunities (which is the part I think has great potential for extension/connection, but alas a post for another day).

When working with middle school students there is often a concern about being different. Obviously, this old man isn’t too concerned about fitting in anymore, but an alternative to measuring their own faces is to have students gather pictures of other people’s faces and measure those. One thing I like about this idea is that students can measure the attributes lots of different ways- measure from a hard or paper copy, measure electronically in the paint application on the computer using pixels as units (see next month’s blog post about specifics), measure using metric or English standard (or both if you want to look at the data from a bivariate perspective). It’s pretty easy to imagine students taking their own pictures to measure and makes sense to me, but both options are great ways to get students measuring.

I know this isn’t an original idea because artists have been doing this for centuries, but I think it is time mathematics teachers got in on the action.

In adding to the conversation regarding the use of the number line for modeling and reasoning quantitatively, what kinds of modeling on the number line would support the high school student?

For showing multiple representations of Real Numbers, the number line serves as a model for locating those Irrational Numbers and displaying their relationship with more familiar Rational Numbers. The locations of these Irrational Numbers can also be completed using constructions.

See Irrational numbers such as pi, e,√5, ∛6, …

For solving functional equations and inequalities, the number line model supports using reflections, slides, and change of scales for those solutions. Additionally and importantly, this conceptual understanding works for all function families (linear, quadratic, exponential, etc.) and thus enables students to think holistically in solving equations from those same functional families.

For solving the absolute value inequality: |2x – 1| > 5

The question remains, if the number line provides a visual model for students to display and operate with numbers and variables, why do we not see a number line in every classroom where it is prominently displayed, used, and referred to at all times?

The Council of Chief State School Officiers (CCSSO) and the National Governors Association Center for Best Practices (NGA Center) recently released the draft Common Core State Standards (CCSM) for English/language arts and mathematics for review and feedback. The website for accessing those standards is http://www.corestandards.org/. These standards define what students should know and be able to do from K – 12 with the hope that high school students graduate with the knowledge and skills needed to be productive in the 21st century.

Kentucky was one of 50 states who participated in the development along with national organizations, educators, and other content experts.

For mathematics, there are eight Standards for Mathematical Practice that provide a process of thinking mathematically about concepts and applications of those concepts. They include:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critiques the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

In my review of the interface of these mathematical practice standards interfaced with the content of number, algebra, geometry and measurement, probability and data there are several things that occurred to me as I connected recent experiences as a consultant working with teachers in a hybrid algebra research project and mathematics/science statewide project. They include how we provide opportunities for students to model with mathematics for making sense of the problem solved and to reason quantitatively.

The number line is one of the best models to use in mathematics in that it allows for students to get an image in their head as to where the numbers are, how that number compares to other numbers, and understanding operation with numbers and variables. Additionally, movement on the number line supports the kinesthetic learner by allowing manipulations while solving the problem. The use of various applets assists in these investigations.

For modeling with mathematics, the use of the number line to model operations, solutions, and Real Numbers is a standard of practice from grades K – 12. No matter what grade, the writers of the Standards support the use of the number line to assist students in visualizing relationships (size, distance from zero, equivalence, inequality), identifying quantities, and operating with numbers and variables.

For showing multiple representations of rational numbers, the number line serves as a model for locating ½ and displaying the various forms of that fraction: ½ = .50 = 50% = two quarters, five dimes =

Also, an illustration from http://www.green-planet-solar-energy.com/fractions.html provides another way to use the number line for conceptual understandings of our rational number system.

For showing operations with rational numbers, adding whole numbers such as 5 + 4 and to be able to illustrate that the commutative property allows one to change the order of the numbers being added 4 + 5 and get the same answer.

(NCTM, Illuminations)

For visualizing and modeling the multiplication of 4 X 3 where the student shows movement along the number line of four groups of three.

(NCTM, Illuminations)

Investigations with the model of three groups of four, or 3 x 4 to determine if the answer is the same as 4 X3 is the logical extension? Is the operation commutative?

(NCTM, Illuminations)

For multiplication of integers, such as 3 X (-2) one could show three groups of moving two units in a negative direction.

Whereas the model for testing the commutative property for this same problem (-2) X 3 is different and requires students to understand the negative being opposite of. For this example the model is the opposite of two groups of three units to the right.

The question remains, if the number line provides a visual model for students to display and operate with numbers and variables, why do we not see a number line in every classroom where it is prominently displayed, used, and referred to at all time?

Our next discussion will focus on expanding the use of the number line to help students create a visual schema for solving equations and inequalities.