# Intersection of Math & Photography

Geometry plays an integral role in the composition and interpretation of photographs. With mobile devices becoming more and more ubiquitous, photography is perhaps one of the most accessible disciplines in which students can explore and apply the concepts that they learn in the classroom.  Ask a student to define parallel lines and they may say something like “lines that never touch,” and in their head they are visualizing this:

I wonder if students would have a different perception and appreciation for mathematics if they saw it as a tool to create a story. What if students, equipped with a camera or their mobile device, were asked to go find and analyze a scene with parallel lines?  Perhaps, they would take a picture like the train tracks above, where even though we know the tracks never touch, they appear to intersect as they disappear into the depth of the landscape. What properties of parallel lines help generate perspective? Why do converging lines seem to create optical distortion?

Although you probably will not see any discussion about photography in your math textbooks, lines are used by photographers to create mood and elicit emotion in their work.  What kinds of lines are apparent in the photograph below? How do parallel lines and intersecting lines help the photographer tell their story?

Horizontal lines typically tend to portray a sense of stability and consistency. Using horizontal lines, for example, photographers can convey a feeling of rest  or the message that there is a lack of change in a scene. Objects that appear horizontally in a photograph can serve as a dividing line in the composition or provide an anchor to the picture’s subject. Vertical lines, on the other hand, help to give interpretation to the mood.  These lines elicit powerful emotions and convey strength, often providing a sense of length or height. The photos below use both horizontal and vertical lines, creating a contrast between stability and strength.

Diagonal lines draw the viewer’s attention to the subject of the picture. Photographers use diagonal lines to create a point of interest. Depending on the direction of a diagonal line, it can even portray movement.

Tell your Geometry teacher that a line is curved and he might pull his hair out, but curved lines in photography can have an even greater impact on composition. Consider the railing that moves from the lower right across the center of the scene below. Although the line is not straight, it is parallel to the far bank and helps to frame the movement of the river towards the point where it vanishes.

Finally, angles formed by intersecting lines can also make a scene more complex depending on the viewpoint of the camera. The relationships between these lines and the angles they form can help students give meaning to the mundane problems of parallel lines cut by a transversal.

Photographers and artists enjoy what they do because they use their work to tell a story. I wonder how we can create opportunities for students to use mathematics to tell their story.

Photos by Thong Vo, Marija Hajster, Demi Kwant, and John Canelis.

# Appropriate Use of Technology

The drastic increase in the use of technology in our everyday lives over the past decade has altered our habits and interactions significantly. As with many changes throughout history, our adaptation in the math classroom seems to be lagging behind the rest of society. We continue to limit the impact of technology on student learning by continually treating the classroom as a secure testing site! The potential of graphing calculators to enhance and extend students’ conceptual understanding is seldom realized if they are not used appropriately. Standard for Mathematical Practice 5 implores us to consider all available tools.

Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.

Using a graphing calculator for mere computation and generating an occasional graph to be copied onto a student paper does not provide any insight into the underlying mathematics of a classroom task!  So what are appropriate uses of graphing calculator technology?  Among other reasons, graphing calculator technology should be used to

• extend beyond what could be done without technology,
• make connections among concepts as well as to real world phenomena,
• and incorporate multiple representations.

A Tool

Why do we buy tools? Besides Father’s Day gifts, we get a tool when we need to perform a task that could not be accomplished without it. A graphing calculator allows us to dynamically repeat a process multiple times within a short period in order to look for patterns or repeated reasoning. Here is one such example.

Determine the effect on the graph of f(x) by replacing it with g(x)=f(x+c).

The calculator allows students to explore what happens by changing the value of c, as well as confirm their conjecture by testing it with different functions f. This process would take an impractical amount of time if students had to graph each individual case separately.  Even more powerful, using a dynamic calculator such as Desmos, allows students to visualize these changes live. (Click each image for samples.)

Capitalizing on Media

Our access to digital media continues to increase daily.  From photographs to videos to live streams, students’ ability to not only access media, but create it as well, has added new ways of engaging students in their own learning.  In its simplest form, students can explore the key features of a function, by trying to generate a graph that matches a structure in a photograph.  In minutes, a student can import a picture from their phone or the web into a graphing utility to determine the effects of changing the coefficients of the equation.

Multiple Representations

Understanding the connection between verbal, numerical, algebraic, and graphical representations of a concept will enhance student’s understanding and their willingness to tackle novel problems. Looking at different forms of a quadratic function, for instance, gives meaning to the structure of each representation.