Intersection of Math & Photography

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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:

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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?

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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.

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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.

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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.

Inequalities: All Rules Are Not Created Equal

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While working with students in several Algebra 1 classrooms this week, I noticed several prevalent misconceptions that contributed to errors in solving and interpreting inequalities. These misconceptions might be rooted in over generalizations of previously learned material instead of adapting learned concepts to new situations. Some of the misconceptions that contribute to students’ confusion regarding inequalities include:

  • Treating inequalities the same as equations
  • Thinking that the solution set to an inequality must be an inequality
  • Lack of understanding of the symbolic meaning of inequalities

Inequalities are the Same as Equations

Knowing how to solve equations can help in manipulating inequalities, but saying that an inequality is the same as an equation causes problems when interpreting solutions.  A student may be able to find a symbolic answer, but be unable to check whether or not an element is in the solution set. Consider an example that is often answered incorrectly:

Which of the following is a solution to the inequality 5x + 7 > 12?

  A) -1     B) 0     C) 1     D) 2

I have found that many students can easily manipulate the inequality to arrive at  x>1, only to select choice C, completely disregarding the inequality symbol.

Other examples require considering the structure of an expression (SMP7). In response to (x+3)(x-4)>0, students who have experienced solving (x+3)(x-4)=0, have no problem stating that (x+3)>0 and (x-4)>0, but may overlook that (x+3)<0 and (x-4)<0 will also generate solutions to the inequality.  In these types of problems, creating a visual representation of the inequalities can help students determine if their answer makes sense or if it should be revised.

The Solution Set Must Be an Inequality

Perhaps due to a lack of exposure to different types of inequalities, I found that some students tend to overlook that solutions could take several forms, particularly when solving a system of inequalities. Consider the following systems:

A) x – 2 ≤ 4 and x – 2 ≥ 4

B) 1+ x > 4 and 3x – 6 < -12

In the first case, the solution is actually an equation (x = 6).  In the second example, there is no solution since there are no numbers that satisfy both statements.  In both cases, graphing the inequalities simultaneously on a number line can help students visualize the problem, trying to find where the inequalities overlap.  

Symbolic Meaning of Inequalities

When reading x>3, one student told me that this was “x is less than 3” because she had been taught that the “pointy part points to the smaller number,” and since it was pointing towards the three, then it must be “less than.” More than other misconceptions that surfaced, this was among the most rudimentary.  Not understanding the fundamental meaning of the inequality symbols also contributed to some students thinking that if 2 < x < 4, then the only solution must be 3. This prevented students from understanding the concept of infinitely many solutions. In this case, asking students to generate examples of inequalities in real contexts would help them construct meaning and verbally analyze solutions to inequalities.

Addressing Misconceptions
Inequalities play an important role in students’ understanding of equality. Often, students’ misconceptions are grounded in their misapplication of previously learned concepts. For this reason, it is important to expose students’ preconceptions before attempting to build on their prior knowledge. Additionally, when students understand inequalities visually, they are more likely to perform algebraic manipulation accurately. Using examples of inequalities applied in context helps to make learning more meaningful and sustainable. Lastly, developing conceptual understanding of the symbolic meaning of inequalities, rather than relying solely on procedural techniques and “rules” that eventually expire, might have a greater impact on students’ transfer of concepts to other topics.