# 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.

# Inequalities: All Rules Are Not Created Equal

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.

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.

# An Unnecessary Distributive Mess!

Teacher: What’s the first thing we do when we see parentheses?

Students: Distribute!

Delighted Teacher: Yes! Distribute!

Add this to the long list of well-meaning techniques taught to students that create an inefficient road block later on.  Blindly using the distributive property before considering the structure of an expression may be a very inefficient technique.

“Cluttering heads with specialized techniques that mask the important general principle at hand does the students no good, in fact it may harm them.”  ~Jim Doherty in Nix the Trix by Tina Cardone and the MTBoS

Consider the equation: 5(4 + x) = 25.

Wouldn’t it be more efficient to rewrite this equation as 4 + x = 5 than 20 + 5x =25?!  The goal is to isolate the variable not to clutter it even more with an increased potential for errors.  Compound the problem even more with a literal equation such as:

Solve for n:  a(n + ab) = c

And what if there were fractional coefficients? An unnecessary distributive mess!

Standard for Mathematical Practice 6 states that mathematically proficient students are efficient in their calculations. The National Research Council’s report Adding It Up includes efficiency as an integral part of procedural fluency.  Students should see mathematics as a tool, not a crux.  When students are able to use the structure of an expression as part of their decision making, they develop a facility in problem solving that transcends the silly techniques that only serve to limit us.

Let us strive to equip students with the skills and habits that will make them thinkers, rather than robots.  Machines become obsolete over time, but thinkers can adapt to new and different situations, creating new innovations along the way.

# Purpose of Homework

As with any assignment, the question of whether homework is essential depends on its purpose. Homework that helps to activate prior knowledge and prepare students for an upcoming lesson serves a much different purpose than one that is purely busy work. One well thought out problem may be enough to help students apply what they have learned in a different context, although it might not be conducive to developing fluency in a particular skill. An assignment that simply practices a skill learned in class may help develop confidence and procedural fluency in younger students, but be perceived as mundane and promote apathy towards mathematics in older students. The impact of homework can be both positive and negative depending on the nature of the assignment, the age of the student, and the time it takes to complete.

In considering whether or not homework should be graded, we might first revisit the purpose of grades in general. Grades should be reflective of a student’s level of mastery towards specific learning targets. To this end, homework should either not be graded or should be weighted so minimally that it will not misrepresent a student’s level of mastery. On the other hand, there are instances where students will not complete homework if it is not graded. In this case, one might contend that either homework has not been purposeful or has not contributed to student learning.

Like so many other things in education, there is probably a right time and a place for homework, and finding it, is a juggling act that must be purposeful and strategic if it’s going to have any impact on student learning.

Other questions to consider:

•  We spend a lot of time talking about differentiated instruction. Should homework assignments be differentiated and tailored based on students’ readiness level?
• Grades are summative in nature. Is homework summative or formative?
• Feedback has been shown to promote student progress. How can we provide feedback to students on homework without assigning a grade?

# What’s the Price of the Cup?

When students learn about a concept by using it in a realistic context, they are more likely to understand and remember what they have learned.  In this activity, students gather data regarding the number of ounces and price of drinks at two fast food restaurants or coffee shops. They represent this data on a scatter plot, and describe how the variables are related. Using a regression model, students calculate the cost of a drink that is 0 ounces, in an attempt to determine the price of an empty cup.  Based on their findings, students explain why an empty cup is not free, and why it costs more at one place than the other.

The task is structured to follow a mathematical modeling cycle, requiring students to identify relevant variables, formulate a model, use the model to answer questions, and interpret their findings considering possible revisions to the model along the way.

In what ways might you improve or enhance this activity?

# 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.

In what ways do you encourage students to make decisions about the tools they use, an more importantly, when it is appropriate to use those tools?

# The Power of Counterexamples

Counterexamples can be used to dispel misconceptions, deepen conceptual understanding, and help students construct viable arguments.  Standard for Mathematical Practice 3 states:

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples.

In the traditional sense, a counterexample is a specific case that shows that a statement is false. First, let’s take a look at the impact of using a counterexample to a student’s misconception.  Consider the following scenario.

Mr. Gebra is teaching a lesson on simplifying radical expressions. He models the following examples before assigning students some problems to complete independently.

Mr. Gebra notices that several students have the following response to the first problem on their assignment.

Several questions come to mind…

• What misconception could possibly be leading students to this incorrect response?
• How might our choice of model problems contribute to students’ misconceptions?
• Could we avoid misconceptions by anticipating students’ thinking and strategically using counterexamples?

Secondly, we can also incorporate the use of counterexamples in learning tasks to help students explore the truth of their conjectures.  Students benefit from generating counterexamples to a false statement.  Consider the following statement:

Division makes numbers smaller.

In determining weather this statement is true or false, students might generate counterexamples. Presented with a falsehood, students can then “fix” the statement to make it their own conjecture.  How would you revise the statement to make it true? Students can test the validity of their conjectures and critique each other’s reasoning.  It is through this process, that opportunities for learning emerge.

Benefits of using counterexamples include:

• Deepens conceptual understanding
• Deemphasizes the reliance on tips and tricks
• Reduces or eliminates misconceptions
• Useful tool for constructing viable arguments
• Provides a nonthreatening method for critiquing the reasoning of others
• Encourages students to practice using the language of mathematics
• Discourages the use of mathematically incorrect overgeneralizations or imprecise vocabulary