Blog #18

Provide opportunities for students to make connections between different math topics and see the big picture of mathematics.  In blog #10, I wrote about the importance of interdisciplinary connections to demonstrate how math is interconnected with other subjects and the real world.  This week, for my last blog of the semester, I want to touch again on connectedness, but this time within math itself.  All too often, students see math topics as stand-alone silos of information.  Once the assessment is completed, students believe they can “move on” to the next topic and have no need to recall what they just learned (what we sometimes refer to as “flushing”).  As math teachers, we understand the sequential nature of mathematics.  We know the topics that students master today will form the foundation upon which they will build in future chapters and courses.  Therefore, it is important that we make the connectedness of mathematics evident to our students.  This will provide a starting point for learning new topics, will increase understanding of math from a “big picture” perspective, and will hopefully avoid “the flush” at the end of each chapter or unit.  Here are a few ideas for how to emphasize the interconnectedness of math in natural ways that make sense and should be easy to implement.

Remember when…

As a way to “hook” students for a new topic, I like to recall previous knowledge that they should have already learned.  For example, when learning about rational expressions, I like to ask students “Remember when…you learned how to add fractions in 6th grade?”  Then I will have an addition of fractions problem and an addition of rational expressions problem (or whatever operation you want to focus on) side by side and explain that we will use our knowledge of fractions to learn about rational expressions.  I ask them to share the first step for adding the fractions.  This is a great opportunity to do some “just-in-time” review, if necessary.  But most students at least realize that they have seen the process for fractions before and can begin to recall what to do.  Once we establish the first step for fractions, then I move to the rational expression and we attempt to do the “same” step in this new context.  Then we determine the next step with fractions and try to apply that to rationals, continuing until we reach a simplified answer.  Students realize that adding rational expressions is not terribly difficult because they already “know” the process.

Another example of “Remember when…” I like to use is during the unit on conic sections.  In order to determine characteristics for the graph of a conic section (circle, parabola, ellipse, or hyperbola) the equation needs to be in standard form.  But sometimes the equation is given in a general form, with what appears to be a mess of x and y terms all over the place.  In order to graph the conic, or even determine which conic the equation represents, it would be helpful to rewrite the equation into the standard form.  So I ask students for ideas…what do they have in their skills toolbox that would help us go from the general form to the standard form.  Usually at least one student sees that we need to use the Completing the Square process.  Then we walk through the steps, again with “just-in-time” review as necessary, to rewrite the equation in a more useful form. 

These examples show that emphasizing the interconnectedness of math does not need to change the lesson plan for the topic.  It simply takes being intentional about pointing out the math skills that students already have which they can apply to the new idea.

Different Representations

Another way to demonstrate the interconnectedness of math is to emphasize different representations of the math concepts students are learning.  The four representations I focus on are:

• Verbal Representation:  Verbal representation involves expressing a math problem using words.  This method helps clarify the problem’s meaning, context and parameters.

  • • Example:  Portia currently has $25.  She receives a $5 allowance each week for completing certain chores around the house.  Portia is saving for an Apple Watch SE.  The one that she wants is $249.  How long will it take Portia to save enough money?

    • Significance:  Verbal representation allows students to understand the problem in real-life terms and enhances their ability to communicate mathematical ideas clearly.

• Numerical Representation:  Numerical representation uses specific numbers and values to convey information about the problem.  This representation uses data points, measurements, or constants.

  • • Example:  From the verbal example above, a numerical approach could be to make a table.
      
      Number of Weeks Total Amount Saved

0 25

1 30

2 35

3 40

Students can continue this table until they can see how many weeks it would take to reach at least $249.

• • • Significance:  Numerical representation provides concrete values, which can help some students perform calculations and analyze relationships between quantities, especially if they are struggling with an abstract concept.

• Algebraic Representation:  Algebraic representation involves using symbols and variables to create equations that describe the relationships in the problem.

  • • Example:  Continuing with the example above, students may recognize that $25 is what Portia started with, and then she gets an additional $5 each week to add to her total.  So the Total Saved and the Number of Weeks are related in the following way:  T = 5W + 25,  where T is the total saved and W is the number of weeks.

    • Significance:  Algebraic representation allows students to manipulate equations to find unknowns, facilitating problem-solving and deeper understanding of mathematical relationships.

• Graphical Representation:  Graphical representation involves creating visual models, such as graphs or diagrams, to illustrate the relationship between variables.

  • • Example:  Using the equation from the algebraic representation, or by plotting points from the numerical representation, students can graph the relationship between total amount saved (dependent variable) and number of weeks (independent variable).  The graph will show things like y-intercept (where Portia started), the slope of the line (how “quickly” Portia is saving), and an overall idea of long it will take to get to the $249 goal.

    • Significance:  Graphical representation helps students visualize mathematical concepts, understand trends, and identify solutions or characteristics of equations at a glance.

Even when a lesson in the textbook may be focusing on one of these representations, try to interject some of the other representations where it makes sense.  By showing students how these four representations are connected to each other, students gain a more comprehensive understanding of the concept.  Each method reinforces the others and provides different insights, enabling students to approach problems from multiple perspectives.  Emphasizing these different representations encourages flexibility in thinking and enhances problem-solving skills, preparing students for real-life applications of mathematics.

Problem Solving/Mathematical Modeling

I have said it before, and I am obviously going to say it again.  The best way to demonstrate the interconnectedness and richness of mathematics is to provide students with authentic problem-solving opportunities.  Especially as students progress through the math curriculum and courses, the types of problems that they can reasonably handle will likely require more than one mathematical concept to solve the problem.  Additionally, authentic problems allow students to choose the types of representations they want to explore.  Here is one example that may seem simple, but has many layers to it:

Designing a Garden

You are a landscaper who has been hired to design a rectangular garden.  Based on which plants the owner wants to include in the garden, she has determined she needs 400 square feet of space.  She also wants to put a fence around the garden to protect her plants from being eaten by animals.

Questions (Round 1)

1. If you want to minimize the amount of fencing used to keep the cost as low as possible, what should the dimensions be?
Students can make a table of possible dimensions with corresponding perimeter (geometry).  Or they could determine expressions to represent the two sides of the rectangle and write an equation for how perimeter is related to the sides.  Students can graph the equation (graphing rational equations) to find a minimum point, understanding that the negative part of the graph is not useful in this context.

2. What assumption are you making in order for this to be the best answer?
The solution to the first question is to create a 20 ft by 20 ft garden with 80 ft of fencing.  However, this assumes that the homeowner has space for a 20 ft by 20 ft square for their garden.  Students can submit their answers this far to receive feedback.

Questions (Round 2) – provided after the answer to Round 1 is determined

1. You go to the hardware store to buy the fencing for the garden.  You discover the store has a special if you buy 100 feet of fencing.  This special price is actually cheaper than buying the 80 feet that you determined in Round 1.  Can you still meet the homeowner’s requirement for 400 square feet if you buy the 100 feet of fencing and use all of it?  If so, what would the dimensions of the garden be?  If not, explain why not?
Students can look on their tables or at their graphs to find that an x value of 10 still creates 400 square feet of garden.  The dimensions would be 10 ft by 40 ft for a perimeter of 100 ft.

2. Is this the largest garden you can design with 100 ft of fencing?  If so, explain how you know.  If not, what are the dimensions of the largest possible garden you could create?
Again, students can choose to approach this numerically with a table or graphically with an equation (graphing quadratic equations).  They will realize that the numerical approach is potentially more work, which can be a great opening for a discussion about advantages and disadvantages of different approaches in different contexts.

This problem does not seem very complex at first glance, but asking students to think about the problem from two perspectives adds depth to their understanding.  It is a problem that allows different approaches for students to explore.  And in the end, students will use geometry, rational equations and quadratic equations to answer the questions. 

Thank you, all, for coming on this journey with me this semester.  I hope that you have found something in these blogs that helped you clarify your own thinking, provided you with examples of how to accomplish something you have been contemplating, or gave you the confidence to try something new.  I will be back in January with more content.  Until then, may the final days of 2024 go smoothly, and may you find time to rest and recharge during your winter break.  

Happy Holidays!! 🎄🎊