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What Happens Next Tuesday (transferable math strategies)?

What happens when something unexpected occurs? It is a question that I reflect on quite often, not just in teaching mathematics but in all types of situations. My son, who is autistic, is an extreme follower of rules. More than anyone I know, established routines govern his life and provide him comfort. However, when habits are broken, learning opportunities happen.

A prime example of a routine for him is the path we take to school or the store. If we take a different route, it will typically lead to frustration. At this moment, I have to make a choice as a parent:

  • Offer him a new routine or rule to make sense of the change. 
  • Allow him to struggle with strategies he has used in other situations to make sense of what is going on.  

The quick fix is to offer a new rule such as “this is the path we will now take on Tuesdays” instead of struggling with “it is ok that we take this route this one time, we have to be flexible, use some of your strategies.” The problem with the quick fix is that while it works now, what happens next Tuesday? Will I remember to go the new way?

I tell that personal anecdote as it is similar to some of the thoughts I have been having about the Algebra 1 curriculum that is currently under construction at Desmos. The unexpected happens for students in Algebra when they first encounter quadratic equations. Students spend most of middle school mathematics learning about linear patterns through proportional reasoning and constant rates of change. However, when teaching quadratic equations, there is a potential tendency to jump to various rules and define new situations for students (some of which do not work in every case). Here are just a few:

  • F.O.I.L. (First, Outside, Inside, Last)
  • Factoring Rules with Boxes or X’s
  • Standard Form
  • Factored Form
  • Vertex Form

Not saying that defined structures, rules or formulas are not an important part of Algebra. The current unit Desmos has released addresses the last three and a slew of vocabulary words related to quadratics. The build-up of conceptual understanding through patterns and visual representations is where the unit shines. It allows students to use familiar strategies instead of rules to make sense of quadratics.

For mathematicians, analyzing patterns and exploring visual representations are two transferable strategies when encountering a new or unexpected situation.

Analyzing Patterns

Devoting time to exploring the pattern of quadratic functions can be a helpful step in transitioning from linear to quadratic thinking. Students come to Algebra from years of pattern explorations in earlier grades and this should feel somewhat familiar to them, but with a new twist.

At the beginning of the unit students are asked to build a pattern and then extend it to step ten with some Desmos interactive magic.

Exploring Visual Representations

After exploring the pattern of quadratics, students have space to play with the graph. Understanding the visual representation of a quadratic in the form of a graph is pivotal to developing connections to various types of quadratic equations.

Students are encouraged to manipulate the graph of quadratic functions using the vertex, concavity, and the concept of symmetry to create some visual art.

There are a variety of approaches to teaching any mathematics content, but are we overloading students with a set of rules that may not work next Tuesday? Consider anchoring students in analyzing patterns and exploring visual representations as transferable mathematical strategies to lean on in any situation.

Side note: If you have not made some parabola art yet, highly recommended, it is a lot of fun.

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Parabola Zapper: Beauty in the Misses

One of my favorite parts of my role at Desmos is having the opportunity to engage in math with teachers. Recently, I took one of our new (and free) Algebra 1 lessons, Parabola Zapper, for a spin with several groups of math educators located in various parts of the country. Whenever I’m teaching a new lesson, the unexpected often excites me. The surprises in this lesson came from the variety of responses to the activity. In particular, the missed zaps from the teachers.

Connection to the x-axis

Early in the lesson, students (teachers in this case) have to “zap” the parabola by finding points on the graph. However, the incorrect zaps or misses provided just as much insight into student thinking as the correct ones. The equation yields students some clues to the location of the parabola. I observed several students input the value of (1,0) or (-3,0) from looking at the equation. The misses here along the x-axis demonstrate a level of understanding about the connection between the factored form of the equation and the x-intercepts.

Plugging in a point into the Parabola Zapper

Finding the Vertex Visually

Similarly, one of the students had a couple of misses surrounding the vertex. The location of the misses are signs that the student used visual understanding of the graph rather than computation to locate the vertex. Building connections, particularly early in the lesson, is more important than finding the exact location of the x-intercepts or vertex on the first try. The beauty of the misses helps shape students’ thinking about a vertex, where it is located, and why it might be potentially significant.

Varying Approaches

As a Curriculum Consultant and former teacher myself, I enjoyed seeing multiple approaches to finding the parabola.  While some students focused on the vertex and x-intercepts to light up their parabola, others used the concept of symmetry.  Given the time, it would be interesting to further dive into these two approaches.

What approach will your students take? Click the link to try it out. 

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Practice vs. Play

Processing through my dissertation data has reinforced my belief that students need to play or “tinker” with mathematics. Play is important because students are allowed to control part of the learning narrative. According to Seymour Papert, “I am convinced that the best learning takes place when the learner takes charge.” Play can take a variety of forms through manipulatives, projects, or a well-constructed task. Or it can be a simple shift in changing a practice problem into a more open question or having students interpret a mathematical situation.

Open Problem

Recently, I created a Desmos activity involving polynomial addition. Practicing adding and subtracting polynomials is a skill that is needed in algebra as it is part of manipulating the language of algebra, which can be useful in problem-solving. However, practicing a skill without exploration or connections can restrict student understanding to procedural fluency or a programmed series of steps. Programming without context is limiting at best. Below is an example of a practice problem with no context, useful in developing skills but can curtail student thinking and student input.

In the same activity, before the practice problem above, I made a slide where students move tiles into different piles to create a given sum (seen below). The shift of having students create the problem by playing with digital manipulatives offers opportunities for different strategies. There is cogitative strength in constructing mathematical connections. Students can make connections among manipulatives, algebraic notation, and quantities through play.

Interpreting A Situation

Functions are essential tools in mathematics as they allow for the construction of models to explore relationships. Defining functions and then having students practicing determining if something is a function or not or even classifying the type of functional relationship (linear, quadratic, exponential) is a valuable skill. Yet, functions without context limit student mathematical understanding. George Pólya encouraged teachers to have students interpret mathematical situations and restate them in their own words as a strategy for solving problems. Play is one way students interpret situations and can tell a mathematical story.

Below is a slide I created as part of exploring the sketch feature in Desmos. The premise is simple, “What happens when you sketch?” Students can play with different types of sketches and determine what happens. Feedback through technology helps shape student understanding through the multiple ladder racers on screen. The speed at which the racers climb, if they are going up or down, where they start on the ladder, which racer is in first place, and if multiple racers appear on a single ladder all depend on the sketches students create. This type of visual feedback is one example of how using technology to play can create meaningful learning conditions for students. When students interpret a situation, they give meaning or purpose to mathematical concepts.

While practice can support student learning in mastering basic skills and applying knowledge, play can create the conditions for student driven discovery and for student centered learning.

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Creating Desmos Lessons for Algebra

Introduction

In preparation for the school year, I was approached by one of our Algebra teachers to create a set of Desmos lessons for both face-to-face and online instruction. It is important to note that these lessons are not perfect, and I would recommend looking through the more polished mathematical experiences found at teacher.desmos.com before the ones that I have created. I post the lessons I build to contribute to the math teacher community and receive feedback to improve my craft. Following is an outline of the first unit and some features of the lessons.

Overview of Unit 1

The set of nine lessons in this unit is a review of some of the skills students will need to succeed in Algebra. The first three lessons include order of operations, combining like terms, and the distributive property, which are algebraic tools that will be useful throughout the course. The next set of three lessons provides practice in these skills in solving equations and inequalities. Students also encounter other algebraic ideas such as modeling of equations and dividing by a negative coefficient with inequalities. The unit concludes with applying the skills reviewed in the first six lessons to “word-problem” like situations and literal equations. The transition from the calculation of values to the manipulation of variables is a hallmark of student mathematical progression in algebra.

Lesson Layout

Along with building the unit from computation of expressions to manipulating variables, there are few other guiding ideas behind the lessons I created. Understanding how the lessons are structured may be beneficial if you use them in your classes.

Introduction Slide

I like to begin each lesson with a slide that is open for interpretation and is accessible to everyone in the class. Too often, teachers start class with a standardized test question that may discourage some students or turn them off to the math lesson. How would you feel as a student if you missed a problem two minutes into the class starting?

In the first video below, students see an animation of a red and yellow counter meeting and disappearing. Students are asked what do you notice? There is no right or wrong answer to the question. It is there to provoke thought and set up the lesson for the day. In the second example, Mario jumps a gap between two hills, and students can choose how they would like to express what they are observing. I love the idea of students having options on how to share their mathematical thoughts.

Red and Yellow Counters
Mario Jumping

Visual Models

One of the benefits of using a technology such as Desmos is connecting mathematical ideas to visual models or animation. In the distribution lesson, students can visualize an area model by seeing the distribution of a quantity or variable across multiple added values. This type of visualization provides a conceptual level of understanding to support the algebraic process. Visual models also provide meaningful feedback that may help anchor mathematical knowledge. For example, in the make it balance lessons, students receive feedback in the form of the model balancing or tipping one way or another based on student response. This type of visual feedback is more purposeful and powerful than just right or wrong.

Distributive Property
Visual Feedback

Analyze

A key feature of being a mathematician is analyzing a situation and discovering what is going on. I like presenting students an answered mathematical problem and having them determine if it is correct or not and why. I will often throw in a wrinkle that they may not have encountered yet to push their thinking a little bit further. For example, shown below, I ask students to put a series of steps to solve a problem in order, but an alternative way to solve the problem appears when they do. I ask them, would this also work? Students often believe there is only one way to solve a problem in mathematics. I try to think about ways to push students to move away from this mode of thought by having them analyze a variety of related mathematical situations.

Conclusion

The creation of these lessons are an attempt to support our Algebra 1 teachers in this time of uncertainty by having a set of mathematical experiences that can be taught virtually or in person. I am happy to share anything that I create to support the mathematics teacher community. If you choose to use them, please let me know how it goes and how they can be improved. You can find the entire collection here: https://kurtsalisbury.com/algebra-1/

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Exploring the Desmos Curriculum

What makes a quality math curriculum? Our school district has been fortunate enough to be selected to pilot the Desmos Curriculum, based on the work from Illustrative Mathematics and Open Up Resources. In preparation for the upcoming school year, I have spent some time going through the lessons. When it comes to curriculum, NCTM (2014) states that “An excellent mathematics program includes a curriculum that develops important mathematics along coherent learning progressions” (Principles to Action, p. 70). Coherent learning progressions value connecting content in meaningful ways across grade levels, units, and individual lessons. The linking of mathematical ideas is one of the many aspects of the Desmos Curriculum that stands out.   

Connecting Mathematical Ideas

In middle school mathematics, developing an understanding of linear relationships is a focal point for student learning. In particular, connecting the ideas of similar triangles, proportionality, unit rate, and slope serve as a  foundation for all things linear (Boaler, Munson, Williams, 2020). Water Slide, a previous activity that Desmos has redone, is one of my favorite lessons that I have worked through thus far. The experience weaves similar triangles, proportional relationships, and unit rates to build a conceptual understanding of slope.

 The lesson begins with students altering triangles to create a “smooth water slide.” What I love about the start of this lesson is the visual relationship between the triangles and slide void of any mathematical language—a welcoming entry point to making sense of slope. The premise is simple, adjust the triangles to make the slide work.  Students are successful when they can create triangles the satisfy the conditions of making a smooth slide.  Feedback is visual and engaging and gives purpose to creating similar triangles.

Bumpy Versus Smooth Slide

The lesson continues by layering numerical values to illustrate the proportional relationships among triangles and the shared unit rate.  Students enter values and see how they impact their slide, only when they create triangles with the same unit rate will their slide work.   Desmos offers little direct instruction for students but instead guides them to discover the relationships among the triangles and numbers as mathematicians.

Exploring Proportional Relationships

The learning progression’s final piece has students practice computing slope to foster procedural fluency from the conceptual understanding the lesson developed in the initial screens. Even at this point in the lesson, Desmos defines slope only as steepness. It does not try to bog students down with formulas or complex academic language.  Students build their understanding first and create a personal working definition that supports their mathematical identity. They can then apply this definition to the practice problems that follow the water slide experience. As students reflect on the learning experience and apply their new knowledge, they can see how all the mathematical ideas used in this lesson: similar triangles, proportional relationships, unit rate, and slope connect as they continue to explore the story of linear relationships. A well-constructed set of mathematical connections, such as those in this lesson, is an indication of a quality curriculum.

Practicing Calculating Slope

On a final note, I lead a group of teachers through this lesson as it is essential for teachers to experience lessons as students.  At one point in the lesson, students can create a slide with a slope they choose.  It was interesting that all the teachers tried to build the most harrowing slide possible.  I don’t know what this necessarily says about our math teachers, maybe they have been around middle schoolers too long.  However, if the lesson was able to spark the curiosity of teachers who already know the story of slope, our students will be captivated.

Teacher Created Slides

Resources

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all.  Reston, VA :NCTM, National Council of Teachers of Mathematics.

Boaler, J., Munson, J., & Williams, C. (2020).  Mindset mathematics: Visualizing and investing big ideas, grade 8.  Hoboken, NJ: Jossey-Bass.