Category Archives: Class Activity

Action-Consequence Advantage!

Using Technology to Make Math Stick

How might we enable students to grasp mathematical concepts and make their learning durable?  One approach is to use the sequence of Action-Consequence-Reflection in lesson activities:

  • Students perform a mathematical action
  • Observe a mathematical consequence
  • Reflect on the result and reason about the underlying mathematical concepts

The ACTION can be on a graph, geometric figure, symbolic algebra expression, list of numbers or physical model.  Technology can be used in order to have a quick and accurate result or CONSEQUENCE for students to observe.

The REFLECTION component is the most important part of this sequence; without this, students might not pay attention to the important math learnings we intended for the lesson.  They might remember using calculators, computers, ipads, or smartboards, but not recall what the tech activity was about.  And if they did learn the concept in the first place, the process of reflection helps make the learning stick—it is one of the cognitive techniques shown to make learning more successful.*

Students can reflect in many ways: record results, answer questions, discuss implications with classmates, make predictions, communicate their thinking orally or in writing, develop proofs and construct arguments.  The intended (or unexpected) learnings should be summarized either individually or as a class in order to solidify the concepts, preferably in a written form. **

In my one-on-one work with students, we often fall into the “procedural trap” in which my students just want to know “what to do” and don’t feel that the “why it works” is all that important (I’ve written about this before here). Also, our time is limited with many topics to cover.  But this past week, I was able to sneak in some Action-Consequence-Reflection with two students because they had mastered the prior material and were getting ahead on a new unit. It was a great opportunity to have them discover a concept or pattern for themselves, far better than simply being told it is true.

For each of these, I used a simple REFLECTION prompt:  What do you observe?  What changes?  What stays the same?

Student #1: Polynomial Function End Behavior, Algebra 2 (or PreCalculus)End Behav QWe used a TI-84+CE to investigate the polynomials.  We began with the even powers on a Zoom Decimal window, and my student noticed that higher powers had “steeper sides”.  I asked “what are the y-values doing to make this happen?” and we noticed the y-values were “getting bigger faster”.  We then used the Zoom In command to investigate what was going on between x = –1 and x = 1, and noticed that higher powers were “flatter” close to the origin because their y-values were lower.

 

Why did this happen? Take x = 2 and raise it to successive powers, and it gets bigger. Take x = ½ and raise it to successive powers and it gets smaller. We confirmed this with the table:

Setting the Table features to “Ask” for the Independent variable but “Auto” for the Dependent made the table populate only with the values we wanted to view. Fractions can be used in the table, and we used the decimal value 0.5 for clarity. Another observation was that the graphs coincided at three points: (1, 1), (0, 0) and (–1, 1).

In fact, none of this was what I “intended” to teach with the lesson, but it was mathematically interesting nonetheless, and my student already had a deeper appreciation for the graph’s properties. We moved on to the odd powers, and the same “steep” vs. “flat” properties were observed:

AC end beh 6

 

Then I asked my student to consider what made the graph of the even powers different from the graph of the odd powers, and what about them was the same, finally getting around to my “lesson”.  He noticed that even powers had a pattern of starting “high” on the left and ending “high” on the right, while odd powers started “low” on the left but ended “high” on the right.  We made predictions about the graphs of x11 and x12, to apply our understanding to new cases.† Then we moved on to the sign of the leading coefficient: when it is negative, our pattern changed to even powers “low” on the left and “low” on the right, with odd powers “high” on the left and “low” on the right.

All of this took just a few minutes, including the detour at the beginning that I wasn’t “intending” to teach.

Student #2: Interior and Exterior Angles of a Triangle, Geometry

Int Ext 1

We began with a dynamic geometry figure of a triangle which displayed the measurements of the three interior angles and one exterior angle.  I asked my student: “What do you notice?” and “what do you want to know about this figure?”  We dragged point B around to make different types of triangles.

This motivated my student to wonder about the angle measures; he was familiar with Linear Pair Angles, so he noticed that ∠ACB and ∠BCD made a linear pair and stated they would add up to 180°.  I then revealed some calculations of the angle measures, which dynamically update as we changed the triangle’s shape (the 180° in the upper right is the sum of the 3 interior angles).  What changes? What stays the same?

Int Ext 6

We noticed that both sums of 180° were constant‡ no matter how the triangle was transformed, but the sum of the 2 remote interior angles kept changing, and in fact, matched the measure of the exterior angle.  My student recorded his findings in his notebook, and then I asked, “can we prove it?”.  It was easy for us to prove the Exterior Angle Theorem based on his previous knowledge of the sum of the interior angles and the concept of supplementary linear pairs.  This student loved that he had “discovered” a new idea for himself, without me just telling him.

Even though these concepts are relatively simple, I feel that using a technology Action-Consequence activity made the learning more impactful and durable for my students, and I believe it was more effective than just telling them the property I wanted to teach. It took us a few more minutes to explore the context with technology than it would have to simply copy the theorems out of the book, but it was worth it!

 


Notes and Resources:

*For a full elaboration of cognitive science strategies for becoming a productive learner (or designing your teaching to enhance learning), see Make It Stick: The Science of Successful Learning by Peter C. Brown, Henry L. Roediger & Mark A. McDaniel, 2014.  Website: http://makeitstick.net .

**Written summaries allow students to Elaborate and Reflect on the learning, two more of the cognitive strategies.  In addition, by insisting that students record the results of the work, the teacher sends the message that the technology investigation comprises important knowledge for the class.

†Making predictions is a way to formatively assess my students’ understanding.  It also is a form of Generation, another cognitive strategy that makes the learning more durable.

‡This is an example of an invariant, a value or sum that doesn’t change, which are often important mathematical/geometric results.

Here are the two activities discussed in the post. Note that the Geogebra file for Power functions has the advantage of having a dynamic slider, but students won’t view the graphs at the same time, so won’t notice the common points or the graph properties between –1 and 1.

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Rational Functions

We are studying Rational Functions, and I was looking for technology activities which would help students visualize the graphs of the functions and deepen their understanding of the concepts involved.  Previously, I had taught algebraic and numerical methods to find the key features of the graphs (asymptotes, holes, zeros, intercepts), then students would sketch by hand and check on the graphing calculator.  I wanted to capitalize on technology’s power of visualization* to give students timely feedback on whether their work/graph is correct, and avoid using the grapher as a “magic” answer machine.  I also wanted to familiarize students with the patterns of rational function graphs—in the same way that they know that quadratic functions are graphed as “U-shape” parabolas.

Here are three ideas:

Interactive Sliders

Students can manipulate the parameters in a rational function using interactive sliders on a variety of platforms (Geogebra, TI-Nspire, Transformation Graphing App for TI-84+ family, Desmos).  Consider the transformations of these two parent functions:

eq1 to become  eq3and

eq2 to become    eq4

Each of these can be explored with various values for the parameters, including negative values of a.

Here are screenshots from Transformation Graphing on the TI-84+ family:

Another option is to explore multiple x-intercepts such aseq5.

This TI-Nspire activity Graphs of Rational Functions does just that:

screen-shot-2017-02-22-at-11-48-07-am

In a lesson using sliders, on any platform, I use the following stages so students will:

  1. Explore the graphs of related functions on an appropriate window.  Especially for the TI-84+ family, consider using a “friendly window” such as ZoomDecimal, and show the Grid in the Zoom>Format menu if desired.  Trace to view holes, and notice that the y-value is indeed “undefined.” capture-6
  2. Record conjectures about the roles of a, h, and k and how the exponent of x changes the shape of the graph.  This Geogebra activity has a “quick change” slider that adjusts the parent function from  eq1  to eq2.capture-geogebra-rationals
  3. Make predictions about what a given function will look like and verify with the graphing technology (or provide a function for a given graph).

A key component of the lesson is to have students work on a lab sheet or in a notebook or in an electronic form to record the results and summarize the findings.  Even if your technology access is limited to demonstrating the process on a teacher computer projected to the class, require students to actively record and discuss.  The activity must engage students in doing the math, not simply viewing the math.

MarbleSlides–Rationals

A Desmos activity reminiscent of the classic GreenGlobs, MarbleSlides-Rationals has students graph curves so their marbles will slide through all of the stars on the screen.  If students already have a working understanding of the parent function graphs, this is a wonderful and fun exploration.

The activity focuses on the same basic curves, and it also introduces the ability to restrict the domain in order to “corral” the marbles.  Users can input multiple equations on one screen.

I really liked how it steps the students through several “Fix It” tasks to learn the fundamentals of changing the value and sign of a, h, k and the domains. These are followed by “Predict” and “Verify” screens, one where you are asked to “Help a Friend” and several culminating “Challenges”.  Particularly fun are the tasks that require more than one equation.

On one challenge, students noticed that the stars were in a linear orientation.

marbleslides_-rationals22-question

Although it could be solved with several equations, I asked if we could reduce it to one or two.  One student wondered how we could make a line out of a rational function.  Discussion turned to slant asymptotes, so we challenged ourselves to find a rational function which would divide to equal the linear function throw the points.  Here was a possible solution:

marbleslides_-rationals22

Asymptotes & Zeros

Finally, I wanted students to master rational functions whose numerator and denominator were polynomials, and connect the factors of these polynomials to the zeros, asymptotes, and holes in the graph.  I used the Asymptotes and Zeros activity (with teacher file) for the TI-84+ family.  It can also be used on other graphing platforms.

Students are asked to graph a polynomial (in blue below) and find its zeros and y-intercept.  They then factor this polynomial and make the conceptual connection between the factor and the zeros.  Another polynomial is examined in the same way (in black below).  Finally, the two original polynomials become the numerator and denominator of a rational function (in green below).  Students relate the zeros and asymptotes of the rational function back to the zeros of the component functions.

capture-4

I particularly liked the illumination of the y-intercept, that it is the quotient of the y-intercepts of the numerator and denominator polynomials.  We had always analyzed the numerator and denominator separately to find the features of the rational function graph, but it hadn’t occurred to me to graph them separately.

A few concluding thoughts to keep in mind: any of these activities can work on another technology platform, so don’t feel limited if you don’t have a particular calculator or students don’t have computer/internet access.  Try to find a like-minded colleague who will work with you as you experiment with technology implementation, so you can share what worked and what didn’t with your students (and if you don’t have someone in your building, connect with the #MTBoS community on Twitter).  Finally, ask good questions of your students, to probe and prod their thinking and be sure they are gaining the conceptual understanding you are seeking.


NOTES & RESOURCES:

*The “Power of Visualization” is a transformative feature of computer and calculator graphers that was promoted by Bert Waits and Frank Demana who founded the Teachers Teaching with Technology professional community.  More information in this article and in Waits, B. K. & Demana, F. (2000).  Calculators in Mathematics Teaching and Learning: Past, Present, and Future. In M. J. Burke & F. R. Curcio (Eds.), Learning Math for a New Century: 2000 Yearbook (51–66).  Reston, VA: NCTM.

All of the activities referenced in this post are found here.  More available on the Texas Instruments website at TI-84 Activity Central and Math Nspired, or at Geogebra or Desmos.

For more about the Transformation Graphing App for the TI-84+ family of calculators, see this information.

GreenGlobs is still available! Check out the website here.

Where Are You? I’ll Meet You There

guy-magnifying-glass

I was so proud.  I had created a great technology activity to use in my Algebra 2 class, complete with a well-thought out lab sheet for students and their partners to work through and document their learning.  It was an exploration of slopes of parallel and perpendicular lines, with students being guided to “discover” the concepts involved.*

Questions for assessment involved different levels of cognitive demand, including creating their own sets of equations, paying attention to mathematical structure, and writing an explanation of their process.  The graphing calculators were ready and the students worked diligently through the class period.  The lesson was a success—everyone demonstrated their understanding of the mathematical objectives.

So what was the problem?  I asked a few students on their way out of class if they enjoyed the calculator lab activity since it was different from our “regular” routine.  They told me: “It was fine.  But Mrs. Campe, we all already knew about slopes of parallel and perpendicular lines.”

I had failed to properly pre-assess my students’ understanding of the concepts. I had wasted a full class period to cover something they had already mastered, when, instead, I could have been moving forward or exploring some other problem more deeply.  I didn’t check where my students were in their understanding before launching into my “great” activity.

Similar things can happen in my one-on-one work with students.  Since I am not in their classroom with them, when students arrive for a work session, I have to rely on them to tell me what their lesson and unit topics are.  Sometimes I go down a path that veers away from what they have done in class.  Some students resist conceptual explanations, wanting only the quickest route to the answer.  I have to push them to realize that learning the “why” behind a procedure helps them understand when and how to use it, and the conceptual background makes their learning more durable and leads to more success in math class.**

So what have I learned from these situations?

1.  It is vitally important to pre-assess and utilize formative assessment to know where my students are.  Class time is at a premium and I want to use it wisely.

2.  Don’t rely on students’ self-report of their understanding; require them to demonstrate their capabilities by doing problems, explaining a process, and answering “why” questions.

3.  Don’t use technology just because I have it.  It must further the lesson objectives and enhance student understanding.  The same warning goes for “fun” or “cool” lesson activities.

4.  Reflect on your lessons: ask yourself what went well and what needs improving so mistakes don’t get repeated.  And discuss with your colleagues, local and virtual. You will find lots of support in the MTBoS; one teacher commented to another on Twitter just last night: “Thx! I am always looking to improve my teaching!”

Mistakes, obviously, show us what needs improving. Without mistakes, how would we know what we had to work on?     

Peter McWilliams

 


Notes & Resources:

The technology lab activity on Parallel and Perpendicular Lines is here.  It was written for the TI-84+ family of calculators, but any graphing technology may be used.

*This lab activity is a “Type 1” investigation structure in that it guides students toward the desired mathematical knowledge, in contrast to a “Type 2” inquiry which encourages more open exploration.  Both types of lesson structures are effective, so match the level of exploration with your objectives.  More about this in McGraw, R. & Grant, M. (2005).  Investigating Mathematics with Technology: Lesson Structures That Encourage a Range of Methods and Solutions. In W. J. Masalski & P. C. Elliott (Eds.), Technology-Supported Mathematics Learning Environments: 67th Yearbook (303-318). Reston, VA: NCTM.

Another dimension useful in analyzing a lesson is type of teacher questioning.  “Funneling” questions guide students through a math activity to a predetermined solution strategy, while in “Focusing” interactions, the teacher listens to students’ reasoning and guides them based on where they are and what strategies they are employing, rather than how the teacher might solve the problem.  More in Herbel-Eisenmann, B. A. & Breyfogle, M. L. (2005). Questioning Our Patterns of Questioning.  Mathematics Teaching in the Middle School, 10(9): 484-489.

**Connecting new knowledge to what you already know (elaboration), building conceptual structures (mental models) and practicing what to do when (discrimination skills) are among the strategies for successful learning discussed in Make It Stick (Brown, Roediger & Mc Daniel, 2014).  See this website for more.

As a final thought, my title is misleading, because I don’t just want to meet the students where they are and stay there, I want to plan for appropriate challenges to take them beyond their current understanding.  There is great value in productive struggle, and choosing lesson components within the students’ “Zone of Proximal Development”.

Body Benchmarks!

Body Benchmarks: Analyze the Data and See How You Measure Up!

I first used this activity in an Algebra I class when I wanted students to have a hands-on experience gathering data and modeling it with a linear function. Later, I used this as an “all-ages” station at our elementary school’s Family Math Night. Let’s explore this problem situation and look at low-tech and high-tech methods to analyze this real-life data.

Here are the Instructions: [link to file]

  1. Write your name on the Record Sheet.
  2. Use the measuring tape to measure your forearm in inches from elbow to tip of middle finger.
  3. Record forearm measurement on the Record Sheet next to your name.
  4. Use the measuring tape to measure your height (if you don’t know it, in inches.)
  5. Record height measurement on the Record Sheet next to your name.
  6. Place a mark on the Group Graph that corresponds to your Forearm Length (Horizontal Axis) and Height (Vertical Axis).

There are some interesting discussions that can occur during data collection and recording. Why did we use inches? How precise should our measurements be (to nearest inch, half-inch, quarter-inch)? Are there some measurement activities that are better served by the metric system? Why did we choose to put Forearm Length on the X-axis and Height on the Y-Axis (is one variable clearly the independent variable and the other depends on it, or does it not matter)? What scale did we use on the large graph paper and did I graph my data point accurately? Is there any point that is clearly an outlier? [Not all of these topics came up in every class setting; are there other conversations you experienced or think are important for the teacher to orchestrate?]

Picture5

Once the data is collected, the group graph shows a positive correlation that could be modeled by a linear function. A low-tech way to find a line of best fit is to graph the data on graph paper, and use a ruler or dry spaghetti/linguine pasta to approximate the line with this criteria: “Follow the trend of the data points, and have about half the points above and half below the line”.

Then select two points (must they be data points? Or just graph grid intersections?) and find the equation of the line (point-slope form or slope-intercept form?). Don’t stop at the equation… what does this formula do for us? Can we predict someone’s height if we know their forearm length? Can you describe the shape of the graph, and how does this relate to the equation? What other questions do you want to ask about this situation?

When technology is available in the classroom, we can use a high-tech approach to analyze the Body Benchmarks data. On the TI-84+ family of graphing calculators (including 83+, 84+, and the color devices 84+C and 84+CE), enter the data into L1 and L2. Then turn on the StatPlot and choose an appropriate window. [What else do you want to discuss with your students… do you have them choose a window or use ZoomStat? Are the students to be responsible for knowing the key presses?]

Then we can analyze the data using the options in the StatCalc menu. I used go directly to the LinReg choice to perform a linear regression. Then I recently discovered that at the bottom of this menu, there is option D:Manual-Fit. This is a high-tech version of the dry pasta best fit line! Note that this feature is available on ALL the TI-84+ graphing calculators, even though my images below are of the color devices.

When Manual-Fit is chosen, you are prompted to designate a location to store the equation. Press ALPHA-TRACE and select Y1. Then arrow down to “Calculate” and press ENTER.

On the graph, move the cursor to place the first point to model the line and press ENTER. If desired, the STYLE of the line can be changed by pressing GRAPH and choosing a new color or line style. Then move the cursor to the second point and press ENTER.

Now, you may have noticed that the line I have chosen is a bit below my set of data, although my slope is a reasonably good fit. BEFORE you are done, you can edit the two parameters M and B in the y = Mx + B equation. Simply enter the new value into the highlighted parameter. When you are happy with your line of fit, press GRAPH to select DONE.

Materials:

Body Benchmarks Instructions and Record Sheet [click here for file]

Body Benchmarks Data that I used in these screenshots [click here for file]

Body Benchmarks Questions [click here for file]

Step-by-step overview of basic STAT EDIT, STAT PLOT, and STAT CALC functions [click here for file]

Extensions: What other measurements can you take and analyze? Consider armspan, foot length, hand size, and head or wrist circumference.


For more information on hands-on lab activities that generate linear data, see the book: Algebra Experiments 1 Exploring Linear Functions (1992) by Mary Jean Winter & Ronald J. Carlson. [link]