Piecewise Functions

Piecewise Functions

We started a unit on piecewise functions in Algebra 1 with the following leveled learning progression:

Level 4: I can sketch a graph of a piecewise-defined function given a verbal description of the relationship between two quantities.

Level 3: I can interpret key features of a piecewise-defined function in terms of its context.

Level 2: I can determine the domain and range of a function given a context.

Level 1: Using any representation of a function, I can evaluate a function at a given value of x, and I can determine the value of x for a given value of f(x).

We started with an opener to ensure that students were successful with Levels 1 and 2 so that we could reach our target (Level 3) during the lesson.

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Questions 1-4 gave us evidence that most students could evaluate a function at a given value of x and determine the value of x for a given value of f(x) using any representation of a function.

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Questions 5-6 gave us evidence that our students needed more support determining the domain and range of a function given a context.

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Without showing the results from the opener that are pictured above, we talked all together about the context, reading the graph, but not explicitly discussing the domain and range. When we sent the question as a Quick Poll, we saw evidence that more students could determine the domain and range of a function given a context.

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We probably could have spent more time on Level 2 in class. But we didn’t. Instead, we had to provide additional support for Level 2 outside of class, through homework practice, zero block, and after school help.

To open our discussion of piecewise functions, we showed this picture from the front of the Jackson airport parking garage.

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What do you notice? What do you wonder?

Students wrote down a few observations individually, then shared their thoughts with a partner. We selected some for our whole class discussion. In particular, it was helpful that one student specifically said, “pay depends on time”.

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How much would you pay for a certain number of hours of parking?

Similar to an idea from the Internet Plans Makeover, we asked students to choose a number between 0 and 24. If you park that many hours, how much will you pay?

We asked students to check work with a partner before submitting. The result wasn’t quite as disastrous as when we tried the Internet Plans Makeover.

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Showing the grid helps some.

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And having a whole class discussion about the stipulations of the sign helped even more.

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Did anyone choose a non-whole number?

What would happen if you parked for 1.5 hours?

Or 2 hours and 20 minutes?

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We also used the Math Nspired lesson Dog Days or Dog Years with good success. What we are still trying to decide is which comes first … the structure from the Dog Days or Dog Years lesson about creating piecewise functions? Or the less structured conceptual introduction from the cost of parking at the airport? I’m not sure it’s wrong (or even better) to start with either one. But we still wonder, as the journey continues …


Posted by on March 30, 2015 in Algebra 1


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Solving Right Triangles

Solving Right Triangles

One of the NCTM Principles to Actions Mathematics Teaching Practices is to build procedural fluency from conceptual understanding. We began conceptual understanding in our lesson on Trig Ratios. So we started our Solving Right Triangles putting-it-all-together lesson with some “Find the Error” (what’s wrong with the procedure) problems from our textbook.

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Students worked a few minutes alone before sharing their thoughts with their teams. What can you find right about the given work? What is wrong about it? How can you correct the given work?

We started our whole class conversation with #1. Students began to recognize how many options there were for correcting the given work. The Class Capture feature of TI-Nspire Navigator gave students the opportunity to see and compare each other’s calculations. They looked at and compared trig ratios for complementary angles D and F.

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We moved to #2. What error did you find?

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The triangle isn’t right. I don’t think we can use sine, cosine, and tangent for triangles that aren’t right.

Not yet … but eventually you’ll be able to.

And then BK said, “But I worked it out.”

If I haven’t learned anything else, I’ve at least learned to hear my students out. I hadn’t noticed what he’d done when students were working individually, so I didn’t know what we were about to get into. “How did you do that?”

I drew in an altitude from angle B and made two right triangles.

And so he did.

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He recognized half of an equilateral triangle, so he used 30-60-90 triangle relationships to get enough information to write a ratio for tangent of 55˚.

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This is what happens when students learn mathematics steeped in using the Standards for Mathematical Practice. Students practice make sense of problems and persevere in solving them. They practice look for and make use of structure. Even in what seems at first glance like simple, procedural problems.

And so the journey continues, with the practices slowly becoming habits of minds for my students’ learning, seeing glimpses of hope more often than not …

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Posted by on March 29, 2015 in Geometry, Right Triangles


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The Circumference of a Circle

Thanks to Andrew Stadel’s CMC-South session, we started our lesson this year with a focus on construct viable arguments and critique the reasoning of others.

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Create an argument for comparing the height and circumference of the bottle.

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Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility. (Thanks to Andrew for this idea in particular. I’ve had students partner with others with opposing arguments on many occasions; I had not thought about the importance of partnering with others with the same argument to make your argument stronger. In the session I attended, we shared our argument with someone who answered like we did at least twice.)

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

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Uh-oh. There were apparently some pretty good convincers from the height < circumference argument. I thought fast about what to do next. I didn’t want to immediately call on someone right or wrong to share her argument with the class – I wasn’t ready for the individual/partner thinking to stop.

So without resolving the first solution, I showed another picture.

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And sent another Quick Poll.

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Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility.

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

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I could go with a whole class discussion based on these results.

CK reflected on this task in a Math Practices journal: “My first instinct was to say, ‘yes, the height is greater than the circumference’, because just looking at the can gave me the impression that the circumference was not very much. Then I was told to prove my argument, so I drew a diagram. …” (I think it’s interesting that CK chose to reflect on SMP1, make sense of problems and persevere in solving them, for this task, even though I emphasized SMP3, construct viable arguments and critique the reasoning of others, in class. The practices complement each other so well.)

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We went on to think a little more about pi, using some data that students had measured at home and submitted via a Google doc and some data through the automatic data capture feature of TI-Nspire.

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Based on feedback from students, I think this will be the last year for our What is Pi? lesson in its current form. We are getting students in high school who have learned math with the standard: CCSS-M.7.G.B.4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. And so our students are now coming to us with some understanding of the formulas for the area and circumference of a circle, unlike before.

I’ve recently learned that several of my geometry students wish that we weren’t learning the geometry the way that we are. They like their previous math classes better because they didn’t have to always think about why.

We are trying to change the habits and practice of how students learn mathematics. Focusing on the Standards for Mathematical Practice has required me to think through and plan learning episodes differently than before. Focusing on the Standards for Mathematical Practice requires my students to interact in those learning episodes differently, even though some don’t prefer to. And so the journey continues …


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Circumscribed Angles

Circles: G-C. A. Understand and apply theorems about circles

  1. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

When I first googled circumscribed angle, I was led to Khan Academy. I’ve never specifically called an angle a circumscribed angle before, although of course it makes sense that an angle formed by two tangents to a circle from the same point outside of a circle can be called a circumscribed angle. I had always heard it called an angle formed by two tangents.

Suppose you are given that the measure of arc ADC is 260˚. What do you know?


How were you taught to calculate the measure of angle B?

How have you taught students to calculate the measure of angle B?

How have your students figured out to calculate the measure of angle B?

I’ve always thought of this angle as one other in the set of angles with vertices outside of the circle whose measure is half the difference of the outer intercepted arc and the inner intercepted arc.

Practicing look for and make use of structure gives my students and me flexibility in what I see and how I can calculate the measure of angle B.

I sent students a Khan Academy question on central, inscribed, and circumscribed angles to see what they could do. The auxiliary lines are drawn because of the central and inscribed angles, but students still practiced look for and make use of structure.

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Even though most got it correct, we still had a whole class discussion so that students could share how they got the angle measure.

Some worked it the way I had been taught.

Some saw the kite.

Even those who saw the kite didn’t think through calculating the angle measure exactly the same way.

At least one thought of the kite as a quadrilateral inscribed in a circle and said that the opposite angles were supplementary. Yikes! I’m glad that (incorrect) thinking was made visible.

Why are the opposite angles supplementary?

This item does give me hope that while I might not “cover” every standard (especially the next year or two), teaching mathematics using the Math Practices is worth our time. Even during an assessment, my students can figure out some of what they need to know by practicing look for and make use of structure and look for and express regularity in repeated reasoning. What #AskDontTell opportunities can you provide your leaners the next time you’re together?


Posted by on March 16, 2015 in Circles, Geometry


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Tangents to a Circle

Is there anything intuitive about tangents drawn to a circle from the same point outside of a circle?

I sent this Quick Poll to my students without telling them anything.

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I watched and listened as students talked with each other.

They drew diagrams, some to scale, and some not, and decided that the tangent segments drawn to the circle from the same point outside the circle were congruent.

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How do you know?

BK and his team knew for sure because they constructed the diagram using dynamic geometry software.

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Why do the tangent segments have to be congruent?

Students practiced look for and make use of structure. What do you see that isn’t pictured?

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Many students drew in some diameters.

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What do you see that isn’t pictured?

Some students recognized that a radius drawn to a point of tangency will be perpendicular to the tangent.

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What do you see that isn’t pictured?

A kite!

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Why is it a kite?

The radii are congruent.

Segment AC is congruent to itself.

The triangles are right, so angles B and D are congruent.

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We got ∆ABC congruent to ∆ADC by HL.

Then the tangent segments are congruent because the triangles are congruent.

And then back to the dynamic geometry software to make more sense of the diagram we had been given.

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What kind of #AskDontTell opportunities are you providing the learners in your care this week?


Posted by on March 15, 2015 in Circles, Geometry


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Trig Ratios

Trig Ratios

How do your students experience learning right triangle trigonometry? How do you introduce sine, cosine, and tangent ratios to them?

NCTM’s Principles to Actions includes build procedural fluency from conceptual understanding as one of the Mathematics Teaching Practices. In what ways can technology help us help our students build procedural fluency from conceptual understanding?

Until I started using TI-Nspire Technology several years ago, right triangle trigonometry is one topic where I felt like I started and ended at procedural fluency. How do you get students to experience trig ratios?

I’ve been using the Geometry Nspired activity Trig Ratios ever since it was published. Over the last year, I also read posts from Mary Bourassa: Calculating Ratios and Jessica Murk: Building Trig Tables about learning experiences for making trigonometric ratios more meaningful for students. Here’s how this year’s lesson played out …

We first established a bit of a need for something called trig (when they finally get to learn about the sin, cos, and tan buttons on their calculator that they’ve not known how to use). I showed a diagram and asked how we could solve it. We reserved “trig” for something they couldn’t yet solve.

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We use TI-Nspire Navigator with our TI-Nspire handhelds, and so I can send Quick Polls to assess where students are. Sometimes Quick Polls aren’t actually so “quick”, but these were, along with letting students think about what we already know and uncovering a few misconceptions along the way (25 isn’t the same thing as 18√2).

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Next I asked each student to construct a right triangle with a 40˚ angle and measure the sides of the triangle.

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I sent a Quick Poll to collect their measurements.

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Then we looked at the TNS document for Trig Ratios. Students can take multiple actions on the diagram. I asked them to start by moving point B. What do you notice? We recorded their statements for our class notes.

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Then I asked them to click on the up and down arrows of the slider. What do you notice?

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What ratio of side lengths is used for the sine of an angle?

You all constructed a right triangle with a 40˚ angle and recorded the measurements. What’s true about all of your triangles?

  • The triangles are all similar because the angles are congruent.
  • The corresponding side lengths are proportional.
  • We know that sin(40˚) is always the same.
  • So the opposite leg over the hypotenuse will be the same?

Will it? We sent their data to a Lists & Spreadsheet page and calculated a fourth column, opp_leg/hyp. What do you notice?

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Of course their ratios aren’t exactly the same, but that’s another good discussion. They are close. And students noticed that one entry has the opposite leg and adjacent leg switched because the leg opposite 40˚ is shorter than the leg opposite 50˚.

We didn’t spend long looking at the TNS pages for tangent and cosine … students were well on their way to understanding a trig ratio conceptually. They just needed to establish which side lengths to use for cosine and which to use for tangent.

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There’s a reason that #AskDontTell has been running through my mind as I have conversations with my students and reflect on them. Jill Gough wrote a post using that hashtag over two years ago: Circle Investigation – #AskDontTell.

What #AskDontTell opportunities can you provide your students this week?

[Cross-posted at T3 Learns]


Posted by on March 12, 2015 in Geometry, Right Triangles


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The Center of Rotation

This is the first year we have tried Identifying Rotations from Illustrative Mathematics.

△ABC has been rotated about a point into the blue triangle. Construct the point about which the triangle was rotated. Justify your conclusion.

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This reminds me of the Reflected Triangles task, which we have used now for several years.

I got a glimpse of students working on the task using Class Capture. I watched them make sense of problems and persevere in solving them.

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We looked at all of the auxiliary lines that LJ made, trying to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Jarret’s work, who used technology to perform a rotation, going backwards to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Justin’s work, who rotated the given triangle about A to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Quinn’s work, who knew that if R is the center of rotation, then the measures of angles ARA’, BRB’, and CRC’ must be the same.

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Students took those conversations and continued their own work.

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The next day, Jared shared his diagram. What can you figure out about the relationship between the center of rotation, pre-image, and image looking at his diagram?

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In my last two posts, I’ve wondered what geometry looks like if we start our unit on Rigid Motions with tasks like these instead of ending the unit with tasks like these. Maybe we will see next year, as the #AskDontTell journey continues …


Posted by on March 10, 2015 in Geometry, Rigid Motions, Uncategorized


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