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A few years ago, I began requiring my students to earn 100 problem solving points (PSPs) each quarter, instead of requiring each student to complete a project of my choosing. PSPs are a way for students to show me what they have learned or solved outside of class.

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These have gone through several iterations over the past few years, but here is what I have now that seems to work. How students earn their PSPs is their choice. Sometimes a student will ask a question in class for which a) we don’t have time to get into the answer or b) I don’t actually know the answer. When that happens, I throw the question back to the class as an opportunity to earn PSPs.

Do some research to find out the significance of Euler’s line, or the shape of the faces of a dodecahedron (we were meeting in the cafeteria, and I didn’t have a model with me), or who first start using the radical symbol. Students then email me what they find, along with their source. What do you do with the calendar insert from the Mathematics Teacher? Those problems provide another way for students to earn PSP. Some students enjoy working calendar problems.

I post the current calendar in my room – and have previous calendars in a binder for students to “check out” to work. On our class Canvas site, I post enrichment opportunities for earning PSPs in each unit – many times they are from articles I’ve read in the Mathematics Teacher – or activities that I read about on blogs and tweets. I link to some. And I link to some that would be beneficial for students but that we aren’t going through explicitly in class.

I also post general enrichment opportunities for earning PSPs. Someone recently shared about a, so I posted a link to that for students with a note “Provide evidence to your instructor that you have watched this short TED Talk for up to 5 PSP”. Students will send me an email with their reflections on watching the video. For the, I have a note “Send a solution to a problem on this site to your instructor for PSP”. We have been paying more attention to Mindset this year in class, so I have an opportunity for students to explore GRIT: Angela Duckworth says that the key to success is GRIT.

Watch her TED Talk. Then determine how much GRIT you have. Then email your instructor a reflection with a response to at least one of the following prompts: I like I wish I wonder I will After reading, I posted the following for my students: is a site with plenty of opportunities for PSP! You can DO: -work on a puzzle -solve a problem -struggle with a problem Turn in your work on the puzzle/problem.

You can MAKE: -recreate a piece of math art -create your own artwork inspired by the original work Turn in your artwork. You can WATCH: -watch a video & leave a comment on the post about what you learned and/or questions you have Turn in a screenshot of your comment on the post.

You can READ: -read about a mathematician and compose a few questions you’d like to ask this person. Turn in your two questions. You can PLAY: -play a math video game, then write a critique of it (likes, dislikes, suggestions, etc.) Turn in your critique of the game. The possibilities for PSPs are endless. Some students read an article on mathematics or technology and share what they learned.

Some students share a website that they have found which is helpful for learning more about mathematics. Some students do constructions using TI-Nspire.

Some students write. Some students work cryptograms and problems of the week or do math history research from. In an effort for encouraging students to take the PSAT, I do give PSP for the math section of the PSAT during the second quarter (not a 1:1 ratio of math score:PSP earned). And for seniors, I give PSP one time for the math section on either the ACT or SAT. Another teacher had this idea, and I have continued it. All students get some choice about what they find interesting enough to explore for PSPs. Student can earn some of their points in class by answering bellringer questions and Quick Polls correctly.

At the end of the quarter, I total up all of their points and multiply by some scale factor that gives each student around 25-50 PSPs, depending on which quarter it is. I usually start out the year with letting them earn up to 50 from class – and decrease that number as the year goes on. I like that they can earn some of these points through class questions, because that gives them some incentive to not only be in class, but also be active participants in class. Using these points for PSPs instead of entering each assignment in the gradebook separately takes off the pressure of having to get every question correct. We are in the process of learning, and we don’t already know it all – there is plenty of leeway for students to earn PSPs and make mistakes.

I’ve wondered from time to time whether I should stop requiring PSPs, but each time I ask students to reflect on their experience with PSP, they insist that I should keep them going. And so I do, as the journey continues.

We started a unit on dilations last week. Our I can statements: Unit 6 – Similarity Level 1: I can identify, define, and perform dilations. G-SRT 1 Level 2: I can determine the similarity of two figures using similarity transformations. G-SRT 2, G-SRT 3 Level 3: I can prove theorems about triangles. G-SRT 4, G-C 1 Level 4: I can solve for and prove relationships in geometric figures using similarity criteria. G-SRT 5 And the standards: Similarity: G-SRT Understand similarity in terms of similarity transformations G-SRT 1.

Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity G-SRT 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Circles: G-C Understand and apply theorems about circles G-C 1. Prove that all circles are similar.

This lesson needed rewriting from last year. We used part of the Geometry Nspired activity, but it just wasn’t all sequenced as it should have been. So we tried again.

We started with a task that I learned about in an NCSM session –. I gave the students a piece of wax paper and a straightedge to determine which rectangles were similar to rectangle a.

I sent a poll to find out what students thought. One student asked whether I was grading the poll. Everyone stared at him.

I’m planning to have a conversation with him about what formative assessment really is sometime this week. I told them I wasn’t going to show them the correct answers yet (even though I have them marked in the screen capture below). Our unit is on dilations. What is your previous experience with the term dilation?

Students paired to talk with each other. I heard “eyes” and “enlargement”. Next we thought about what we need for a dilation. If we were going to dilate the given hexagon what do we need to know?

Someone suggested that we need a number. He called it a factor. Okay, so we need a number to know how big or small to make the image. We’re going to call that number a scale factor. What else do we need for a dilation? I have some cartoon dilations on the wall of my classroom.

We looked at those. What do they all have in common? (Have you seen a cartoon dilation before? We used to do them in class.

Not anymore, although I do think they give students a good understanding of dilations. We did these cartoon dilations with compasses and straightedges – not grids.) What do the lines have in common? They all intersect at a point. Okay, so we are going to call that point the center of dilation. Using our dynamic geometry software, we performed a dilation of the hexagon.

We clicked on the hexagon, we clicked on a point outside the hexagon for the center of dilation, and we typed in a number for the scale factor. I asked students to explore a few ideas. What happens when the center of dilation is on, inside, or outside of the figure? What happens when the scale factor changes? What if the scale factor is negative? I am not always so specific when I give students time to explore, but I tried it this time to see what happened. I set the timer for 4 minutes.

I used the Class Capture feature of TI-Nspire Navigator to see what they figured out. We had a conversation about what happens when the center of dilation is on the figure compared to outside. We had a conversation about what happens when the center of dilation is inside the figure. When is there a reduction? When the scale factor is a fraction. Does everyone agree?

What if the scale factor is 5/2? Not just any fraction – a fraction less than 1. Does everyone agree? –3 is less than one, but it isn’t a reduction. Fractions between –1 and 0 and between 0 and 1. What happens when the scale factor is negative? The figure is reflected.

We didn’t describe what is the line of reflection. Next up: some measurements. On this page, I want you to play with the slider. If I give you two of the measurements on the page, what can you do to determine the 3 rd? Which figure is the image and which is the pre-image?

How do you know? If the scale factor on the first picture were 0.5, which figure is the image and which is the pre-image? On the next page, we began to think about how many copies of segment AY it takes to make segment DE. What if we change the scale factor to 3? What if we change it to 0.5? We also began to think about how many copies of ∆AYM it takes to make ∆EDT.

What if we change the scale factor to 3? We are going to define two figures as similar if there is a dilation (and if needed, a set of rigid motions) that would map one figure onto another. Can you show that AYMDET?

After students explore on their own, I made CA the Live Presenter so that she could share what she did. She rotated ∆DET using S to control the angle of rotation. Then she translated ∆DET by vector YC. Then she undid the dilation by changing the scale factor to 1. Did everyone do the same? No – some undid the dilation first; others rotated and undid the dilation without a translation.

How can you show the two figures are similar? With a reflection and a dilation. We moved to paper. After a few minutes, a student shared how they dilated the triangle. Did everyone work it the same way?

Students saw all kinds of proportional relationships, and many thought about slope. What if we want to dilate points A and B about O using a scale factor of 3? We started on paper. And moved to the dynamic software. That was it for the first day.

We came back to a few problems the next day to continue our work on dilations. One figure is a dilation of the other.

Where is the center of dilation? Some students emphasized that we don’t only consider the vertices, but every pair of corresponding points on the pre-image and the image. How can we show the two figures are similar? What Standards for Mathematical Practice did the students have the opportunity to employ during our lesson on dilations? The dynamic software definitely provided them the opportunity to look for regularity in repeated reasoning. Does the lesson still need some work?

Is it better than last year? And so the journey continues. I’ve been asking my students to reflect on each unit as we finish so that I can have some feedback on what changes to make for next year. I collect their responses through an assignment in Canvas.

I can use inductive and deductive reasoning to make conclusions about statements, converses, inverses, and contrapositives. I can use and prove theorems about special pairs of angles. I can solve problems using parallel lines. I can prove theorems about parallel lines. Which Standard of Mathematical Practice did you use most often in this unit? The top 3 responses: Make sense of problems and persevere in solving them.

Construct viable arguments and critique the reasoning of others. Look for and make use of structure. Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts? Many students recognized vertical angles, parallel lines, and the triangle sum theorem as something they had used before. However, they also recognized that we were learning why those relationships worked – and not just that they worked.

One student replied: None of these were repeats so to speak. I already knew what parallel lines and the different types of angles were, but I had no idea why or how or how you use them so much in math. I never sat there and thought, “Oh, we learned this last year, I don’t have to pay attention.” Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit?. I really like 3A Logic for this lesson. It allowed me to think through things and use “logic” to solve problems. Interactive activities were particularly helpful as they visually showed the justification of postulates and theorems such as the Angle Addition Postulate, the Definition of Vertical Angles, etc. Logic in the beginning helped out with the whole unit because that’s mainly what everything linked to.

All of the lessons were helpful, but I found that the logic lesson was particularly helpful. It helped me to understand what a proof is and get introduced to them. Proofs were very important to meeting the learning targets for this unit. The symbolic logic was helpful in the case that if I could figure out one part, it could lead me to unlocking other parts, kind of like a puzzle. What have you learned in this unit?.

I have learned how to prove things that I already knew were true. I have learned all the objectives. I am also better at solving proofs and proving theorems. I learned about triangles and theorems about those as well as the way I can draw in Auxiliary Lines to find angles in problems. Do not procrastinate on homework. I’ve learned that math isn’t always numbers and equations; it can be proving things through logic and reasoning and postulates and many other things that I really didn’t know existed before this. I learned WHY parallel lines are parallel and WHY the alternate exterior and alternate interior and corresponding angles are the same and WHY all of the angles of a triangle added together equal 180 degrees.

I also learned how to apply this knowledge in geometry and how it can help us in real life. I have learned that I cannot give up when problems are complicated. I also learned that my way isn’t always the only way. I think these reflections are good news as the journey continues. PARCC finally released a.

I couldn’t wait to click on it to see what we are up against. The evidence statement: Construct, autonomously, chains of reasoning that will justify or refute geometric propositions or conjectures. The most relevant standards: G-CO.D: Make geometric constructions 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. The sample item: One thing I like about this task is that it makes students pay attention to the steps of the construction and not just blindly follow the directions without thinking about the results.

As we have been doing constructions this year, our focus has been to ask what is congruent at the end of each step. What segments, angles, arcs, and triangles are congruent as a result of the construction?

CCSS-M has made me change that. In the past, constructions were a checklist of steps. I didn’t make my students think about what they were doing. This particular construction provides students a great opportunity to look for and make use of structure.

That mathematical practice is listed as additional. The two practices that PARCC highlights for this task are construct a viable argument and critique the reasoning of others and attend to precision. So here is what disappoints me about this task. The geometry standards have been written to define congruence and similarity in terms of transformations. Two figures are congruent if there is a rigid motion (or set of rigid motions) that maps one figure onto another. Two figures are similar if there is a dilation (and if needed, set of rigid motions) that maps one figure onto another.

The scoring information for this task requires that students prove the resulting triangles congruent by SSS and the angles congruent by CPCTC. That feels like old-school geometry to me. So I gave the task to my students. I replayed the that I found online while they worked quietly for 10-15 minutes. According to the scoring guide, students get 1 point if they reason that the compass settings made congruent segments.

They note that the third side is congruent by reflexive. Will students get the point if they don’t note that the third pair of sides is congruent by the reflexive property? Students get 1 point if they reason that the triangles are congruent by SSS. Students get the 3 rd and final point if they reason that the angles are congruent by CPCTC, thus producing an angle bisector. From the notes: “The reasoning must include the triangle congruency statement and how the steps in the construction form the pairs of congruent sides.” I haven’t scored all of my student work according to the guidelines. I don’t think I am.

But I want to share some of the results. Out of 60 students, I had about 10 who proved or attempted to prove that the triangles noted in the PARCC solution were congruent by SSS.

I had about 5 who talked about points on a perpendicular bisector being equidistant from the two endpoints of a segment and points on an angle bisector being equidistant from the two sides of an angle. I had a few who recognized that ∆ABC was isosceles and used properties of isosceles triangles to show that the angle had been bisected. And I had at least 40 students who looked for and made use of structure.

They saw a kite. And they used the properties of a kite to justify that the ray was an angle bisector. So what happens now? If this were the AP exam, I would be confident that my students would receive credit for their responses.

Or at least some credit – I recognize that everyone didn’t give a step-by-step statement/reason response, and I recognize that everyone didn’t state why the figure was a kite. As the journey continues.

The, G-CO.3, says: A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.” (and I’m sure others – he was just the first one from whom I read it) call this the inclusive definition of a trapezoid. I have been posing this definition to my students as a possibility for several years now. What would happen if we defined a trapezoid as a quadrilateral with at least one pair of parallel sides? Then a parallelogram is also a trapezoid.

Our always/sometimes/never fill in the blank statements of “A parallelogram is a trapezoid” or “A trapezoid is a square” change from never to sometimes. But since our textbook defined a trapezoid as a quadrilateral with exactly one pair of parallel sides, we did too. This year, we defined a trapezoid as PARCC will. My colleagues and I have been thinking about the implications of this definition on our deductive system. On a side note, I’ve been looking for real CCSS geometry resources.

I usually first look at where the unit on transformations is located. If it’s not 1st or 2nd, then that is a clear sign to me that the text hasn’t really been edited for CCSS-M.

I’ve been forgetting to look at how they talk about trapezoids. Not that every CCSS-M textbook resource has to define trapezoids like PARCC does, but are they even suggesting the inclusive definition as a possibility? Our Venn Diagram arrangement of the quadrilaterals has been deleted. Several of our practice problems have had to be rewritten: Old: By definition, what is a quadrilateral with exactly one pair of opposite sides parallel? New: By definition, what is a quadrilateral with at least one pair of opposite sides parallel? Old: By definition, what is a quadrilateral in which both pairs of opposite sides are parallel?

New: By definition, what is a trapezoid in which both pairs of opposite sides are parallel? Technically, we could have kept the old question, but we changed it since we are trying to emphasize the new definition. Old: Which statement is NEVER true? Square ABCD is a rhombus. Parallelogram PQRS is a square. Trapezoid GHJK is a parallelogram.

Square WXYZ is a parallelogram. Trapezoid EFGH is an isosceles trapezoid. New: We deleted this question completely. But now that I look at it more closely, I guess we could have changed parallelogram in choice C to kite. Old: Two consecutive angles of a trapezoid are right angles. Four of the following statements about the trapezoid could be true. Which statement CANNOT be true?

The two right angles are base angles. The diagonals are not congruent. Two of the sides are congruent. No two sides are congruent. Exactly two sides are parallel.

New: We added the word Exactly at the beginning of the question stem. Old: Write the number of each of the five figures in the appropriate region of the diagram. New: We didn’t have to change the questiononly the solution. Other conversations we had were about how to define other quadrilaterals.

If we define a trapezoid as a quadrilateral with at least one pair of parallel sides, then shouldn’t a parallelogram be a trapezoid with both pairs of opposite sides parallel? We have always defined a rectangle as a parallelogram with four congruent angles, a rhombus as a parallelogram with four congruent sides, and a square as a parallelogram that is both a rectangle and a rhombus. Those definitions still seem to work. But are they the best definitions? Do we change our definition of kite from a quadrilateral with two pairs of consecutive congruent sides to a quadrilateral with at least two pairs of consecutive congruent sides?

And if so, then should we define a rhombus as a kite with four congruent sides? How do we define an isosceles trapezoid in this deductive system? A trapezoid with congruent legs no longer seems to work. What about a trapezoid with congruent diagonals?

Or a trapezoid with congruent base angles? Do we even talk about bases and legs of a trapezoid anymore? Another question I have had is whether K-8 teachers know about this inclusive definition of trapezoid.

Does it bother students for a definition to change midway through their years of school? Or do we need to start a campaign to inform teachers about this definition who might not be reading the PARCC Evidence Statement Table for Geometry EOY? Just for the record, I think my students are okay with this definition. They have actually been more flexible in their thinking than the teachers.

Last year, my 2 nd grader came home with this question on a worksheet, which I shared with my students. Unfortunately, the intention was to mark one response even though my daughter marked more than one response. Can you imagine the distress of having a 4 th figure that is a non-special trapezoid with directions to “Mark the trapezoid”?

As far as exploring properties of a trapezoid, I really like the inclusive definition of a trapezoid. We built a trapezoid using our dynamic geometry software. We recognize that two pairs of consecutive angles are supplementary. But as we move the vertices to observe what stays the same and what changes, we recognize that there are times when all pairs of consecutive angles are supplementary.

A trapezoid can be a parallelogram. We also used our dynamic geometry software to observe what happens when we create a midsegment of the trapezoid. What is true about the midsegment of a trapezoid? It is parallel to the bases. It cuts the trapezoid into two smaller trapezoids. How do you know? How could we show that ABNM is also a trapezoid?

And what is true about the length of the midsegment compared to the bases? It is half of the sum of the bases.

How can we show that the length of the midsegment is always half of the sum of the bases? I am glad that this conversation was started elsewhere, and that I easily found it through Google.

The graphic organizer on Mr. Chase’s blog posts has been most helpful in thinking through this change. I’ve also found grades 4 and 5 tasks from Illustrative Mathematics: It’s nice not to be alone as the journey continues. So kites aren’t specifically listed in CCSS-M. Congruence G-CO 11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. But look for and make use of structure is one of the Standards for Mathematical Practice.

And the Mathematics Assessment Project task has kites in it. And my students haven’t thought about properties of kites before.

So they enjoyed thinking about a figure that was mostly new to them. We used the Math Nspired activity as a guide for our exploration. I gave students about three minutes to play with with this page.

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Then I sent a Quick Poll to assess what they had observed. What else do you notice about the kite? Some students noticed that diagonal KN is a line of reflection for the two triangles that it creates.

Some students noticed that diagonal IG decomposes the kite into two isosceles triangles. Since the kite is two isosceles triangles, we can deduce even more properties. What if we construct both diagonals? We get two pairs of congruent triangles. The diagonals are perpendicular.

One of the diagonals bisects the other. I get why kites aren’t explicitly listed in the standards but might still show up in the tasks. But I still want to provide my students the opportunity to think about the structure of a kite while they’re not alone on an assessment. I’ve had conversations with teachers about some of the Geometry Nspired documents giving away too much of the math. One teacher thought that students should construct the kite instead of it already being made.

I think it would be great to have time for my students to construct every one of the quadrilaterals according to their definition and then observe the resulting properties. But I’ve decided I’m not going to spend the time that it takes to do that. Instead, my students constructed the parallelogram from the definition. We proved those properties – and then we observed properties of other quadrilaterals that were already constructed. I’m not convinced that full-time modeling is the way to go in my classes yet, but I want to get to modeling. And if I’m going to get there, I can’t do everything else. A week after this lesson, students successfully worked through the Floor Plan.

And so, whether or not we have the best plan, the journey continues. CCSS-M Congruence G-CO 11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. We have defined a parallelogram as a trapezoid with both pairs of opposite sides parallel. Some of our students know properties of parallelograms before getting to our high school geometry course. But they have not thought about why the properties are true. We started with a blank Geometry page of a TI-Nspire document.

The first task was to construct a parallelogram using the definition. Note: Specific instructions for construction a parallelogram can be found in the Geometry Nspired activity. Students used several geometry tools – segment and parallel, and then they used the polygon tool to go over the parallelogram. Some hid the parallel lines and some did not.

So what do you notice about the parallelogram? I gave students several minutes to explore and use measurement tools to verify their thinking.

I used the Class Capture tool of TI-Nspire Navigator to watch what they were doing. And I walked around the room to listen to their thoughts.

Then we talked together. We know that opposite sides of a parallelogram are parallel because of our definition. That is how we constructed the parallelogram.

What else happens as a result of that definition? The first observation by WA was that consecutive angles of a parallelogram are supplementary. So we can observe using our technology that consecutive angles are supplementary.

Can we prove that it is always true using our deductive system? The lines are parallel, and so consecutive interior angles are supplementary. The second observation by HK was that opposite angles of a parallelogram are congruent. So we can observe using our technology that opposite angles are congruent. Can we prove that it is always true using our deductive system?

Two angles supplementary to the same angle are congruent to each other. On a side note, the sequencing of these two observations worked out nicely. It wasn’t deliberate. It was lucky that the second observation followed from the first. It would be interesting to see how students would have proved opposite angles are congruent without first noting that consecutive angles are supplementary.

The third observation by BE was that opposite sides of a parallelogram are congruent. So we can observe using our technology that opposite sides are congruent. Can we prove that it is always true using our deductive system? We started with a picture of what we know is true. How can we use this information to prove that the opposite sides are congruent? Someone (not me) suggested that we draw diagonal BD.

How does that help? What do you see now that you didn’t see before? Congruent triangles. Alternate interior angles are congruent because the lines are parallel. BD=BD by reflexive. We have two congruent triangles by AAS.

Once the triangles are congruent, the remaining corresponding parts are congruent. The fourth observation by CM was that the diagonals bisect each other. How do you know? She knew because she used the midpoint tool to find the midpoint of each diagonal and noted that both midpoints were the same as the point of intersection of the diagonals. So we can observe using our technology that the diagonals bisect each other. Can we prove that it is always true using our deductive system? More congruent triangles.

Different congruent triangles, but congruent triangles nonetheless. My plan for the day was for students to prove properties of a parallelogram.

We started with a blank TI-Nspire geometry page and a definition. And the students proved the properties. But it was their work and observations that got us there. And so the journey continues.