Parallel and Perpendicular Sides

I noticed the following question on the PARCC Grade 4 Mathematics EOY Spring 2014 release practice test. I am not sure why this question struck me as questionable. It must have gone through quite a bit of review to have made it to the release.

I wonder what it means for a side to be parallel. I wonder what it means for a side to be perpendicular.

How many perpendicular sides would you say a rectangle has? How many pairs of perpendicular sides would you say a rectangle has?

PARCC defines a trapezoid as a quadrilateral with at least one pair of parallel sides.

How many parallel sides would you say the following figure has?

Would you say that the figure has 3 perpendicular sides?

I recently read through Wu’s Teaching Geometry through the Common Core Standards paper where he says “Perpendicularity is one of two special relationships between two lines. The other is parallelism.” He warns us to limit our talk of parallelism to lines, since there are many segments that do not intersect for which the lines that contains them do intersect.

Wu goes on to define a parallelogram as a quadrilateral with opposite sides parallel, and so I know that we mean the lines that contains the sides. But somehow the language of this question strikes me as suggesting a side can be parallel on its own. Can someone convince me that it is okay as written?

And so the journey to attend to precision continues…

 

PARCC Geometry Sample Item

PARCC finally released a sample high school geometry item.

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.

13. 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 animation of an angle bisector construction 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.

So I wonder. As the journey continues …

 

Trapezoids

The PARCC Evidence Statement Table for Geometry EOY, G-CO.3, says: A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”

Usiskin (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?

A. Square ABCD is a rhombus.

B. Parallelogram PQRS is a square.

C. Trapezoid GHJK is a parallelogram.

D. Square WXYZ is a parallelogram.

E. 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?

A. The two right angles are base angles.

B. The diagonals are not congruent.

C. Two of the sides are congruent.

D. No two sides are congruent.

E. 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 question…only 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.

Why I hate the definition of trapezoids

Why I hate the definition of trapezoids (again)

Why I hate the definition of trapezoids (part 3)

I’ve also found grades 4 and 5 tasks from Illustrative Mathematics:

What is a trapezoid? (Part 1)

What is a trapezoid? (Part 2)

It’s nice not to be alone as the journey continues ….