Is This a Rectangle?

One of our learning intentions in our Coordinate Geometry unit is for students to be able to say I can use slope, distance, and midpoint along with properties of geometric objects to verify claims about the objects.

G-GPE. Expressing Geometric Properties with Equations

B. Use coordinates to prove simple geometric theorems algebraically

1. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

We recently used the Illustrative Mathematics Task Is This a Rectangle to provide students the opportunity to practice.

We also used Jill Gough’s and Kato Nims’ visual #ShowYourWork learning progression to frame how to write a solution to the task.

How often do we tell our students Show Your Work only to get papers on which work isn’t shown? How often do we write Show Your Work next to a student answer for which the student thought she had shown her work? How often do our students wonder what we mean when we say Show Your Work?

The Show Your Work learning progression begins to help students understand what we mean when we say Show Your Work. I have seen it empower students to ask each other for feedback on their work: Can you read this and understand it without asking me any questions? It has been transformative for my AP Calculus students as they write Free Response questions that will be scored by readers who can’t ask them questions and don’t know what math they can do in their heads.

We set the timer for 5 minutes of quiet think time. Most students began by sketching the graph on paper or creating it using their dynamic graphs software. [Some students painfully and slowly drew every tick mark on a grid, making me realize I should have graph paper more readily available for them.]

They began to look for and make use of structure. Some sketched in right triangles to see the slope or length of the sides. Some used slope and distance formulas to calculate the slope or length of the sides.

I saw several who were showing necessary but not sufficient information to verify that the figure is a rectangle. I wondered how I could steer them towards a solution without telling them they weren’t there yet.

I decided to summarize a few of the solutions I was seeing and send them in a Quick Poll, asking students to decide which reasoning was sufficient for verifying that the figure is a rectangle.

Students discussed and used what they learned to improve their work.

It occurred to me that it might be helpful for them to determine the Show Your Work level for some sample student work. And so I showed a sample and asked the level.

But I didn’t plan ahead for that, and so I hurriedly selected two pieces of student work from last year to display. I was pleased with the response to the first piece of work. Most students recognized that the solution is correct and that the work could be improved so that the reader knows what the student means.

I wish that I hadn’t chosen the second piece of work. Did students say that this work was at level 3 because there are lots of words in the explanation and plenty of numbers on the diagram? Unfortunately, the logic is lacking: adjacent sides perpendicular is not a result of parallel opposite sides. Learning to pay close enough attention to whether an argument is valid is good, hard work.

Tasks like this often take longer than I expect. I’m not sure whether that is because I am now well practiced at easing the hurry syndrome or whether that is because learning to Show Your Work just takes longer than copying the teacher’s work. And so the journey continues …

 

Squares on the Coordinate Grid

I’ve written before about Squares on the Coordinate Grid, an Illustrative Mathematics task using coordinate geometry.

CCSS-M G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

How do you provide opportunities for your students to practice I can look for and make use of structure?

How do you draw a square with an area of 2 on the coordinate grid?

It helped some students to start by thinking about what 2 square units looks like, which was easier to see in a non-special rectangle.

What’s true about the side length of a square with an area of 2?

How could we arrange 2 square units into a square?

How do you know the figure is a square? Is it enough for all four sides to be square root of 2?

CC made his thinking visible by reflecting on his learning after class:

“Now drawing the square root of two exactly on paper is nearly impossible unless you know how to use right triangles.”

 

Reflected Triangles + #ShowYourWork

I’ve used the Illustrative Mathematics task Reflected Triangles for several years now. This year students practiced “Show Your Work” (from Jill Gough) along with coming up with a correct solution.

Level 4: I can show more than one way to find a solution to the problem.

Level 3: I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.

Level 2: I can find a correct solution to the problem.

Level 1: I can ask questions to help me work toward a solution to the problem

Which of the directions can you follow exactly to construct the line of reflection? Are there any that need clarification?

Students wrote first and then constructed their lines of reflection. I used the Class Capture feature of TI-Nspire Navigator to watch. We’ve spent longer on this task in the past, but this year, I only had about 5 minutes for a whole class discussion. Whose work would you select for the whole class to see?

We started with Kaelon’s construction.

How is Holly’s construction different? Can you tell what Holly did?

How is Phillip’s different? Can you tell what he did?

We are well on our way towards learning how to “describe or illustrate a solution in a way that the reader understands without talking to me”, as the journey continues …

 

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.

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.

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.

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.

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.

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.

Students took those conversations and continued their own work.

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?

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 …

 

The Line of Reflection

I’ve blogged about the Illustrative Mathematics task Reflected Triangles before. I really like that it asks students to determine the line of reflection given the pre-image and image instead of determining the image given the pre-image and line of reflection.

△ABC has been reflected across a line into the blue triangle. Construct the line across which the triangle was reflected. Justify your conclusion.

How would you construct the line of reflection?

Work on your mathematical flexibility to come up with more than one way to construct the line of reflection.

I used Class Capture to monitor student work and selected some students to share their approach with the whole class.

Many students approached the task like Paris:

What can we learn about the relationship between the pre-image, image, and line of reflection from the additional line in Max’s diagram?

I’ve always used this task towards the end of our unit on Rigid Motions, but I am thinking about using it earlier in the unit next year. Maybe the task itself could be an #AskDontTell approach for students learning what we want them to learn about the relationship between the pre-image, image, and line of reflection. We will see next year, as the journey continues …

 

Dilating a Line

CCSS-M G-SRT.A.1. Verify experimentally the properties of dilations given by a center and a scale factor:

1. 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.
2. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

CCSS-M G-C.A.1. Prove that all circles are similar.

It probably shouldn’t surprise me that my students have a difficult time dilating lines and circles. Most of the objects that we transform in mathematics are polygons, and we often think about what happens to them in terms of their vertices, neglecting to note what’s happening to the other points on the polygons or the sides of the polygons.

We use the Illustrative Mathematics task Dilating a Line. For an animation of dilating a line, see The Mathematics Common Core Toolbox.

Suppose we apply a dilation by a factor of 2, centered at the point P, to the figure below.

Can you anticipate misconceptions that students will have?

As I was monitoring student work, I saw several approaches that I wanted the class to see. I took some pictures and thought carefully about whether to start with one that was correct or one that was incorrect.

I decided to start with a student who had used her ruler as a measuring tool and not just as a straightedge. FS drew the line that contains PA past A, measured PA, and then marked off that same measurement for AA’. She did the same for PB and BB’ and the same for PC and CC’. She hadn’t yet drawn the line through A’, B’, and C’. What will be true about that line compared to the line through A, B, and C?

More than one student said the line would double. What does it mean for a line to double?

SC had a similar approach, except that she didn’t use her ruler to measure, she used her compass to measure. She demonstrated how she measured PA and marked AA’ the same length.

Then I showed BB’s work. What do you think?

What do you like about her work?

What do you wonder about her work?

BB notes that A’B’ is double AB in her answer to part (c), but her diagram isn’t convincing.

What do you think about BK’s work?

What do you like about his work?
What do you wonder about his work?

One of the other explorations was “How would you dilate circle C by a scale factor of 3?”

Here’s the work of one student.

Which made me realize I had not specified that A was the center of dilation.

And the last exploration was “Given any two circles, can you always find a dilation that maps one circle onto another?”

Which we didn’t get to that day.

I’ve written about Dilating a Line before, but the experience of the task changes each year, as the journey of teaching and learning continues with a different community of learners …

 

Angle Bisection and Midpoints of Line Segments

As we finished Unit 2 on Tools of Geometry this year, I looked back at Illustrative Mathematics to see if a new task had been posted that we might use on our “put it all together” day before the summative assessment.

I found Angle Bisection and Midpoints of Line Segments.

I had recently read Jessica Murk’s blog post on an introduction to peer feedback, and so I decided to incorporate the feedback template that she used with the task.

The task:

What misconceptions do you anticipate that students will have while working on this task?

 

What can you find right about the arguments below? What do you question about the arguments below?

Student A:

Student B:

Student C:

Student D:

Student E:

Student F:

Student G:

Student H:

Student I:

Student J:

The misconception that stuck out to me the most is that students didn’t recognize the difference between parts (a) and (b). I’ve wondered before whether we should still give students the opportunity to recognize differences and similarities between a conditional statement, its converse, inverse, contrapositive, and biconditional. We decided as a geometry team to continue including some work on building our deductive system using logic, even though our standards don’t explicitly include this work. We know that our standards are the “floor, not the ceiling”. We did this task before our work on conditional statements in Unit 3, and so students didn’t realize that, essentially, one statement was the converse of the other. Which means that what we start with (our given information) in part (a) is what we are trying to prove in part(b). And vice versa.

The feedback that students gave was tainted by this misconception.

Another misconception I noticed more than once is that while every point on an angle bisector is equidistant from the sides of the angle, students carelessly talked about the distance from a point to a line, not requiring the length of the segment perpendicular from the point to the line and instead just noting that that the lengths of two segments from two lines to a point are equal.

It occurred to me mid-lesson that maybe we should look at some student work together to give feedback. (This happened after I saw the “What he said” feedback given by one of the students.)

I have the Reflector App on my iPad, but between the wireless infrastructure in my room for large files like images and my fumbling around on the iPad, it takes too long to get student work displayed on the board. A document camera would be helpful. But I don’t have one. And I’m not sure how I’d get the work we do through the document camera into the student notes for the day. So I actually did take a picture or too, use Dropbox to get the pictures from my iPad to my computer, and then displayed them on the board using my Promethean ActivInspire flipchart so that we could write on them. And then a few of those were so light because of the pencil (and/or maybe lack of confidence that students had while writing) that the time spent wasn’t helpful for student learning.

Looking back at Jessica’s post, I see that her students partnered to give feedback, since they were just learning to give feedback. That might have helped some, but I’m not sure that would have “fixed” this lesson.

So while I can’t say with confidence that this was a great lesson, I can say with confidence that next year will be better. Next year, I’ll give students time to write their own arguments, and then I’ll show them some of the arguments shown here and ask them to provide feedback together to improve them. Maybe next year, too, I’ll add a question to the opener that gives a true conditional statement and a converse and ask whether the true conditional statement implies that the converse must be true, just so they have some experience with recognizing the difference between conditional statements and converses before we try this task.

And so the journey continues, this time with gratefulness for “do-overs”.

 

Sum of Angles in a Triangle

I’ve talked before about providing students an opportunity to look for and make use of structure while trying to prove the Triangle Sum Theorem. This summer I ran across a task on Illustrative Mathematics with a proof of the Triangle Sum Theorem using transformational geometry. The task is scaffolded quite a bit, and so while I didn’t give my students the task as-is with the scaffolded instructions, those questions ended up playing a big role in our whole class conversation. I asked the question the same way I have before – I just knew because of reading through the task and learning from the task that I wanted my students to recognize another way to prove the Triangle Sum Theorem.

We talked for a moment about our Learning Progression for look for and make use of structure. In geometry, we often have to ask, “What do you see that’s not pictured?” We often have to draw auxiliary lines to help make sense of a figure.

I gave students a diagram of a triangle. And I gave them the following words:

Given: ∆ABC

Prove: m∠A+m∠B+m∠C=180°

I then asked the class to think back through our progression in building our deductive system. What do we know? What have we proved? What have we allowed into our systems as postulates? They thought back through our units of study:

Triangles – medians, altitudes, angle bisectors, perpendicular bisectors

Vertical angles are congruent; Angle & Segment Addition Postulates

Parallel Line postulate & theorems – corresponding angles, alternate interior angles, …

Transformations – reflections, rotations, translations

I set the timer for 3 minutes, moved the Learning Mode clip to “Individual”, and watched (monitored) as they thought. Some sat for three minutes thinking without drawing anything. Some drew in an altitude for the triangle. Some composed the triangle into a rectangle or a non-special parallelogram… not all the same way.

     

Students continued to work alone for several more minutes before they even noticed that the timer had finished. I couldn’t believe what I saw besides the traditional responses. On SC’s paper, I saw three triangles: the original, and two images of the original triangles that had been rotated about the sides. I’ve had students proving the Triangle Sum Theorem for years now and never once has someone thought to transform the triangle by rotating it. I asked students to share their work with their team. I listened. And I asked a few probing questions, especially to SC. SC needed to cut out a triangle congruent to the image so that she could describe the resulting rotations. Her first thought was that the triangle had been rotated around one of its vertices.

     

I drew a few of the diagrams that students had created on the board & asked students to take a few more minutes to see if they could justify the Triangle Sum Theorem using one of the diagrams.

Then we talked all together. Several students had used a rectangle to show why the sum of the measures of the angles of the triangle has to equal 180˚. (A few tried to use the interior angles of a quadrilateral summing to 360˚ in their reasoning, so we talked about being unable to use that, since it is really a result of what we are trying to prove.)

Others used the side of the rectangle parallel to the base of the triangle showing that alternate interior angles congruent and then used the Angle Addition Postulate to finalize their result.

Next we moved to the diagram of the rotated triangles.

How can we describe the rotation that resulted in the triangle on the left?

Students suggested rotating 180˚ about the midpoint of segment AB. Point A = Point B’, and Point B = Point A’. We loosely used 1’, 2’, and 3’ to name the angles in the image. We know that ∠1’ is congruent to ∠1 because a rotation preserves congruence. And so then we know that segment A’B is parallel to segment AC since alternate interior angles are congruent.

Similarity, we can rotate the triangle 180˚ about the midpoint of segment BC. Point B = Point C’’ and Point C = Point B’’. We loosely used 1’’, 2’’, and 3’’ to name the angles in the image. We know that ∠3’’ is congruent to ∠3 because a rotation preserves congruence. And so then we know that segment A’’B is parallel to segment AC since alternate interior angles are congruent.

So how do we know that A’’, B, and A’ are collinear?

Because if they are, then A’A’’ is parallel to AC, and m∠1’+m∠2+m∠3’’=180, which means m∠1+m∠2+m∠3=180.

For one of the first times in class, we actually used the parallel postulate to explain why A’’, B, and A’ are collinear (through a point not on a line, there is exactly one line through the point parallel to the given line). We are still studying Euclidean geometry, after all.

We are always running out of time, and so I was just using the rotation tools on my Promethean Board ActivInspire software in our conversation.

Next year, we will add our dynamic geometry software to help verify and make sense of our results.

In my last CMC-S session yesterday I gave participants just a few minutes to come up with a way to prove the Triangle Sum Theorem using transformations. Of course I was expecting the rotation solution. I’m not sure when I’ll ever quit being surprised by solutions I don’t expect. One participant suggested that we translate ∆ABC using vector AB. We labeled the resulting image ∆A’B’C’. We know that ∠1 ≅∠1’ because a translation preserves angle congruence. Then BC’ is parallel to segment AC because corresponding angles are congruent.

Similarly, we translated ∆ABC using vector CB. We labeled the resulting image ∆A’’B’’C’’. We know that ∠2 ≅∠2’’ because a translation preserves angle congruence. Then BA’’ is parallel to segment AC because corresponding angles are congruent.

A’’, B, and C’ are collinear by the Parallel Postulate since there can be only one line through B parallel to segment AC. ∠3 ≅∠B’’BB’ because vertical angles are congruent. The Angle Addition Postulate gets us m∠2’’+m∠B’’BB’+m∠1’=180, and then with substitution and the definition of congruent angles, we can conclude that the sum of the measures of the angles of the triangle is 180˚.

And so the journey continues … ever grateful for resources like Illustrative Mathematics, that push me to keep learning – and keep me pushing my students to make connections that we haven’t previously been making, and ever grateful for the educators who attend conferences like CMC-South eager and willing to learn alongside other educators.

 

Locating a Warehouse

We changed the Learning Mode to individual. Where would you place a warehouse that needed to be equidistant from all three roads? (From Illustrative Mathematics.)

Students started sketching on paper, and I set up a Quick Poll so that we could see everyone’s conjecture at the same time.

We changed the Learning Mode to whole class. With whom do you agree?

I didn’t intend for us to talk in detail at this point. I wanted students to be able to test their conjecture using their dynamic geometry software. But we had done that the day before for Placing a Fire Hydrant (post to come), and class was cut short during this lesson because of lock-down and evacuation drills. So we did talk in more detail than I had planned. Is the point outside of the triangle equidistant from the three roads? One student vehemently defended her point: I drew a circle with that point as center that touched all three roads. (We have been talking about the distance from a point to a line.) How do you know that the roads are the same distance from the center? They are all radii of the circle. They are perpendicular to the road from the center.

Could a point inside the triangle of roads be correct? If so, which? We started drawing distances from the points to the lines. Some points were about the same distance from two of the roads but obviously to close to the third road. What’s significant about the point that will be the same distance from all three sides of a triangle? Several students wondered about drawing perpendicular bisectors. Another student vehemently insisted that the point needed to lie on an angle bisector. Would that always work?

Are you going to let us try it ourselves? Well of course! So with about 12 minutes left, students began to construct.

With about 3 minutes left, I made a student the Live Presenter who showed us that the angle bisectors are concurrent, and used the length measurement tool to show us that the point is equidistant to the sides of the triangle.

With about 2 minutes left I made another student the Live Presenter who had made a circle inside the triangle. How did you get that circle? What is significant about the circle? It’s inscribed. The center is the where the angle bisectors intersect. So we call that point the incenter. It’s the center of the inscribed circle of a triangle, and the point of concurrency for the angle bisectors. How is this point different from the circumcenter?

And with 1 minute left: Do you understand what we mean when we say that every point on an angle bisector is equidistant from the two sides?

And so while I have some record of what every student did during class through Quick Polls and Class Capture and collecting their TNS document once the bell rang, my efforts at closure are foiled again. Maybe one day I’ll actually send one of the Exit Quick Polls that I have made for every lesson.

 

Squares on a Coordinate Grid

I was excited to find a new Illustrative Mathematics task using coordinate geometry.

CCSS-M G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

In the picture below a square is outlined whose vertices lie on the coordinate grid points:

The area of this particular square is 16 square units. For each whole number n between 1 and 10, find a square with vertices on the coordinate grid whose area is n square units or show that there is no such square.

As a precursor to the task, I included the following statements on the bell work for students to discuss in their groups before we had a brief class discussion.

The three sides of a right triangle can all be even.

The three sides of a right triangle can all be odd.

Last year, I heard Linda Griffith talk about giving a part of this task to some students in Arkansas. I began with my students the way she began with hers. Each student had a sheet of graph centimeter graph paper and a straightedge. Near the top left corner, draw a square with an area of 1 square centimeter.

Challenge accepted, although some students drew their square in the top right corner.

Next, I want you to choose a point, which can be above your square on even on your square, and I want you to dilate your square by a scale factor of two.

This took longer, but it was a good reminder of what we need for a dilation.

What happened? What can you tell me about your image?

It has an area of 4 square centimeters.

How do you know?

I counted the squares.

Someone else noted that the similarity ratio is 1:2, so the ratio of the areas is 1:4.

What will happen if you dilate your original square by a scale factor of 3.

We will get a square with an area of 9.

And so they did.

Now. Here is our goal for this lesson: For each whole number n between 1 and 10, find a square with vertices on the coordinate grid whose area is n square units or show that there is no such square.

So far, we have 1, 4, and 9. What do you know about those numbers?

They’re perfect squares.

Yes. So now I want you to draw a square with an area of 2 square centimeters. I’d like for you to work by yourself for 2 minutes, and then you can share what you’ve found with your group.

I watched while they worked. I saw many students approximating √2 on their calculator. I saw several students who had made a rectangle with an area of 2 square centimeters. I saw one student who had immediately thought of 45-45-90 triangles and had drawn a square with an exact area of 2. Everyone was doing something, even if they were using approximations.

After two minutes, I told students they could work together now, and that I had two reminders: I have asked you to draw a square. And I want it to have an exact area of 2 square centimeters.

I heard great conversation. I asked a few of those who had approximated the side length of their square how they knew the side was √2. Linda Griffith told a great story last year about some of her students: they decided to put “not drawn to scale” next to their diagram, as they had seen on one too many of the diagrams from their geometry class. Several others made the 45-45-90-connection for an isosceles right triangle with a leg of length 1 cm to get the desired square. I listened to one group who realized they had confused whether a square is always a rectangle or a rectangle is always a square take their rectangle and compose its parts a different way to get a square.

I decided to have them share first. It occurred to me after they started talking that I should video their explanation. I caught part of it.

I love that these two took their rectangle of area 2 and rearranged it to make a square of area 2.

Next I asked the student who had immediately thought of 45-45-90 to explain her thinking.

She related her work to the Pythagorean Theorem.

And finally one other student shared who had composed his square differently from the girls with the rectangle.

Now that we have a square with an area of 2, what other square areas can we easily get?

Of course a dilation by a scale factor of 2 will give us a square with an area of 8.

What side length does that square have? 2√2.

So what is next? We still need squares with areas of 3, 5, 6, 7, and 10.

What could we do to get 5?

Several students simultaneously thought about 3-4-5 right triangles. So what does that give us? An area of 25, which we can get with oblique side lengths from the 3-4-5 triangle or horizontal/vertical side lengths of 5 cm.

It isn’t really 5 we need. What can we do to get √5 for a side length?

Students continued working, many coming up with a 1-2-√5 triangle from which to draw a square with an area of 5.

If we can get 5, can we get 10?

I was expecting to hear 1²+3²=10, and I did hear that. But I also heard (√5)²+(√5)²=10, which I didn’t hear as loudly because I wasn’t expecting to hear it. You would think I’d have learned by now to pay closer attention to what my students actually say. What I am learning, though, is that it takes time to process student thinking for a task that isn’t “cookie cutter”, and I don’t always do that quickly in class, especially when the bell is about to ring. We ended with a discussion of more than one way to get a square with an area of 10 – and I left 3, 6, and 7 for the students to finish exploring outside of class.

So I would have liked to talk about why 3, 6, and 7 don’t work. We didn’t get there this year.

But we did make it farther than last year as the journey continues …