Pythagorean Theorem Proofs

CCSS-M G-SRT.B.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.

We’ve been teaching our CCSS Geometry course for three years now, and this is the first year that we have been able to spend more than a little class time on proofs of the Pythagorean Theorem. (Our students are coming to us knowing more mathematics than three years ago. Our students are coming to us more willing to take risks and use the Standards for Mathematical Practice than three years ago. We are making progress just in time for our legislators to decide that collaborating with other states to write standards and assessments was a bad idea.)

We started with the Mathematics Assessment Project formative assessment lesson (FAL) on Proofs of the Pythagorean Theorem. This FAL is one that includes student work. Students focus on SMP3: construct a viable argument and critique the reasoning of others.

As students practice look for and make use of structure, I asked them to share what they noticed and wondered.

Then we looked specifically at a diagram drawn to scale, and students noted what they knew to be true (and why).

As we started to examine the student work proofs so that students could critique the proofs, SC asked to go back to the previous page. I wonder what will happen if we reflect the outer right triangles about their hypotenuses into the center square.

What do you think will happen?

The triangles will make a square.

I think I’ve said before that technology slows me down in the classroom. Students notice and wonder more than they did before, and the technology gives us the chance to see what happens so that we can make sense of why it happens mathematically. I am not the only expert in the room. The student who gets mathematics without technology is not the only expert in the room. Our use of technology increases our confidence and lifts all in the room to experts. And so the journey continues …

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Graphing Stories

How do you and your coworkers effect change in the classroom? Two years ago, I asked our principal whether we could schedule a geometry class during first block with four teachers and about 25 students. It was our first year to implement our new CCSS Geometry standards, and we needed to try it together. I have learned over the years that it doesn’t hurt to ask – he might say no, but he might also said yes. Well, he said yes, and as you can imagine, sharing a class together has been important for us both as learners and as teachers. As one of the teachers reflected recently, “Participating in a class all year with a team of teachers is the best professional development I have ever had.” Last year, our Algebra 2 team had a shared class to implement their new standards, and this year, our Algebra 1 team shares a class. I visit as often as I can.

We are building our course as we go, using all sorts of resources. We are using the framework from EngageNY, and we are using some of the activities and tasks in their lessons. We have started with a unit on Graphing Stories.

 

On the first day, we used Growing Patterns from an NCTM Article Coloring Formulas for Growing Patterns.

 

Lesson Goals (written with Jill Gough this summer):

Level 4: I can represent the number of tiles in a figure in more than one way and show the equivalence between the expressions.

Level 3: I can represent the number of tiles in a figure using an explicit expression or a recursive process.

Level 2: I can apply patterns to predict the number of tiles in a later figure.

Level 1: I can describe the pattern and draw a figure before and after the given figures.

 

How do you see the pattern growing?

How many tiles are in H(5)? H(100)? H(N)?

Some students completed a table of values, and some students drew a graph.

 

On the second day, we used a Mathematics Assessment Project formative assessment lesson, Interpreting Time-Distance Graphs, with a focus on rate of change.

Lesson Goals:

Level 4: I can calculate average rate of change from a graph and a table.

Level 3: I can calculate average rate of change from a graph or a table.

Level 2: I can match a distance-time graph with a story and a table.

Level 1: I can annotate a graph using away from home and towards home, how fast and how slow.

 

On the third day, we used Graphing Stories Video 2 to begin the lesson.

Lesson Goals:

Level 4: I can create a believable story for a given graph.

Level 3: I can calculate average rates of change for an elevation graph.

Level 2: I can create an elevation graph from a video, labeling the axes with appropriate units of measure.

Level 1: I can identify time intervals for each piece of an elevation graph.

Level 1: I can calculate the slope of a line.

Whose graph is believable?

We used additional scenarios from Engage NY Algebra 1, Module 1, Lessons 1-2.

 

There were more videos and scenarios on Day 4 from Engage NY Algebra 1, Module 1, Lessons 3-4.

The lesson from Day 5 comes from David Wees’ webinar at the Global Math Department on Strategic Inquiry.

 

We have read Timothy Kanold’s blogpost on Leaving the Front of the Classroom Behind. And we are trying. And we are most grateful to our administrators for letting us try it together before we have to try it alone.

 

2D Representations of 3D Objects

G-GMD.B. Visualize relationships between two-dimensional and three-dimensional objects

We used the Mathematics Assessment Project formative assessment lesson on 2D Representations of 3D Objects.

As with other MAP lessons, students start with a pre-assessment (and they are given an opportunity to revise their work at the end of the lesson).

Next we moved to a class problem: The cylinder is full of water. This water flows out through a pipe at the bottom of the cylinder. Imagine looking down on the cylinder as the water flows out of it. Draw the shape of the surface of the water at five different levels. Will the radius of the shape change?

(The student who answered yes was including the pipe in his answer.)

And another: Draw the shape of the surface of the water at five different levels. Again, imagine looking down on the bottom cylinder. As the bottom cylinder fills with water, what is the shape of the surface of the water? Draw the shape of surface of the water in the bottom cylinder, at five different levels.

Student teams gets sets of cards and match eat set of shapes of the surface of the water with a top (water flowing out) or bottom (water flowing in) container. They draw in the shapes that are missing.

The MAP lessons include deliberate instructions to students on how to work together:

Some students used held models of 3D objects to help them visualize how the water level affected the size and shape of the cross section.

Some students didn’t need the models.

Teams worked well together, ensuring that everyone was participating and agreeing on the solutions.

 

The MAP lessons also include deliberate instructions for how to share work with other teams when it is time to do so:

I am realizing more and more that we have to deliberately teach our students how to work together. Speaking of which, have you read Elizabeth’s blog post on Rooting Out Blind Spots in the Language of Group Roles in Complex Instruction-based Group Work? I am excited to learn alongside Elizabeth and incorporate some of her ideas for team work this coming year with my students.

A few students commented on this lesson in their reflection at the end of the unit.

• The lesson on two dimensional representations of three dimensional objects was helpful because it gave me different visualizations in my head of different objects, which helped me be able to answer questions more precisely.
• Although I enjoyed the popcorn activity, I feel like the most helpful activity was the one where we cut out the cards and thought through the gradual shapes of the water as it emptied the top figure and filled the bottom figure. It allowed me to work with a team to reason out our arguments and working with my peers in a collective effort enlightened me with thoughts and ideas that I had not previously thought of or would have otherwise ventured to discuss.

And so the journey to not only provide opportunities for students to work together but to deliberately teach students how to work together continues …

 

Volumes of Compound Objects

We used the Mathematics Assessment Project formative assessment lesson on Calculating Volumes of Compound Objects. MAP also has a Glasses task that includes student work.

Students calculate the volume of liquid that the glasses can hold, and then determine the height of the liquid in Glass 3 when it is half full.

Minutes before I gave students their handout with the glasses, I decided to cover up all of the measurements on the glasses and ask the students what information would be sufficient for calculating the volume.

 

(I’ve been learning from reading Dan Meyer’s blog about being less helpful.)

radius height 1

height and radious   1

circumfrence, height 1

Height and radius of the cylinder and the volume of the solid holding up the cylinder          1

surface area uf cup part

radius of cup part

height of cup part     1

dimensions, shape    1

area of circle and height      1

Just the radius and the height, but either if the volume is given.           1

height and base        1

radius of circle, height of glass        1

height, radius            1

radius and height of the cylinder   1

height of cylinder and radius          1

radius of base, height of cylinder    1

diameter, hieght, to get volume.      1

height and radius of the cylinder   1

Every thing you need to know for volume of a cylinder  1

 

height of cylinder, height of hfmishpere, and radius       1

height of the straight side and radius        1

radius and height of the cylinder   2

radius and straight side length       1

radius height 1

Radius and height     1

 

Asking what information students needed gave a different spin to mindless calculations with volume formulas, even for volumes of compound objects.

Once students had the needed information, they calculated. I sent a Quick Poll with all four questions at once so that students could work at their own rate without feeling the pressure of a timed poll.

I monitored student responses as they came in, talking directly with teams who needed to revise their work.

Until we got to the question about the glass being half full.

From the student responses, it seemed that the glass was half empty instead.

What do you do when no one has the correct response?

There are times that I have given the correct response and asked students to determine how to get it. I’m not sure how effective giving a decimal approximation of the correct height would have been here. Instead, we looked at the picture. Can you draw where the height will be when the glass is half full? What do you see?

Similar triangles are often elusive for my geometry students. We need more practice.

And so the journey continues, trying to find a balance between giving too much information and too little information …

 

Evaluating Statements about Enlargements

We recently used the Mathematics Assessment Project formative assessment lesson on Evaluating Statements about Enlargements.

I had just returned from NCSM where I heard Tim Kanold’s session “Beyond Teaching for Understanding: The elements of an authentic formative assessment process”. In the session, he suggested that no more than 35% of class should be the teacher leading from the front of the classroom. I was determined to figure out how this played out in my classroom when I got back to school. I also found a blog post where he talks about leaving the front of the classroom behind.

We started with Candy Rings.

When I got the results from the first poll, I knew I was in trouble. Why did you choose “correct”, Amber? Two of the small rings has the same total circumference as one of the large rings. Why did you choose “incorrect”, Ryan? Some of the pieces on the larger ring look broken. Those on the smaller ring are closer together.

Against my better judgment, I asked the next question. After all, construct a viable argument and critique the reasoning of others is how we are learning math, right?

If the price of the small ring of candy is 40 cents, what is a fair price for a large one?

About half of the class used proportional reasoning, deciding that 80 cents was fair. The other half used business reasoning. Some included tax. Some decided that the larger portion should have a bit of a discount. All of them had an argument for why they chose what they did.

 

No one thought that Jasmina reasoned correctly about the amount of pizzas.

And only a few insisted on using business sense to come up with a fair price for a large pizza.

Students then worked with their teams to determine whether their cards with statements about enlargements were true or false. I gave each team 6 cards to evaluate. They reasoned abstractly and quantitatively. They constructed viable arguments and critiqued the reasoning of others. They even had a good time.

This particular class clocked in at exactly 65% peer-to-peer discussion. Even then, the 35% whole group discourse wasn’t just one raised student hand at a time. I used the Quick Poll results to selectively call on students who don’t always raise their hand. They presented their argument and the class decided whether to buy it or not. I was there to facilitate the conversation and move it forward. I’m not sure whether that only counts as “leading from the front of the classroom”.

Either way, the journey to leave the front of the classroom behind continues …

 

Length & Area – Sliding Triangles

We started our unit on Geometric Measure & Dimension with a Mathematics Assessment Project formative assessment lesson on Evaluating Statements about Length and Area.

I chose the card that each group would explore ahead of time. I only gave each group one card, and I had another task ready for them if they finished early. They didn’t. Students chose whether they wanted to explore on paper or using technology.

Each card had a hint, which I held until I felt like the group might need it. While students were working, I monitored their progress.

The second group had Sliding Triangles.

If you slide the top corner of a triangle from left to right, its area stays the same.

If you slide the top corner of a triangle from left to right, its perimeter changes.

They read through their card and drew a few diagrams, but they decided to spend most of their time building the scenario using our dynamic geometry software. This group did not need the hint card.

When it was time for this group to present their work, we sent the Quick Polls so that we would know what students instinctually thought, even though they had not all had time to explore the statements in depth. I am learning to make use of the TI-Nspire Navigator allowing me to send more than one Quick Poll at a time.

I showed the results, but I deselected Show Correct Answer before doing so. I wanted the group to know what their peers thought before they just told them the results.

I used the Live Presenter feature of Navigator to make one of the student’s calculators live on the projector at the front of the room. They grabbed and moved the point that changed the top vertex of the triangle, noting with the square they had constructed off the base that the height of the triangle remained constant.

The area of the triangle will always be the same.

What about the perimeter? The perimeter changes.

Does the perimeter always change? Sometimes the perimeter is the same.

When is the perimeter the same? When one triangle is a reflection of the other about the perpendicular bisector of the base.

And so the journey continues …

 

 

Length & Area: Cutting Shapes

We started our unit on Geometric Measure & Dimension with a Mathematics Assessment Project formative assessment lesson on Evaluating Statements about Length and Area.

I chose the card that each group would explore ahead of time. I only gave each group one card, and I had another task ready for them if they finished early. They didn’t. Students chose whether they wanted to explore on paper or using technology.

Each card had a hint, which I held until I felt like the group might need it. While students were working, I monitored their progress.

The first group had Cutting Shapes.

When you cut a piece off a shape you reduce its area.

When you cut a piece off a shape you reduce its perimeter.

They thought Always for both statements.

Most of them were taking a quadrilateral and cutting off a triangle on the corner.

How could I get them to figure out it was sometimes besides just telling them? I also didn’t want to give this group the hint card because I felt like it gave too much away.

What if you cut off something besides a triangle? Someone cut off a rectangular corner.

What just happened to the area? It got smaller.

What just happened to the perimeter? It stayed the same!

How do you know?

What if you cut somewhere besides a corner?

I went to another group after I asked this question.

By the time I got back to them, they had a great explanation as to why both statements were not always.

I’m sure there was a better way to do this, which might include having students evaluate the statements by themselves before coming to class for the lesson. We could have spent a week on these six cards. But we didn’t.

Before each group presented their work, we sent out a Quick Poll to see what the rest of the class instinctively thought about the statements.

I showed the results, but I deselected Show Correct Answer before doing so. I wanted the group to know what their peers thought before they just told them the results.

The group presented their work, not just giving us their results, instead talking us through their thinking about the statements, and how they arrived at their conclusion.

And so the journey continues …

 

Circles in the Coordinate Plane

CCSS-M G-GPE.A.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

We used part of the Math Nspired activity Exploring the Equation of a Circle and part of the Mathematics Assessment Project formative assessment lesson Equations of Circles 1  for our introductory lesson on circles in the coordinate plane.

 

What do you notice?

-There is a right triangle.

-The hypotenuse is always 5.

What will happen if we trace point P as we move the triangle around in the coordinate plane? What will be the locus of points that it travels?

What can we say about the lengths of the sides of the triangle?

a²+b²=5²

Would it be okay to name the lengths of the legs x and y instead of a and b, since they are horizontal and vertical lengths in the coordinate plane?

So x²+y²=5²?

Yes. So for each (x,y) location of the point P, we can say x²+y²=5². That is how we describe the equation of this circle in the coordinate plane.

Now I’m going to let you play with a few more pages in your TNS document and then answer a few questions. 

Students played.

 

What happens when you translate the center of the circle?

  

And then students answered more formal questions about the equation of a circle to ensure that they had looked for and expressed regularity in repeated reasoning. I collected their responses (this was their bell work for the day, but I decided not to immediately show them the results. I took a glance myself to know how they were progressing).

Next we moved to a few questions from the MAP formative assessment lesson on Equations of Circles.

I am always impressed by the progression of questioning in the lessons. I was particularly interested in how students decided whether the point (5,6) lay inside, on, or outside the circle x²+y²=36.

One student knew that it lay outside the circle because the point (0,6) was on the circle. He reasoned that if (0,6) is on the circle, (5,6) can’t be. Another student drew the point on the grid and recognized that it could not lie on or inside the circle. Another student used the equation to show that the point (5,6) did not lie on the circle.

What if the point is too close to tell from a sketch of the graph?

What does it mean for a point to lie on a circle? Another Quick Poll, with the idea for the question from MAP.

I am learning to ask questions that I think are obvious.

 

We looked at the coordinates that they entered, and we changed to the Graph View. I added a Teacher Equation (two, actually) to show the circle. What does it mean for a point to lie on a circle?

An aside: I was in a grade 7 classroom recently. Students were determining the x- and y-intercepts of lines given the equation. I asked the teacher to send a Quick Poll of the graph of a line and have students drop a point on its x-intercept. She was surprised to find out how many students didn’t know what the x-intercept was, and yet they’re supposedly calculating x-intercepts from equations. (Students had dropped points all over the x-axis, but only a few students had dropped a point at the intersection of the given line and the x-axis.)

Next students completed the Mathematics Assessment Project chart about equations of circles to help quell any misconceptions they might have.

And then another Quick Poll to see how students are doing writing the equation “from scratch”.

Which brings us a good opportunity to attend to precision. This is where, in the past, I might ask

-do you have parentheses?

-is your radius squared?

-do you have (x+5) squared and (y-1) squared?

And all of my students would have nodded.

But with Navigator, the students determine which are correct and what some need in order to be correct. They see whether their response is leveling up to the standard or not. They find out what to do for their next response to level up to the standard.

We revisited the bell work during the last few minutes of class. Students decided whether they wanted to keep their original responses or revise their response after the lesson. A few changed their responses.

And I know who to need extra support as we continue to learn.

And so the journey continues with good evidence of what my students know and what my students still need to know …

 

 

 

Sides of a Rectangle

Sides of a rectangle played out differently this year than last.

The question is tucked away in the Mathematics Assessment Project formative assessment lesson on Finding Equations of Parallel and Perpendicular Lines.

Line segment SP has equation y=2x+3. Find the equations of the line segments forming the other three sides of the rectangle.

Towards the end of our first day in our unit on Coordinate Geometry, I sent a Quick Poll to collect three equations from each student on which the other three sides of the rectangle might lie.

I was easily able to sort the results in the Portfolio Workspace by student so I could tell whether each student was successful.

For example, Chandler has two equations with a slope of 4, and upon further investigation, they represent the same line. He also has the given equation, y=2x+3 as one of the possibilities, although he has written it in an equivalent form.

Daniel’s submissions will work, but Eli has submitted a line that will not be perpendicular to the given line (his slope is ½ instead of –½.)

There were other issues as well. What do you notice about Kelsey’s equations (and several others, whose work I have not shown)?

Kelsey y=−1/2-4       09:57:58.322

Kelsey y=−1/2+4      09:57:58.322

Kelsey y=2*x-3          09:57:58.321

Last year I had my students enter their equations on a Graphs page. I was able to watch them correct their mistakes with Class Capture. I didn’t have to be the expert. This year, I sent a Quick Poll on the second day that was just a little different from the first. I have “Include a Graph Preview” checked, so as students entered the equations they had come up with the day before without graphing technology, the equations are graphed for them.

I watched using Class Capture.

 

Kelsey’s equations turned in to

Kelsey y=−1/2*x-6   09:42:41.031

Kelsey y=−1/2*x+4  09:42:41.031

Kelsey y=2*x-3          09:42:41.031

Chandler realized that graphing y=(4/2)x+(6/2) was the same as y=2x+3. He changed his submission to

Chandler        y=2*x+7 09:45:08.199

Chandler        y=−1/2*x+1 09:45:08.199

Chandler        y=−1/2*x+9 09:45:08.199

Eli submitted

Eli        y=−1/2*x+6  09:41:20.023

Eli        y=2*x  09:41:33.813

Eli        y=−1/2*x-2   09:41:33.813

I did want to have some whole class discussion on how students were working. And I realized later that I should have “allow resubmit” on the Quick Poll before I sent it so that students would be able to revise their answer after submission if needed. So I made a student the Live Presenter.

What do you notice about the equations? What kind of figure is formed? How can we correct the equations?

And then I made another student the Live Presenter.

What do you notice about the equations? What kind of figure is formed? How can we correct the equations?

And so the journey continues … providing opportunities for students to make sense of mathematics, often using technology to help them learn, even if the final assessment question might be calculator off.

 

Inscribed & Circumscribed Right Triangles

Several students noted in their reflection on our circles unit that the performance assessment tasks helped them “combine everything that we learned to find the correct answers to challenging problems”.

We started with a diagram from the Mathematics Assessment Project formative assessment lesson. Inscribing and Circumscribing Right Triangles

and asked students to

1. Figure out the radii of the circumscribed circle for a right triangle with sides 5 units, 12 units and 13 units. Show and justify every step of your reasoning.

2. Use mathematics to explain carefully how you can figure out the radii of the circumscribed circle of a right triangle with sides of any length: a, b and c (where c is the hypotenuse).

The second task provided a bit more structure. Circles in Triangles comes from a Mathematics Assessment Project apprentice task.

Students were given the following:

This diagram shows a circle that just touches the sides of a right triangle whose sides are 3 units, 4 units, and 5 units long.

1. Prove that triangles AOX and AOY are congruent.

2. What can you say about the measures of the line segments CX and CZ?

3. Find r, the radius of the circle. Explain your work clearly and show all your calculations.

I wonder what would have happened if we had asked students to determine the radius of the inscribed circle without as many auxiliary lines already given in the figure? Instead, what if we had used the same figure as the first task?

Another teacher taught this lesson to my students. She saved their work for me because even with the auxiliary lines drawn, students made sense of the structure (and particularly the expressions they wrote for the segment lengths) in different ways.

 

Then they were asked to determine the radius of the inscribed circle for a 5, 12, 13 right triangle, and then they were asked to generalize their results.

This diagram shows a circle that just touches the sides of a right triangle whose sides are 5 units, 12 units, and 13 units long.

4. Draw construction lines as in the previous task, and find the radius of the circle in this 5, 12, 13 right triangle. Explain your work and show your calculations.

5. Use mathematics to explain carefully how you can figure out the radii of the inscribed circle of a right triangle with sides of any length: a, b and c (where c is the hypotenuse).

 

I wonder how technology fits in with tasks like these?
We had a skeleton of a diagram prepared for students who wanted to use it.

Students had to measure themselves for the construction to be helpful in making sense of the mathematics. I find that the technology can be helpful for those who don’t know where to start. What do you notice when the triangle is right?

Can it help them reason abstractly and quantitatively, starting with the quantitative and building to generalization? Can it help them make sense of problems and persevere in solving them, when they don’t know what else to do on paper?

We also had a skeleton of a diagram prepared for the inscribed circle for those who wanted to use it.

What auxiliary lines (or segments) would you construct and measure for the inscribed circle?

The technology can help them make a conjecture about the length of the radius, and then they can go back to the mathematics to help them understand why.

And so the journey continues … Maybe some year I will be brave enough to start our Circles unit with this task and let the mathematics unfold in the context of the task as it is needed.