Tag Archives: CCSS Mathematical Practices

#NCSM14 Art of Questioning: Leading Learners to Level Up #LL2LU

What if we empower and embolden our learners to ask the questions they need to ask by improving the way we communicate and assess?

Great teachers lead us just far enough down a path so we can challenge for ourselves. They provide us just enough insight so we can work toward a solution that makes us, makes me want to jump up and shout out the solution to the world, makes me want to step to the next higher level.  Great teachers somehow make us want to ask the questions that they want us to answer, overcome the challenge that they, because they are our teacher, believe we need to overcome. (Lichtman, 20 pag.)

On Monday, April 7, 2014, Jennifer Wilson (@jwilson828) and Jill Gough (@jgough) presented at the National Council of Supervisors of Mathematics Conference in New Orleans.

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Jill started with a personal story (you’re letting her shoot…) about actionable feedback and then gave the quick 4-minute Ignite talk on the foundational ideas supporting the Leading Learners to Level Up  philosophy.

Our hope was that many of our 130 participants would help us ideate to craft leveled learning progressions for implementing the Common Core State Standards Mathematical Practices.  Jennifer prompted participants to consider how we might building understanding and confidence with I can make sense of problems and persevere in solving them. After giving time for each participant to think, she prompted them to collaborate to describe how to coach learners to reach this target.  Jennifer shared our idea of how we might help learners grow in this practice.

Level 4:
I can find a second or third solution and describe how the pathways to these solutions relate.

Level 3:
I can make sense of problems and persevere in solving them.

Level 2:
I can ask questions to clarify the problem, and I can keep working when things aren’t going well and try again.

Level 1:
I can show at least one attempt to investigate or solve the task.

 Participants then went right to work writing an essential learning – Level 3 – I can… statement and the learning progression around this essential learning. Artifacts of this work are captured on the #LL2LU Flickr page.

Here are the additional resources we shared:

How might we coach our learners into asking more questions? Not just any question – targeted questions.  What if we coach and develop the skill of questioning self-talk?

Interrogative self-talk, the researchers say, “may inspire thoughts about autonomous or intrinsically motivated reasons to purse a goal.”  As ample research has demonstrated, people are more likely to act, and to perform well, when the motivations come from intrinsic choices rather than from extrinsic pressures.  Declarative self-talk risks bypassing one’s motivations.  Questioning self-talk elicits the reasons for doing something and reminds people that many of those reasons come from within. (Pink, 103 pag.)

[Cross-posted on Experiments in Learning by Doing]


Lichtman, Grant, and Sunzi. The Falconer: What We Wish We Had Learned in School. New York: IUniverse, 2008. Print.

Pink, Daniel H. To Sell Is Human: The Surprising Truth about Moving Others. New York: Riverhead, 2012. Print.

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Posted by on April 9, 2014 in Professional Learning & Pedagogy


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We had spent some time using the Math Nspired activity Congruent Triangles to explore sufficient criteria for proving triangles congruent. Our goal is to make two triangles using the given criteria that are not congruent. We are obviously not proving that the criteria works when we can’t do it, as we might not have extinguished all possibilities, but if we can find a counterexample, then we are proving that the criteria doesn’t work.

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I didn’t realize until I began studying CCSS-M that most of our geometry textbooks admit SSS, SAS, and ASA as postulates. With CCSS-M, we prove the triangle criteria using rigid motions. But I digress. I want to write today specifically about SSA.

So we figured out that SSA doesn’t always work. And using our dynamic geometry software, we also figured out that SSA does work when the triangles are right (otherwise known as HL).

After our exploration, I gave students the following diagram and asked whether the triangles were congruent.

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The given information shows SA. Is there anything not marked that we can mark? NP=NP by reflexive. (So it wasn’t that I asked the question and students said NP=NP and then I asked why. It really was that students answered NP=NP by reflexive. They are learning that I don’t just care about the answer. I care about why. And they are beginning to include their justifications as part of their answer.) So now the given information shows SSA. Is that sufficient information to prove the triangles congruent? Most students said no. But I had a few dissenters. They were not convinced that the given triangles weren’t congruent. Their initial argument was that P must lie on the perpendicular bisector of segment MO. But does N also have to lie on that perpendicular bisector? This is the beauty of dynamic geometry software. I don’t have to be the expert. Can you convince me that the triangles are congruent? I’ll give problem solving points to anyone who can. (Note: what problem solving points are will be a future post.)

Some students built the diagram using TI-Nspire.

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Other students drew the auxiliary line segment MO to show first that triangle PMO is isosceles and ultimately that triangle MNO is isosceles.

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We cut out the SSA example that doesn’t work and showed that if NP is one of the congruent sides, then MP and NO would have to be the other pair of congruent sides.


A few days later, I asked about another pair of triangles.

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How would your students answer?

I am thankful for a community of students who feel comfortable dissenting. And so the journey continues …

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Posted by on November 20, 2013 in Angles & Triangles, Geometry


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Approximating Area – Follow-Up

In an attempt to think more about setting goals and formative assessment, I posted my calculus lesson plan last night, and I am now reflecting on what actually happened.

Bell 4-1 (if needed, #6 will be for problem solving points)

My Notes: The questions went well, but they took a bit longer. Students were definitely using the Math Practices during this time. One constructed a viable argument about how the given trapezoid couldn’t exist, as one of its legs was shorter than the height.

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Another student used a CAS command, tangent line, that we had not used before in class to solve the second problem, so I made him the Live Presenter with TI-Nspire Navigator so that he could show it to his classmates.

We heard from several students about their methods for estimating the area on the next two problems.

They had great arguments – some used rectangles, others used triangles and rectangles, others eliminated answer choices using good reasoning. This was a good start to thinking about the area under the curve.

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25 minutes
The Old Pump – problem for students to work alone. Will send QP to collect responses and then have some share their methods.

My Notes: I enjoyed hearing students’ responses for this. While they were working, I selected student work for our whole class discussion, and I sequenced the student work deliberately. One student used the table – multiplied each rate by 10 and added the initial condition to find the total amount of water.

Other students graphed the data and thought about rectangles using the graph. A few students used regression to get an equation for a curve passing through most of the points. Some students found the mean of consecutive rates of change to use for each 10 minute interval. Over half got a reasonable estimate for the area. One even gave his response as an interval of gallons.

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15 minutes
Overview of answering our essential question “what is the area under the curve”, using flipchart as guide.

My Notes: This part went well, and it took longer than expected. But in the midst of our discussion, we began to talk about Riemann sums, and we figured out that for equal subintervals, the change in x (base of the rectangle) will be (b-a)/n. We also talked about unequal subintervals often giving better estimates. Students also decided that trapezoids would be better. The student questions drove the discussion during this section, which means we didn’t cover everything exactly as I had planned.

15 minutes
Practice Riemann Sum Rectangles – Right, Left, Midpoint

Students have choice of paper or TNS document

Use observation, Class Capture, & QP as needed for formative assessment

My Notes: Students needed help knowing what we meant by right rectangles. But we got there without too much difficulty. We spent more time than I expected on right, but we did it well. I asked students to draw left & midpoint outside of class.

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15 minutes
How can we get a better approximation?

My Notes: We had already talked about this because it came up during the overview.


5 minutes
Trapezoidal Rule

Trapezoid_Midpoint.tns & Riemann_Sums.tns

My Notes: We will talk about the Trapezoidal Rule formally next time. Today, we figured out that it will give a better estimate than the same number left or right rectangles, but we didn’t actually work one. We did spend a few minutes on the Riemann Sums TNS document, where students can see how the left/right/midpoint/trapezoids look different. Students put it together that if we found the sum of the areas of an infinite number of rectangles we could get the exact area – and recognized that we would use a limit to do that.

15 minutes
Closure – Plane Crash Application

My Notes: We didn’t have time to look back at the application, but I gave it to them to finish outside of class. But I did ask students to reflect on what they learned before the bell rang. I’ve included some of those reflections below.

5 minutes

I have learned deltax=((b-a)/(n)).

My question is why use rectangles and not trapezoid.

I have learned how geometry works in calculus.

My question is whats anosher way to do pump problem.


I have learned rieman sums.

My question is how to find the limit as the rectangles approach ∞.


I have learned more about approx area under curve.

My question is what mindbloxing thing are we learning next.


I have learned about Riemann rectangles and the difference in rectangle perspective.

My question is how to apply limits to this problem.


I have learned . trapizoids are extremely useful

My question is how to work an equation.


I have learned how to begin estimating area under curve.

My question is how to correctly find the area of curve.


I have learned that trapezoids are effective shapes to estimate area uner a curve.

My question is . how does the derivative relate


I have learned about riemann sums .

My question is i am confused on how to calculate midpoint rectangles.


I have learned that in order to obtain exact area, we must achieve infinite rectangles.

My question is how.


I have learned how to to estimate area .

My question is how to find exact area.


I have learned about how to not estimate area under a curve.

My question is why infinite rectangles doesn’t lead to infinite area.


I’ve sent an email to the student who is confused about midpoint rectangles to stop by during zero period on Thursday. She has already replied that she will come see me then.


The first I can statements for this unit are “I can approximate the area between two curves using left, right, and midpoint Riemann sums”, “I can approximate the area between two curves using the Trapezoidal Rule”, and “I can use an infinite number of rectangles to get the exact area between two curves”. I didn’t share these with students today, as I didn’t want to give what we were doing away too quickly. We definitely moved towards the first statement today. I’ll check where students think they are at the end of class next time as the journey continues …

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Posted by on November 4, 2013 in Calculus


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Approximating Area

I have been challenged this weekend to really think about goals and formative assessment, reflecting on how I change what I am doing mid-lesson (or mid-PD session) based on feedback from my students (or participants).

I have our M-STAR Teacher formative assessment tomorrow during my Calculus class. I have a plan, which I am going to share now. Then I will write a reflective blog post after the lesson to think about whether and how formative assessment during the lesson changed my plan.

The plan:

4-1 Approximating Area Lesson Plan – Jennifer Wilson, AP Calculus

Essential question: What is the area under the curve?

Objectives: Numerical approximations to definite integrals

  • Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.
  • Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral.

Note: Objectives are from the AP Calculus Course Description.

Bell 4-1 (if needed, #6 will be for problem solving points) 25 minutes
The Old Pump – problem for students to work alone. Will send QP to collect responses and then have some share their methods. 15 minutes
Overview of answering our essential question “what is the area under the curve”, using flipchart as guide. 15 minutes
Practice Riemann Sum Rectangles – Right, Left Midpoint

Students have choice of paper or TNS document

Use observation, Class Capture, & QP as needed for formative assessment

15 minutes
How can we get a better approximation? 5 minutes
Trapezoidal Rule

Trapezoid_Midpoint.tns & Riemann_Sums.tns

15 minutes
Closure – Plane Crash Application 5 minutes

While I have thought about some “I can” statements for students, I am not sharing those with students before this lesson. While I want them to be able to say “I can approximate the area between two curves using left, right, and midpoint Riemann sums” and “I can approximate the area between two curves using the Trapezoidal Rule” and “I can use an infinite number of rectangles to get the exact area between two curves”, I don’t want to give that away before the lesson. I want them to figure some of those things out through our exploration. What I’m going to tell them is that we are beginning to explore the area between curves, and that we are going to focus on the mathematical practice “reason abstractly and quantitatively”.

I’ll report back after the lesson to see what happens as the journey continues ….

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Posted by on November 4, 2013 in Calculus


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Day 1 – Embracing a Growth Mindset for Learning

Another school year has begun, and so I want to report on the first day of class. The report won’t make as much sense without knowing what has been on my summer reading list.

Embedded Formative Assessment – Dylan Wiliam

5 Practices for Orchestrating Productive Mathematics Discussions – Smith & Stein

Mindset – Carol Deck

Outliers by Malcolm Gladwell

What’s Math Got to Do with It?: Helping Children to Learn to Love Their Most Hated Subject–and Why It’s Important for America – Jo Boaler

Teach Like a Pirate – Dave Burgess

On top of that, I’ve been taking Jo Boaler’s course on How to Learn Math.

After reading what I have read, I knew that somehow the first day of class had to be different this year.

In the 4th session of Jo Boaler’s course, she talks about a framework for a growth mindset task that they developed at Stanford.

Growth Mindset Task Framework

1. Openness

2. Different ways of seeing

3. Multiple entry points

4. Multiple paths / strategies

5. Clear learning goals and opportunities for feedback.

Consider the following task for a geometry student as students are beginning to think about inductive reasoning.

Finish the sequence: 2,3,5,8,12,17,…

A few years ago I changed this task for day 1 of geometry.

Now the directions are to find and explain at least two ways to finish the sequence.




So what is the difference between the two tasks?

I want students to realize from the first day of class that there will often be more than one way to answer tasks, that we are not all going to see the same thing, and that being able to explain our thinking is important. Even if we arrive at the same solution, we might use different paths to get there. I also want students to realize that part of being a good student is being a good listener, so that we can really begin to get at constructing viable arguments and critiquing the reasoning of others.

I collected student responses to the sequences using TI-Nspire Navigator.


Our discussion encompassed recursive sequences, prime numbers (and composite), the Riemann Hypothesis, RSA Encryption, powers of 2, memory storage for electronic devices, the Fibonacci sequence, and more.

In light of this task, I passed out the CCSS Math Practices Handout that our department made last year and asked my students to reflect on which practices they had already used in class (they suggested make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, look for and make use of structure, look for regularity in repeated reasoning).

We talked about using the practices to do math, and I told them that I will want them to reflect on using a practice at least once each quarter using the CCSS MP journal prompts that I had copied on the back of their handout.

Finally, we had a discussion about fixed and growth mindsets. I sent the following poll and asked students with which statements (from the first chapter of Mindset) they agree.


Dweck suggests that those who agree with the first two statements tend towards a fixed mindset regarding their intelligence and those who agree with the last two statements tend towards a growth mindset regarding their intelligence.

One class of students marked the statements as shown below.


I shared some of the research that Dweck cites about growing your brain bigger (creating synapses) when you make and learn from mistakes. I also told my students that I can look back in my own life and see a change from a fixed mindset into a growth mindset. We are certainly not finished having this conversation, but I hope that my students will begin to realize that they can do something to change their intelligence and embrace the work that will require.

So here’s to another school year as the journey continues ….


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Standards for Mathematical Practice Student Journals

My goal this year has not just been for me to provide opportunities for my students to enter into the Standards for Mathematical Practice – but for my students to begin to recognize when they are entering into the practices. I gave my students a list of the practices along with a short explanation of each at the beginning of the year. I have that same list posted in the front of my room. This language has become part of how we talk to each other about what we are doing in class – and what our goals are for learning in each unit.

Every quarter this year, I asked my students to reflect on a time when they participated in one of the mathematical practices. When we started, I told them I had no idea what these journals were supposed to look like. I gave them a few reflection questions for each practice, but I didn’t know if they were the right questions to ask. It has been an absolute pleasure to read their responses – especially to get to learn about times when they recognized that they were participating in a mathematical practice and I didn’t observe that happening.

I want to share some of those reflections with you.

1. Hannah writes about the mathematical practice “Make sense of problems and persevere in solving them.” She says, “As I was working in class I didn’t even notice that I was using a mathematical practice. I am going to be honest, I don’t like math in the first place, so when I don’t know how to work something I get very discouraged. I kept working, and even though I was one of the last people to finish, I still was proud of myself. I am learning not to give up and to keep going even though I can’t figure it out.


The students were given a matrix of the vertices of a triangle. You can see from Hannah’s work that they were trying to figure out by what matrix they could multiply the given matrix in order to produce the vertices of a certain transformed images.

Hannah continues – “Next time I am struggling with a problem I will think of this incident, take a deep breath, and persevere through it.”

2. Franky writes “This semester’s math class has been one of the most challenging classes I have taken. With that being said, it has also been my most beneficial, and it is one of my favorites. No matter how difficult a problem may be, I have learned to continue trying and persevering, for I should be able to figure the problem out. Franky goes on to describe a problem – and then he says to begin solving “of course, draw a triangle”. I have plenty of students who used to never think about drawing a diagram to make sense of a problem – some of them know it will help but just don’t want to take the time to do it – we are having to change the habits and practice of our students.


Franky continues – “There is no greater feeling than solving a math problem correctly, especially if it is difficult. This class has taught me time and time again to just keep on trying.”


I think it is significant for our students to know that we are going to give them problems for which they must persevere in solving.

3. Erin talks about the practice model with mathematics. While she was cycling she made a connection between linear distance (which we had studied in trigonometry) and how the sensor on her bike was able to tell her the distance traveled.


4. While he was at band practice, Sam made a connection between the band’s formation on the field and the transformations that we were studying in geometry. He writes “I am in the center of a diamond shape. We have to translate across the field.” – Note that this was early in the year and he had not yet heard my “a diamond is a gem and not a geometric shape” lecture.


5. Emilee writes about attending to precision. In her mind, she knows that negative cosecant of an angle is different from inverse cosecant of an angle, but she said negative cosecant, knowing very well she meant inverse cosecant. She writes “My ignorance of precision led to confusion among my table, but I am slowly learning to pay more attention to my words. Saying things, thingys, and whatchamacallit are not acceptable anymore.”


6. Kaci writes about “look for regularity in repeated reasoning”. We figured out that half of a square is a 45-45-90 triangle, and students were trying to determine the other two sides of the triangle given one side length of the triangle. She says “To find the length of the hypotenuse, you take the length of a side and multiply by sqrt(2). The sqrt(2) will always be in the hypotenuse even though it may not be seen like sqrt(2). In her examples, the triangle to the left has sqrt(2) shown in the hypotenuse, but the triangle to the right has sqrt(2) in the answer even though it isn’t shown, since 3sqrt(2)sqrt(2) is not in lowest form. She says, “I looked for regularity in repeated reasoning and found an interesting answer.”


7. Jordan writes about “Construct viable arguments and critique the reasoning of others”. She says, “If you can really understand something you can teach it. Every person relates to and thinks about problems in a different way, so understanding different ways to get to an answer can help to broaden your knowledge of the subject. Arguments are all about having good, logical facts. If you can be confident enough to argue for your reasoning you have learned the material well.”


8. And Franky says that construct viable arguments and critique the reasoning of others is probably our most used mathematical practice. If someone has a question about a problem, Mrs. Wilson is always looking for a student that understands the problem to explain it. And once he or she is finished, Mrs. Wilson will ask if anyone got the correct answer, but worked it a different way. By seeing multiple ways to work the problem, it is easier for me to fully understand.”


These student reflections have been very significant in my CCSS-M journey this year. They give me some hope that my students are at least beginning to understand that how we do and learn math is important to me. These student reflections give me hope as our journey continues …


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Exploring the Equation of a Circle


Expressing Geometric Properties with Equations G-GPE

Translate between the geometric description and the equation for a conic section

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.

CCSS-M 8.G.8

Understand and apply the Pythagorean Theorem.

8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

We used the Geometry Nspired activity Exploring the Equation of a Circle to begin our exploration of circles.

We also used some ideas from the Mathematics Assessment Project formative assessment lesson Equations of Circles 1.

The TNS document begins by having students observe what they know about the given triangle.

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It is a right triangle.

As students move point P, what happens?

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The triangle is still right.

The hypotenuse stays 5.

The legs change length depending on the location of P.

Some students might say that a2+b2=52, if we let a and b represent the legs of the right triangle.

Then we do a geometry trace of P as we move P.

What path does P follow?

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If we let x represent the length of the horizontal length of the leg and y represent the vertical length of the leg, then we can say that x2+y2=52 for this circle. Alternatively, if we let (x,y) represent the coordinates of point P, then we can say that x2+y2=52. Then we explored what happens as we make the radius of the circle shorter and longer.

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After exploring the equation of a circle centered at the origin, we translate the center in the coordinate plane. Now what can we say about the right triangle that is pictured?

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After the exploration, we used the sorting activity in the Mathematics Assessment Project’s formative assessment lesson.

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And then TI-Nspire Navigator provided a good opportunity for formative assessment – and for students to attend to precision.

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My students left class not only with an understanding of how the Pythagorean Theorem is related to the Distance Formula and the Equation of a Circle, but they also got some good practice attending to precision through the formative Quick Poll that I sent and by categorizing circle equations. This was a much better lesson than I have had in previous years of teaching the equation of a circle.

And hopefully next year will be even better as the journey continues …


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Two Wheels and a Belt

Two Wheels and a Belt (and Why I am Convinced That the Standards for Mathematical Practice Must Be How We Do Math)

One of the last tasks that we gave our geometry students this year was Two Wheels and a Belt from Illustrative Mathematics.

A certain machine is to contain two wheels, one of radius 3 centimeters and one of radius 5 centimeters, whose centers are attached to points 14 centimeters apart. The manufacturer of this machine needs to produce a belt that will fit snugly around the two wheels, as shown in the diagram below. How long should the belt be?


The correct answer is 53.42 cm, which several students got, in more than one way.

Some used correct mathematical reasoning to get the correct answer.


Others used incorrect mathematical reasoning to get the correct answer.


It is unfortunate that the incorrect reasoning produced the correct answer. What is more unfortunate is that this incorrect reasoning goes unobserved when our focus is only on answers. When our focus is on how we do the math and not just on what we get as an answer, students and teachers can learn more about mathematics.

As soon as the students began to construct viable arguments and critique the reasoning of others, the misconceptions in the incorrect solution became evident.

And so the journey to help students know and understand mathematics continues …


Posted by on June 18, 2013 in Circles, Geometry, Right Triangles


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Sides of a Rectangle

I used the MAP Lesson Finding Equations of Parallel and Perpendicular Lines as a start to our unit on Coordinate Geometry.

One of the interesting tasks was for students to determine the equations of three other sides of a rectangle given that one side had the equation y=2x+3. I would have never thought to leave off the x- and y-axes had I created the question, but it brought about some interesting responses from my students for us to discuss.


We ran out of time to discuss the problem during the first lesson (imagine that!), so I asked for students to consider the problem outside of class. During the second class, I collected their work, but we did not go over it. The following are some of the responses that I got.

1. Screen Shot 2013-03-25 at 5.56.52 AM     2. Screen Shot 2013-03-25 at 5.57.01 AM

3. Screen Shot 2013-03-25 at 5.57.10 AM

4. Screen Shot 2013-03-25 at 5.57.18 AM     5. Screen Shot 2013-03-25 at 5.57.24 AM

6. Screen Shot 2013-03-25 at 5.57.34 AM

7. Screen Shot 2013-03-25 at 5.57.42 AM

I was very surprised by the progression of responses. It would have never occurred to me that students would have suggested the same line for the two opposite sides of a parallelogram. Several students remembered the slope criteria for perpendicular lines from their algebra class, but, again, more than one used the same line for the other pair of opposite sides of the rectangle.

I kind of liked the generalizing that several students did for the y-intercept of the equations of the sides, but they were not totally prepared for the practice of “reason abstractly and quantitatively”. Did they intend for their values of b or r (see above) to be equal? They used the same constant to represent the values.

The day after I looked at the student work, we revisited the problem as a class. We started with a Graphs page. Only a few of the students that I have this year used TI-Nspire handhelds last year in algebra, so their experience with graphing on TI-Nspire was limited. In fact, most had not thought of graphing the equations as an option. They  graphed the given equation. I showed them the first picture above. Could the same equation work for the opposite side? Some of them had to graph the equation again to realize that it wouldn’t work; others knew immediately.

Students determined a second equation for the opposite side. I used the Class Capture feature of TI-Nspire Navigator to monitor students’ progress – I am able to set Class Capture to refresh every 30 seconds so that I can walk around and discuss the problem with individuals or groups who need help.


As students continued to work, I was pleased to see them troubleshooting their own work. They entered the practice of “attend to precision” on their own – I didn’t have to tell them that lines with slopes of 2 and -2 are not perpendicular.


I didn’t have to tell them that y=-½+3 is the equation of a horizontal line instead of an oblique line. The technology provided them the opportunity to figure that out for themselves.




Even if I ask a similar question on a no calculator assessment later, I want my students learning and making sense of the mathematics with the technology.

And speaking of a future assessment, I’ve been trying to think of a good question to follow up on this task. What if I give them the equation of one side of a square and ask for the rest?

Or the equation of one side of an isosceles trapezoid and ask for the rest? If I remember correctly, what happens with the slopes of the two legs is interesting.

At least I’ll never run out of problems as the journey continues…

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Posted by on March 25, 2013 in Coordinate Geometry, Geometry, Polygons


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A Detour to the Slopes of Parallel Lines

G-GPE 5, under “Use coordinates to prove simple geometric theorems algebraically”, says that students should “Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).”

I read and re-read this learning objective several times before it occurred to me that we weren’t just exploring the slopes of parallel and perpendicular lines to make conclusions about their relationships – we were actually asked to prove the slope criteria for parallel and perpendicular lines. And then I had to think even longer to realize that I don’t remember ever proving that parallel lines have equal slopes. It just seems like something that I have always known.

I recently came across learning objective 8.EE.6 while working on a project.

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for a line through the origin and the equation y=mx+b for a line intercepting the vertical axis at b.

And so luckily it didn’t take me long to make the connection between the progression of how students should think about slope triangles from grade 8 through high school geometry.

We started with the Math Nspired Algebra Slope as Rate activity.


Points A and B are dynamic on this page. In the midst of talking about what we mean by slope, I happened to move the points so that the slope of the line containing A and B was 1.


One of the students noted to the whole class that the slope triangle was a 45°-45°-90° triangle. So we talked about the slope of the hypotenuse of a 45°-45°-90° triangle with horizontal and vertical legs.

It made me think to ask about the slope of the hypotenuse of a 30°-60°-90° triangle with horizontal and vertical legs. So we paused for a moment, and I set up a Quick Poll to ask what is the slope.



You can tell from the poll that I had the diagram drawn elsewhere. We had decided to put the 30° angle for the angle formed by the hypotenuse and horizontal leg. You can also tell from the poll, that no one correctly got the slope. One of the Standards for Mathematical Practice is to reason abstractly and quantitatively. They weren’t quite ready to reason abstractly.

So without giving them the answer, I added one piece of information. I honestly don’t remember now which piece of information it was, but I think it was a possible length of the hypotenuse.


Now that students had a quantity, about half were successful at giving the slope of the hypotenuse.



By this point, I was in a hurry. We had strayed off topic. I know that I should have spent more time on the slope problem to get everyone with us. Instead, I let a student explain how she got her answer and we spent just a little time looking at the responses of the Quick Poll in Navigator to correct some of the incorrect thinking. (I look forward to seeing what students do on their summative assessment when I change the angle formed by the hypotenuse and horizontal leg to 60°.)

And then we moved on. I was trying to get my students to the point of proving that the slopes of parallel lines are equal, and we hadn’t even made it to talking about the relationship between two slope triangles on the same line. So we looked at this page that I had added (with the help of Jeff McCalla). What relationship do two slope triangles on the same line have to each other? Students immediately recognized that the triangles would be similar. How do you know?

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They first asked to move point C to coincide with point A. They had seen these triangles before – and so it didn’t take long to show that the triangles were similar by AA~. And then because I am really trying to change our first instinct for proving triangles are similar           from AA~, SSS~, and SAS~ to thinking about whether one triangle is a dilation (or some series of rigid motions + a dilation) of the other, we talked about that as well. Then we generalized to what happens when A and C are not the same point.

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By this time we were ready to move to parallel lines (or at least I was).

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What Standards of Mathematical Practice did students have the opportunity to enter into with this problem?

  • Look for and make use of structure (Would they think to add auxiliary lines – slope triangles for each of the parallel lines? Where would they place the slope triangles? Did it matter?)
  • Reason abstractly and quantitatively (Were they ready for abstract? Or did we need to start with some numbers?)

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But I failed. We raced through talking about what a proof would look like, but we didn’t really prove it. And the next day, another lesson was calling our name (or at least my name).

This is hard work. And even though I try every day to “ease the hurry syndrome”, I’m not there yet. I keep hoping for next year when I will have the opportunity to try this again with a new group of students. It will be easier then, right?

At least the journey continues ….

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Posted by on March 14, 2013 in Coordinate Geometry, Geometry


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