Tag Archives: Math Nspired

Connecting Factors and Zeros

NCTM’s Principles to Actions suggests Mathematics Teaching Practices for teachers. Two of those are the following.

MTP 1 Establish mathematics goals to focus learning

MTP 6 Build procedural fluency from conceptual understanding

If the goal for students is to use the factors of a quadratic function to determine its zeros, what concepts must students understand to meet that learning goal?

Our team wrote this leveled learning progression for our lesson.

Level 4: I can factor a quadratic function.

Level 3: I can use the factors of a quadratic function to determine its zeros.

Level 2: I can expand the product of two binomials.

Level 1: I can solve an equation in one variable.

Level 1: I can determine the zero(s) of a function from the graph of a function.

We decided to first ensure that students know what a zero is, and we checked this is more than one way on the opener for the day. (See this source for similar Level 1 problems.)

Students had to place a point at the zero of the function.

Almost all students were able to note that the point of interest is where the graph intersects the x-axis.

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Students had to name the coordinates of the zero of the function, which about half could do.

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And then students had to answer a question about a zero in context. A few more than half could do this.

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We decided that students also need to be able to solve an equation in one variable.

Which they could easily do.

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And we also decided that if students are going to meet the learning goal, they are also going to have to be able to multiply binomials. Which you can tell from the results that they could not easily do (Q8 and Q9).

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In the lesson, we started with the zeros of a linear function.

What do you notice?

If I give you a similar equation, can you tell me the zero?

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What do you notice on this page?

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If I give you a similar equation, can you tell me the zero?

We checked in with students using some Quick Polls.

What do you notice about the answers for this first poll?

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(I noticed that not all are x-intercepts.)

Students showed some improvement as we continued.

The answers are all x-intercepts.

We asked questions like …

How can we tell that (-6,0) is the correct choice using the equation?

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We spent a long time on linear functions. Some might think we spent too long.

Then we looked at a quadratic function.

And we related the linear factors to the quadratic visually.

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This is part of a Math Nspired activity called Zeros of a Quadratic Function, where there is a lot more flexibility in changing the factors.

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Our leveled learning progression for the second lesson changed a little:

Level 4: I can factor a quadratic function.

Level 3: I can use the factors of a quadratic function to determine its zeros, and I can use the zeros of a quadratic function to determine its factors.

Level 2: I can rewrite a quadratic function given in factored form to standard form.

Level 1: I can determine the zero(s) of a quadratic function from the graph of a function.

When we checked for student understanding during the opener of the second lesson, we saw that students were able to determine the zero(s) of a quadratic function from the graph of a function.

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Lots of students were at Level 1, determine the zeros when given the graph and the equation.

Not as many were at the target – but definitely more than had reached it the day before.

We have worked to build procedural knowledge from conceptual knowledge in our unit on Zeros and Factors. Our standards say that we want students to “Factor a quadratic expression to reveal the zeros of the function it defines”. The standards don’t say that we want students to factor a quadratic expression just for the sake of factoring.

What opportunities are you providing your students to concentrate on relationships rather than just results?


Posted by on April 27, 2015 in Algebra 1


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In geometry, we often use Always, Sometimes, or Never:

A trapezoid is ___ a parallelogram.

A parallelogram is ___ a trapezoid.

(Be careful how you answer those if you are using the inclusive definition of trapezoid.)

And in geometry, we often use True (A) or False (S or N):

A trapezoid is a parallelogram.

A parallelogram is a trapezoid.

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(Apparently we were talking about squares and trapezoids, not parallelograms and trapezoids, when we were figuring out which had to be true.)

And in geometry, we often use implied True (A):

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So when a few students asked about this question, we asked whether you could draw any parallelogram that doesn’t have four right angles. Since you can, we don’t say that the statement is (A) true.

[Note: the green marks indicate the number of students who answered both and only rectangle and square.]

In geometry, we are still learning the implications of the inclusive definition of trapezoid. Another of our questions was the following.

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And thank goodness, in geometry, I have students who question me, even those with a voice so quiet we have to lean in to hear. “But I can draw a trapezoid that doesn’t have exactly one pair of parallel sides. I don’t think the trapezoids should be marked correct.”

And of course, he is right. In our deductive system, we don’t name any quadrilaterals with exactly one pair of parallel sides.

So how could I recover our lesson and be sure that my students understood both what we mean by (A) true and our inclusive definition of trapezoid?

I asked students to look ahead to a graphic organizer (borrowed from Mr. Chase, who borrowed from, review it, and answer the new question. I took the question from the opener and changed “exactly one” to “at least one”. I asked students to work alone.

Graphic Organizer

Here’s what I got back.

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Without showing them the results, I asked students to talk with their teams and answer the poll one last time. All 31 students answered correctly.

So what next?

I’ve been determined over the past three years to stay away from the quadrilateral checklist. You remember the one, right? This is mine from the first 18 years of teaching geometry. I didn’t complete the list for them – each team had a different figure, and they measured (with rulers and protractors before we had the technology with measurement tools) and figured out which properties were always true. But still – how effective is it to complete a checklist, even when you and your classmates are figuring it out?

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We wanted to get at how the quadrilaterals are the same and how they are different in a way that was more engaging than just showing a few figures and asking students to calculate a missing measurement.

So another Quick Poll to get the conversation going. Students immediately began talking with their teams. For which figure(s) will the one angle measure be enough for us to determine the remaining interior angle measures?

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And decent results.

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We went to figure B. Why isn’t one angle measure enough?

And then figure C. Why isn’t one angle measure enough?

And then figure D. Why is one angle measure enough?

Student justifications included words like “rotation”, “reflection”, “decompose into triangles”, “isosceles triangle”. We talked about how we knew the triangles in the kite pictured were not congruent, and in fact not similar either, when decomposed by the horizontal diagonal. Informal justifications … but justifications, nonetheless … and hopefully ammunition for students to realize they can make sense out of these exercises transformationally without having a list of properties for each figure memorized.

We spent a little more time on rhombi using a Math Nspired document for exploration, after which I sent another Quick Poll:

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How did ten students get 144˚?

The students figured out the error: a misreading of which angle is 36˚ … not a misunderstanding of angle measure relationships in rhombi.

And then more about kites using the same Math Nspired activity, during which time a student asked to be made the Live Presenter so that he could show his concave kite to the class. What properties do concave and convex kites share? (More than I expected. I’m not the what-can-I-do-to-break-the-rule kind of person. But I am surrounded by students and daughters who are.) And I am still amazed that SC asked to be the Live Presenter since that was usually the time that he excuses himself to go to the restroom.

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So what information is enough angle-wise in the kite for you to determine the rest?

We ended with a bit of closure with two final Quick Polls & results that provide evidence of student learning.

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And so the journey continues … always rethinking and revising lessons and questions to get the most out of our time and conversations together.


Posted by on November 5, 2014 in Geometry, Polygons, Rigid Motions


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Proving Slope Criteria for Parallel Lines

What do geometry students need to know so that they can successfully master the following standard?

CCSS-M G-GPE.B.5. 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).

We recently started a unit on coordinate geometry. On the bell work, we put a question or two about calculating slope. Then we played with our version of the Math Nspired document Slope as Rate.

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What do you notice?

What is true about the slope of a line that is increasing or decreasing, horizontal or vertical?

Then a Quick Poll. Can you relate slope to what we have been learning in geometry?

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And another.

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Would it help to reason quantitatively instead of abstractly?

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Not much. 56% of the students now have it, compared to 50% when there were no given side lengths.

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In Transformative Assessment in Action, James Popham says that one of the key “choice-points” in using formative assessment is to determine what performance level of students will cause you to make an “instructional adjustment” (pages 56, 74, 93). He suggests that making a decision ahead of time about the cut-off is more helpful than making that decision on the fly. I didn’t really think about this ahead of time – I’m learning to implement that practice – but because I consider these “challenge” questions that connect slope a bit differently to what we have been learning, I decided not to stop too long on these questions.

What happens when we put two slope triangles on the same line?

They are similar.

How do you know?

One student had her triangles arranged with one inside the other so that it was obvious that the corresponding horizontal and vertical sides were parallel, creating corresponding congruent angles when using the line as our transversal.

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Another student arranged her triangles so that a dilation about the shared vertex of the two triangles was more obvious.

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CCSS-M 8.EE.B.5 6 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.

So let’s think about two parallel lines. What do we know about parallel lines?

They don’t intersect.

Their slopes are the same.

How do you know their slopes are the same?

Our teacher told us.

So guess what we get to do in this class?
Prove it?


Let’s start with two parallel lines in the coordinate plane. I’m going to give you a few minutes to think by yourself about how you might prove that the two lines have equal slope. I’ll watch why you think.

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Many students used the math practice look for and make use of structure, adding auxiliary lines in their diagram.

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Which diagram would you choose to share first? Quite a few students had created parallelograms out of their lines. But we didn’t have much time, so we didn’t talk about the parallelogram as a whole class.

We started with the quantitative reasoning. LB had put points on his lines and calculated the slope of each to show that they were equal. How can we extend that to work for all parallel lines?

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I asked HJ to draw her picture for us.

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What do we know?

Students asked for some labels so that it was easier to talk about the diagram.

It didn’t take long to prove that the triangles were similar.

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How can we show that the slopes are equal?

CCSS-M G-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Since we have shown the triangles are similar, then all corresponding pairs of sides are proportional. Now we have actually proven that the slopes of parallel lines are equal.

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So what about the slopes of perpendicular lines? They are negative reciprocals. They are opposite reciprocals.

Okay. In some textbooks you would see that the product of the slopes of perpendicular lines is –1. Let me give you a minute to think that through.

Now we need to prove it. What do you see?

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Let’s extend the lines. What do you see now?

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It could be helpful to have one of the lines intersect the origin. Then all that is needed is a rigid motion to prove that it always works.

What do you see now?

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You would think that I know by now to pay attention to my students. I was expecting to hear that they saw the right triangle that is highlighted in pink. After all, that is what I saw when I proved the slope criteria for perpendicular lines myself in preparation for the lesson (using the geometric mean relationships that occur because of the altitude drawn to the hypotenuse).

So I was surprised to hear a student say that she saw the blue triangle, which of course is a great example of making use of structure.

So that is where our lesson stopped. We had already spent two days on our one-day lesson on Lines. I had to spend at least a few minutes of the remaining class time on Triangles in the Coordinate Plane.

I offered Problem Solving Points to anyone who wanted to finish the proof outside of class.

One student’s proof is below.

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The good news is that we made it farther in the lesson this year than last. We talked through a proof of the slope criteria for parallel lines last year but students didn’t have the opportunity to dig as deeply into as they did this year. At this rate, maybe next year we will get to do both.

And so the journey continues … being reminded each day to listen to what my students actually say instead of what I expect them to say.


Posted by on March 15, 2014 in Coordinate Geometry, Geometry


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Pythagorean Relationships: The Question That Probes

I don’t have the exact quote, but I read in Embedded Formative Assessment (Dylan Wiliam) that teachers having time to plan the questions that they will ask in a lesson to push and probe student thinking is important. We should ask questions to push students’ thinking forward and to uncover student thinking (and misconceptions).

I have read more about this idea of selecting questions ahead of time in Transformative Assessment (James Popham) and Transformative Assessment in Action (James Popham). Popham suggests that teachers not only need to plan what questions they will ask for formative assessment but also how they will respond when all students answer the question correctly versus the majority of students versus half of the students versus few or none of the students. I think that is ideal. I haven’t decided yet that it is practical for every lesson I teach.

In our dilations unit, we did a lesson on Pythagorean Triples and Pythagorean Relationships. We used part of the linked Math Nspired activities by the same name. The main purpose was to provide students an opportunity to make connection between a primitive Pythagorean Triple and the resulting triangles that can be dilated from that triangle. But at the recommendation of my upperclassman who have already taken the ACT and SAT, we also spent a bit of time providing students an opportunity to determine whether a triangle was acute, right, or obtuse given its 3 side lengths. While I think this concept could be implied from CCSS 7.G.A.2, it will be 2-3 more years before we have high school students who have been through CCSS Grade 7.

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

I started the lesson by asking students what question they might ask if they knew the side lengths of a triangle were 8, 16, and 17. Many asked about classifying the triangle, but a few asked what the area of the triangle would be. The question provided a nice problem solving points opportunity for those who wanted to learn about calculating the area of a triangle given the three side lengths.

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I exported the questions students asked and have included below:

Is it a right triangle? 6

What is the area of a triangle with side lengths of 8, 16, and 17?          1

Does it have a right angle?   2

What type of triangle do these 3 sides make?       2

is it a right triangle   1

The question would be what type of triangle is this        1

is it scalene?   1

what is the question that goes with this information?     1

what kind of triangle is it     1

classify the triangle? 1

what type of triangle is it     2

Is it a pythagorean triple?    2

what kind of triangle is it?   1

is it a pythagorean triple?    1

What type of triangle is it?   3

is it an acute triangle?          1

with these given measurements is there a pythagrian triple?    1

which side is the base?         1

what type of triangle does this three sides make 1

Is it obtuse?   1

is it a right triangle?  8

is this a right triangle?          2

Is the set of numbers a Pythagorean Triple?        1

can we find the area 1

what is the height of the triangle?  1

is it a pythagorean triple      1

What is the height?   1

Which are the legs, and which is the hypotenuse?           1

what is the area?       1

Can the area be found?        1

how do you know what the base is 1

is this a right triangle           1

whats is the base      1

what is the area        1

is it a right triangle ? 1

what type of triangle is it?   2

what is the height? to find area.      1

As students interact with the Pythagorean Relationships TNS document, they record whether the given side lengths form a triangle that is acute, right, obtuse, or nonexistent. Then students look for regularity in repeated reasoning to describe a relationship that is true about the squares of the sides of the triangles.

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For all of the given triangles, a≤b≤c. Some of my students wrote that a triangle is acute when a2+b2>c2. Others wrote that the triangle is acute when c2< a2+b2. Students already knew that a triangle is right when a2+b2=c2 or c2= a2+b2. Students also determined that a triangle is obtuse when a2+b2<c2 or c2>a2+b2. Students already knew that for 3 lengths to form a triangle, a+b>c, a+c>b, and b+c>a.

I sent a Quick Poll to assess student understanding. Our geometry team has a set of pre-prepared Quick Polls for each lesson. Teachers send them as needed in their classrooms. We don’t use every one of them, and we don’t necessarily send the same polls each time we teach a lesson. We practice formative assessment during our lessons to decide which questions to send and use the results to adjust the lesson.

I didn’t want to send three side lengths that formed a right triangle. Nor did I want to send three lengths that did not form a triangle. The first one I came upon happened to be an obtuse triangle. I sent the poll.

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And I was surprised by the results. The students had determined that for a triangle to be obtuse, a2+b2<c2. Why did one-third of them miss the question? I had to think fast. I could have shown them the correct answer. And then I could have worked the problem correctly. Or I could have asked what misconception the students who marked acute had. Would everyone have paid attention to that?

What I did instead was to show the students the results without displaying the correct answer.

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I asked students to find another student in the room and construct a viable argument and critique the reasoning of others. I walked around and listened to their arguments. And I sent the poll again.

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At that point, the students shared what happened to those students who had marked acute the first time. They had only observed that 8+15>18 was true instead of also noting that 82+152>182 was not true. And so it struck me at that moment that I had gotten lucky. Without realizing it ahead of time, I had chosen the right problem to send students to uncover their misconceptions. Had I sent another of the prepared Quick Polls instead that asked students to classify a triangle with side lengths 16, 48, and 50, all of the students would have gotten it correct, but they would have gotten it correct for the wrong reason. For that triangle, both 16+48>50 and 162+482>502 are true, and so the students would have chosen acute even if they had incorrectly used the Triangle Inequality Theorem to decide that.

I am glad that I sometimes get lucky as the journey continues …


Posted by on January 12, 2014 in Geometry, Right Triangles


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We started a unit on dilations last week.

Our I can statements:

Unit 6 – Similarity

Level 1: I can identify, define, and perform dilations. G-SRT 1

Level 2: I can determine the similarity of two figures using similarity transformations. G-SRT 2, G-SRT 3

Level 3: I can prove theorems about triangles. G-SRT 4, G-C 1

Level 4: I can solve for and prove relationships in geometric figures using similarity criteria. G-SRT 5

And the standards:

Similarity: G-SRT

Understand similarity in terms of similarity transformations

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

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

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

G-SRT 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

G-SRT 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity

G-SRT 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

G-SRT 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Circles: G-C

Understand and apply theorems about circles

G-C 1. Prove that all circles are similar.

This lesson needed rewriting from last year. We used part of the Geometry Nspired activity Corresponding Parts of Similar Triangles, but it just wasn’t all sequenced as it should have been.

So we tried again. We started with a task that I learned about in an NCSM session – Hannah’s Rectangle Problem.

I gave the students a piece of wax paper and a straightedge to determine which rectangles were similar to rectangle a. I sent a poll to find out what students thought. One student asked whether I was grading the poll. Everyone stared at him. I’m planning to have a conversation with him about what formative assessment really is sometime this week. I told them I wasn’t going to show them the correct answers yet (even though I have them marked in the screen capture below).

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Our unit is on dilations. What is your previous experience with the term dilation? Students paired to talk with each other. I heard “eyes” and “enlargement”.

Next we thought about what we need for a dilation. If we were going to dilate the given hexagon what do we need to know?

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Someone suggested that we need a number. He called it a factor. Okay, so we need a number to know how big or small to make the image. We’re going to call that number a scale factor. What else do we need for a dilation?

I have some cartoon dilations on the wall of my classroom. We looked at those. What do they all have in common? Lines. (Have you seen a cartoon dilation before? We used to do them in class. Not anymore, although I do think they give students a good understanding of dilations. We did these cartoon dilations with compasses and straightedges – not grids.) What do the lines have in common? A point. They all intersect at a point. Okay, so we are going to call that point the center of dilation.

Using our dynamic geometry software, we performed a dilation of the hexagon. We clicked on the hexagon, we clicked on a point outside the hexagon for the center of dilation, and we typed in a number for the scale factor. I asked students to explore a few ideas. What happens when the center of dilation is on, inside, or outside of the figure? What happens when the scale factor changes? What if the scale factor is negative? I am not always so specific when I give students time to explore, but I tried it this time to see what happened. I set the timer for 4 minutes. I used the Class Capture feature of TI-Nspire Navigator to see what they figured out.

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We had a conversation about what happens when the center of dilation is on the figure compared to outside. We had a conversation about what happens when the center of dilation is inside the figure.

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When is there a reduction? When the scale factor is a fraction.

Does everyone agree?

What if the scale factor is 5/2? Oh. Not just any fraction – a fraction less than 1.

Does everyone agree? –3 is less than one, but it isn’t a reduction. Fractions between –1 and 0 and between 0 and 1.

What happens when the scale factor is negative? The figure is reflected. We didn’t describe what is the line of reflection.

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Next up: some measurements. On this page, I want you to play with the slider. If I give you two of the measurements on the page, what can you do to determine the 3rd?

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Which figure is the image and which is the pre-image? How do you know?

If the scale factor on the first picture were 0.5, which figure is the image and which is the pre-image?

On the next page, we began to think about how many copies of segment AY it takes to make segment DE. What if we change the scale factor to 3? What if we change it to 0.5? We also began to think about how many copies of ∆AYM it takes to make ∆EDT. What if we change the scale factor to 3?

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We are going to define two figures as similar if there is a dilation (and if needed, a set of rigid motions) that would map one figure onto another. Can you show that AYM~DET? After students explore on their own, I made CA the Live Presenter so that she could share what she did. She rotated ∆DET using S to control the angle of rotation. Then she translated ∆DET by vector YC. Then she undid the dilation by changing the scale factor to 1. Did everyone do the same? No – some undid the dilation first; others rotated and undid the dilation without a translation.

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How can you show the two figures are similar? With a reflection and a dilation.

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We moved to paper.

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After a few minutes, a student shared how they dilated the triangle.

Did everyone work it the same way?

Students saw all kinds of proportional relationships, and many thought about slope.

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What if we want to dilate points A and B about O using a scale factor of 3?

We started on paper.

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And moved to the dynamic software.

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That was it for the first day.

We came back to a few problems the next day to continue our work on dilations.

One figure is a dilation of the other. Where is the center of dilation?

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Some students emphasized that we don’t only consider the vertices, but every pair of corresponding points on the pre-image and the image.

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How can we show the two figures are similar?

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What Standards for Mathematical Practice did the students have the opportunity to employ during our lesson on dilations? The dynamic software definitely provided them the opportunity to look for regularity in repeated reasoning.

Does the lesson still need some work? Absolutely. Is it better than last year? Absolutely. And so the journey continues ….

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Posted by on November 25, 2013 in Dilations, Geometry


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Angles in Polygons

So this is one of the topics that I’m hoping we will eventually phase out of our high school geometry course. But for now, we’re keeping it in since our students don’t come to us knowing the sum of the interior and exterior angles of a polygon.

I like the Math Nspired activity Interior Angles of Regular Polygons because it provides students the opportunity to look for regularity in repeated reasoning.

Students start by looking at the central angles of a regular polygon.

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Then they think about the triangles formed by the central angles of a regular polygon and generalize the sum of the interior angles as 180*n-360.

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Then they look at the polygon by drawing diagonals from one vertex and generalizing the sum of the interior angles as (n-2)*180.

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What is the relationship between these two expressions? An opportunity to look for and make use of structure.

When I first saw this activity, I worried about there being two ways to get to the interior angle sum. Will our teachers think we should only give them one way to do it? Won’t it confuse students for there to be more than one option? But who are we to decide which way makes sense to our students? I heard one of our geometry teachers saying that some of her students preferred subtracting out the central angles…that method made sense to them. I am proud of our teachers providing our students with multiple entry points to making sense of this formula instead of choosing one over the other. I’ve heard another teacher remark recently that some of our students can’t handle there being more than one way to do something. I think that is an unfortunate consequence of our students being raised in our education system. We must change the perception of students (and many of their parents) that there is only one way to get to the right answer.

We also use the Sum of Exterior Angles of Polygons activity, which gives students the opportunity to make sense of why that sum is always 360 degrees. I start by having students draw a triangle with angle measures and then one exterior angle at each vertex. What is the sum of the exterior angles? What do you predict for the sum of the exterior angles of a quadrilateral? Try it. Draw a quadrilateral with angle measures and then one exterior angle at each vertex.

Then we move to the technology to confirm our findings.

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I am glad to work with teachers who are willing to step outside their comfort zone as the journey continues ….


Posted by on November 20, 2013 in Geometry, Polygons


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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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The Center of a Circle

From NAEP, 1996, Grade 12 Mathematics:

This question requires you to show your work and explain your reasoning. You may use drawings, words, and numbers in your explanation. Your answer should be clear enough so that another person could read it and understand your thinking. It is important that you show all your work.

Describe a procedure for locating the point that is the center of a circular paper disk. Use geometric definitions, properties, or principles to explain why your procedure is correct. Use the disk provided to help you formulate your procedure. You may write on it or fold it in any way that you find helpful, but it will not be collected.

The NAEP results were not hugely promising in 1996.

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I’ve posed this problem to my students for several years now with some success. I first give them a paper circle for playing. Then I pass out circular lids and let them play for a few more minutes. And then we talk.

As you can imagine, the first suggestion is to fold the circle in half once and then fold it in half again. Which is great. As long as the circle can be folded.

So then we restrict our investigation to circles that cannot be folded.

How can we determine the center of the circle if the circle can’t be folded?

I had some interesting answers to this question this year. We didn’t spend as long on it as usual. We had more whole class discussion earlier. But we still learned something from each other.

In one class, a student suggested that we draw a chord, and then construct a perpendicular to the chord at one of the endpoints, creating a right triangle. (We had previously explored Tangents to a Circle from Geometry Nspired.)

And then, even better, before suggesting that we could then construct the midpoint of the hypotenuse of the right triangle (which of course, would be easy at this point), he suggested that we repeat the process – and that the intersection of the hypotenuses of the two right triangles would be the center of the circle.

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I did this lesson with another class last week, and they started with the usual suggestion of folding the circle in half two times. We moved to TI-Nspire. They suggested I draw in the diameter. But how would I know that it was a diameter?

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They agreed that I wouldn’t. So we started with a chord. And we made the chord so that it was obviously not a diameter.

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Someone suggested that we construct a tangent at one of the endpoints of the chord. Now if we had started with a tangent, it wouldn’t have been bad to construct a perpendicular to the tangent throught the point of tangency. But we hadn’t started with the tangent. We had started with the chord. I had no idea where this might go, but I was up for trying.

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Then someone suggested that we construct another tangent. Another tangent where? At the other endpoint of the chord.

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So now we have two tangents. Do we have a diameter? We don’t, obviously. But then they asked to move an endpoint of the chord. Make it so the tangents are parallel to each other. How do we know when they are parallel?

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When two parallel lines are cut by a transversal, consecutive interior angles must be supplementary.

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In both classes, we ended up discussing a lot more geometry than if I had passed out the instructions for constructing the center of a circle and asked them to follow the steps.

We still thought about the traditional construction of the center of a circle. How can we use this chord to get the center? Someone in the class suggested finding the midpoint of the chord.

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How can the midpoint help us to get to the center of the circle? They decided they needed a hint. So we looked through the menu to see if anything jumped out at us.

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And something did. Perpendicular bisector.

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And from there, we all agreed that now that we had a diameter, the center was no trouble at all.

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Of course we really didn’t need the midpoint first if we were going to construct the perpendicular bisector of the chord. But I am learning to listen to my students – and to use their suggestions even when I don’t first see where they are going to lead us. That is how I teach and learn mathematics every day with my students. And that is why I look forward to tomorrow’s classes, when the journey continues…


Posted by on March 24, 2013 in Circles, Geometry, Tools of Geometry


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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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The Area of a Square


I first remember asking this question seven years ago, which was pre-TI-Nspire Navigator; at the time, I used Promethean Activotes (a form of “clickers”) with my students. The clickers only worked with multiple choice questions, but now that I use TI-Nspire Technology, I am not limited to collecting results electronically for only multiple choice questions.

This seems like an ordinary geometry problem. But seven years ago when I gave the problem, I realized by using the time stamp on the electronic results that Emmalee had solved the problem correctly in just a few seconds. Emmalee was one of my quiet students about whom I am speaking when I say that the TI-Nspire Navigator system gives every student in my classroom a voice. Without using a response system, I would have never realized how quickly Emmalee had solved the problem and known to ask her how she worked the problem. I’ve been asking students for years whether anyone solved a problem differently than the others, but Emmalee would have never volunteered that information to us.

I actually didn’t start our discussion by asking Emmalee how she had solved it. I started with some of the students who successfully solved the problem in the longest amount of time. You see, just like you, I knew something about Effective Classroom Discourse, even though I didn’t know how to name what I was doing. After some discussion, we realized that most of the students recognized that the square could be decomposed into two 45°-45°-90° triangles. They went from the given hypotenuse back to the leg using the generalizations that they had made about the side lengths of Special Right Triangles. Once they found the leg, some rationalized it and some didn’t, but they all squared it to get the area of the square. At that point, I asked Emmalee how she had solved the problem. Emmalee looked up and said, “We were just talking about how to calculate the area of a rhombus. Since a square is a rhombus, I used that area formula instead.” The area of a rhombus is half the product of the diagonals, and it had taken Emmalee less than five seconds to calculate ½*10*10. Emmalee had entered the practice of look for and make use of structure long before we even had the Standards for Mathematical Practice. Emmalee’s explanation was so compelling that I realized I needed to provide my students the opportunity every year to think about this problem the way Emmalee had. I’ve probably asked this problem for most of the years that I taught geometry, but it was seven years ago that the problem was memorable – it was then that I realized the problem could be solved quickly because of the technology that we were using in the classroom. Many students before Emmalee might have worked it the same way, but that discussion had never come up in my classroom – maybe other students were quiet, too, and I hadn’t the technology to give them a voice at the time.

This year I had to find a different way to include the problem. We don’t really have a unit on area anymore. Unfortunately, we haven’t had a day to make sense of area formulas like we have in the past. Every once in a while, in passing, we will talk about an area formula. So most of my students didn’t explicitly know the area formula for a rhombus. I put a sequence of questions on a recent bellringer to see what would happen.

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So as we were talking about these problems, we had to spend some time developing the formula for the area of a rhombus. The students first recognized that one diagonal of a rhombus decomposes the rhombus into two isosceles triangles – we generalized the formula for the area of the rhombus that way. (I was surprised that they didn’t first notice that we could decompose the rhombus into four right triangles.) Then we talked about the area of the square – and I had to lead them through the practice of look for and make use of structure. Now that you know how to calculate the area of a rhombus, would you have calculated the area of the square differently? They thought and talked and made sense of the structure of the square. And they figured out that they could calculate ½*6*6 for the area of the square.

These are small steps in getting them to extend their reasoning to shaded area problems:

  • A circle is inscribed in a square with a side of 12 ft. What is the area of the region between the square and the circle?
  • A square with a side of 12 ft is inscribed in a circle. What is the area of the region between the square and the circle?
  • A square with a diagonal of 10 ft is inscribed in a circle. What is the area of the region between the square and the circle?
  • A circle is inscribed in a square with a diagonal of 12 ft. What is the area of the region between the square and the circle?

And so the journey continues ….


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