# Tag Archives: TI-Nspire

## Seven Circles … Again

We tried Seven Circles I from Illustrative Mathematics a few weeks ago. At the end of class one day, I showed students the diagram and what question they might explore with it. I collected their responses using an Open Response Quick Poll and have shown the results below.

What does this figure have to do with geometry?            1

if we connected each top vertex of the triangles, will it make a hexagon?         1

whats the area of all the circles       1

Why are the circles in this shape?  1

what are the circles forming?          2

what is the area of all of the circles            1

why are we looking at circles?         1

are the spaces between the circles triangles?       1

what are the seven circles forming?           1

do all the circles have the same diameter?            1

do all the circles have the same diameter  1

Are the circles’ diameters the same?          1

Can the measures of the triangles that can be drawn through circles be calculated quickly  1

why are all the circles touching?     1

Why are the circles in that certain arrangement?            1

Can you find the area for that?       1

Is there a way to solve non 90° triangles? (With sin, cos, tan, or the other trig functions)      1

Can the circles be mapped onto each other with a rigid motion?           1

when you look at the image what do you see?      1

are the 6 figures that look like triangles in the gaps of the circles considered triangles since their sides arent straight    1

what are the circles for        1

are all of the circles congruent to each other?       1

What is the significance of the circular pattern?   1

How can you find the measurement of each circle           1

What are the triangular looking spaces in between the triangles called?          1

Some students were interested in the space between the circles. Other students wondered whether the circles were congruent. The task is given below. My students felt like it was pretty obvious that this could work with 7 congruent circles. I gave them different sized coins so that they could play. What if the circle in the middle is not congruent to the others? Will this work for 6 congruent circles? Or 8 congruent circles?

After students played for a few minutes, I sent them a TNS document that a friend made to explore this task. I used Class Capture to watch while students used the technology to make sense of the necessary and sufficient conditions for 6 circles and 7 circles in the given arrangement. Who had something interesting to discuss with the whole class?  Many students saw the regular pentagon or regular hexagon with vertices at the centers of the outside circles and used that to make sense of the mathematics. While I was watching them, I was trying to figure out how we should proceed as a class. We started with Claire’s work. What do we know? We saw a dilation. We saw central angles of a regular pentagon. We saw isosceles triangles, which we bisected to make right triangles. We saw an opportunity to use right triangle trigonometry. We looked for and made use of structure. We reasoned abstractly and quantitatively.

And before the bell rang, we looked back at the picture with 7 circles and recognized that the 30-60-90 triangles require that the radius of the center circle equal the radius of the outer circles. We only touched the surface of what we can learn from this task. Last year, we didn’t even do that. Last year, I shared the task with students during their performance assessment lesson, but we spent all of our time on Hopewell Triangles. This year, we got to it, but I know that our exploration could have been better. We began to answer what are the necessary and sufficient conditions for 6 circles. And in the process, we came across an argument for why the 7 circles must be congruent. But we didn’t really solve the conditions for 6 circles.

I wanted to write about this as a reminder that we are all learning. In this journey, I am finding good tasks out there to try with my students. And I am more confident about some than about others. Even though I don’t know exactly how the tasks should play out in the classroom, I am going to keep trying them. And I’m not going to throw out the task just because we didn’t get as deep into the mathematics as I wish we had. I will try again next year as the journey continues …

Posted by on February 24, 2014 in Circles, Geometry, Right Triangles

## Angles in Circles

Circles: G-C

Understand and apply theorems about circles

2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

We used part of the Geometry Nspired activity Secants, Tangents, and Arcs for this lesson.

I am not convinced as to whether we should still discuss and make sense of angles formed by two chords, two secants, a secant/tangent, and two tangents. Part of me thinks maybe this is one of those topics in our infamous mile-wide-inch-deep curriculum. We had a lesson on it anyway. For two reasons. 1. CCSS-M is the floor of our curriculum, not the ceiling. 2. I asked my junior and senior students whether I include it, and they said yes.

In the spirit of deliberately having my students estimate before we explore, I sent a poll. 23% of the students got it correct. So maybe this is another reason to include this topic: the calculations are apparently not intuitive.   We moved to our dynamic geometry software to explore. Was anyone correct in their thinking?

After students explored for a few minutes on their own, I sent another poll. One minute in, 68% of my students have the correct answer. But neither show the students nor stop the poll. I wait. Technology helps me to “ease the hurry syndrome”. (Or at least makes me). One minute and 40 seconds in, 90% of my students have the correct answer.

We formalize our conjecture. And then I ask the question a different way.  What happens when an angle is formed by two secants? Before our exploration, I sent a poll. Again, not very intuitive as to how to calculate the angle measure. 3 students have it correct, 10%.  This time I asked the students who got it correct to tell us what they did.

One said that he realized the angle would be smaller than that formed by two chords with the same intercepted arcs. Instead of adding the two arcs, he subtracted and then divided by 2. Did anyone else work it a different way? Another student excitedly said that he had the same reasoning as the first student.

We moved to the technology to verify the conjecture and formalize the results.

And a final poll, to check for understanding. 81% correct.  Of course, by this point, class was ending. I looked back through my poll results after class to see whether the same students were missing polls all day … to determine who is going to need extra support for this lesson. We had spent the class period exploring only two types of problems. 15 years ago, I would have “gotten through” all of my examples. But I know for sure, because of my assessment results, that not as many students owned the material as I had thought while they were politely nodding their heads “in understanding”. And I know for sure, because of my juniors and seniors who assured me that I should continue to teach this lesson, that learning by exploring helps them remember.

And so the journey continues …

Posted by on February 18, 2014 in Circles, Geometry

## The Similarity Ratio

The Similarity Ratio

How would your students solve the following problem? (After they discuss their love or hatred of clowns, that is.) A clown’s face on a balloon is 4 in. high when the balloon holds 108 in.3 of air. How much air must the balloon hold for the face to be 8 in. high?

My students try to take the given measurements and make a proportion out of them. For several years now, I have tried to figure out a way to help students not just understand that the areas and volumes of two similar figures are not proportional to their side lengths (or perimeters) but know how to apply that concept to solve problems.

This year we started with the Soccer Ball Inflation video on 101 Questions.

Then we explored what happened with similar rectangles – their similarity ratios, perimeters, and areas.    We summarized the results: And then moved to right triangles.  And then we worked problems similar to the clown problem, uncovering misconceptions, figuring out the incorrect thinking that occurred for incorrect responses. And the results on the summative assessment were better than usual. Last year, on a problem like the clown problem, less than 50% of students answered it correctly. This year, 72% answered it correctly.

And so, the journey continues, where some years the results are better than others …

Posted by on January 26, 2014 in Dilations, Geometry

## Similarity Theorems

Similarity Theorems

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.

I have struggled some to know which additional theorems we should include when teaching G-SRT 4. So, at least for now, my best source for whether to teach something that we have traditionally included has been to ask my juniors and seniors who have recently taken a high stakes standardized test for college entry and scholarships. I get that my only reason to teach something shouldn’t be whether or not it is “on the test”, but I don’t want my students to be at a disadvantage when they go to take the test.

We started the Dilations unit by showing that two triangles are similar when there is a dilation and if needed, set of rigid motions, to map one figure onto another. In our lesson on Similarity Theorems, we discuss similarity criteria for triangles in terms of the traditional similarity theorems: AA~, SAS~, SSS~. We formatively assess that students can use the similarity theorems.

Are the figures similar? If so, why?

If the figures are similar, then what is the value of x?

Students always do well with the first triangle but not with the second. Asking the second uncovers student misconceptions.

We briefly explored 3 other theorems dealing with similar figures.

1. A midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. What is the relationship between the midsegment and the third side of the triangle? We used part of the Geometry Nspired activity called Triangle Midsegments for this exploration.

This is an example of where technology makes me slow down. It would be so much faster for me to tell students that the midsegment is parallel to the third side of the triangle and that it is half the length of the third side of the triangle. But I have found that students remember the information longer when they make sense of it: from constructing the midsegment to observing what happens as the triangle changes.

We formatively assess to be sure that students can use what they have observed. And then we prove the theorem. Where would you start?

What Standard for Mathematical Practice would be helpful?

2. The Side-Splitter Theorem.

Observe what happens.

Formatively assess to be sure students can use what they observed. And then prove the theorem. Where would you start?

What Standard for Mathematical Practice would be helpful?

We used look for and make use of structure. The auxiliary line gives us something two cases of “a line parallel to one side of a triangle divides the other two proportionally”, and then we can use the parallel lines and transversal to show that the two triangles are similar to each other by AA~. 3. The Angle Bisector Theorem.

Observe what happens.

Formatively assess to be sure students can use what they observed. And then prove the theorem. Where would you start?

What Standard for Mathematical Practice would be helpful?

What auxiliary lines would be helpful? We actually didn’t get to completely prove The Angle Bisector Theorem…our time ran out. But I think that’s okay. We covered the Angle Bisector Theorem because my former students suggested it is helpful. And because we have taken care all year to observe what happens and prove most of what happens, my students realize that the result isn’t magic.

I started reading The Joy of X this morning. I love Steven Strogatz’ quote at the bottom of page 5: “Logic leaves us no choice. In that sense, math always involves both invention and discovery: we invent the concepts but discover their consequences. … in mathematics our freedom lies in the questions we ask – and in how we pursue them – but not in the answers awaiting us.”

And so the journey to ask more questions continues …

Posted by on January 26, 2014 in Dilations, Geometry

## 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. 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.

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. 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. 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. 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

## 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. 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. 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. 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. 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 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 …

Posted by on November 4, 2013 in Calculus

## 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 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 ….