# Tag Archives: reason abstractly and quantitatively

## The Circumference of a Cylinder

We talked about pi earlier this week in geometry, and we used Andrew Stadel’s water bottle question to start.

I’m not one to pull of the wager that Andrew used (unfortunately, my students will agree that I am a bit too serious for that), but we still had an interesting conversation.

Compare the circumference and height of the water bottle. Here’s what they estimated by themselves. Then they faced left if they thought height > circumference, straight if =, and right if height < circumference. (I saw Andrew lead this at CMC-South year before last … I certainly didn’t think of it myself.) They found someone who agreed with their answer, and practiced I can construct a viable argument and critique the reasoning of others. Next they found a second person who agreed, and practiced I can construct a viable argument and critique the reasoning of others again. (By this time, we decided it was easier to raise 1, 2, or 3 fingers based on answer choice rather than turn a certain direction as it was a challenge for some to see someone turned the same direction.) Finally, they found someone who disagreed, and practiced I can construct a viable argument and critique the reasoning of others. I sent the poll again.

It didn’t change much. So without discussion, I sent a poll with a bit more context … a cylindrical can holding 3 tennis balls. Would the can of tennis balls help them reason abstractly and quantitatively? Apparently not.

Here’s what they thought by themselves. And here’s what they thought after talking with someone else. The clock was ticking. I still wanted us to talk about pi. I asked someone who correctly answered to share her thinking with the rest of the class to convince them. And we used string to show that the water bottle circumference was, in fact, longer than its height.

I intended to follow up with this Quick Poll. But I was in a hurry and forgot. Maybe next year. You can find more number sense ideas from Andrew here.

I’ll look forward to hearing about how they play out in your classroom, as the journey continues …

## An Infinite Number of Rectangles

We have started our unit on the definite integral for a few years now with Lin McMullin’s The Old Pump.

I love watching students work without yet having developed Riemann Sums. Many use areas of rectangles to approximate the amount of water in the tank, but even then, they don’t all do it the same way. That work leads us to developing the idea of estimating area between a curve and the x-axis using Riemann Sums and the Trapezoidal Rule. And then we are finally ready to determine the exact area between a curve and the x-axis using a Riemann Sum with an infinite number of rectangles.

We practice reason abstractly and quantitatively throughout these lessons. Once we’ve thought about numerical approximations for area between a curve and the x-axis, we spend some time writing a Riemann Sum to represent area and evaluating its limit as the number of rectangles approaches ∞. I want them to be able to go backwards, too. So we start with a limit, and I ask them what definite integral will have the same value.

Which is apparently not as difficult as the groans suggested when I first gave it to them.  But we are always working on our Mathematical Flexibility, and while I was pleased that everyone can get a definite integral, I was disappointed that they all did it the same way. Jill Gough has provided us with a leveled learning progression for Mathematical Flexibility. Can you write another definite integral for which the area can be calculated using the given limit?

It took a while. But students made progress. Some made use of the symmetry of the graph of y=x2 to write a second integral. Some figured out that translating the parabola and the limits of integration one unit to the right would result in a region that has the same area.

Those were the types of answers I was expecting. But I also got answers I wasn’t expecting.

Some of my students were on the path to Level 4 of reason abstractly and quantitatively, beginning to generalize the idea of translating the parabola and the limits of integration c units to the right, resulting in a region that has the same area. They didn’t quite make it, as their limits were shifted to the right c but their parabola shifted to the left c. I was still impressed by their jump to Level 4, finding connections between pathways.

Our TI-Nspire CAS software let us check our results and helped us attend to precision.

And so the journey continues … learning more from my students and our technology every day about mathematical flexibility.

Posted by on February 11, 2015 in Calculus

## Circles and Squares

As we finished up our unit on Geometric Measure and Dimension, we used a task from the Mathematics Assessment Project called Circles and Squares. This task is no different from many other tasks in scaffolding the work that students will likely need to do in order to answer the final question – what is the ratio of the areas of the two squares. Instead of going straight to the questions, I just showed students a picture of the diagram and asked them to complete the prompt “I wonder …”. if the right triangle inside the smaller square is half of the larger right triangle.         1

what is the side length of either square or t he diameter\\radius of the circle           1

if the area of the sectors in the circle have the same area as the curved triangle thing          1

what the similarity ratio is of the small square to the large square       1

if there is a proportional relation between the area of the big square and the area of the little square, regardless of the radius of the circle   1

if circles and squares can keep going into each other infinitely  1

what are the dimensions of the two squares?       1

cibrcles           1

is the small square similar to the large square     1

what the ratio of small to big square is      1

if the area of the circle not encompassed by the smaller square is the same as the bigger squar     1

how the radii of all three shapes are related        1

if the squares are simular.   1

if the area of all of the shapes are related 1

is the area between the smaller square and circle equivalent to the area between the larger square and circle   1

if the smaller square is half the larger square      1

what is the radius of the smaller square compared to the larger square?        1

why there isnt another circle          1

what we could possibly do with all of the possible calculations  1

if the shapes are similar anc how they relate to each other        1

Is the square scale facator 1.5?       1

if the smaller square is proportional to the larger one.    1

why the smaller square is significant.        1

how the areas of the shapes relate to each other 1

if we can figre ot the areaof the space between the shapes        1

if the the smaller square has ((1)/(2))the area of the largest? 1

why there is a circle represented in between 2 squares 1

do all these have the same center  1

I’m not sure what I expected, but I am still always surprised how often what students wonder is tied to our learning goals for the lesson. Next I asked them to estimate the ratio of the area of the smaller square to the larger square. And then I gave them the handout, which had the scaffolding questions from the Mathematics Assessment Project. As students worked, I sent Quick Polls to assess their work. Students kept working with their teams after submitting their responses. I only stopped them for a whole class discussion when I felt like their responses needed that. After the first question, I went to the table of the three who didn’t answer correctly to find out their thinking. After the second question, I stopped them for a moment to ask why the “Quick Poll grader” had marked both of the first two responses correct. After the third question, I deselected “Show Correct Answer” and asked the teams to decide which expressions we should mark as correct, equivalent to 1:2.

This task provided students a good opportunity to both reason abstractly and quantitatively and look for and make use of structure.

We neglected to go back to the class estimates to discuss how they had done. There’s always next year, right?

And so the journey continues …

## Length & Area Cards

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

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

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

The third group had Rectangles.

Draw a diagonal of a rectangle and mark any point on it as P. Draw lines through P, parallel to the sides of the rectangle. The two shaded rectangles have equal areas.

Draw a diagonal of a rectangle and mark any point on it as P. Draw lines through P, parallel to the sides of the rectangle. The two shaded rectangles have equal perimeters.

They read through their card and drew a few diagrams, but they decided to spend most of their time building the scenario using our dynamic geometry software.

When it was time for this group to present their work, we sent the Quick Polls so that we would know what students instinctually thought, even though they had not all had time to explore the statements in depth. I am learning to make use of the TI-Nspire Navigator allowing me to send more than one Quick Poll at a time.  I showed the results, but I deselected Show Correct Answer before doing so. I wanted the group to know what their peers thought before they just told them the results.

I used the Live Presenter feature of Navigator to make one of the student’s calculators live on the projector at the front of the room. They grabbed and moved the point that changed the point on the diagonal. What do you see? A lot of rectangles.

What is true about the rectangles? Some have a diagonal.

What does the diagonal buy us? Triangles. Congruent triangles.

So the shaded areas are always equal.

What about their perimeters? When are they equal? When P is the midpoint of the diagonal, we can show that all four rectangles are congruent.

The fourth group had Medians of a Triangle.

If you join each vertex of a triangle to the midpoint of the opposite side, the six triangles you get all have the same area.

This group spent most of their time building the scenario using our dynamic geometry software.

The students used technology to see that the statement was true. Then they talked about why. Technology helps us see things that we might not see at first glance. Technology shows us that something is true (or not). Even though we still have to make sense of why it is true (or not). Technology makes more problems accessible to more of my students. I didn’t get to hear this group’s presentation. Another teacher in my department had my students on this day. But this group spent a lot of time looking for and making use of structure. How do we know that the brown triangle has the same area as the yellow triangle?

How do we know that the triangle formed by the green, orange, and brown triangles is equal to the triangle formed by the blue, pink, and yellow triangles?

How do we know that the triangle formed by the green, orange, and blue triangles is equal to the triangle formed by the brown, pink, and yellow triangles?

The fifth group had Square and Circle.

If a square and a circle have the same perimeter, the circle has the smallest area. This group mostly reasoned quantitatively. They didn’t use the technology, but we looked at a square and circle that had been built with the same perimeter so that we could have that dynamic feel to make sense of their results.   The last group had Midpoints of a Quadrilateral.

If you join the midpoints of the sides of a quadrilateral, you get a parallelogram with one half the area of the original quadrilateral.

We’ve talked before about the result of joining the midpoints of the sides as a parallelogram, but we had never discussed the areas of the figures.

I enjoyed looking at the work this group did using their technology to make sense of the statement.      Why is this statement always true? Last year a teacher in our department had several students wonder if they would do cards like these again in class. She asked them why they wanted to do more. The students answered, “Because they were like puzzles. It was fun to figure them out.”

And so the journey continues …

## Evaluating Statements about Enlargements

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

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

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

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

If the price of the small ring of candy is 40 cents, what is a fair price for a large one?   About half of the class used proportional reasoning, deciding that 80 cents was fair. The other half used business reasoning. Some included tax. Some decided that the larger portion should have a bit of a discount. All of them had an argument for why they chose what they did. No one thought that Jasmina reasoned correctly about the amount of pizzas. And only a few insisted on using business sense to come up with a fair price for a large pizza.  Students then worked with their teams to determine whether their cards with statements about enlargements were true or false. I gave each team 6 cards to evaluate. They reasoned abstractly and quantitatively. They constructed viable arguments and critiqued the reasoning of others. They even had a good time.  This particular class clocked in at exactly 65% peer-to-peer discussion. Even then, the 35% whole group discourse wasn’t just one raised student hand at a time. I used the Quick Poll results to selectively call on students who don’t always raise their hand. They presented their argument and the class decided whether to buy it or not. I was there to facilitate the conversation and move it forward. I’m not sure whether that only counts as “leading from the front of the classroom”.

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

## Coordinate Geometry Proofs

G-GPE.B. Use coordinates to prove simple geometric theorems algebraically

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

How would you give students ownership of this standard?

We had been reasoning quantitatively and were ready to move into reasoning abstractly. Where would you conveniently locate each figure in the coordinate plane?

Some students put a vertex of the square at the origin. Others put the center of the square at the origin. What about a parallelogram? What about the coordinates for point P in this parallelogram? I loved seeing how students wrote their responses in different ways. You can see how they were thinking about calculating the coordinates of the P, which led to good whole class discourse. How would you locate a kite in the coordinate plane?

Several students showed us what they did on our interactive whiteboard. And one made me realize that I need to make my kite less special. The almost right angle for the “top” angle of the kite (when oriented “normally”) led to a response that a non-right angle might not have.

We reasoned abstractly to show that the diagonals of a rectangle are congruent. Some students used the distance formula. Others used the Pythagorean Theorem. We reasoned abstractly with triangles. But we still need more work. Only 20% of students were successful on this question on their summative assessment. At least we get a do over for next year, right? And so the journey continues … learning and revising.

## Area Formulas

We started a unit on Geometric Measure and Dimension a few weeks ago. During the first lesson I asked whether students had made sense of the formulas for the areas of a triangle, parallelogram, and trapezoid, or whether they had just memorized them without understanding. Since they said the latter, we took a few minutes to make sense of the formulas.

We used Area Formulas from Geometry Nspired for demonstration. We moved B and H. What do you notice about the given rectangle and parallelogram?

Their bases have the same length.

They have the same height.

What is true about their areas?

They are equal.

How do you know? What is the formula for the area of a parallelogram? We moved B and H. What do you notice about the given parallelogram and triangle?

Their bases have the same length.

They have the same height.

What is true about their areas?

They are not equal.

What relationship does the area of the parallelogram have to the area of the triangle?

It is twice the area of the triangle.

The area of the triangle is ½ the area of the parallelogram.

How do you know?

So what about the area of a trapezoid?

I didn’t show them the next page in the TNS document.

I showed them a general trapezoid and asked them to work for a few minutes alone to make sense of how to calculate the are of the trapezoid. I walked around and watched.

Whose would you choose for a whole class discussion?

I asked DT to show us her picture first. She had decomposed the trapezoid into two triangles using a diagonal. We were able to use the distributive property to show that the area of the two triangles was the same as the textbook trapezoid formula. As you can see in the bottom right, another student had decomposed the trapezoid into three triangles. We didn’t take the time to show that the area of the three triangles was the same as the textbook trapezoid formula.

I asked BA to show us her work next. She had decomposed the trapezoid into a parallelogram and a triangle. Again, we were able to show how the area she calculated was the same as the textbook trapezoid formula. I asked MA to show us her work next. She had recomposed the trapezoid into a rectangle that had a base equal to the median of the trapezoid and a height the same as the trapezoid. Again, we were able to show how the area she calculated was the same as the textbook trapezoid formula. I asked BE, whose work I somehow missed photographing, to explain what he did last. Unknown to him, his work was like that shown in the TNS document.  I teach a high school geometry class. Every day I wonder whether I waste my students’ time going back and making sense of concepts that they’ve been using for a while mathematically. I struggle to know what is our best use of class time. Surely there is a way that my students could have done this outside of class … would it have been effective? We were at five ways and counting to make sense of the formula. I’m not sure that is trivial. My students were reasoning abstractly and quantitatively, even if the math content was not an explicit part of our standards. And so the journey continues …

## Inscribed & Circumscribed Right Triangles

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

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

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

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

The second task provided a bit more structure. Circles in Triangles comes from a Mathematics Assessment Project apprentice task. Students were given the following:

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

1. Prove that triangles AOX and AOY are congruent.

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

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

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

Another teacher taught this lesson to my students. She saved their work for me because even with the auxiliary lines drawn, students made sense of the structure (and particularly the expressions they wrote for the segment lengths) in different ways. Then they were asked to determine the radius of the inscribed circle for a 5, 12, 13 right triangle, and then they were asked to generalize their results. This diagram shows a circle that just touches the sides of a right triangle whose sides are 5 units, 12 units, and 13 units long.

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

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

I wonder how technology fits in with tasks like these?
We had a skeleton of a diagram prepared for students who wanted to use it. Students had to measure themselves for the construction to be helpful in making sense of the mathematics. I find that the technology can be helpful for those who don’t know where to start. What do you notice when the triangle is right? Can it help them reason abstractly and quantitatively, starting with the quantitative and building to generalization? Can it help them make sense of problems and persevere in solving them, when they don’t know what else to do on paper?

We also had a skeleton of a diagram prepared for the inscribed circle for those who wanted to use it. What auxiliary lines (or segments) would you construct and measure for the inscribed circle? The technology can help them make a conjecture about the length of the radius, and then they can go back to the mathematics to help them understand why.

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

## 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

## 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 …