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Tag Archives: similarity using dilation

Notice and Note: Dilations

Are you familiar with Notice & Note: Strategies for Close Reading? Here’s a link to Heinemann’s Notice & Note learning community, and here’s a sample PDF. I wonder whether our Standards for Mathematical Practice are similar to the Notice and Note literary signposts.

It’s not enough to just read a text. We want students to read for understanding and comprehension. The literary signposts help students with close reading of a literary text.

Similarly, it’s not enough to just explore math with dynamic graphs and geometry. We want students to explore for understanding and comprehension. The math practices help students learn how to interact with a mathematical problem or concept … and what to notice.

 

Last week, we explored dilations.

What do you need for a dilation?

A figure, a point (which we’ll call the center of dilation), and a number (which we’ll call the scale factor)

We used our dynamic geometry software to perform a dilation.

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About what things might you be curious as you explore dilations?

(I thought of Kristin when I used the word curious.)

What happens when the center of dilation is inside the pre-image?

What happens when the center of dilation is on the pre-image? (on a side, on a vertex)

What happens when the scale factor is between 0 and 1?
What happens when the scale factor is negative?

How do the corresponding side lengths in the pre-image and image relate to each other?

 

I asked students to practice look for and express regularity in repeated reasoning as they explored the dilation. Do you know what it means to look for and express regularity in repeated reasoning?

Find a pattern.

Yes. Figure out what changes and what stays the same as you take a dynamic action on the dilation. Begin to make some generalizations about what you notice.

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And don’t just notice, but actually note what you’re thinking.

The room got quiet as students noticed and noted their observations about dilations. I monitored student work both using Class Capture and walking around to see what students were noting.

(I promise I’ve tried to make it clear to students that dilation has 3 syllables and not 4 … but we do live in the South.)

Eventually, they shared some of their findings with their team, and then I selected a few to note their observations for the whole class.

BB showed us what happened when he perfomed a dilation with a scale factor of -1. He had noted that it was the same as rotating the pre-image 180˚ about the center of dilation.

SA talked with us about when the dilation would be a reduction. She had decided it wasn’t enough to say a scale factor less than 1 or a fractional scale factor but that we needed to say a scale factor between 0 and 1 or between -1 and 0.

FK showed us that when she drew a line connecting a pre-image point and its image, the line also contained the center of dilation.

PS noted that when the scale factor was 2, the length of the segment from the center of dilation to a pre-image point equaled the length of the segment from the pre-image point to its image.

When the scale factor was 3, the length of the segment from the center of dilation to a pre-image point equaled one-half the length of the segment from the pre-image point to its image.

 

We next determined a dilation and set of rigid motions would show that the two figures are similar.

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Translate ∆DET using vector EY.

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Rotate ∆D’E’T’ about Y using angle D’YA.

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Dilate ∆D’’E’’T’’ about Y using scale factor AY/D’’Y.

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Then we looked at dilations in the coordinate plane. I knew that my students had some experience with this from middle school, and so I sent a Quick Poll to see what they remembered.

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Due to the success on the first question, I changed it up a bit with the second question.

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But I wonder now whether I should have started with the second question. If they could do the second question, doesn’t that tell me they can also do the first?

I’ve rearranged the polls to try that the next time I teach dilations.

 

We ended the lesson with a triangle that had been dilated. Where is the center of dilation?

And so the journey continues, with hope that noticing & noting will make a difference in what students learn and remember …

 
3 Comments

Posted by on November 17, 2015 in Dilations, Geometry

 

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Running around a Track II

I gave a different class of students Running around a Track II recently.

I started by showing them the same picture of the start of the 2012 Olympic Women’s 400 M race.

Track

I changed the prompt, though. What do you wonder?

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I wonder why they start at different lines instead of at the same place.           1

I wonder who 1st realized that the smaller of the concentric circles would be shorter           1

I wonder whats special about this picture?           1

I wonder why if you are running on the outter circle its the same distance as running on the inner circle 1

I wonder what the distance is that they are running       1

I wonder what event they are running?    1

I wonder why tracks are ovals        1

I wonder if each one of those arcs are similar by using dialations         1

I wonder what the exact distance of all of the lanes are  1

I wonder what is happening.           1

I wonder if the circles are cocentric            1

I wonder how big the race track is as a whole.      1

I wonder why all of the starting points are in different places   1

I wonder what the ratio of the innermost ring is to the rest of the rings… or are they all congruent?           1

I wonder why the runners start at different starting positions. 1

I wonder why there are flags that are the same   1

I wonder if its better to be closer up on the outside lane than farther back on the inside lane         1

I wonder if that’s the olympics        1

I wonder which lane is the shortest distance        1

I wonder how they decided how far back to put the first runner          1

I wonder how I can figure out how to find the different measures of each of those arcs of the circle          1

[And from my cynic:] I wonder what question you’re going to ask about this image.   1

I wonder why are there so many americans         1

I wonder what shape the track is   1

I wonder why there are 3 people from us and why they are not starting from same side?   1

I wonder who the runners are        1

I wonder if the distance in the inner circle is different than the distance in the outer circle even if they start in differert places           1

I wonder if all of the starting points are marked off by distance or angle.        1

I wonder why the track runners are spaced apart rather than running all together. 1

I wonder what the arc measure of the track is?    1

Next, I sent out a Quick Poll of a question from a released ACT that students should be able answer as a result of the lesson. Who already knows how to solve this? Who knows how to solve it quickly, since the ACT is timed? I removed the choices, just to see how students answered without them. We didn’t talk about the results. I told them we would revisit the question at the end of the lesson. Just less than half of the students had it correct initially: 14/31.

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Before we started analyzing what was happening with each lane, I asked the students whether the lane lines were similar. Just for the record, I wouldn’t have thought to ask this question on my own. It was suggested in the commentary for the task on Illustrative Math, so I thought I would try it.

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I’m not sure I really had a backup plan for this formative assessment check. Not one student got this correct? How should we proceed? Understand that at this point, the students don’t know if everyone is correct or no one is correct. There is a great feature of Navigator that lets me “show correct answer” or not. While I monitor student responses with my projector screen frozen, I decide whether I will show the correct answer.

What does it mean for two figures to be similar?

Several students wanted to answer. I called on F.K., because F.K. loves dilations, and I knew that she would explain similarity in terms of transformations.

Two figures are similar if there is a sequence of transformations including a dilation that will map one figure onto the other.

Okay – so can we describe a dilation that will map one lane onto another lane? What would be the center of the dilation?

At this point, we moved to the technology. I had created the track on a Graphs page of TI-Nspire. Mainly because I wanted to prove that I could without asking my technical author friends Bryson and Jeff. It’s not beautiful, but it is functional. I didn’t even use this document in the class with Running around a Track I, but maybe it would be helpful now.

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I have a slider set up to change the radius of the circular part of the track. Are the lanes of the track similar? Does the technology help you see it?

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What happens if we dilate the straight part of the track about the center of the track by a scale factor of 1.1? And then let’s dilate the circular part of the track using the same center. (Bryson and Jeff could probably figure out how to make this happen all at once – my track is in 3 separate parts.) Is there a dilation to show that the lanes of the track are similar?

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The straight part of our track is a problem. There is no line that contains the center of dilation and the endpoint of each straight part of the track.

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One of the students wondered if each one of those arcs are similar by using dilations. Now if we are just talking the arcs, then yes, they are similar using a dilation about the center of the arc – but not using the center of the track.

The technology helped us make sense of whether the lane lines are similar.

Now back to the real task. This task is a bit different – it has less scaffolding than the first task. Students jump right in to calculating the perimeter of the track 20 cm inside lane 1 and 30 cm inside lane 2 so that they can determine how far ahead the runner in lane 2 needs to be ahead of the runner in lane 1. I sent a poll to collect the results.

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It was disastrous, and the bell was about to ring.

How in the world could I recover the lesson?

We went back to the picture, and ended class by talking in more detail about the diagrams.

We started the next class calculating the perimeter of the paint on the most inside lane.

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Even then, we didn’t have everyone with us. Don’t you love that 5 of my students just assumed the inside lane was 400 m without doing any calculations?

We went back to the questions on Running around a Track II, though and really did make progress.

To close the lesson, I sent the Quick Poll from the start of class:

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We are up to 80% correct, from just under 50% at the beginning of the lesson.

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And with choices, we have 90% correct. I finally showed the students the results and we talked about the misconception for choice C.

I think it is interesting to ask students which practices they used when working on a task.

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And which would you choose if you could only choose one Math Practice?

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I have shared before that my goal isn’t just to provide opportunities for my students to use the Math Practices in class – but also for them to recognize when they are using them. I ask my students to write a journal reflection each quarter on using a math practice.

N.R. writes about this task: In class today and yesterday, we worked on a problem about the track of the 400 meter dash in the Olympics. While working on this problem, we used the math practice of modelling with mathematics. We applied what we learned about circles in this unit to figuring out how far apart runners in different lanes have to start in order to run the same distance and to end at the same area in the straightway section. When solving this problem, I had to use the equations for circumference and perimeter and combine them. Once I finished working the problem, I decided that the runner in lane 1 has to start 7.037 meters behind the runner in lane 2. I also found that the runner in lane 2 has to start 7.666 meters in behind the runner in lane 3. This problem has helped me to be very attentive to detail. In this problem I had to be very careful that I worked everything correctly and completely.

 

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