Working with triangle dilations

Today we moved back to work with Triangle Dilations. This work starts with students constructing dilations. Students grabbed their compasses and started working right away. Having worked through constructing dilations last class really helped. There were no questions and students were able to work through the initial constructions.

Listing proportions were a bit more troublesome. I had to clarify what was being asked. I decided to have students focus on the relationship of segments, for example, AB = 2A''B'', etc. Students could understand relationships expressed this way and it tied into the idea of scale factor. I'll wait until we go through responses to get into how to take these relationships and re-express them as proportions. I also am waiting to discuss their findings on the proportion relationships until next class.

The review of angle relationships was interesting. Students recognized that we had worked with these relationships but quite a few struggled with naming one or the other of the relationship names. I thought it was good review to keep the vocabulary up front with the class.

The next section on creating the described dilation seemed to be going well with the class. As I walked around, there really wasn't any questions about this section and the drawings appeared to be correct. Most of the class was either working on this section or just completing this section when class ended.

I asked students to complete the remainder of the questions as homework. I also told them that they should use a ruler to complete any remaining constructions.

The class worked throughout the entire period. I was encouraged by the work I saw produced and the small amount of support that the students needed to complete the tasks.

Next class we'll discuss responses to the questions. I want to flesh out the side relationships that exist between the pre-image and image in a dilation. I also want to get students to think about the side relationships in terms of a proportional relation.

Drawing dilations using a compass

Today we looked at the ideas connected with drawing dilations and finding centers of dilations. Class started by checking on responses to the Photocopy Faux Pas questions. Students seemed comfortable with identifying scale factors and finding corresponding side lengths.

One issue that needed clarification was the idea that a scale factor is actually a numeric value. This came about on the first problem in which the scale factor used was 2. When asking the class the scale factor, students replied that the scale factor doubled. While this was the result of using a scale factor of 2, double was not the scale factor.

Another issue that presented itself was that students did not understand the idea of center of a dilation. The scenarios presented in the last set of questions baffled students. The common thinking was the center of dilation was between the pre-image and image figures. I didn't press the issue but knew that we needed more work to build the understanding of center of a dilation.

I also checked their understanding about the issue of lengthening the rubber band chain that was used in the last class. Some thought the image would get larger but there was little confidence in the responses.

The next investigation I was planning on using was Triangle Dilations. Since this investigation required students to construct dilations and know centers of dilation, I didn't want to tackle this investigation just yet. First, I wanted students to learn how to construct dilations using a compass and practice a bit more with scale factors and work with centers of dilation.

A colleague had used the Drawing Dilations exploration from the On Core Mathematics curriculum. This exploration explains how to use a compass to draw dilations and provides work with using dilations in a coordinate plane. The dilation constructions require the construction of segment bisectors, which is a nice tie-in to previous geometric construction work. By the end of the exploration and work, students have a much better understanding of the center of a dilation.

This work took the entire class. Students had some basic questions and some needed reminding about segment bisector construction. The most difficulty occurred with dilations in the coordinate plane. Students repeatedly wanted to add the scale factor rather than multiply the scale factor. I tried to relate the idea of a scale factor to a continuous expansion or contraction. This seemed to help.

By the end of class students were feeling very comfortable with the compass constructions and understood how to apply scale factors in the coordinate plane. More importantly, they also had a better understanding about the center of dilations and what would happen with using longer rubber band chains to draw dilations.

When I asked the class what would happen if we used a longer rubber band chain, the class confidently stated the image would become larger. When I asked the class where the center of dilation is for the photocopy scenario, students recognized that the upper left corner was the center of the dilation. With these ideas solidified, I will proceed to the Triangle Dilations investigation next class.

Introducing similarity - dilations and scale factors

Similarity was introduced today. I used the Photocopy Faux Pas investigation package. Here is a run-down of how the lesson unfolded.

First, I told students we were going to look at similarity. Since this was the first look, we would work through an investigation to better understand the concepts.

I held up a copy of the first two pages of the packet (copied 2-sided). I told students not to be concerned with the questions on the back side for now. I asked students to read through the text one time to help establish context. Basically, I wanted to have students read through the text like they would a story. I then told them that they needed to read through the story a second time, highlighting or annotating key words, math ideas and pertinent information.

I passed out a copy to each student and let them read through and annotate. I had to remind students that this was not a group activity at this point and to read through the text twice: one time for context and one time for detail. I walked around to check on student progress and to gauge their level of engagement with annotating the text. Overall, this piece progressed smoothly.

Next I told students to imagine they were the boss. I asked the class to think about why things went wrong and how they could be corrected. With this perspective in mind, I asked the class to complete four of the questions on the back-side of the page (skipping question 2 for now). I told students to answer to the best of their ability given their current understanding.

I had tables share their responses to the fifth question and then we did a class share out. The list included things like initial position, starting size, desired end size, size of paper, etc. Students were clearly thinking about relevant issues.

We then went back to question two. This step required use of a rubber band stretcher. A colleague found instructions for a Two Band Stretcher Activity. I handed out the first page instructions to each table group and then either page 3 or page 4 to every student. I passed out two rubber bands to each student and demonstrated how to tie the rubber bands together. After the second pass through the tying instructions there were only 3 students who didn't have their rubber bands tied. Table mates helped these individuals so everyone was set.

I explained the process of having the knot of the rubber bands trace along the figure. They were not to look at what their pencils were doing. At this point the class started tracing their figures. Some students complained that their rectangle drawing went off the paper. My response was, "Well, what happened in the photocopy situation?" Students realized the same issue arose.

I asked the class to consider what they were seeing and to go back through their responses to the other four questions. I wanted the class to consider whether any of their responses changed based upon the rubber band drawing results.

I then passed out pages 3 and 4 from the Photocopy Faux Pas investigation packet. I briefly connected dilations and scale factors to image manipulation in software packages. I discussed the meaning of scale factor and then had students work on determining scale factors and side lengths.

As the class drew to a close I asked students to think about where the center of dilation was for the photocopier. At this point, a student asked what impact having a longer rubber band would have on the image drawing. It was a good question that was asked right when class ended. I asked students to think about how they would respond to that question as well.

Tomorrow we'll address those questions, discussion their ideas on the location of centers of dilations described in the problems and then move into the next investigation piece.

A colleague said that her students were still confused about the idea of doubling an image size. Did this mean doubling side lengths of did it mean doubling area. It will be interesting to see how my class thinks about the idea of doubling as we move forward.

A final look at quadrilaterals in the coordinate plane

The common unit assessment we're using includes a couple of coordinate proof problems. Since we hadn't worked in the coordinate plane for a week or so, I decided it would be helpful to revisit this work before moving to similarity.

A colleague had put together four problems that covered this material. Each problem gave coordinates for a different quadrilateral. In the first problem, they had to calculate the slopes of the sides to show opposite sides were parallel. In the second, students had to calculate the length of the sides to show that opposite sides were congruent. In the third, students had to show that diagonals bisected each other. In the fourth, they used protractors to show that opposite angles were congruent.

This work went fairly well. Some students stilled had questions about calculating side lengths or slopes. For the most part they seemed comfortable with the tasks. For a couple of students, they really just wanted a formula to use.

I pointed out that they should try to work with what they know. If a problem didn't have pieces they know, think about how they could break the problem down into pieces that they knew. For example, the distance formula is essentially the Pythagorean theorem used for a specific purpose. However, problems often don't provide the right triangle dimensions needed. Students need to ask themselves how they could create a right triangle to use the Pythagorean theorem, since they typically know this theorem well. The idea started to click for some of these students and they were able to proceed ahead on their own.

I did pass out a quiz and told students it was a take-home quiz. The quiz emphasized concepts as opposed to solving problems for x. I allow notes to be used on quizzes and tests, so I felt there wouldn't be much difference in results by having the quiz completed at home. This also saved me a class period for investigation and instruction versus assessment.

Tomorrow we start similarity. Today, another colleague worked through the first lesson I am using tomorrow. She was really pleased with the engagement and learning that took place. I'm excited to see how the lesson goes in my class.

Wrapping up work with quadrilaterals

Today's class concluded our work with kites and trapezoids. Students, generally, felt that the assigned problems were some of the easiest they had so far. There were still some points of confusion that needed to be addressed, especially with regard to problems involving kites. These centered around realizing that one pair of opposite angles had to be congruent and that one diagonal was the perpendicular bisector of the other diagonal.

To conclude the work with quadrilaterals and begin paving the way for work on similarity, I used pages 3 and 4 of the Parallelism Preserved investigation. Students seemed to breeze through identifying quadrilaterals and, when asked, were able to justify their responses with the specific characteristics they used.

The next three questions caused a bit of confusion, but students were able to make reasonable arguments as to when parallelism was preserved. The final problems reviewed triangle congruence theorems. As we move into exploring triangle similarity theorems, the review will be helpful.

We'll check responses next class and have an assessment. I'm still deciding whether to make this a partner quiz or a take-home quiz. I intend on using the Understanding similarity in terms of similarity transformations tasks to work through the unit on similarity. There will be a summative assessment over quadrilaterals and similarity once this unit is complete.

Kites and trapezoids

The class wrapped up working with parallelograms today. The class spent about 20 minutes completing questions and discussing problems. We then went through responses. I displayed a page of responses from a randomly chosen student and then the class commented or questioned the results. There were some good discussions and some ideas were clarified and some misunderstandings identified.

We then dove into working with kites and trapezoids. I had three examples of each on the board. For the trapezoids, I was sure to include one example of an isosceles trapezoid. I then had students refer to the quadrilateral properties grid and asked them to consider the examples and try to complete the grid columns for kites and trapezoids.

I walked around and answered questions students had. I also asked students their reasoning and thinking about their entries. The main focus I was trying to have students consider the properties that parallelograms always have and what this would mean for kites and trapezoids. Students seemed to struggle with the connections but finally started to realize that if a parallelogram always had diagonals that bisected each other then kites and trapezoids could never have this property.

The properties that parallelograms sometimes have were a bit trickier. For example, all side lengths are congruent is always true for a rhombus; kites and trapezoids can never have all side lengths congruent. On the other hand, a rhombus always has diagonals that are perpendicular but this does not preclude a kite from also always having perpendicular diagonals.

I referenced traditional kites that students may have used as children to make connections to the diagonal structures that are present in a kite.

I passed out a sheet that contained definitions and theorems concerning kites and trapezoids. This sheet also provided a couple of worked examples and a few problems. Their homework is to complete the problems for next class, which will provide additional problem and work practice.

Assessing work with parallelograms

At this point, parallelograms have been worked through from a number of different perspectives. I wanted to assess informally students' understanding of parallelogram properties and the relationships of different parallelograms. The Guess My Parallelogram packet fit my needs perfectly.

Students spent the entire class working through the problems. As I walked around the class, students were having good mathematical discussions about the questions. I did have to encourage some students to refer to the quadrilateral property grid for help. There were very few questions.

One thing I liked about the problems was that it included some review of using a compass and straight edge to construct segment and angle bisectors. As we move to dilations, we'll be using a compass and straight edge to construct the image of a dilation.

Students were still working on the questions at the end of class. I asked that they finish any non-construction questions for next class. I'll give the class about 15 minutes at the start to complete any questions and constructions and to discuss their answers with their table partners. We'll then go through responses as a class.

We'll be moving on to kites and trapezoids next.

Working with parallelograms

Today we finished working through the Parallelogram Conjectures and Proofs problems and a few additional problems pulled from an old textbook. Students had actually worked through many of the problems at home and seemed to be understanding what they were doing.

On the new set of problems, students were recognizing that because they had a parallelogram that opposite sides were congruent and, therefore, the expressions denoting side lengths had to be equal. The same went with opposite angles. I was pleased to see this recognition happening.

We did go through answers for the Parallelogram Conjectures and Proofs problems in class. The last set of questions related to notation went well, although there were a couple of minor notation issues that had to be addressed. Problem 10 on the second page was an issue for students.

 Students were able to find values for x and y but not z. I had a couple of minutes to play around with it in front of class and started to suspect that there wasn't enough information to solve for z. I asked students to play around with different relationships over the weekend to see if they could come up with anything.

Over lunch, I set up missing angles as variables and established a 4 x 4 matrix to represent the system of equations. The matrix is singular, so the system I set up does not have a solution.

This will be a good point of discussion. At what point do you start to suspect that there isn't a solution or the statement is not true. I can pull in a couple of historical references (5th postulate or general solution for the quintic) to tie into this point.

Rectangles, rhombuses, and squares

This class started with confirming properties for rectangles, rhombuses, and squares. Students were able to confirm at home that rectangles had congruent diagonals and that rhombuses had perpendicular diagonals. We updated the quadrilateral property grid with information for rectangles, rhombuses, and squares. (See the Jan. 6th post for information on the quadrilateral property grid.)

At this point, a student asked what the official definition of a rhombus was. I asked the class to look at their grid and to identify how rhombuses differ from parallelograms. There are two characteristics that parallelograms sometimes have and rhombuses always have: 1) all sides are congruent and 2) diagonals are perpendicular. I told the class that you could define a rhombus based on either of these two characteristics and then demonstrate (prove) that the other property had to be true. It's easier to think of rhombuses in terms of side lengths than in terms of perpendicular diagonals, so that is how a rhombus is defined: A rhombus is a parallelogram that has all sides congruent.

Of course, this definition requires knowing the definition of a parallelogram, etc. There is a large amount of layering that goes into mathematical definitions as we progress through the curriculum and I'm not sure that students always have that foundation to really understand the depth and intricacies for these definitions. That is where building conceptual knowledge helps to bridge the gap and make the definitions more understandable.

With the grid complete, I wanted students to work through a series of investigative problems and activities. As mentioned previously, I am drawing resources from the Simi Valley Unified School District Common Core Mathematics geometry site.

I had students work through Parallelogram Conjectures and Proofs. This work allows students to express reasoning, make use of properties that have just learned, revisit triangle congruence theorems as part of their justifications, work with angle relationships formed by intersecting lines, and explore connections to congruence under rigid motions.

Most students were able to get through the first ten tasks. I left the remainder as homework. Students still struggled with expressing their reasoning but are getting a little bit better. Many needed to be reminded about looking for congruent triangles to then prove that corresponding parts were congruent. Once they got through the first task using this idea they were able to proceed through the others without much assistance.

We'll continue working through these tasks. The plan is to use Guess My Parallelogram next class.

When do parallelograms have congruent diagonals or perpendicular diagonals?

Today we continued our investigation of parallelogram diagonals. Specifically, we were trying to answer two questions:

1. Do parallelograms with congruent diagonals share any common characteristics?
2. Do parallelograms with perpendicular diagonals share any common characteristics?

These questions were driven by the fact that sometimes parallelograms have perpendicular diagonals and sometimes they have congruent diagonals.

Students used whiteboard grids or graph paper to help them investigate these situations. I instructed them to work visually and use rulers and protractors in their search. Once they thought they had a parallelogram that met one or both of the conditions, they were to verify through mathematics that their figure worked. To do this they would need to use the distance formula to calculate diagonal lengths and calculate slopes of diagonals to determine if they were negative reciprocals. In addition, I told the class they would need to calculate the diagonal midpoints to check that the diagonals bisected each other, which would confirm that their figure was actually a parallelogram.

I walked around to check on progress. A few students needed some additional direction but most jumped in to tackle the job. A couple of times I had to ask students what they were seeing: Did the diagonals look congruent? Did the diagonals look perpendicular? I then asked these students how their parallelogram could be altered to get closer to either of these goals.

Toward the end of class, we were able to share that rectangles and squares had congruent diagonals and that squares and rhombuses had perpendicular diagonals. Since not every student had reached the same point, I assigned the task of verifying these results through example as homework.

I wanted to check on understanding and asked students what the official definition of a rectangle was and what the official definition of a rhombus was. I was pleased when the definitions offered up began with "A rectangle is a parallelogram that has..." or "A rhombus is a parallelogram that has..."

We'll finish up the property grids for rhombuses and rectangles next class. My intent is to work through some practice problems and then work through an investigation that will allow me to assess how well the class understands their work with parallelograms.

Investigating properties of diagonals in parallelograms - continued

We continued looking at diagonals in parallelograms. Students didn't do well over the weekend in finding midpoints and segment lengths, so I got the coordinates for the endpoints of a diagonal for one students and quickly ran through how to find a midpoint and the length. After a few clarifying questions, I released the class to find the lengths and midpoints for the diagonals of their parallelograms. Many students drew new figures and most were hesitant or struggling with the calculations, so this took a lot longer than I had hoped.  By the end of class, students were successful in computing their midpoints and their diagonal lengths. We reviewed the results and concluded that the diagonals of parallelograms always bisect each other. We also concluded that the diagonals are sometimes congruent.

At this point we have found for parallelograms that:

1. Opposite sides are always parallel
2. Opposite sides are always congruent
3. Opposite angles are always congruent
4. Diagonals always bisect each other
5. Diagonals sometimes are congruent
6. Diagonals sometimes are perpendicular
7. All sides are sometimes congruent
8. All angles are sometimes congruent

For the next class I intend to have students explore what types of parallelograms have perpendicular diagonals and what types of parallelograms have congruent diagonals. I'll use this to connect to when all sides are sometimes congruent and when all angles are sometimes congruent to flesh out the properties of rectangles and rhombuses.

Investigating properties of diagonals in parallelograms

Today we focused on three statements regarding parallelogram diagonals:

1. Diagonals are perpendicular
2. Diagonals are congruent
3. Diagonals bisect each other

Students drew a parallelogram on a grid and then were asked to determine whether or not each statement was true for their parallelogram. I reminded students that they would need to find the slopes of the diagonals and then multiply these values together to see if the product equaled -1. For the second statement they needed to calculate the length of each diagonal using the distance formula/Pythagorean theorem. For the third statement, they would need to find the midpoint of each diagonal and see if the midpoints were the same.

At this point I let the class work. I knew there would be issues since students, generally, are not comfortable finding slopes, lengths, and midpoints. I walked around and helped students, reminding them of how slopes were calculated. This process took almost the entire class. I had intended this to be a work day with the expectation that students would need help. Having the statements as drivers for why to calculate slopes, lengths, and midpoints definitely helped. It wasn't a matter of why were we calculating these values but, rather, how do I go about calculating these values.

We shared out findings about the slope relationships at the end of class. A few students had diagonals that ended up being perpendicular while the majority did not. We were able to conclude that parallelogram diagonals are not always perpendicular but they can be perpendicular sometimes. Many students had not finished calculating the lengths or midpoints for the diagonals. I asked them to attempt to finish these calculations for next class.

We'll continue practicing work with parallelograms before diving into the sub-categories of rhombuses, rectangles, and squares.

Parallelogram properties - an initial investigation

Today we dove into looking at parallelograms. To start things off, I had students grab small whiteboards that have an xy-grid on them, dry erase markers, and a straight edge. On the board, I projected a grid system and drew two pairs of parallel lines that intersected with each other and labeled the intersection points as A, B, C, and D.

I asked students to draw two pairs of intersecting parallel lines on their boards and label the intersection points. I noted that we wanted to focus on the figure defined by the intersection points, the parallelogram ABCD.

The task was for students to use their knowledge of parallel lines and intersections with transversals to identify everything they could about the properties of the parallelogram. I told them to reference their notes if they couldn't remember the different properties and theorems we had previously worked with.

As I walked around the room, students had their figures drawn and were actually remembering or recovering from their notes different facts and theorems they could use. I talked with different groups, asking what they had found. Sometimes, I had to redirect the focus to the actual figure since they were focusing on external characteristics. Sometimes they would make a claim about parallelograms that were not necessarily true all the time. In these cases I asked if that was always true or if they could think of a situation when that might not happen.

At the end of the 10 minutes, I had students share out what they found:

1. The opposite sides of a parallelogram are parallel
2. The opposite interior angles are congruent
3. The opposite sides are congruent

After the last two statements I wrote, "why?"

I then referenced the quadrilateral properties grid that I passed out last class. We worked through the parallelogram column. From the three statements posed above, we were able to work through all but the last three characteristics, which all dealt with diagonal properties. These properties deal with diagonals being perpendicular, bisecting each other, or being congruent.

I asked the class how they could tell if two line segments or lines were perpendicular. No one remembered. I told them that we would need to use slope. Both of these topics are from Algebra I, so I know they have been taught these ideas. I briefly went through a few ideas, such as slopes of perpendicular lines are negative reciprocals, slopes of parallel lines are equal, and how slope measures relative changes in height (y) versus length (x). I used mountain slopes and skateboard ramp steepness to help connect these ideas.

I then asked students to calculate the slope for one pair of parallel lines that they drew in order to determine if, in fact, the drawing they made was a parallelogram. With a bit of assistance here and there, most students seemed to recall how to do this.

For those students that didn't remember or struggled, I encouraged them to go to Khan Academy to review and practice, as I know there are lessons on calculating slopes, as well as working with slopes of parallel and perpendicular lines.

Next class we'll focus on the diagonals of parallelograms and their properties. We'll be calculating slopes, midpoints, and lengths for these, so we'll have some good algebra review, which may prove to be a struggle for some. 

Starting second semester - polygon classification and quadrilateral properties

Today was the first break back from winter break and the official start of second semester. I have all the same students in class, so I was able to briefly go through reminders of rules and dive into content.

Our first unit this semester wraps up work with polygons and quadrilaterals. We'll then connect the polygon work to similarity.

To start things off, I showed students a biology classification chart. Almost all the class is taking biology this year, so it is something they are familiar with.

I mentioned how there are many different versions of these. No single version is the "right" version; they just reflect differing opinions of what to focus on when classifying organisms.

With this setting the stage, I asked students to create their own version of a classification chart for polygons. I briefly listed a few items, but not too many, as I wanted students to revisit their notes to draw out different vocabulary and topics. I had students work individually and then in groups to consolidate their work. I had several students present their work. Some focused on number of sides, some focused on side characteristics and some focused on angle structures. Students could ask questions or comment on the presented charts, although there were few questions asked or comments made. It was a good way to reengage their brains after a two and one-half week layoff.

Next, I told students we were going to spend time focused on quadrilaterals. I passed out a grid of quadrilaterals across the top and characteristics along the side. I asked students to complete the grid in pencil, anticipating that changes would need to be made as we moved through these figures. My intent is to use this as a way for students to self-assess on their knowledge and perceptions of quadrilateral properties.

I also had students draw examples of each quadrilateral on the board to verify that they knew what each figure should look like.

Next class we'll work through properties of parallelograms.