Wrapping up first semester in geometry

At this point, we are coming up on finals. I am not introducing any new material but trying to give students a chance to work through a variety of problems. Since the focus has been on building conceptual knowledge, students need some time to practice using the building blocks they have learned.

As for the quiz, about 1/3 of students did a fair to good job explaining the diagonal formula. One student did an exceptional job. The rest did not put thought into the situation and simply said that n represented the number of sides. Period. The biggest issue in the explanations was explaining why n multiples (n - 3).

I'll be back to go through progress second semester starting in January.

Diagonals in Regular Polygons - Part 2

I started off the second day of looking into diagonals in polygons by writing two questions on the board:

1. What is the number of diagonals in a regular 50-gon?
2. What is a formula to calculate the number of diagonals in a regular n-gon?

With that, I told the class I was awaiting their answers.

Of course, this led to a lot of grumbling, staring, and not a lot of activity. I used this as an opportunity to tell the class that every one of them had the knowledge and ability to answer these questions. They needed to think about what was going on and use what they already knew to answer the questions. I also told them I wouldn't be asking them these questions if I didn't believe they could answer them.

The class settled down and started working. A few students still were sitting there looking distracted. I talked to each one about what they were doing. I told them that avoiding the problem was not going to help find a solution and re-directed them to start writing down what they knew and drawing figures to help them consider the dynamics that were occurring as we moved from a square to a pentagon to a hexagon.

After a while, students started observing and making notes of patterns. Most saw that the difference in the number of diagonals increased by one each time. A few students actually wrote out the sequence all the way to the 50-gon in order to determine the number of diagonals. One student noticed that the number of diagonals connected to a single vertex increased by one each time, i.e. in a square a single diagonal is connected to each vertex, in a pentagon two diagonals are connected to each vertex, etc.

It took a while for students to get to this point. I then tried to push them to answer the second problem. Here, students really struggled. A couple of groups were able to write out an informal recursive formula. No one was able to create the closed formula, although a couple of students did look up the formula online, saying D = n (n - 3) / 2.

I focused on the recursive formula first. Students don't get to see recursive formulas very often and they aren't presented in a more formal setting. I wanted to contrast the recursive formula with a closed formula in the hope of contrasting that the closed formula is a function solely based on the number of vertices versus a function that is based on the preceding value.

I took what students had and re-wrote it using subscripts: D_n = D_n-1 + n-2. We discussed what this showed, how to use it, and how you could work through each step to get to an answer, say for D₅₀. I then said that, theoretically, every recursive formula could be re-written as a closed formula. Sometimes the conversion could be complex, but it could be done.

We turned our attention to the closed formula. I told everyone that if they couldn't explain the formula they couldn't use the formula. I then pushed them to consider what the n represented, why was three being subtracted from n, why were these two values being multiplied together and then divided by 2?

Students immediately said that the n in the formula represented the number of sides. I drew a pentagon and its diagonals on the board. I asked if n is the number of sides, that does n - 3 represent? I erased three of the sides. Students were baffled. I told the class the formula should connect back to reality.

After thinking about it some more, most students still believed that n still represented the number of sides. I asked, if this was the case, why are you multiplying the number of sides by three less than the number of sides? Again the class had no response.

I asked what else n could possibly represent. A few students said it could be the number of vertices rather than the number of sides. I told the class that when faced with evidence that contradicts or doesn't support their assumption of n representing the number of sides, that, perhaps, they should rethink their assumptions rather than ignore the evidence.

We focused on a single vertex. How many diagonals could not be drawn from this vertex? Well, you can't draw a diagonal to the vertex itself. You also cannot draw a vertex to the two adjacent vertices. This means that there are n - 3 vertices where diagonals can be drawn. For how many vertices total can this happen? Well, there are n vertices in total. So, how many total diagonals can be drawn? Well, you have to add up n - 3 each time for every vertex. What's a shorthand way to do repeated addition? Multiplication! So n (n - 3) is counting the number of diagonals that can be drawn.

But why do we divide this number by 2?

At this point time was running out on the class. I told students to focus on the physical process that was occurring here. Some started to see that by the time you went around all the vertices that you actually had drawn to diagonals between each pair of non-adjacent vertices.

I told the class we would have a quiz on Friday. The quiz is one question: explain how the formula n (n - 3) / 2 models the number of diagonals in a regular polygon.

I know that students can look up the result online and that is fine with me. I want them to see that formulas aren't just these arbitrary results. I want them to see and understand that mathematics is simply a way of describing reality. And that by describing reality, we can make use of the math to answer questions that would be difficult or impossible to answer otherwise, such as how many diagonals does a regular 500-gon contain?

Diagonals in regular polygons - a first take

To preface this post, I didn't get as far as I hoped to as our school had a lock-down drill today that took up half the class period.

I asked students to consider how many diagonals were contained within different regular polygons. I drew a triangle, a square, and a pentagon on the board and drew the diagonals within each. As I drew, I defined a diagonal as a line segment that connects two non-adjacent vertices. I wrote the number of diagonals below each figure.

I told the class they were going to investigate the number of diagonals and look for patterns. Specifically,. they were to draw and determine the number of diagonals present in a hexagon, heptagon, and octagon. They were to use the patterns they were seeing to help determine the number of diagonals in a 50-gon. Finally, they were to use what they did to determine a formula for calculating the number of diagonals in any regular polygon.

I asked students to work on this task individually, what I call "engaging in individual think time." I re-visited some of the norms for working on an individual: explore, identify characteristics, connect to other work they may have done, and to check their work. I asked them to work individually for 5 minutes and then they could work with their table groups.

As I walked around the room, I could see that students were struggling with the basic idea of diagonals and that they wanted to jump straight to drawing the octagon and its diagonals.

I wrote the definition of a diagonal on the board and told students they needed to draw all the diagonals for all of the figures. I continued to walk around and help point out where diagonals were missing.

Once students started to discuss what they were seeing there was some progress. We confirmed that a hexagon has 9 diagonals and a heptagon has 14 diagonals. It took a while for students to confirm the heptagon value. Some students had counted 10 diagonals. I asked students to consider if this value made sense given all the information they had. The diagonal pattern developed so far was 0, 2, 5, 9. Does it look reasonable that the next diagonal count would be 10? Students admitted that 10 didn't look right.

Some students had counted 16 diagonals for the octagon. I repeated my question of does that look like a reasonable value? Students are supposed to try correctly counting diagonals in an octagon and to try to use this pattern to determine the number of diagonals in an enneagon and a decagon.

There continues to be a distinct lack of effort when it comes to pushing thinking and true problem solving. Next class may not progress very far as my intention is to push the class until they identify patterns that make sense to them and can then model their patterns using mathematics. If all goes well, students will push through and be able to calculate the number of diagonals in a 50-gon and have a formula they can use for any n-sided regular polygon.

Meaningful formulas for interior and exterior angle sums

We had district testing last class, so it's been almost a week since we last looked at polygons, exterior angle values, and interior angle values. I figured I would need to spend some time easing into where we left off with the interior angle table for regular polygons and looking for a pattern for sums of interior angles in regular polygons.

I asked the class to pull out their tables and discuss what they had seen in their table groups. I projected a table on the board and asked for the class to share what they were seeing. Students mentioned that the size of the interior angle decreases as the number of sides increase and another stated that the interior angle size increases as the number of sides increase.

I wanted the class to be thinking about formulas and representations that model the work they are doing. I had thought about the process they were following to find angle measurements:

1. Exterior angle measurement = 360^o / number of sides/angles
2. Interior angle measurement = 180^o - Exterior angle measurement
3. Sum of interior angle measurements = number of sides/angles • Interior angle measurement

A colleague had shared some practice worksheets, some of which contained formulas. Specifically, the formula for the sum of interior angle measurements was given as S = 180(n - 2). It struck me that a formula like this would not make any sense to my class given the approach we had taken. And while the two formulas are equivalent, one might not be as meaningful as another.

This connected to a discussion we had in the math department about whether students should simplify. I noticed that we tend to set up arbitrary rules to always re-work expressions and equations to the "simplest" form. Yet, in simplifying the expression, we may have made the connection to what the math is modeling more complex.

I wrote the three statements on the board and asked the class how they might be able to shorten how to write out these equations. There is nothing wrong with writing equations like this except that it is a lot of writing and can be a bit cumbersome.

Students discussed this at their tables. Some students just wanted to use actually values, some wanted to just write the equations as they were. A few suggested using variables.

I re-wrote the equations as

1. EA = 360^o / NS
2. IA = 180^o - EA
3. TIM = NS • IA

and then defined

• EA = exterior angle measurement
• IA = interior angle measurement
• TIM = total interior angle measurement

I explained that defining abbreviations enables me to use terms repeatedly without having to write out the whole definition each time. This is what happens when mathematicians write papers. Time is spent up-front defining terms and establishing notation that is then used throughout. I emphasized to the class that their formula should be meaningful to them. The mathematics is modeling reality and there should be a connection to the original problem situation.

I mentioned that sometimes they may see functional notation employed. Rather than writing TIM = NS • IA, they might see f(NS) = NS • IA. But that this may just use a generic variable label such as f(n) = n • IA.

At this point I could tell students were starting to become a bit confused or indifferent. I forged ahead believing I could bring everyone back to the same point.

I then pointed out that since IA = 180^o - EA, we could substitute and get f(n) = n • (180^o - EA). However, EA = 360^o / NS, so we could substitute further to get f(n) = n • (180^o - 360^o / NS) or f(n) = n • (180^o - 360^o / n).

This last formula solely depends on how many sides the figure has. But we could now simplify the formula by using the distributive property to end up with f(n) = 180^o (n - 2). This is the "traditional" formula that is given in geometry texts. However, how meaningful is this formula.

I asked the students to look at the table showing sums of interior angles. I asked students to focus on how the sum changes. They noticed it is going up by 180^o each time. Why does this happen? Students were stumped. I asked them what connections could they make to 180^o? Could they think of any figures that they associate with 180^o? Students said that straight lines and triangles come to mind. Could they make connections with what they know to this situation?

After some thought, one student suggested that by adding two sides to the previous figure, another triangle was being added to the figure. I drew a triangle, a square, and a pentagon on the board. I placed an external vertex and drew the two new sides. It was clear to see that, in fact a new triangle was being added to the previous figure. This explained the increase of 180^o each time.

But what about the formula. I asked students to pick a vertex and draw line segments to each of the non-adjacent vertices in the figure. For a triangle there were none, for a square there was one, and for a pentagon there were two. The results were one triangle, two triangles, and three triangles. Subtracting the number of triangles from the number of sides resulted in a constant value of two.

So, if I took the number of sides n and then subtracted 2 from it, I could determine how many triangles were contained in the figure. We had now circled back to the traditional formula. At this point, students could see why 180^o  was being multiplied by the number of sides minus two.

I wanted students to focus on creating meaningful formulas. The point being, that the math models the reality. When presented with a formula, they need to make connections back to the situation that the formula is modeling. There should be direct connections that they can make that will help make the formula more understandable. It isn't a matter of memorizing a formula, it is a matter of making sense of a formula that will help make the situations more understandable, not less.

With that, I asked students to capture their thinking about what they can do to make formulas more understandable.

Next class, we'll explore the number of diagonals and I will challenge the class to come up with a formula that describes the situation and makes sense to them. And then, we'll practice working through a series of problems where they can use different angle relationships to answer questions, in preparation for their final exam next wee.

Regular polygons and exterior/interior angle theorems

Today, we continued looking at properties of polygons. I needed to cover regular versus irregular polygons, the exterior angle theorem for polygons, and the interior angle theorem for polygons.

To start things off, I gave the class a sheet of regular and irregular polygons. I asked them to name the polygons and to discuss whether they thought they were regular polygons or not. The sheet I used was an Identifying Regular and Irregular Polygons worksheet from Common Core Sheets - Shapes page. This was the first time I have used the Common Core Sheets site but I will probably use it again since there are a lot of options for sheets, reviews, and tests/quizzes.

I gave students time to work through naming the polygons, working with those students who needed help on an individual basis. Students discussed their ideas about which were regular polygons and which were irregular polygons at their tables and then we did a class share out.

On one board I wrote the title Regular Polygons and on another board I wrote the title Irregular Polygons. Students shared their ideas about what they though made a polygon regular or irregular. For regular polygons: common, well-known shape; all sides congruent.  For irregular polygons: zi-zag shape, obtuse angles.

From here, I tackled the idea of regular polygons had congruent sides. I drew two hexagons, one regular and one irregular. The sides were congruent in each. Students could recognize the regular polygon but then wrestled with what else was needed to define regularity. Finally, a student said that all the interior angles were also congruent. Bingo, we had a definition for a regular polygon: A regular polygon is a polygon that has all sides and all interior angles congruent.

I next tackled the obtuse angle idea for irregular polygons. I asked students to look at the figures of regular octagons and decagons that they had on their worksheet. They could see that having an obtuse angle did not affect the condition of irregularity. We instead turned to the definition of regular polygons to guide the definition of irregular polygons: An irregular polygon is a polygon that has at least non-congruent side or angle.

Since we never formally defined what a polygon was, I took time to draw three closed figures on the board. The first had two straight sides and a curved side. The second had four sides, two of which crossed over each other. The third was a polygon. I asked students which of the figures were polygons. Students generally thought the first two figures were not polygons. They were correct for the first figure and they had noted it had a curved side.

The second figure was a bit trickier since it was a complex polygon. I pointed out how this was a closed plane figure formed by line segments (straight line segments) that intersected at single points. From this definition, the second figure did meet the criteria of being a polygon.

With that settled, I asked the class what an exterior angle was for a polygon. I drew a pentagon on the board. I asked students to think about how exterior angles were formed in triangles. A student said the exterior angle would be formed by extending a side. We discussed how many exterior angles could be formed and I drew side extension. I mentioned to the class that I liked to make the extensions look like a pin wheel.

We then moved into an investigation of exterior angles. Students drew different convex polygons and measured their exterior angles using a protractor. They then added their angle measurements together to find the exterior angle sum. I walked around helping students with using their protractor and with drawing exterior angles properly.

Students saw that the exterior angles were always summing to 360^o. I then said we could use this information to answer questions such as, "What is the measure of exterior angles for a regular 18-gon?" Students came up with the answer and I checked by asking other students why they responded 20^o. I also told students they should be able to answer, "How many sides does a regular polygon have if its exterior angles are all 18^o?

From here I moved to interior angles. If the exterior angle is 18^o, what must the interior angle measure? Students realized that the angles formed a linear pair and they just needed to subtract the angle measurement from 180^o. I checked for understanding with other students and then moved to the next investigation.

I asked students to create a table that listed the interior angle measurement, the exterior angle measurement, and the sum of the interior angles. I asked the class to complete this table for polygons from triangles through decagons.

Some students were confused and asked for clarification. I used the triangle as an example. The exterior angles for a triangle summed to 360^o so what does the size of an exterior angle have to be? Next, knowing that the exterior angle is 120^o, what is the size of the interior angle? Knowing you have three interior angles, what is the sum of the interior angles. I then told students to do this for all the other polygons in the table.

Students were just finishing up this task when class ended. Their homework is to complete the table, if they hadn't finished, and then look for patterns in the sum of interior angles. Next class we'll look at what they come up with and then explore why the relationship is present.

This is a reverse from how I have taught the interior angle formula in the past. Most books break the inside of a polygon into triangles and then count how many triangles are present to determine how many times you multiply 180^o.

My approach this year has a better feel. It ties together the exterior angle theorem to the size of interior angles which directly leads to the sum of the interior angles. We can then explore the result that the sum increases by 180^o each time. The triangle breakdown helps to explain or justify the pattern that is found versus force fitting the pattern because of how we are breaking down the inside of a polygon.

Introducing polygons and quadrilaterals

Our next unit focuses on quadrilaterals in the coordinate plane. There is some introductory work with polygons that also needs to be covered.

To start things off, I made use of Unexpected Riches from a Geoboard Quadrilateral Activity described in Rich and Engaging Mathematical Tasks Grades 5-9. Students were asked to construct as many quadrilaterals as possible on a geoboard and record the results. As students worked through their constructions, I encouraged them to be creative and think beyond just squares and rectangles.

Excluding rigid transformations, there are 16 shapes that can be created. As students worked, I called out how many different groups had found and told them to keep pushing. Students got to 16 shapes but some were either rotations or reflections of other shapes. Across the room, I did see all 16 shapes.

I displayed the shapes and gave students a chance to compare what they had completed. I numbered the 16 figures and wrote out the numbers for four of the figures on the board. I told the class these four figures all had something in common. I asked the class to describe what the common trait was.

The four figures were the only concave quadrilaterals. Students described how they looked like arrow heads. I pushed them for more clarification. They struggled some with how to describe what was going on. There were incorrect statements about all angles being acute, and the like, which I threw back at the class for verification. The discussion centered around the vertex which defined the concavity of the quadrilateral.

Finally, a student noted that the interior angle of the vertex was obtuse. I wrote down a definition:
A quadrilateral is concave if it has at least one interior angle that is obtuse.

I then noted that the other quadrilaterals were convex quadrilaterals. I wrote down a definition:
A quadrilateral is convex if all its interior angles are less than 180^o.

I then asked the class if a concave triangle can exist. There was some good discussion around this idea. One student noted that he could draw a triangle with an obtuse angle and so it should be called a concave triangle. Other students objected since there wasn't any indentation in the figure and this would require a fourth side to accomplish that result.

I turned back to the definition of a concave quadrilateral and said that we might need a more precise definition. That having an obtuse angle was not enough to define concavity. I also note a couple of the convex quadrilaterals that had obtuse angles. At this, a student suggested that is should state the angle is greater than 180^o and not just an obtuse angle. I re-wrote the definition:
A polygon is concave if it has at least one interior angle greater than 180^o.

This provided a precise mathematical definition for concavity that works for more than just quadrilaterals. The discussion also built upon students' prior knowledge of interior angles of triangles and a natural transference of this idea to quadrilaterals.

With these ideas established, I moved on to polygon names. Most of the standard names through hexagons were known, as was octagon. Students did not know a 7-sided figure was called a heptagon, nor that a 9-sided figure was called an enneagon. With some connections to common words, such as 10 years is called a decade, students came up with a 10-sided figure is called a decagon.

I went through the naming of 11-sided through 19-sided figures as well, only to show how these names were built upon from earlier names, especially figures 13-19.

I then asked students to construct a convex and a concave pentagon on their geoboards. Next, I assigned table groups different figures, hexagon, heptagon, octagon, decagon and asked them to construct a concave figure on their geoboards. It was fun to see their creativity. For example, one group constructed what looked like a crown for the concave decagon.

I concluded with having students record their thoughts and capture definitions in their notebooks.

I still need to cover the ideas of exterior angles and regular polygons. Once these are done, I will move on to look at the sums of exterior angles, the sums of interior angles, and the number of diagonals in polygons.

Working with triangles

For the past two classes, the focus has been on working with triangles. Students are starting to improve on names and definitions but still have a long way to go. Although there were only two students who could name vertical angles before, after working through problems in which vertical angles are used, more students can identify the angles.

I have noticed a tendency for students to work through identifying congruent parts and then not to take a step back and look at what they had shown to be congruent parts. As a result, they finish finding congruent pieces and then just start guessing which congruence theorem applies. I keep telling students to take a step back and look at what they have now identified as congruent. This helps a little, but I am not confident in the transference. Unless students can do this on their own, they will be lost.

The problems the class has been working on cover a wide variety of topics in order to get them prepared for our upcoming test. I have been trying to push them to be able to write proofs on their own without any extra help. There seems to be slight progress on this front. Their issues are tied to what I just previously discussed; they get lost in the details and never take a step back to look at the whole picture.

Perhaps it will help if they are stopped periodically in their work and asked what they have demonstrated so far and what they still need to demonstrate? I need to check research to see if there is anything that will help on this front.

We are going to finish up working through these problems and then we'll take the test. Since the test doesn't cover mid-segments and related topics, I am skipping the topic in order to catch up with the rest of the geometry team.  I was planning on using the fire hydrant and warehouse placement investigations as outlined in the NCTM teacher's blog. It's unfortunate I don't have the time right now to fit these in; I think they would be productive investigations.

Up next, quadrilaterals and coordinate geometry proofs. I'm starting to look around at what's available from an inquiry-based approach. I sure hope there are some good pieces I can use.

Working with triangle congruence

Today, we started by reviewing what we had learned so far about triangle congruence. I listed out SSS, SAS, and SSA and indicated which worked and which didn't. I then said the focus so far had been on looking at triangle sides. What happens when you focus on angles instead.

I wrote out the three equivalent statements for angles: AAA, ASA, and AAS. I felt this laid out some symmetry to the situation that would help students better relate to why we would investigate these options. I gave students some time to try these.

I walked around to see how things were progressing. There was the usual help with how to use a protractor but I was also amazed at how many students would just guess rather than try drawing the figures need. I asked how they came to their conclusion and the said they just guessed. What was the basis for the guess? It just looks like it wouldn't or would work. How can you know without having any foundation for your guess?

I encourage students to actually draw out their figures and measure their angles and sides. If nothing else, maybe they'll get better at using a protractor. The classroom discussion that resulted was mixed since there were still a decent number of students who just thought about it to draw a conclusion. We finally did get out that ASA and AAS worked but that AAA did not work.

We then practiced working through some congruence problems and a congruence proof. The proof gave the statements and students were to justify the statements with a reason. There was still a lot of struggle with the reasons although I am seeing some improvement.

The last two steps of the proof involve using a vertical pair and one of the triangle congruence theorems. Students didn't know what the name for the vertical pair was and couldn't articulate that because the angles were a vertical pair they were congruent. It's really difficult to write or complete proofs if you don't know that vocabulary and the properties of things.

Since no one seemed to remember or have this in their notes, the homework for tonight is to find out what the name of angles are when they are formed by two intersecting lines. They are also to identify the properties of these angle.

Tweeting triangles and triangle congruence

Last class I had students draw a triangle. and asked them to consider what were the fewest parts of a triangle that could be communicated so that the triangle could be copied exactly. I set this up as follows:

First, draw what you consider to be the most beautiful triangle. Measure and record all of the side lengths and angles for your triangle. Be as precise as you can.

Second, you have just created what you consider to be the most beautiful triangle ever created. You are so excited about your triangle that you want to share it with the world. You go to your Twitter account to send a tweet and realize that the small number of characters you can send may be a problem.

Undaunted you decide to see what the fewest number of sides and angles of your triangle that you can tweet and still have followers be able to re-draw your beautiful triangle exactly as you have drawn it. Your task is to determine the fewest number of sides and angles of your triangle that you can tweet. Think about this problem and decide what you would do. 

I gave students a couple of minutes and then have students pair up and see if their method works. I walk around to see what they come up with. I had students share out their findings, which follow:

• give 2 angles and 2 sides
• an equilateral triangle with the side length
• right or isosceles triangle and two side lengths
• 2 congruent side lengths and 1 angle

Many students had elected to draw isosceles or right triangles. As I walked around, one group had concluded that you just need 3 pieces of information about the triangle, deciding that AAA and SSA would also work.

I decided to tackle this by having the one group state that just three parts of the triangle were needed. I wrote this on the board. I focused on the right or isosceles triangle and two side lengths first. I didn't want to tackle right triangles yet, so I asked students to focus on the isosceles triangles.

I asked the class to re-create an isosceles triangle I was thinking of, the congruent side lengths were 5". After pondering this, students realized that there wasn't enough information. Without the angle formed by the congruent sides, they could not replicate the triangle. So, we basically needed to know the two sides and the angle formed by those two sides.

Does this work for any triangle? I asked students to try this out. There was still some question about the process working, mostly due to  students' poor use of protractors and rulers (although they are much better than they were). I asked them to all draw a triangle that had side lengths of 3" and 4" with the angle formed by these sides being 70^o. When they compared their drawings they realized that they had drawn congruent triangles. I wrote under the three parts side-angle-side and the abbreviation SAS. At the end of the line I wrote works.

I asked if the order given for the two sides and angle mattered. What would happen if you were told that a side of 3" joined a side of 4" and at the other end of the 4" side there was a 70^o angle? Students determined that this didn't work. I wrote side-side-angle SSA and then didn't work on the board.

I next used the equilateral triangle as a springboard. In this situation, you are essentially communicating the lengths of all three sides. Would the idea of communicating all three side lengths work for any triangle? I gave them triangle side lengths of 3", 4", and 6" and asked them to draw the triangle to see if they had the same triangle. After the typical struggles and my referencing back to what was going on when they copies angles using a compass and straight-edge, the class decided that providing three sides worked. I wrote side-side-side SSS and works after it.

We were running out of time so I asked the class to consider the analogous situations of ASA and AAS, which flipped the SAS and SSA relationships. I'll use next class to work through these and also bring to attention of the entire class the use of AAA.

Introducing congruent triangles

Last class was devoted to introducing and discussing congruent shapes. I started the introduction by asking students to draw a simple figure, perhaps using a straight edge of graph paper. I then asked the class to make an exact copy of the figure they drew.

With these in hand, I asked the class how they knew that the figures were exactly the same? This led to a discussion about characteristics that each figure needed. The class boiled it down to each figure having the same number of sides with matching side lengths and that all the angles had to match in measurement.

I then tied this back to rigid motions. With rigid motions we are creating a pre-image and image that are congruent. I referenced how we labeled both images and then proceeded to show different pre-image and image figures from rigid motions placed on a graph.

The task for the class was how to mark the figures to show which pieces were congruent. There was a bit of hesitancy at first, but a student came to the board and appropriately marked the congruent sides. Another student then marked the congruent angles. This happened with the second pair of figures as well.

The third pair of figures presented a slight wrinkle. As before, a student marked congruent sides and then congruent angles. The pair presented were isosceles triangles. As the rest of the class looked at the markings, a couple of hands shot up. One girl asked whether two sides should actually be marked congruent. Others in the class agreed, noting that the triangle was isosceles and should have two congruent sides. (They were using actual side length to determine the congruence.) Then another student said that the base angles should also be marked as congruent.

I was pleased with students recognizing the anomaly in the last pair. I was also pleased with how comfortable students were with marking congruent sides and congruent angles.

I mentioned that what they had identified was that corresponding parts of congruent triangles are congruent. This was the first mention of what is commonly abbreviated CPCTC. I didn't dwell on this, but wanted to point it out in case they ran across this terminology.

I gave students some practice problems, identifying corresponding parts or writing congruence statements. I used the practice sheet problems that showed congruence statements as the guide of how to write a congruence statement. They readily picked it up and seemed comfortable working through the problems.

I next started the thinking about how much information is needed to convey congruence between triangles. I used a tweeting triangle lesson that I have used over the years.

The tweeting triangles lesson outline is posted on my website. I'll have students try their ideas next class and we'll, hopefully, build all of the triangle congruence shortcuts.

Why do students struggle with geometric proofs?

We've spent a couple of days working through puzzle type problems that involve angle relationships, both triangular and angles formed by lines (intersecting, parallel, etc.). Students have been engaged in working through these though they still struggle to different degrees. There is a tendency to not see what is given, to not think about what is given actually means, and to not use what has been solved as a piece of the next solution. After working through challenging problems for a couple of days they are doing better with these.

The other piece that continues to challenge students are geometric proofs. Since we've been working on problems involving the exterior angle theorem of a triangle, I thought this would be a simple proof for students to tackle.

I presented the scenario, with diagram, and asked them to describe how they could prove the following theorem: The measure of an exterior angle to a triangle equals the sum of the measures of the two non-adjacent interior angles of the triangle.

Walking around and talking with different groups and then discussing as a class showed they were thinking about the proof in a proper way. Students described that the exterior angle and the adjacent interior angle formed a linear pair and were supplementary. They explained that the three interior angles summed to 180^o. They said you could substitute and subtract to show the result. They could say all of this but they couldn't write it out.

As I walked around, I could see that students would leave steps out. How do you know that those angle measurements sum to 180^o? Why do you say these two angle measurements equal the exterior angle's measurement?

Some of the struggle comes from a lack of understanding the vocabulary. You cannot write a proof if you don't know what a linear pair is and what properties it possesses. Part of it is that students do not have experience in writing step-by-step detailed instructions. Writing a proof is akin to writing a computer program. You cannot skip steps or the computer won't understand what you want done. Proofs are the same. You can't skip a step or the reader is left scratching their head as to how you got from step A to step B.

We're told that students should have a level of proficiency in writing proofs by this time in the semester. I don't think that is realistic. Without having the background of writing detailed step-by-step instructions, students do not have any basis upon which to draw.

It is necessary to build this basis of writing detailed instructions that can lead to proof. I will be experimenting with some ideas. For example, have students construct a figure with compass and straight edge and then write instructions so that another student can replicate the figure. We are moving into congruence next. so I'll be able to test this out to see if it helps.

Triangle properties

I tried to use workstations and have students investigate different triangle properties: isosceles triangle base angle theorems, exterior angle theorem, and correspondence of angle length to side length. These investigations came from the Discovering Geometry book and included using compass, protractor, and patty paper.

The investigations went okay. I particularly liked the exterior angle theorem investigation and the correspondence of angle size to opposite side length. For whatever reason, students got bogged down with the two isosceles triangle base angle theorems (two congruent sides results in two congruent angles and the converse).

I was expecting these investigations to go quicker than they did. After collecting sheets to see what students captured, it appeared that many of them did not pick up on key properties. I have to hope that as we continue to work with these properties that they will make sense. I intend to keep referencing back to these investigations, which I hope will reinforce the importance of making sense of what is happening during an investigation.

I intend to have students work through the exterior angle theorem and then complete an angle puzzle that a colleague found online.

Working with triangles

The next unit in geometry covers triangles. I started by using a graphic organizer to see what students already know and correct any misconceptions. The organizer lists names of triangles (scalene, isosceles, equilateral, obtuse, etc.) down one side and properties (no congruent angles, 2 congruent angles, 3 congruent sides, 1 right angle, etc.) across the top. I had students fill out what they could, discuss in groups and then discuss as a class. I asked the class to complete this in pencil as I knew there would be corrections.

This led to some good questions and discussions about whether certain properties could exist. We worked through these questions and misconceptions and then used the information to either name or create described triangles.

The next class started by working through a couple more problems using naming to identify triangles. This class turned into a long, drawn-out affair because students couldn't find the side lengths of a triangle given its coordinates. I was baffled by this. I finally drew a single line segment on the graph and asked how they would determine the length of the segment. Students readily said that you would create a right triangle and use the Pythagorean theorem. It became apparent that students did not recognize triangle sides as line segments. Once we got over this hurdle we were able to complete the remainder of the practice problems (with some struggles on computation). Unfortunately, this took the entire class.

Today we worked through proving the interior angle sum theorem for triangles. I started with three progressive problems: the first gave to values and then asked the measure of the third angle, the second gave one value and two expressions for angle measure and asked for the value of x, the third gave three expressions for angle measurement and asked for the value of x. I anticipated that moving from the first to the second would give some students pause, which it did. Once they realized that the same interior sum property held, they were able to move on.

I then asked students what properties or theorems could be used to help prove that the interior angle sum theorem was true. After briefly discussing in groups we did a share out. Many of the groups thought if we could somehow relate the interior angles to a straight line it would help. (Many students were thinking along the line of unfolding the triangle.) One group suggested using a protractor and compass. A couple of groups thought the triangle angles all had to be less than 180^o and that perhaps that could be made use of.

I drew a line parallel to one side. I could relate this to their thinking about somehow relating the angles to a straight line. I instructed them to look at the angle relationships when a transversal crosses a pair of parallel lines. How are the angle relationships connected.

This again was a struggle for the class. I kept reminding students to draw things out and make connections. Finally, a couple of students drew out the triangle, extended the sides to lines and then drew a parallel line. They started to mark congruent angles and saw the alternate interior angles connected to angles in the triangle.

I wrote out the first few steps in the proof and asked students to try and finish a two-column proof based on this start. It will be interesting to see if any are able to complete the thinking.

Geometric proofs - initial wrap-up

Today concluded the introduction to geometric proofs. Students continue to struggle with breaking down their thinking in discrete steps that lead to a conclusion. At this point, students cannot write even simple geometric proofs at the level of detail needed. I did ask them, given the steps and a list of possible reasons, if they could match reasons to the steps. Most did not think they could do this. I provided a very simple proof that had one given and four steps, two of which were the result of the segment addition postulate and two from substitution. It took table groups 10-15 minutes to reason through this but they did come to a correct conclusion.

Tomorrow is a review day before the unit assessment.

Working through geometric proofs

Today was a continuation of working through proofs. It was a mixed bag as to how students fared on their own. Most didn't get very far but at least I could see that many had actually attempted the next proof.

I had the class work in groups discussing what they did. I walked around and helped get students going in the right direction. Many students could see a statement was true and could articulate general reasoning. They fell short in providing step-by-step instructions or providing the "why" for their reasoning.

As the class continued to struggle through the proofs, I could see that they were beginning to grasp better the detail and supporting reasons needed for a proof. I found that describing a proof as something similar to giving detailed instructions that also included why each step in the instructions was given seemed to help. For a couple of students who take computer programming, I likened writing a proof to writing lines of code. You need to provide the computer all of the instructions and you cannot leave out any instructions.

The plan is to wrap-up proofs next class and then work through a review sheet. I will assess the class the middle of next week. The assessment will be a common assessment that the geometry team uses coupled with an additional assessment where students have to explain and illustrate the connections between reflections, translations, and rotations.

Rotations, perpendicular bisectors, and constructions

Trying to catch up three days in this post.

Most of the class did not come up with any ideas about how to find reflection lines or the center of rotation. A couple of students had an idea about finding a circle that passed through the pre-image and image points and looking at where the center of the circle was located. It was an interesting idea. I pointed out that any sized circle could be used and demonstrated how this would work. I didn't pursue the relationship of centers of circles of the same size that passed through the associated points. Could be an interesting investigation. Perhaps during a circle-focused unit, but not now.

I had anticipated this and modeled problem-solving thinking. I basically walked through thinking that rotations and reflections were somehow connected. Maybe I could start with reflections that I could create and see where this led me. After reflecting all three points I saw that the reflection lines intersected. No big surprise there, except that all three intersected at the same point. This was interesting. I had the class check this out and they found that their reflecting lines also intersected at a single point.

In taking a step back, I was able to remind students that the segment connecting pre-image and image points and the reflecting line were perpendicular to each other and the reflecting line bisected the segment between pre-image and image point. The reflecting line was a perpendicular bisector.

With this, I transitioned to how do we construct perpendicular bisectors. I gave students a series of constructions: copy segment, copy angle, segment bisector, perpendicular line, perpendicular line, parallel line through point not on a line, angle bisector, equilateral triangle, 30^o angle, and 45^o angle.

I walked around and helped students struggling with these and let those who were on it to just go. Overall the class stayed engaged and worked through these constructions. I provided little direct instruction except for those who were struggling. Since each construction had two versions, I was then able to have struggling students practice on the second instance to be sure they go it.

From there we moved into geometric proofs. I provided a sheet of definitions, postulates, and theorems (angle addition, segment addition) that could be used. The class was given a series of proofs to work through. The first five had written plans, so students were to follow the plan and write the proof. I walked through the first proof with the class and they worked through the second proof.  Their homework was to work through the next three proofs that had plans written out.

We'll continue with proofs next class.

Investigating generalizations

After investigating connections between rotations and reflections I started wondering about generalizations. I felt this would be a good way for students to practice working with rotations and gain some better insight into the difference between a procedure working in specialized situations or working in general.

I posed the question to the class whether the methods they used in the investigation (connecting line segments created from two different pairs of points between the pre-image and image) would always identify the center of rotation. I also asked if the two reflecting lines always provide a way to determine the angle of rotation and the center of rotation.

Tables investigated and discovered that the process of connecting two line segments as per the investigation did not work in general. This is a valuable lesson because too often students learn some procedure and start applying without regard to when it is appropriate to apply.

The second piece led to the conclusion that the center of rotation is at the point of intersection of the two reflecting lines and can be used to help determine the degree of rotation.

I then posed the reverse of this, given a pre-image and image figure, how do you determine the reflecting lines and from them identify the center of rotation and the degree of rotation.

I intend to use this as a springboard back to constructions. I am hoping that students make some connection to bisectors (perpendicular bisectors in particular).

Also, my intention is to use MS Paint to investigate dilations. I can use the coordinates of the figure and then change the size by percentage and look at the new coordinates. I think this will bring dilations to life and provide a more understandable setting as to what is going on and why.

Connecting reflections and translations - proving relationships

Today did not progress as I hoped. Students did not make any progress on why the three segments connecting pre-image to double-reflected points were all the same length. There is a distinct lack of initiative and strong desire to be told what to do that can be disheartening.

I decided to try work through the problem as a proof. Unfortunately, not thinking this through, I tackled all three segments at once, which resulted in extensive repetition and loss of the central issue. Doing this again, I would prove that one segment was equal to twice the distance between the parallel lines and then say that the exact same argument would apply to the other two segments.

As part of this proof, we needed a working definition of what it means to reflect a point. One student suggested a reflection was a flip of 180^o over a given line. I drew an example that did not have the two flipped points equally distant from the reflection line. Students realized that the definition needed to include that the two points were the same distance from the reflection line. While the definition could be tightened up a bit, the critical aspect of equal distance apart was what was needed for the proof.

After plodding starting to plod through the proof, I could see that it was going to be too lengthy and confusing doing all three segments. I stopped along the way knowing that the pieces of thinking about what was given and how they relate had been communicated.

I then asked students to address how they knew if the three segments were parallel. Again, there was not much activity going on. Was it Monday doldrums or just a lack of interest? I told students to review angle relationships associated with parallel lines and transversals. At this point I had to ask the class about different angle relationships and their associated names.

The problem now made use of the reflection definition; a flip of 180^o equated with the lines being perpendicular. Now, all the angle pairs are 90^o and the converse relationships are in play.

The class ended with students tasked with determining how to position or to determine the location of the parallel lines given a specific reflection. As I walked around the room asking groups their thoughts before the end of class, I could see there was a lack of understanding about the concept of generalizing results. Yet something else that needs to be addressed.

Making arguments in geometry and the need for vocabulary and notation

Today we continued looking at the connection between reflections and translations. Students struggled with explaining why the three segments connecting double-reflected images had to be parallel.

I had students talk things over in their groups. As I walked around asking what they were thinking, most groups expressed that they knew in their heads what they wanted to say but couldn't express it well. I told them to not worry about how rough or awkward the communication was, but to try to get their thoughts out.

The results were interesting. Most groups expressed conjectures or possible theorems about the relationship. For example, one group stated if the segments connecting the reflected points were parallel then the result would have to be a translation. Another group stated that if the segments were not parallel then the result could not be a translation. These indicated that the students were trying to think about the situation in a mathematical way.

One group said they were focused on the distance between the reflected figures and the given parallel lines. They weren't sure how to proceed with their thoughts but felt that the distances would be relevant to concluding the results were translations. Another student thought that the first reflection flipped the figure 180^o. The second reflection over a parallel line would then flip the figure back and therefore it would be a translation. This was a good intuitive way of thinking about what was happening.

I used this as an opportunity to discuss maths vocabulary and notation. The class understood how they were struggling to communicate what they were thinking and seeing. This is why vocabulary and notation was developed. Sometimes it took centuries to create but the need to express and communicate thoughts drove the vocabulary and notation. Students were able to better appreciate the value of learning maths vocabulary and notation.

I then asked students to focus on the idea of distance. They've been working on this aspect of reflections and know that a reflected point is the same distance away from the line of reflection as the original point. I want to see if students can use this result to determine that all three line segments must be equal. I'll see what they come up with next class.

Reflecting over the line y = 2x and connecting translations to reflections

Another short post. I had students try reflecting over y=2x. This was a struggle because students wanted the reflected points to behave nicely, i.e. reflect onto nice coordinates, which they weren't. I had to pass out tracing paper so that students could see that the image points they were drawing were not actually the reflection points. On the positive side, they stayed with the effort for over 30 minutes.

Students could see that the reflection points were not behaving well and that the nice symmetry of reflecting over a horizontal line, a vertical line, y=x, or y=-x was gone. Based on where they were, I determined that I needed to move on from this for now. Trying to investigate reflections over y=-2x or y=2x+3 would not be productive.

Since this was a block period, I did a quick brain break to re-energize the class and then moved on to translations. I had a triangle on a coordinate plane and asked what the translation of this figure by translating four points to the right and 3 points down would look like. I told students to guess at the meaning if they weren't sure what a translation was.

Students easily performed this task. We wrote out the notation (x, y) --> (x + 4, y - 3) as suggested by the class. I asked what the general expression would be for translating the x by a points and y by b points. One student suggested (x, y) --> ( x ± a, y ± b), which I hadn't expected. We tried a couple of more translations that I provided through expressions and then started to explore connections between translations and reflections.

I asked the class what connections they could think of between reflections and translations. No ideas were forth coming. I told the class we were going to explore this. The Slide Me Now activity from Navigating through Geometry provided the basis for this investigation. Students completed the first four questions during class. Their homework is to complete questions 5 & 6. I told the class to use what they know about parallel lines, transversals, and the angle relationships they learned to help them.

Next class I'll see what they come up with. I'm going to try to push this to begin looking a geometry proofs. I plan on completing the first two questions in the extension section of this investigation as well.

I'm thinking to revisit the reflection over y=2x and looking at why the segment connecting the pre-image and image points must be perpendicular to the line of reflection. This could lead into relationship of slope to perpendicular lines. I might also look at how to construct a segment bisector or perpendicular line as offshoots of this. I'm still debating which direction I want to take this piece.

Reflecting over lines that are not vertical or horizontal

Things got back on track today. I had a brief discussion at the start of class. I explained that I expected them to struggle, that I expected them to discuss the problem, and that I expected them to ask questions of others or me. What I will not abide is intellectual laziness; the class sitting around waiting to be told exactly what steps to take. I related this to later in life when they are trained in a job. The training doesn't cover all situations. The expectation is that an employee will be able to apply their training to the situation at hand.

With that we explored reflections over the line y = x. I asked students to conjecture what would happen in this situation. I instructed students to close their eyes and picture the grid with the line y = x on it. I asked them to imagine what would happen as an object was reflected over this line. We discussed their thoughts, with many thinking about how the coordinates might go from positive to negative. One girl conjectured that this would be equivalent to reflecting over the x-axis and then over the y-axis. The students were ready to start exploring.

Some students did not know what the line y = x looked like and I had to assist them to get things started. The other thing that kept cropping up was that students would still end up reflecting as if it was over a vertical line or horizontal line. Once they got this down, they were able to see the result that the pre-image point (x, y) ends at the image point (y, x).

I asked the class to consider what if the line was y = -x. Students still felt that the x and y values would flip. A couple thought, in addition, that the coordinates would become negative. After exploring the results, the class saw that (x, y) --> (-y, -x).

We had time to start exploring further. I asked what would happen when reflecting over the line y=2x? Again, some students weren't sure how to graph the line. I reminded them how to create a table and then they were off exploring. The class ended at this point, so we'll pick up where we left off last time.

I really like this exploration as it is bringing back algebra review that students obviously need. Hopefully the energy level and effort will continue.

Reflecting over a vertical line

This will be a brief post. I continued with students trying to determine the relationship between reflected points and the line over which they points were reflected.

I encouraged students to write their results on the board (vertical line used, pre-image coordinates, image coordinates). After getting a half-dozen results on the board, I asked students to look for patterns and connections. Not much was happening, so I told students to get out of their seats and take a closer look for connections.

Students returned to their seats and some discussions began. Different students started interacting with other groups. Some students continued to try new values. I let the struggles continue and periodically asked what the connection was between the coordinates and the line used.

Finally, two girls went to the board on their own and were arguing/discussing the relationship. Their discussion centered on the idea of midpoints. After they had satisfied themselves, I asked the two to share their discussion with the rest of the class. There was a lot of discussion and questions about this idea, but gradually the class started to come together with the idea.

I checked their understanding by giving two x-coordinates and asking what line was reflected over. I then assigned them the task of coming up with an expression that could determine the image coordinate given the line of reflection and the pre-image point.

This entire exploration took 45 minutes, but they started to figure things out.

Introducing rigid transformations in geometry

Today I introduced rigid transformations. We started by discussing transformations and students were able to identify translations, reflections, rotations, and dilations (although they only described them and didn't name them).

I had mini-whiteboards with axes on them and I started the students off by plotting some points. This helped me see who might be struggling with coordinates and helped students refresh their memory on coordinates and plotting points.

Satisfied that students could correctly plot points, I proceeded to introduce reflections. I decided to use NCTM's Navigating through Geometry sequence for transformations. The first activity deals with reflections over the y-axis.

The beginning prompts dealt with drawing and tracing to create a reflection. Students began to have issues when they were asked to drawn a segment between two points they had selected. The notation seemed to baffle them, which was disheartening given how much time we had spent on notation and understanding what structure and information is given in prompts.

After moving beyond this hurdle, students were to identify how the two points (pre-image and image) were related to the line of reflection. Students readily recognized that the two points were the same distance from the reflection line, lying on opposite sides of the line.

The next piece became a problem. Students were to draw a triangle and then draw its reflection based upon what they had just discussed. Students wanted to fold the paper and trace out again or seemed totally lost. I realized that using the whiteboards and grids could help. I moved students to using these and they were able to proceed without much incident.

We discussed what was happening with the coordinates and students could more readily see that the pre-image point (x , y) became the image point ( -x, y) when reflecting over the y-axis (line x = 0).

At this point I deviated from the activity slightly by focusing on reflections over vertical lines. I asked students to pick a different vertical line, such as x = 2, and reflect their pre-image over this line. I told students to focus on what is happening to the x coordinate now. How does reflecting over a different line change things.

The class struggled mightily on this task. Many wanted to give up and several asked me to just tell them what the result was. I finally told the class that I expected them to make reasonable conjectures based upon what they were seeing. I told them if it takes the class two weeks to work through this issue then it will take two weeks.

Their homework is to come up with some conjectures. I'll see what they bring to class.

My hope is that we can understand the mathematics of reflections over vertical lines and the resulting expressions that describe these reflections. It should then be fairly easy to conjecture about the results we would see reflecting over horizontal lines and verifying the results.

My goal is for students to tackle lines that are not horizontal or vertical. I'll start with y = x and y = -x and then have them tackle a line such as y = x + 2. From there, I hope to look at more general lines such as y = 2x - 3.

After these explorations, I'll revert back to using more of the Navigating through Geometry activities.

Introducing algebraic proofs

Today I introduced algebraic proofs as a lead in to geometric proofs.

I started by having the expression, "Oh yeah, prove it!" on the board. I then asked students what it means to prove something. The discussion brought out using evidence to demonstrate a theory or statement was correct.

I wanted to have students consider rules and properties they work with when calculating or working with expressions. The idea was to pull out some of the properties that we would use as building blocks for algebraic proofs. This turned out to be a bit tougher than I expected.

I did provide an example:

For any two real numbers a and b, if a equals b then b = a. The meaning is shown in writing by "If a = b, then  b = a." (This is the symmetric property of equality.)

It may work better to start with something even more simple such as the reflexive property, a = a.

Students slowly started putting things on the board, such as the additive equality, a x 0 = 0, PEMDAS (an acronym for order of operations), and a couple more. I then pointed out how these are accepted properties and rules. I used order of operations as a way to explain that the accepted order guarantees that any two people making a calculation from an expression will reach the same result.

I next put up the following nine properties:
   1) addition property of equality
   2) subtraction property of equality
   3) multiplication property of equality
   4) division property of equality
   5) distributive property
   6) substitution property
   7) reflexive property
   8) symmetric property
   9) transitive property

I provided an example for the first property of writing it symbolically: if a = b then ac = bc for any other value c.

I asked students to write expressions for the other eight properties. I told them they could use their cell phone to search for assistance.

The students completed most of the other properties.  These properties are the building blocks that we use for algebraic proofs.

The first piece I worked with was from an algebraic proof worksheet that a colleague found online. I talked through the first proof, asking the class at each step what allowed us to write the statement. I then wrote in the appropriate property. I told the class to use Q.E.D. to show that their proof was concluded.

Students were then turned loose on the next two proofs. I checked with students to verify they completed the second proof correctly. The remaining proofs were assigned as homework, although some students were able to finish these before leaving class.

Geometric constructions continued

The second day of constructions started with students practicing duplicating an angle. This embedded constructing segments of the same length, so it was good practice for both constructions introduced on the first day.

Some students were still struggling a bit with the congruent angle construction. As I worked with these students, I could see the light bulb going off as to what they were doing and why they were doing it.

I then presented the next challenge. I drew a line segment on the board and labeled it as segment AB. I then placed a point C on the board such that C was not on the segment. I told students their task was to construct a line parallel to segment AB that passed through point C.

I let students play around with this for about 5 minutes. Most were stymied. I told them to think about what they knew about parallel lines and angle relationships. This got a few students moving. I told students to look in their notes to see what the relationships were. Some said they had no notes on this topic. As I looked through their notes, I pointed out where this information was recorded.

More and more students started to realize they needed to make use of corresponding angles, yet were reluctant to draw a new line (the transversal) to help. I pushed them to think about how they would duplicate the angle without adding any new lines. Most came to realize that they indeed had to draw a transversal.

With the transversal drawn, many students still struggled with how to proceed. I encouraged them to think about the process they used for duplicating angles. Tentatively, students started to create the congruent angle they needed. I noticed many students wanted to fall back on using a protractor or making use of markings on the compass as to the width of the compass opening. I had to reinforce that the compass was the measuring device and that any markings on the device was irrelevant.

As I walked around, I noticed one girl had done something entirely different. I asked her to explain her method to me. She said she had used the compass to measure the separation between points A and C. She then went to point B and drew an arc the same distance. Next she measured the distance between points A and B. Placing the compass on point C, see drew an arc that intersected with the first arc she had drawn and labeled the intersection point D.

She said that point D was the same distance away from B as C was from A. She drew the segment CD and said this was parallel because C and D were the same distance away from the segment and so every point in between will also be the same distance. She had, in effect, constructed a parallelogram. I congratulated her on her thinking. She said it just made more sense to her based on one of the definitions of parallel lines that we had examined.

I had a student present the construction of the parallel line by using corresponding angles. We discussed this construction and students asked questions for clarification. I could tell that many still felt shaky on this construction. I then had the one girl share her parallel points construction. The reaction from the class was this was so much simpler and easier to understand. I had to agree.

I spent the last few minutes of the class going over a couple of problems from a quiz I gave last week. I was disappointed in students not thinking about the information given to them. I discussed two problems in particular. One, the problem stated a ray was an angle bisector and students had to justify the equation they set up to solve the problem. The second involved midpoints; many students wrote on the quiz that they didn't remember the midpoint formula.

This disturbed me because I hadn't asked them to remember formulas but, rather, to use reasoning and common sense to find solutions. These problems are perfect examples of why students do not perform well on state and national assessments. They have been hammered with memorizing formulas that have no meaning to them versus understanding the facts of the situation and using some basic knowledge to construct a solution. Hopefully the class will become more comfortable with this approach as we move along.

The geometry team has a common assessment scheduled for next week that covers the first unit. I will spend the two class days before working with students to get better at identifying the relationships they are seeing. Their algebra skills seem adequate to solve the equations that result in the problems, it's the recognition of the relationships that appears to be the problem.

I was to introduce algebraic proofs as part of this unit. There is no way that I can do this justice prior to the assessment, so I'll hold off assessing this piece until later.

Introducing Geometric Constructions

After devoting a class to just practicing work with angle relationships and midpoints, I needed to move on to geometric constructions. I wanted to take a more inquiry-based approach to this topic.

My first challenge was thinking about what exposure students may have had to compasses. I know that I have seen a lot of movies with sailing ships and the chart scenes normally included the use of a compass. I decided to look up a clips that showed the use of a compass in ship navigation.

To start class, I held up a compass and asked students if they had ever seen this before. When I asked where, a couple of students said it was something pirates used. Perfect! I asked the class what pirates would use the compass for. Most shrugged their shoulders. A few replied that they could draw circles with them.

I then showed the first clip I found. It's short and has no sound but shows someone using a compass with a chart. I asked the class what the person was doing with the compass. The class responded it appeared the person was using it to measure distance. I re-emphasized the idea of using the compass to measure distance.

It was now time to show the second clip. This clip shows how the compass is used to measure distance and make markings with arcs.

I then drew a line segment on the board and labeled it as segment AB. I marked a third point C on the board. I told the class the challenge was to make an exact copy of AB so that AB = CD by using the compass as a distance measuring device and a ruler solely to draw straight lines.

I then let students struggle through the challenge. Some students were done quickly. Checking their work, I asked how they copied the line. These students said they used the ruler. I told them that wasn't allowed. The only device they had for measuring distance was the compass. I told them to think about what they saw on the video.

Slowly students started to get the idea. I did have to walk around a lot and talk through how the compass could be used to measure distance with quite a few groups. After everyone had the general idea, I asked them to draw another segment and then make a copy that was congruent. This time students seemed to get what they needed to do.

I next drew an angle on the board and labeled the vertex A. I drew a second point B and told them the challenge was to make an exact copy of angle A so that the measures of both angles were the same. Again, they were to use only the compass and straight edge.

Students worked on this for 15 plus minutes. I walked around and checked on their work. Many students had drawn two angles with both angles having side segments that were the same length. I asked how they knew the angles were the same measure. They were stumped by this. Others had measured the separation of the rays but hadn't considered that they weren't measuring the width of the angles from equivalent points.

After about 15 minutes, several students were honing in on some productive ideas. One student in particular said he though he had it. He went through his process and it was exactly what I would have shown if I were giving step-by-step instructions. I had him share his method with his group before letting him share it with the class.

As a class, we discussed why this process duplicated the angle and related it back to the video and distance measurement done in the clip. I asked students to try using this to duplicate their angle.

By this time, almost the entire class was comfortable with duplicating the length of a line segment. They weren't as comfortable with the ideas of using the different lengths to ensure the angles were congruent.

Next class, we'll work on duplicating another angle and then I'll turn them loose on trying to construct a parallel line through a point not on the given line.

Overall, I was pleased with the outcome of this class. Students gained a better understanding of how to use a compass to measure distance, how to use arcs as markings, and how to construct congruent line segments. They also were exposed to how to put these ideas together to construct congruent angles.

Making connections in geometry

The last few classes have been focused on simple explorations of angle relationships, such as linear pairs, vertical angles, and angles formed by transversals cutting across parallel lines. I continue to reference the parallel parking scenario to motivate where and how these angles come about.

I've had to work in algebraic expressions as values to help prepare the class for questions they may see on common assessments. I have been hard pressed to come up with scenarios that would naturally generate these expressions; it's something I'll need to work on.

For one set of practice problems, students indicated they were getting stuck on some problems. Rather than working through specific situations, I asked students to step back and focus on the angle relationships they are seeing and how they relate to each other. I told them not to worry about the values they were given for different angles. I drew two intersection lines and asked them to tell me the different relationships they saw in the four angles. I then gave a couple of expressions for two angles and asked them how this fit into the situation and could be used. The students that said they were stuck said this helped. I told them to keep working on the problems and we'll discuss them next class.

My purpose is to have students focus on the pieces and how they can be put together to answer questions. By breaking problems down into the components they know and understand, I believe students can piece things back together and become better problem solvers.

Today's class wrapped up with a question tied back to the parallel parking situation. I told them we are turning 45^o in relation to the original position of the parking car. As the car backs up at this angle it forms an angle with the curb going away from the car. I asked what was the size of the angle.

It was interesting that students concluded the angle would be 135^o. I asked them how they knew this. Many stumbled around with rather unconvincing arguments. I asked them to focus on the angle relationships they were seeing. Some students started to recognize that the 45^o angle was a corresponding angle to the angle paired with our angle of interest and that these two angles formed a linear pair. Perfect. Students were using geometry but didn't realize why the angle had to be 135^o. I'm hopeful I can push them further into asking themselves why things work.

What does bad notation buy you?

As I was teaching my geometry class on Friday, I went into the need for good notation and labeling as ways to understand and begin to problem solve. Labeling items helps to identify different characteristics and enables our brains to start to begin absorbing pertinent facts. Hopefully, the brain makes connections to similar situations or relates one idea to another to begin the problem solving process.

Notation is an often overlooked weapon in the problem solving arsenal. We teach lots of notation to be memorized but too often overlook how important good notation is to solving problems and making math useful.

I have a couple of examples that I can think of where notation actually made mathematics more useful and expanded its role in the world. The first being the use of Hindu-Arabic numerals and the use of 0 in writing numbers.

The Romans had an expansive empire that lasted for centuries but did little in the advancement of mathematics. Yes, they were wonderful engineers and ruthless conquerors but they did little new in the way of mathematics. One reason that has been hypothesized is that Roman numerals hinder mathematics.

Let's try this. Don't convert these values, try to use them as a Roman would. What is the answer to the following addition problem?

XCIV + LVI = ?

It's not the easiest problem to work out, is it?

Now, what about this problem?

94 + 66 = ?

There is quite a difference in working through these problems. One yields and answer of CLX, the other answer of 160. Oh, wait, they are the same value but the connection between the problem and result are not as clear when adding with Roman numerals. 

The use and spread of Hindu-Arabic numerals led to a rapid expansion of mathematics in the western world.

Flash ahead to the 1600's and early 1700's in Europe. The calculus wars raged during this period and while most of Europe adopted Leibniz's notation, England stuck with Newton's. The weaker notation that Newton developed slowed mathematical progress in England while mathematics across Europe flourished, expanding mathematics influence across many fields.

The next time you work with notation or have students work and understand notation, think about whether the notation is flexible and useful or cumbersome.

Mathematics should help students to easily model situations and to make their lives easier not more difficult. Good notation can help in this effort.

Labeling and notation in geometry

In my last post, I concluded with Day 7. The next two days started with a focus on labeling and notation. I used an expanded model of the parallel parking situation.

Referencing what students had proposed before (using A, B, C, and D to indicate the vertices of the rectangle) I labeled the blue cars vertices. I also label two lines. I asked students to consider how they could label the graph so that they could identify and distinguish between each car (the three rectangles). Many of the students struggled with this task. As I walked around to different groups I kept asking how they could label different things on the graph to help identify which car was which.

Eventually, students started to label either the cars or the vertices of the cars. We discussed why labeling was important. Besides simple identification, the process makes you think about what you are looking at and what you consider to be important features. Labeling is an early stage of evaluation and analysis. Forcing students to think about what to label and how to label helps them to better see and understand what they are working on.

I had groups present their labeling ideas. Some good ideas and ways to label came out from these presentations, including making use of subscripts. One group used ABCD for the vertices of each car but then included subscripts to identify if it was car 1, 2, or 3. This way they could not only identify the car but the corresponding corners for each car. I liked the thinking that went into this labeling.

After we had everything labeled, I asked students to identify, using proper notation, 3 segments, rays, lines, and angles. I also asked them how they would label the plane on which this graph sat. This was their homework assignment.

The next class, I started with looking at the results. Again, many students were stuck or struggled. Students worked in their groups while I put up the five categories on the board. I asked students to write up on the board what they had come up with. Much of the notation was missing and students were simply writing letter combinations. 

Once we had several entries under each heading, I asked students to reference their graphic organizers for notation and comment on or correct what they say. Students slowly started going to the board to add arrows or missing lines above the letter pairs.

Soon we had corrected the notation and students started asking questions, such as what happens if you reverse the letters for a ray or does it matter which two points you pick on a line to use as the label. These were really good questions and discussions that showed students were resolving issues they had with notation.

At this point, the graph was getting a bit messy. I was trying to move one of the cars and getting lines or points caught in the selection. I told students we should move to working with a simpler model at this point so we could more easily focus on specific characteristics.

Again, what I noticed was the class did not have any issue with looking at a graph that had fewer lines and items on it or focusing on a figure not drawn on a graph. The transition seemed to be more natural because of where we started.

I am now transitioning to covering what may be considered more traditional geometry topics, such as the angle addition postulate, which is where I went first.

I used measuring angles formed between fingers and then had students measure the angle formed by their thumb and pinky finger. I should have emphasized to students we are creating a model of our hand by having segments represent fingers. As it was, students drew outlines of their hands and measure angles from these outlines. The sums were off quite a bit in some cases but students were still getting the idea of adding angles together to find the angle of a bigger angle.

I went through some definitions and then we worked through some examples. As we progressed, students were fine as long as the values given were numeric. As soon as values were given as expressions, they froze.

I asked them what the angle addition postulate stated, which they were able to tell me. I wrote this out and asked for each angle, what was the measurement given. I wrote these out under the generic postulate and then they started to understand. Of course they struggled a bit with solving the resulting equation but most were able to work through their issues either on their own or with their group.

I will continue working through the first introductory topics in this vein, referring back to the parallel parking problem as an anchor and reference.