Choose a lesson you have taught on your own. Evaluate it.
1. Identify objective and give a brief overview
2. Was the objective met? Justify your conclusion with specific evidence.
3. What worked well? Explain.
4. What improvement would you incorporate next time. Explain.
I taught my pre-calculus classes how to find the zeros of polynomials with real coefficients. I started my class by showing the answers to the homework on an overhead projector and then we immediately went to a Warm up exercise where I make sure that the students knew how to find the complex conjugate of nonreal complex numbers. When I handed out the paperwork, I heard some of the students say that they liked to get handouts from the teacher. They did not sound sarcastic so I think that they actually liked it. Looking back at the lesson, I think that the objective was met because by the end of the class the students were able to find the rest of the zeros of a polynomial with real coefficients when they are given one nonreal zero, find the equation of the polynomial given only the zeros and also describe the x-axis with respect to the number of zeros and their multiplicity.
ReplyDeleteI chunked the lesson into pieces where I explained the theory – but I did forget to tell them to write it down specifically though I did see that mostly the students were writing down the theory. After going over a problem on the board where we needed to solve for the rest of the zeros knowing one of them, I then made the students complete a problem on their own and made sure they were on track by walking around. I then showed them how to calculate the polynomial given the zeros and made them do a problem while walking around. The next chuck was the discussion of how we know how many zeros we have when looking at a polynomial and where the x axis intercepts the figure. I again had a problem ready for the students to do on their own and I even had a bit of a discovery problem for them using the calculator and seeing what happens when you change the constant on a polynomial equation.
I found that the handouts work really well as well as letting the students do a problem for themselves after I show them how it is done. I think I could have improved the lesson plan by leaving more room for notes and even telling them what to write down. I should have left a space for the major theorem and even written the name of the theorem on the page and just had them write in the definition. For some of the students who need more step-by-step instructions, I could have also had an overhead of the procedure for solving these types of problems or I could have worded by explanation and overview such that I was saying the procedure with numbers. I did find that I needed to really keep a close eye on the time so that I finished everything on time. For my first class, I did not really have enough time for the activity where we used our graphing calculators to se what happens when the constant on a polynomial changes. In the second class, I was able to cut down some of the Warm Up section on proofs and instead used that time to finish the lesson on time and with a detailed list of what we did today and what the students should have taken away from this activity. I am still finding it hard to come up with real world examples of where you would use factoring polynomials with real coefficients. Any ideas would be greatly appreciated.
Hey Anne. Great start.
ReplyDeleteRemember, you can fudge the application and explain to kids that you are giving a simplified version of a real life situation. I often create nice looking sales and cost functions and subtract for a profit function. Each function uses something like number of employees as an independent variable (domain).
Google polynomial modeling for some ideas:
http://www.jamestown.k12.nd.us/school/highschool/carpenter/ia67.pdf
http://www.glencoe.com/sec/math/algebra/algebra2/algebra2_08/other_calculator_keystrokes/pdfs/346_347_Alg2_6-5_873831_CFX.pdf
The objective of the lesson this week was to determine if 2 quantities or ratios were proportional. The students have been learning ratios, rates & unit rates and are now moving into proportions.
ReplyDeleteIn daily math, the students are shown 2 groups of 5 fractions and are asked to determine which does not belong and explain why they think it doesn't belong, as well as what is the next in the pattern and why. This exercise gets them thinking in terms of equivalent fractions, that all equivalent fractions can be simplified to the same number, and that they are all proportional.
The students met the daily learning objective. I gave them a pop quiz (to turn in) of 4 questions to determine if the ratios given were proportional. All students got 100. They were also able to demonstrate their understanding of proportions on the board and explain their methodology to the rest of the class.
The students worked together in pairs or threes to solve the problems. I explained before they started that all of them needed to do the problems and be ready to show them on the board & explain, as I was going to select the students from the name cards I had them do the 1st day I taught class. The students really enjoyed this and a number of them were disappointed not to be chosen to do problems on the board. As I mentioned before, having them explain their process to the class not only helped them & their peers to understand, but also helped me to understand better their thinking process because it was a long time ago that I was 12. Our closure activity of using equivalent fraction tiles or dominoes was also a big hit with the students, they enjoyed the manipulatives and also designed some really neat patterns with the triangle shaped pieces. This further reinforced the objective of finding proportional quantities.
Next time I do this type of activity, I would incorporate the timer. I gave 3 problems at a time and waited for most to finish. The ones who didn't finish were work-avoiders, even knowing they could be picked to go to the board. One did in fact get chosen and had to have another student help him at the board. A timer (maybe on the promethean board) would offer at least a visual reminder that they don't have the entire class period to do these problems. The more I teach with different methods and strategies, I can really see the benefit of the timer in class. But I also think it needs to be a timer the students can see, not just a little digital timer that beeps when time is up for that particular activity.
Obj-prove triangles similar using AA, SSS and SAS shortcuts.
ReplyDeleteTransitioned from similar polygons, asking students to recall those requirements.
PK-asked about and discussed congruency shortcuts as lead in.
This was the second time I taught this lesson. First time I demoed and explained AA, gave practice worksheet, circulated, helped individually and reviewed harder problems w/ whole class. Did same routine for SSS and SAS together.
Second time I incorporated bingo. Distributed blank bingo sheets and had students randomly number the boxes 1-24, leaving space for the answers. I renumbered the practice worksheets 1-24. Then I would call out a number, the students would solve the problem and put the answer in the respective box. They quickly got to work, trying to solve all of the worksheet problems even before I called the question #. The first class (w/o the game) still did the problems, but the bingo class seemed a lot more motivated to complete it quickly. We still reviewed each problem as a whole class during the game. This way they got feedback on their answers and I could assess learning by questioning the students.
I sort of winged it w/ the bingo, so next time I would give more instruction on the game.
we closed by discussing as a group a compare/contrast of similarity vs congruency shortcuts.
this was a block class, and it seemed to go by very quickly.
I am hoping to get more mileage out of this, but it may not fit all objectives as well as this one. Also, I suspect the novelty may fade quickly.
Objective was to be able to divide by a fraction. The key skill was to recognize to change it into a multiplication problem using the reciprocal of the divisor. This lesson was just after a couple of lessons on multiplying fractions and mixed numbers which was a skill they were now well-versed.
ReplyDeleteMy Do Now was a handout of individual work starting with a practice of some multiplication with mixed numbers (a warm up). Then there was a “What doesn’t belong and why?” problem (my subtle homage to Don Imus) with 4/2, “4 div 2”, “4 x ½” and finally the culprit “4 div ½”. The students had enough prior knowledge to know the first three evaluated to 2 so most indeed could “guess” it was the last one that didn’t belong. But none knew why other than it couldn’t be the same as “4 div 2” (not bad though). The Do Now had one last question which asked them to write different equivalent expressions for “4 / ⅓” with a hint to use the previous question as a template. Most produced “4 div ⅓” but no one produced the “4 x 3”. I didn’t really think they would, but it was worth the try.
My lesson grew directly from the Do Now. We discussed together with questions like what “4 2” means (how many 2s fit in 4). We used pies on the Smartboard to see there were two “two pies” in the “four pies”. We then talked about “4 div 1” then went to “4 div 1/2” to see how many ½ pies fit in 4 pies. Now they could understand more fully why “4 div 1/2” didn’t belong.
We then discussed together the notion that 4 div 2 = 4 x ½ and talked about 2 and ½ and the relationship they shared (“flipped” and multiplied by each other would be 1). They wrote down the definition of reciprocal and came to understand it as “flipping”. A whole number was already improper fraction and its reciprocal could be “the flip”.
ReplyDeleteWe ended the “discussion” part of the lesson with the “4 div ⅓” from the Do Now, going back to the pies and seeing how many 1/3 pies fit in 4 pies (12). They then volunteered (I was hoping they would), that 4 div ⅓ was indeed equal to 4 x 3. Fractions could flip just like whole numbers could flip. We could now write down “dividing by a number was the same as multiplying by its reciprocal”.
The rest of the lesson was practice for the students identifying reciprocols of whole numbers and proper fractions, followed by practice of modifying division by a number into multiplication by reciprocal. Finally practice dividing by fractions were being solved.
I could see the lesson objective was being met through the practice I was observing. The students definitely could employ the skill and they even expressed relief that division by a fraction was just a multiplication problem they already were comfortable with (much more comfortable with than adding/subtracting fractions). I think the build up of the concept and jointly discovering the “trick” worked well with the students. In that sense “chunking” of sorts was effective.
I would however modify the plan for much more early practice for the students themselves. Practice with specifying reciprocals earlier in the lesson would help break up the “discussion” more. While I was very conscious of leading the discussion via questions and keeping the discussion interactive, all students probably needed to be engaged earlier with practice of some parts of the chunking. I could do a better job of having practice be included in each chunk.
ReplyDeleteFinally, to really round out the full picture, I would like to find better ways of helping students recognize in word problems when the problem represents a division problem vs. a multiplication problem. Later assignments and even assessments revealed some real confusion by many students in recognizing one problem fro the other in word problems. To be honest, I think in some sense I produced students that were good marksmen and could use the gun well, but they weren’t doing very well of recognizing when they needed to use it. They weren’t (and in some respects still aren’t) seeing the forest for the trees. The idea of a problem involving finding out how many of one size fits into another size (like the pies) was a division problem was not very effective. Many are still having problems recognizing multiplication situations from division situations. Are there any good tips?
The lesson I chose to evaluate is the sixth grade “Exponents” lesson.
ReplyDeleteThe objectives were: use exponents to express numbers, and write expressions containing exponents in standard form. The title of my lesson was the “Super power number”.
The initiation was me as the teacher, attempting to hook the students with the idea of discussing super heroes and their special powers, and how the power of the super power number is to multiply whole numbers.
After that I moved to defining and identifying the whole number as the Base, and the super power number as the Exponent. I gave few examples and first handout to assess the kids. Next we went over the three ways/forms of writing exponents (Expanded, Exponential and Standard). I gave few examples and the second handout to assess the kids. The closure was a review of the lesson with information about next lesson, “We will talk on the super power friends of the super power number: Zero power, First power, Squared, Cubed, Base 10, and Y to the power of X.”
My objectives were met. The handouts helped me to assess who seemed to understand the concept. Whoever did not seem to understand, the handouts showed me exactly what I needed to review. The topic of the lesson helped me to create a comfortable environment for the kids to set the stage for them to be highly engaged in the lesson. I believe the students had as much fun as I did relating the numbers to superheroes.
Creating a safe environment for the kids helped me to get them engaged and to have great responses to my questions. Those responses gave me a live assessment on how the lesson was going and it made it clearer to me when to move on.
For the next time I teach “Exponents” I came up during this lesson with a better idea (I think). I would try to use paper cutting technique of multiple identical images, I believe it is called paper garland, to illustrate the “Power” of the exponent and how it works (expanding and grouping the paper cutting). I also learned that I need to show a base and exponent before I give their definitions. Some numbers can have different bases to express the standard form (16, 64, and 81). The kids taught me the “flip flop” numbers (two to the fourth power and four to the second power). Next time it would be better to handle it earlier in the lesson, and make sure that the “flip flop” numbers are not always equal. There are many other little things which I would do differently next time, but I wanted to keep it brief by sharing this example. After all we are blogging, not writing a whole reflection which could be difficult to read due to length.
Next week is the algebra midterm, hence I only taught algebra for three days followed by three days of Pre Calc. My cooperating teacher did the review of the algebra material I had not taught. It's funny, but hearing her summarize all of the material covered over several chapters was useful to me. I guess I don't learn things until they jump out at me. But (finally) I think I believe in working backwards from the bigger objective into daily lesson plans. I had been looking at the current chapter and working forward. It turns out my school has a master plan. Not everything in the book is taught, and certainly everything isn't taught the way the book presents it.
ReplyDeleteFor the next two weeks of lessons I've asked for the department "standardized test". They agree ahead on the exact questions. I think the real value is now understanding more clearly what the goal is. I also was given an explanation on why the material is presented in the order it is. For example, next week is a small part of one chapter and then off to something in a totally different section
We gave an algebra test on the material I presented. It was quite interesting evaluating the tests. I'm particularly pleased to see one student (likely through no consequence on my part) do much better than he had all semester. I plan to say that I was quite impressed with his continued efforts given his current (failing) status. It seems that he's doing something rather difficult and should be recognized for such.
And since you all have seen me get tangled up in the micro teaches, you could probably have predicted the next item. I gave a Trig word problem to the Pre-Calc class that chewed up 30 minutes. (Everyone ended up getting a solution, there was plenty of debate and time for one on one help. No one lost interest.) This actually wasn't intentional. The cooperating teacher mentioned that these students would never see such complexity this year (something about solving for four unknowns in a single problem....). In my mind the real issue is how did they do on the homework. Nothing was going to be as tough as what we did together. If they come in Monday having done teh problems and saying they got it, great. If not it may be time for plan B.
I also put out a challenge for the class to create a very difficult problem. One student responded with a very clever and challenging Trig problem. For his efforts he was awarded the 'prize'. The question went back out to the whole class as an optional extra credit exercise called "Math Off" (along with a couple of problems I found). Completely optional. This is not an honors class, but these kids are unusually motivated. They scored very high on this week's pop quiz, so I hope they go for the challenge.
Continuing in random order, the Pre Calc lesson was all problem solving. We stressed [1] finding the real question, [2] draw a great diagram .... then start working. There is nothing like a great diagram!
Opps ... I should read the instructions first.
ReplyDeleteI'll go back to Pre Calc. The lesson was applying basic Trig functions to solving word problems that involved elevation/depression and what I call Navigational terms (heading, bearing, course, compass directions). The students knew the Trig ratios. The challenge was setting up a proper problem. The math is almost trivial to them. I started the class with pictures an diagrams without any math. We all did a number of compass heading conversions. Alternate interior angles was covered (but I should have stressed it more as a useful tool)
I believe the first two days went well as evidenced by the quiz results which were quite good.
There was one person in class who missed the concept of angle of depression (per the quiz results). In retrospect I could have made an effort to get everyone in this rather small class to participate. This person did not. However, not really knowing who is generally strong and who isn't was a real disadvantage. I've worked with the Algebra kids more and I'm starting to know which ones I need to spend extra time.
I also let the students do the initial problems with much less guidance than I would next time. Doing a few very, very simple problems quickly would have left much more time for everyone to start their homework and provided more time to walk aroud to check progress.
Yesterday I taught a lesson in my 7th grade pre-algebra class and the objective was to connect the area of the net to the surface area of a rectangular prism. A net is a two-dimensional pattern that can be folded to form a three-dimensional figure. Therefore the idea was to build on the students’ prior knowledge of area and perimeter of two-dimensional figures and use that to discover strategies how to find the surface area of a box (rectangular prism). Overall, I think the lesson went well and the objective was met. Students were asked to design nets on the inch grid paper, cut them out and form the prisms. They were presented with a box with specific dimensions and it was up to them what particular net design they would come up with. It was actually a great activity, they realized that some designs would not work and they were able to fix it. Some students needed some assistance with getting started and understanding the dimensions of each face. What worked well I think was that after they had a design they had to calculate the area of each face and add it together. Then they folded their nets and they were asked if the area of the individual faces had changed? What about the total area or surface area? What if we turn the rectangular prism sideways does the surface area changes? How does the total area of the net compare to the surface area of the rectangular prism? Why does it make sense that this two measures are the same? … and so on. These were some of the discussions that we had. If I plan this activity next time I have to be more aware of time limits and clean up after. Some of the better students were done very quickly with their designs and on the spot I had asked them to come up with another more creative design of the net and I was very impressed what creative ideas they had- imagination at work! If only I had thought of picking up the camera from my desk….
ReplyDeleteI taught nets to 7th grade. My objective was to have them create a 2-d net for a 3-d object and identify a 3-d object from a 2-d net. I used a cereal box earlier in the week to discuss surface area so I continued with a net of the same cereal box. I had previously cut out nets of different shapes - cubes, cones, triangular prisms, cylinders etc. I handed these out to the students and had them try to guess what they were and then assemble them. Then switch with another student. The cylinder and cone were the big surprises for them. I next had them work individually paper to sketch a net of a door stop. The custodian had rounded up several for me and made a few more for me the night before the lesson. The students had the tangible wedge to use to create the net. Once I verified (by walking around and checking each student's net)that they got the net for the wedge correct, I gave them a worksheet that moved from easier to more difficult activities. Some included identifying a net of triangular pyramid, drawing a net for cylider, identifying net for cube. Next they had 6 possible nets for cubes and had to tell which ones would actually make a cube - 3 of 6 would make a cube. I had cut-outs of these 6 so they could do it hands-on if they needed it. I regrouped the class to make a net for a rectangular prism that was 6x5x3 and we did it as a group activity.
ReplyDeleteI tried this lesson 2 times before I got it right. The first time through, I found I needed more hands on activity. The second time I found I needed to move more slowly and provide more guided steps. I took my lunch break to back up and began more simply and it worked much better for my last class.
The student's results showed they understood the 2-d to the 3-d. They had a harder time moving from 3-d to 2-d but most were catching on by the end of the period. They will need more time on this. One boy explained his strategy of picturing rotating the box across the paper. One of my more challenging girls was inspired to go home and make a net for one of the 6 cubes and see if it really didn't make a cube.
For this topic, the hands-on clearly made it easier for the students to see the transition from 2-d to 3-d and back. They really enjoyed it and could move at their own pace.
This lesson took a lot of prep and something like this can't be done all the time but it was fun.
The objective was the student will be able to solve a two-step equation.
ReplyDeleteYes, the objective was met. I had the students do classwork which they then entered into their senteos, providing them and also me with instant results on how well they understood the concept.
I explained the concept using th Friends and Family with the infectious x in the hospital room. I though the students really responded to this strategy. I also showed them SADMEP (PEMDAS backward) - the students decided they like the Friends and Family concept better. I used this concept in further lessons - multi-step and variables on both sides.
I think the Friends and Family concept can be taken further. When I went onto variables with both sides, I said the variables had chicken pox and all had to be put in the same room. I have been using a do now daily to test their knowledge and I am always walking around the room checking their work and making corrections if needed. I can see how the pop quiz would work in situations like this. However, there is really only so much time in the day. Two days a week, my classes are only 35 minutes long, so the time constraints are tight.
Last week I had to to teach grade 9 about systems of equations and since I was aware that the least I spoke to them the better my class was my Do Now had the students expressing an algebraic situation about the total ages of a man and his son as well as the difference of the ages to a mathematical expression. I had resistance but nonetheless some wrote the equations and so I weas able to launch into the lesson albeit very slowly. What I've taken from that class is that I have to ensure that the wording of my expressions are similar to those used in textbooks. I've learnt that it so essential to have a test run with someone of how I phrase my questions to avoid ambiguity and the pop quiz is a terrific thing.
ReplyDeleteI was able to have them use their graphing calculator but like a bad luck the smartboard didn't work and so the smartview of the calculator was not available. Anyway in the furture I'll have in a printout the steps with picture to input the formula in the calculator. The only hands on activity I could find was the use of the graphing calculator to graph the equations. I have the same lesson to do again so I'm open to any suggestions of any additional activity
Oh I must admit that I had 1 shining moment in my class on linear equations.. I've been trying to explain that steps ought to be shown ALL the time in the homework and 1 particular student just last week methodically did so. I'm encouraging them by rewarding methodical work with additional points.
ReplyDeleteTwas beautiful.
Sharon - Good news! Small steps in the ight direction.
ReplyDeleteI had an interesting week. My ½ calculus class was traded away and now I have a statistics class. So I have five classes, which makes for a very full day. Like Nadeen, I seem be to obsessed with trying to make things perfect and sometimes try to cram too much info into a single lesson. So I have backed down a bit form the amount of info I will try to cover. I guess you learn from your experiences. Most of the classes have been going well but I have had a couple of clunkers. When I think about the clunkers afterwards it seems there is a common thread. Preparation is a vital key.
ReplyDeleteI had a functions lesson in which I was anticipating that students might have difficulty fully grasping. As a result I spent a lot of time trying to scaffold and chunk the information for them. My geometry classes had an upcoming quiz so I thought their day was going to be review work, and I guessed that their lesson did not need much prep, just respond to their questions. Well those were difficult periods. As a result of not being fully prepared for geometry I stuttered and stammered and they were not happy and I was becoming frustrated as well. When looking back at the most productive lessons and comparing them to the less productive lessons it is obvious that that my preparation was the common thread.
Lesson learned. Do your preparation and homework before entering the classroom, you will feel more confident and the students will sense it. The more preparation I gave to lesson planning, the better it flowed.
April 8, 2011 3:29 PM
The lesson I chose was an introduction to solving absolute value equations. My Do Now reflected their previous work with absolute values. I continued the lesson by giving them absolute values with variables and had them try it on their own. From there, the class generated steps to solving absolute value equations. They will use these steps to solve for future equations.
ReplyDeleteMy objective for the lesson was that TSW solve absolute value equalities. To assess the class, I evaluated their work when they were doing independent practice. I was able to work one on one with several students to meet my objective. Based on the class work I was able to determine that the objective was met, however not mastered.
I liked how the students came up with the steps themselves. It really made them think about the process and what to do with it. I felt that I made them do some higher level thinking by assigning them this task. I also gave a lot of wait time during this process and didn’t call on the first student that raised their hand. That way I was able to give the other students time to think of the following step.
I would like to do a better assessment of this during the lesson. I found it hard to do assessment during an introductory lesson because I was so focused on introducing the concept to them. I think next time I should maybe do a POP QUIZ after a class example to see if they can do it on their own. That way I don’t send them home with work they don’t know how to do.
The lesson that I choose was scatter plots.The student will plot a scatter graph using a graphing calculator. We had previously used plotted basic scatter plots on graphs on paper. The lesson took longer than expected and needed an additional period to complete as I did too much review before getting into the lesson. I also had cumbersome instructions that were hard to follow. The night after the first lesson, I revised the instructions to make it easier for the students to follow including clearing out the lists and data. Because the students are in 8th grade, they do not have their own TI calculators so they were not able to bring them home and practice. It also was challenging as many students pushed the incorrect keys and got errors. I needed to give more specific instructions about what the students needed to do when I was guiding them through the examples. After reviewing the lesson on the second day, the students had a chance to do problems in class independently. Most students were able to calculate the data using a scatter plot and interpret it correctly.
ReplyDeleteRelevancy worked well for me. I also used a question to discuss if all data is true which I tied into data not always being correct and that even though you have 2 variables that looked consistent; there may be more variables that may affect the outcome.
I used more relevant data than was included in the textbook that the students were more interested in. I used the UConn Huskies stats after they had won the championship( like we did in class and with height and weight of the Celtics). I used Kemba Walker as an example of when data does not always match. Someone could assume that a taller person would be a better player but Kemba Walker, MVP, was only 72 inches tall vs. many of his taller teammates. I used another example as a platform to talk about when data may or not be true. We cannot always trust everything that we read, especially on the internet. I used this question as part of my do now and then had pairs discuss their answers together and then we had a dialog with the class. "Mr. Jones gave a math test to all the students in his high school. He made the startling discovery that the taller students did better than the short ones. His conclusion was that 'as your height increases, so does your math ability'.
I also used the TI calculator poster that was helpful to show the students where the keys were. I might also try to use the smart view for the TI instead of using the overhead. For the future, I wish that they would be able to take the calculators home and practice the problems at home.
The next time I do a lesson with the calculators, I will make sure that the instructions that I provide are very simple and easy to follow. I also want to make sure that the students are clear on that the graph of the line of y=mx+b is the same as when they do the linear regression for the scatter plot that shows y=ax+b because that caused some confusion as well.