Anyway, totally apropos, this totally cracked me up today.
Wednesday, May 22, 2013
If only all right triangles were this cute...
Hi, I'm teaching trig to my ("low" math group) 9th-graders, and loving it. We only had time for 1 day of trig lesson before we will have to concentrate on reviewing for the end-of-year test (after which we'll come back and do some outdoors trig/angles of elevation and depression type of stuff, leading into complicated trig word problems). The kids were awesome at basic trig! They learned how to apply sine, cosine, and tangent correctly and consistently in one 80-minute period. Rock stars, these kids. They use the tactile trick to figure out which side is opposite, adjacent, and hypotenuse, and then they use cross-multiplication to consistently and correctly solve for x. For now, since we're only learning the whole of right-triangle trig in one day, I am going to give them the acronym SOH-CAH-TOA on the exam and just require them to remember what the acronym stands for. Eventually, they'll have to memorize SOH-CAH-TOA, obviously.
Anyway, totally apropos, this totally cracked me up today.
Anyhow, this week is a killer. I keep trying to get ahead, but it seems impossible, as I am pulled in all directions as a teacher, a department chair, and a person soon to move across a big pond. I wonder when the next sigh of relief will come. Hopefully, there is one scheduled before July...
Anyway, totally apropos, this totally cracked me up today.
Sunday, May 12, 2013
My Shiny New School Next Year!
This year, job search had been particularly challenging for me, since I was juggling planning for my wedding and the long-distance interviews with schools that typically hire people only after having met them in person. But, in the end, I couldn't be happier to say that I will be moving in July to a great school in Seattle!
Most exciting for me is that I'll be part of a fabulous math department. They are highly collaborative, and they love hanging out with each other on a daily basis. These math teachers are also highly reflective/self-improving, and were especially commended for this during their school's recent accreditation cycle. The school as a whole has a very special culture of sustainability, which is seen through things like the staffers taking all the kids 3 times a week to scrub down the whole school, including cleaning bathrooms and compost bins. Their kids do this (surprisingly) gladly and learn to protect their school environment. They also, on Fridays, serve food in the cafeteria to recycle leftovers from the week, as part of their sustainability theme. They're in the process of building a new green building that is solar-powered, collects rain water, and has an energy counter. When I visited the school, I loved seeing the seamless integration of student art into every corner of the beautiful, historic building. Everyone I had met -- including the class of Grade 11 students that interviewed me -- had asked me tough questions, and I tried to answer them thoughtfully to the best of my ability. It seemed to work out OK, because in the end they had decided to offer me a job on the spot, at the end of the long interview day!! (They said that they don't typically do this, but they had already gone through most of their candidates and were pretty confident that I was the best fit for what they were looking for.) Considering that at that point, I had already fallen in love with the school, I am really glad this was the outcome, because otherwise I would have probably felt totally crushed over a rejection.
I would have been happy to commute for a long time to work at a school like this, but it turns out that this school is actually in a fabulous location, right downtown! I'll be able to walk to work easily, which is an amazing perk for both Geoff and me.
I am very happy with this job-search outcome, and I look forward to a fabulous year! Geoff and I plan to stick around Seattle for a while (probably until our babies grow up to an age where it's appropriate for us to take them abroad), so I'm extra glad that I've found a school that I think I would be happy to stay at, for the duration of that whole time.
PS. By the way, as part of my interview day I had done two demo lessons for them. Despite having run out of time, I really liked the trig lesson that I planned (bit.ly/ferrisTrigWS), and I think that in the future, when I start teaching PreCalc again, it could be fleshed out into a multi-day technology project for the kids. During the demo, since we didn't have time for each kid to build their own ferris wheel animation, I simply pulled this up bit.ly/ferrisTrig to show what is possible, given their understanding of circular modeling and parametric equations. In a multi-day project, we'd start with analyzing / building a ferris wheel together, and then from there on they would create their own story involving circular rotation and minimizing / maximizing distances, and model accordingly.
PPS. An example of why this school is a great fit for me is that they were actually amused and delighted that I had negotiated with them for a better demo lesson topic. For some other schools, that could have been a deal breaker, but for them, they liked that I had a preference and an opinion about the relative boringness of topics, and they also liked that I tried to choose a less-dense topic that allowed me to showcase different ways to engage students instead of requiring me to stand at the board for most of the period. My kind of people!
Most exciting for me is that I'll be part of a fabulous math department. They are highly collaborative, and they love hanging out with each other on a daily basis. These math teachers are also highly reflective/self-improving, and were especially commended for this during their school's recent accreditation cycle. The school as a whole has a very special culture of sustainability, which is seen through things like the staffers taking all the kids 3 times a week to scrub down the whole school, including cleaning bathrooms and compost bins. Their kids do this (surprisingly) gladly and learn to protect their school environment. They also, on Fridays, serve food in the cafeteria to recycle leftovers from the week, as part of their sustainability theme. They're in the process of building a new green building that is solar-powered, collects rain water, and has an energy counter. When I visited the school, I loved seeing the seamless integration of student art into every corner of the beautiful, historic building. Everyone I had met -- including the class of Grade 11 students that interviewed me -- had asked me tough questions, and I tried to answer them thoughtfully to the best of my ability. It seemed to work out OK, because in the end they had decided to offer me a job on the spot, at the end of the long interview day!! (They said that they don't typically do this, but they had already gone through most of their candidates and were pretty confident that I was the best fit for what they were looking for.) Considering that at that point, I had already fallen in love with the school, I am really glad this was the outcome, because otherwise I would have probably felt totally crushed over a rejection.
I would have been happy to commute for a long time to work at a school like this, but it turns out that this school is actually in a fabulous location, right downtown! I'll be able to walk to work easily, which is an amazing perk for both Geoff and me.
I am very happy with this job-search outcome, and I look forward to a fabulous year! Geoff and I plan to stick around Seattle for a while (probably until our babies grow up to an age where it's appropriate for us to take them abroad), so I'm extra glad that I've found a school that I think I would be happy to stay at, for the duration of that whole time.
PS. By the way, as part of my interview day I had done two demo lessons for them. Despite having run out of time, I really liked the trig lesson that I planned (bit.ly/ferrisTrigWS), and I think that in the future, when I start teaching PreCalc again, it could be fleshed out into a multi-day technology project for the kids. During the demo, since we didn't have time for each kid to build their own ferris wheel animation, I simply pulled this up bit.ly/ferrisTrig to show what is possible, given their understanding of circular modeling and parametric equations. In a multi-day project, we'd start with analyzing / building a ferris wheel together, and then from there on they would create their own story involving circular rotation and minimizing / maximizing distances, and model accordingly.
PPS. An example of why this school is a great fit for me is that they were actually amused and delighted that I had negotiated with them for a better demo lesson topic. For some other schools, that could have been a deal breaker, but for them, they liked that I had a preference and an opinion about the relative boringness of topics, and they also liked that I tried to choose a less-dense topic that allowed me to showcase different ways to engage students instead of requiring me to stand at the board for most of the period. My kind of people!
Thursday, April 25, 2013
Mythical Form
My 7th-graders have been doing some lovely exploration and estimation activities on circles. It took a few days, but I think it was well worth our while, as it helped the abstract formulas make sense to them.
My students today were boggled by the fact that if pi has different digits that go on forever, that means that either the diameter or the circumference is a quantity with also digits that go on forever. That means that we have a "measurable" (ie. finite) quantity that is, in fact, not truly measurable. Trippy, eh? For a moment there, I felt the beauty of abstract math peek its head into our Grade 7 class. The kids now think the circle is a mythical, awe-inspiring form.
My students today were boggled by the fact that if pi has different digits that go on forever, that means that either the diameter or the circumference is a quantity with also digits that go on forever. That means that we have a "measurable" (ie. finite) quantity that is, in fact, not truly measurable. Trippy, eh? For a moment there, I felt the beauty of abstract math peek its head into our Grade 7 class. The kids now think the circle is a mythical, awe-inspiring form.
Tuesday, April 23, 2013
Last Week of IB Test Prep 2013
It's full-on test-prep season, and this year I feel very satisfied with how the test prep went for my Grade 12s (who are off on their study leave this week and have requested for just one last voluntary class session with me on Friday), as well as for my Grade 11s (who are starting their mock IB exams tomorrow).
Some things that I've done throughout the year that I found helpful:
Some things that I've done throughout the year that I found helpful:
- Sequence of repetitive quiz prep/practice, building up to a fairly complex quiz. I did this with my Grade 11s throughout the year, and I found it immensely helpful in repetitively drilling into them ways to think about incorporating graphical analysis into algebraic processes flexibly. I also did this with my Grade 12s regularly throughout this year, in order to go back and fill in some of their procedural gaps from last year. The Grade 12s have said to me that these quizzes have been very helpful, and more importantly, as they began to do mixed review this spring, I didn't feel like they had really any major gaps from last year yet to be filled or reviewed.
- Review packets organized by topic for Grade 12s, spiralling back through topics from last year. This year, instead of waiting until the spring to do review for old topics, I started handing out monthly review packets in August and giving detailed written feedback as the packets were handed in to me. I felt that these packets were very useful for me to have a written dialogue with each kid to get them thinking just a little bit further on each studied topic, and the threat of contacting their parents when they laxed on the completion meant that the kids were responding and at least doing some amount of review during the year instead of waiting until April to think about those old concepts.
- Weekly lunch time review sessions for Grade 12s starting in January, where they just did full-length old exams. Each week, I would pass out either a new calculator exam or a new non-calculator exam paper (I alternated which type to give them), and we would go over the previous week's exam paper problem-by-problem. The effect of this was that the motivated kids had a chance to try mixed problems on a regular basis, well before we finished learning all the topics in the IB syllabus. So, they got used to looking at full-length papers and feeling that sense of anxiety/uncertainty in their stomach during February, instead of during April. This was immensely helpful in building the confidence of those motivated kids over time.
- During the final weeks of concentrated old-exam practice during class, I asked the Grade 12s to identify orally at the start of each class the most common mistakes they tend to make within each topic. (ie. in circle sector problems, not using the correct radian mode; or in solving equations, forgetting that you can solve a complicated equation by simply graphing for intersection) This list helped to provide them with some mental focus even as they sat down for a mixed-problem practice session.
- Skimming over/discussing the last semester's mock exam problem-by-problem with my Grade 11s, right before the end of our last class before their new mock exams. Although we had gone over these problems immediately after January, they were more focused now that the stakes were up again. Taking a fresh look at old problems after a few months helped them to focus on thinking about access points into each old problem that they had struggled with, in order to encourage them 1. to go back and revisit the last semester's mock exam and topics during their review 2. to think strategically and flexibly about how to approach each problem type during the test 3. to see how far along they have come in building confidence within those old topics.
Friday, April 19, 2013
Quadratic Function Project Brainstorm
I'm brainstorming / laying out my end-of-year plans for my 8th-graders. After their end-of-year exam in late May, we will close grades, but we will still have about 3 or so weeks of instruction, which is enough time to do something very rich and not have to coordinate with other classes (since we use the May test to do placement for Grade 9). Last year, I used this extra time to let the 8th-graders define their own math projects, which were plenty of fun, but I wasn't entirely happy with the rigor of their mathematical results. This year, I'm toying with the idea of doing an exploratory quadratic functions unit. (Technically, quadratic FUNCTIONS are a Grade 9 topic for us, but previewing it in Grade 8 is always beneficial.)
I'm thinking of making it largely exploratory, since by then pacing won't be much of an issue and I can let them really take the time to develop their conceptual understanding of quadratic functions, which is the essential access point to a lot of higher-level algebra analysis down the road.... The timing is tight (as it was last year with my other end-of-year projects), but I think it's still doable and has a lot of potential!!!
Let me know what you think. Is it an OK approach for intro to quadratic functions / basic function transformations?? This is based on my rumination about a different way to think about flexible factorization of quadratic functions.
Day 1: Developing the understanding of how to graph y = x2 + bx.
Plan - In pairs, kids will be given y = x2 + 2x, y = x2 + 5x, y = x2- 3x, y = x2 - 7x. to graph on the calculator. They will sketch results in their notes, recording all intercepts in the form (x, y), and writing a one-sentence hypothesis about what the graph of y = x2 + bx will look like.
Then, they will be reminded that in a function, we can solve for the x-intercept(s) by setting the height of the point, y, equal to zero. They will algebraically show that their hypothesis works for all b values.
Day 2: Developing the understanding of how to graph y = ax2 + bx, which is a more general version of the quadratic function.
Plan - In pairs, kids will be given y = x2 + 6x, y = 2x2 + 6x, y = 3x2 + 6x, y = 12x2 + 6x. They will again sketch graphs, noting x-intercepts, and then make a hypothesis about the effect of the leading coefficient on the graph. They will show how to solve for the x-intercepts using algebra only.
Day 3: Developing the understanding of how the graph is affected by the sign of its leading coefficient.
Plan - In pairs, kids will be given y = -x2 + 6x, y = -2x2 + 6x, y = -3x2 - 6x, y = -12x2 - 6x. They will again sketch graphs, noting x-intercepts, and then make a hypothesis about the effect of the leading coefficient on the shape of the graph. They will show how to solve for the x-intercepts using algebra.
As part of Day 3, they will do some matching between equations and pictures of graphs and to justify their choices orally.
By the end of Day 3, they should also be able to explain in writing how to graph y = ax2 + bx.
Day 4: Developing the understanding of the effect of the constant term c.
Plan - In pairs, the kids will put in a function like y = x2, and look at its table in the calculator. They will be asked to generate a second function that would increase all y-values by 1. They will prove their new equation works, by showing the values of both functions side by side in the calculator (y1 and y2), and copying down the table values. Then, they will write down the formula for the new function.
They will then look at the graphs of the two functions to determine "what happened" visually to the original graph when the equation got changed that way.
They will keep playing around with this idea, translating upwards and downwards and checking both the table and the graph to observe/verify the effect of c.
By the end of Day 4, they will be given a graph of two functions. One of the functions will have an accompanying formula, and they will be able to see visually what happened to the points on the graph. They will then need to "guess" at the equation of the other, vertically shifted function, and verify it in the calculator.
Day 5: Putting the algebra pieces altogether
Plan - One partner of the pair will have a set of sequenced instructions, to be given to their partner one step at a time. The first step will sound like, "Sketch a graph of y = x2 - 9x, labeling x-intercepts with values." Then, after that has been successfully completed, the next instruction will be given: "Now sketch the result of shifting that graph vertically up 4 units, labeling the resulting images of those original points you knew." After that has been done, the partner gives the third instruction: "Now, write the formula for this new graph." Once the partner is finished, they verify their results using the graphing calculator's graphing and table features and write a brief explanation of how they checked their results. Then, they switch, and the new partner has instructions that has to do with a downwards facing function like y = -x2 - 6x, and repeat a similar sequence of instructions to generate a new graph, a new / related equation, and to verify all results against the calculator.
Both partners will then work together to complete problems starting with functions of the form
y = ax2 + bx and translating those graphs vertically to get new graphs.
By the end of Day 5, they should be able to explain the connection between y = ax2 + bx and
y = ax2 + bx + c, and explain how to use this connection to graph any standard-form quadratic function quickly in under 1 minute.
Day 6: Practicing/drilling the connection between quadratic function equation and graphs
Plan - In pairs, they will start with a function y = x2 - 9x + 1, highlight the first two terms, sketch that function using dashed lines, and then sketch in the "real" final function using solid line. They will repeat this a few times with different functions, until they can fluidly graph any y = ax2 + bx + c function. On this day, they'll also learn to visualize the axis of symmetry and to write its equation by inspection of graph.
Day 7: Going backwards from a graph to an equation
Plan - In pairs, they will be given one quadratic graph with two "nice", symmetric integer points being emphasized on the graph, one of the points being on the y-axis. They will be asked to sketch using dashed lines what this function would look like if you shifted those two points down to the x-axis, and be asked to write the function equation of both graphs. They will practice this a few times.
At the end of Day 7, they will be given a quadratic graph whose two "nice", symmetric integer points are both not on the y-axis. This tests them to see if they can figure out that the translated graph would have an equation that looks like y = (x - m)(x - n) + p instead of y = x(x - n) + p
Day 8: Playing around with the idea of adjusting "a".
Plan - In pairs, they will import Dan Meyer's basketball photo into GeoGebra. We will discuss as a class the need to find a modeling equation in order to fully predict whether the ball will make it into the hoop. From there, they will choose two nice integer points, write the equation, and graph. If they notice that the curve goes through those two points but doesn't have the correct steepness desired in order to fit the photo, then they will create a slider value in GeoGebra and toggle the value of "a" until they get a good "fit" around the graph, and record their results.
As a class, we will then go over the idea of solving for "a" using an unused point (x, y) and link it to solving for the y-intercept in linear functions. They will solve for "a" this way to compare analytical results against the technology results.
Day 9: Modeling Individually
Plan - Following a discussion of examples of parabolic applications, each pair will find and import their own photos of "real-life" parabolic shapes from the web. They will then model the function in Geogebra both using technology and using algebraic analysis.
Day 10: Creating posters
Plan - Each pair will create two posters, one with the modeled functions overlaying the photos, and one poster explaining the general process of graphing y = ax2 + bx + c and the general process of fitting an equation to a parabolic graph.
Day 11: Practice presentations
Day 12: Math fair for other classes / parents?!
I'm thinking of making it largely exploratory, since by then pacing won't be much of an issue and I can let them really take the time to develop their conceptual understanding of quadratic functions, which is the essential access point to a lot of higher-level algebra analysis down the road.... The timing is tight (as it was last year with my other end-of-year projects), but I think it's still doable and has a lot of potential!!!
Let me know what you think. Is it an OK approach for intro to quadratic functions / basic function transformations?? This is based on my rumination about a different way to think about flexible factorization of quadratic functions.
Day 1: Developing the understanding of how to graph y = x2 + bx.
Plan - In pairs, kids will be given y = x2 + 2x, y = x2 + 5x, y = x2- 3x, y = x2 - 7x. to graph on the calculator. They will sketch results in their notes, recording all intercepts in the form (x, y), and writing a one-sentence hypothesis about what the graph of y = x2 + bx will look like.
Then, they will be reminded that in a function, we can solve for the x-intercept(s) by setting the height of the point, y, equal to zero. They will algebraically show that their hypothesis works for all b values.
Day 2: Developing the understanding of how to graph y = ax2 + bx, which is a more general version of the quadratic function.
Plan - In pairs, kids will be given y = x2 + 6x, y = 2x2 + 6x, y = 3x2 + 6x, y = 12x2 + 6x. They will again sketch graphs, noting x-intercepts, and then make a hypothesis about the effect of the leading coefficient on the graph. They will show how to solve for the x-intercepts using algebra only.
Day 3: Developing the understanding of how the graph is affected by the sign of its leading coefficient.
Plan - In pairs, kids will be given y = -x2 + 6x, y = -2x2 + 6x, y = -3x2 - 6x, y = -12x2 - 6x. They will again sketch graphs, noting x-intercepts, and then make a hypothesis about the effect of the leading coefficient on the shape of the graph. They will show how to solve for the x-intercepts using algebra.
As part of Day 3, they will do some matching between equations and pictures of graphs and to justify their choices orally.
By the end of Day 3, they should also be able to explain in writing how to graph y = ax2 + bx.
Day 4: Developing the understanding of the effect of the constant term c.
Plan - In pairs, the kids will put in a function like y = x2, and look at its table in the calculator. They will be asked to generate a second function that would increase all y-values by 1. They will prove their new equation works, by showing the values of both functions side by side in the calculator (y1 and y2), and copying down the table values. Then, they will write down the formula for the new function.
They will then look at the graphs of the two functions to determine "what happened" visually to the original graph when the equation got changed that way.
They will keep playing around with this idea, translating upwards and downwards and checking both the table and the graph to observe/verify the effect of c.
By the end of Day 4, they will be given a graph of two functions. One of the functions will have an accompanying formula, and they will be able to see visually what happened to the points on the graph. They will then need to "guess" at the equation of the other, vertically shifted function, and verify it in the calculator.
Day 5: Putting the algebra pieces altogether
Plan - One partner of the pair will have a set of sequenced instructions, to be given to their partner one step at a time. The first step will sound like, "Sketch a graph of y = x2 - 9x, labeling x-intercepts with values." Then, after that has been successfully completed, the next instruction will be given: "Now sketch the result of shifting that graph vertically up 4 units, labeling the resulting images of those original points you knew." After that has been done, the partner gives the third instruction: "Now, write the formula for this new graph." Once the partner is finished, they verify their results using the graphing calculator's graphing and table features and write a brief explanation of how they checked their results. Then, they switch, and the new partner has instructions that has to do with a downwards facing function like y = -x2 - 6x, and repeat a similar sequence of instructions to generate a new graph, a new / related equation, and to verify all results against the calculator.
Both partners will then work together to complete problems starting with functions of the form
y = ax2 + bx and translating those graphs vertically to get new graphs.
By the end of Day 5, they should be able to explain the connection between y = ax2 + bx and
y = ax2 + bx + c, and explain how to use this connection to graph any standard-form quadratic function quickly in under 1 minute.
Day 6: Practicing/drilling the connection between quadratic function equation and graphs
Plan - In pairs, they will start with a function y = x2 - 9x + 1, highlight the first two terms, sketch that function using dashed lines, and then sketch in the "real" final function using solid line. They will repeat this a few times with different functions, until they can fluidly graph any y = ax2 + bx + c function. On this day, they'll also learn to visualize the axis of symmetry and to write its equation by inspection of graph.
Day 7: Going backwards from a graph to an equation
Plan - In pairs, they will be given one quadratic graph with two "nice", symmetric integer points being emphasized on the graph, one of the points being on the y-axis. They will be asked to sketch using dashed lines what this function would look like if you shifted those two points down to the x-axis, and be asked to write the function equation of both graphs. They will practice this a few times.
At the end of Day 7, they will be given a quadratic graph whose two "nice", symmetric integer points are both not on the y-axis. This tests them to see if they can figure out that the translated graph would have an equation that looks like y = (x - m)(x - n) + p instead of y = x(x - n) + p
Day 8: Playing around with the idea of adjusting "a".
Plan - In pairs, they will import Dan Meyer's basketball photo into GeoGebra. We will discuss as a class the need to find a modeling equation in order to fully predict whether the ball will make it into the hoop. From there, they will choose two nice integer points, write the equation, and graph. If they notice that the curve goes through those two points but doesn't have the correct steepness desired in order to fit the photo, then they will create a slider value in GeoGebra and toggle the value of "a" until they get a good "fit" around the graph, and record their results.
As a class, we will then go over the idea of solving for "a" using an unused point (x, y) and link it to solving for the y-intercept in linear functions. They will solve for "a" this way to compare analytical results against the technology results.
Day 9: Modeling Individually
Plan - Following a discussion of examples of parabolic applications, each pair will find and import their own photos of "real-life" parabolic shapes from the web. They will then model the function in Geogebra both using technology and using algebraic analysis.
Day 10: Creating posters
Plan - Each pair will create two posters, one with the modeled functions overlaying the photos, and one poster explaining the general process of graphing y = ax2 + bx + c and the general process of fitting an equation to a parabolic graph.
Day 11: Practice presentations
Day 12: Math fair for other classes / parents?!
Wednesday, April 17, 2013
Totally Silly but Works
I made up a totally silly call-and-response thing this year for practicing exponent rules (after we did the initial exploration, obviously, so that they could understand why the rules work). It's mad cheesy, but the kids totally remember the rules now!! The hardest part is keeping the clapping going, but I'm not sure if it's because of my students being totally off-rhythm in general or what (they're super suburban kids).
So we clap, step from side to side, and I say, "8B, are you ready?" and they chant, "Yeah, oh yeah!"
And then I call on a random kid, "Nora, are you ready?" and she chants, "Yeah, oh yeah!"
and then I call out one of the following: "Power times power", "Power to a power", or "Power over power" while holding up fingers in each hand (up to 5, obviously, in each hand) to represent the original exponents we're working with.
Depending on which one I call out, the kids need to reply with, "You gotta add them up!" "You gotta mul-ti-ply!" or "You gotta can-cel out!" in a sing-song voice, and that kid I named would then have to say the answer (resulting exponent) immediately after. (For example, if I am holding up 3 fingers and 5 fingers, and it's "power times power, you gotta add them up!" then the kid would shout out "8!")
And then we'd resume with me calling on the next kid randomly. It's mad cheesy, but it works! Afterwards, they were all loose and happy when practicing exponent rules. Every practice problem I would put on the board, I'd ask them which rule can be applied first or next, and they'd say it back in that sing-song voice, "you gotta mul-ti-ply!"
Go kids for being good sports!! It helps to make a boring topic a little less tedious! Next year, I'll necessarily add dance moves to help our kinesthetic learners. (I already have them. I came up with them after we did the exercise.)
Yup... I've got little shame left. :)
So we clap, step from side to side, and I say, "8B, are you ready?" and they chant, "Yeah, oh yeah!"
And then I call on a random kid, "Nora, are you ready?" and she chants, "Yeah, oh yeah!"
and then I call out one of the following: "Power times power", "Power to a power", or "Power over power" while holding up fingers in each hand (up to 5, obviously, in each hand) to represent the original exponents we're working with.
Depending on which one I call out, the kids need to reply with, "You gotta add them up!" "You gotta mul-ti-ply!" or "You gotta can-cel out!" in a sing-song voice, and that kid I named would then have to say the answer (resulting exponent) immediately after. (For example, if I am holding up 3 fingers and 5 fingers, and it's "power times power, you gotta add them up!" then the kid would shout out "8!")
And then we'd resume with me calling on the next kid randomly. It's mad cheesy, but it works! Afterwards, they were all loose and happy when practicing exponent rules. Every practice problem I would put on the board, I'd ask them which rule can be applied first or next, and they'd say it back in that sing-song voice, "you gotta mul-ti-ply!"
Go kids for being good sports!! It helps to make a boring topic a little less tedious! Next year, I'll necessarily add dance moves to help our kinesthetic learners. (I already have them. I came up with them after we did the exercise.)
Yup... I've got little shame left. :)
Thinking About Factorization Flexibly
I was randomly thinking about this on the way home today and truly fascinated by the teaching possibilities:
Sketching a graph of f(x) = x2 + 6x + 7
is the same as sketching f(x) = x(x + 6) + 7
which is the same as sketching g(x) = x(x + 6) and then shifting g up 7 units.
Since the points (0, 0) and (-6, 0) are on the graph of g, the points (0, 7), (-6, 7) must be on the graph of f. This allows us to quickly see the symmetry line at x = -3 without memorizing x=-b/(2a).
Another way of "partially" factoring f to get the middle term 6x is f(x) = (x + 1)(x + 5) + 2
So, if h(x) = (x + 1)(x + 5), then we can imagine the points (-1, 0) and (-5, 0) from h being translated up 2 units to get the points (-1, 2), (-5, 2) on f.
Another way of "partially" factoring f to get the middle term 6x is f(x) = (x + 2)(x + 4) - 1
So, if j(x) = (x + 2)(x + 4), by thinking about the relationship between j and f, we can deduce that f must have the points (-2, -1), (-4, -1).
Similarly, we can get the partial factorization f(x) = (x + 3)(x + 3) - 2.
If we assume m(x) = (x + 3)(x + 3) and consider the relationship between m and f, we can deduce that (-3, -2) must exist on the graph of f.
So, we can pull together all those points so far to get (0, 7), (-6, 7), (-1, 2), (-5, 2), (-2, -1), (-4, -1), and (-3, -2) as points that must be on f. This way of thinking about graphing quadratics ties together strongly the ideas of factorization and transformation. They're no longer two separate concepts but integrated as one. Since I've never seen this connection in a textbook before, I decided to call it flexible factorization.
One distinct advantage of flexible factorization is that as soon as you are given y = x2 + kx + m, you can quickly factor it partially into y = x(x + k) + m, which allows you to quickly determine two points on the graph, (0, m) and (-k, m) and to find the axis of symmetry at x=-k/2. You can sketch the graph roughly in about 30 seconds for any standard quadratic function (this extends to y = ax2 + bx + c, as it factors into y = x(ax + b) + c, which means that (0, c) and (-b/a, c) are two points on this graph and the parabola opens in the direction as indicated by the leading coefficient "a".)
Of course, this does not mean that the kids won't have to learn the standard analysis techniques, but I think being able to connect factorization with transformation gives them another tool when modeling and thinking about graphs.
I'm going to keep playing around with this idea, possibly turning it into an end-of-year project in Grade 8. Stay tuned!
Sketching a graph of f(x) = x2 + 6x + 7
is the same as sketching f(x) = x(x + 6) + 7
which is the same as sketching g(x) = x(x + 6) and then shifting g up 7 units.
Since the points (0, 0) and (-6, 0) are on the graph of g, the points (0, 7), (-6, 7) must be on the graph of f. This allows us to quickly see the symmetry line at x = -3 without memorizing x=-b/(2a).
Another way of "partially" factoring f to get the middle term 6x is f(x) = (x + 1)(x + 5) + 2
So, if h(x) = (x + 1)(x + 5), then we can imagine the points (-1, 0) and (-5, 0) from h being translated up 2 units to get the points (-1, 2), (-5, 2) on f.
Another way of "partially" factoring f to get the middle term 6x is f(x) = (x + 2)(x + 4) - 1
So, if j(x) = (x + 2)(x + 4), by thinking about the relationship between j and f, we can deduce that f must have the points (-2, -1), (-4, -1).
Similarly, we can get the partial factorization f(x) = (x + 3)(x + 3) - 2.
If we assume m(x) = (x + 3)(x + 3) and consider the relationship between m and f, we can deduce that (-3, -2) must exist on the graph of f.
So, we can pull together all those points so far to get (0, 7), (-6, 7), (-1, 2), (-5, 2), (-2, -1), (-4, -1), and (-3, -2) as points that must be on f. This way of thinking about graphing quadratics ties together strongly the ideas of factorization and transformation. They're no longer two separate concepts but integrated as one. Since I've never seen this connection in a textbook before, I decided to call it flexible factorization.
One distinct advantage of flexible factorization is that as soon as you are given y = x2 + kx + m, you can quickly factor it partially into y = x(x + k) + m, which allows you to quickly determine two points on the graph, (0, m) and (-k, m) and to find the axis of symmetry at x=-k/2. You can sketch the graph roughly in about 30 seconds for any standard quadratic function (this extends to y = ax2 + bx + c, as it factors into y = x(ax + b) + c, which means that (0, c) and (-b/a, c) are two points on this graph and the parabola opens in the direction as indicated by the leading coefficient "a".)
Of course, this does not mean that the kids won't have to learn the standard analysis techniques, but I think being able to connect factorization with transformation gives them another tool when modeling and thinking about graphs.
I'm going to keep playing around with this idea, possibly turning it into an end-of-year project in Grade 8. Stay tuned!
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