Fabulous Component Machines
Thursday, October 3, 2019
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This is a lesson I taught way dorsum inwards early on January. I've pose off writing it because I desire to brand sure I do it justice. Do you lot ever receive got a lesson that you lot are just insanely proud of? This is that lesson for me. (Feel gratuitous to link to your lesson or tell us nigh it inwards the comments! Sharing is caring!)
[Warning: This is in all probability the longest postal service I've ever written. It's also got a ridiculous amount of photos together with videos. I hope you lot volition accept the fourth dimension to read through it together with offering your feedback inwards the comments, though. If you lot were coming to my degree to uncovering me teach, this is the lesson I would desire you lot to see. I know at that topographic point are tons of things I could do improve that you lot tin run across that I can't. I'm also real interested inwards hearing your ideas on how I tin capture the spirit of this activity together with utilisation it to engage students throughout the year.]
Let me set the phase thus you lot tin run across where this falls inwards my Algebra 1 curriculum. At this bespeak inwards the year, my Algebra 1 kiddos receive got reviewed the pre-algebra basics of the existent number system, integer operations, together with the enterprise of operations. We learned exponent rules together with named polynomials. We're experts (hopefully) at distributing together with undistributing (factoring) expressions. Equations receive got been written together with solved. Inequalities receive got been solved together with graphed (in i variable.) Students receive got learned the definitions of relations together with functions. We've also discussed domain together with attain together with independent together with theme variables. We tin graph points on the coordinate plane. It's at nowadays fourth dimension to delve into component subdivision notation, expand on the thought of a component subdivision machine, together with graph functions past times making an input/output table. We haven't discussed gradient or looked at the divergence betwixt linear together with non-linear functions. Those volition come upwardly inwards the side past times side unit.
When nosotros beak over how to differentiate betwixt a relation that is a component subdivision together with a relation that is non a function, I introduced the concept of a component subdivision machine. Humans similar functional machines. We similar to survive able to predict what's going to happen. When nosotros pose breadstuff inwards our toasters, nosotros await toast. If nosotros pose a toaster strudel inwards our toaster, nosotros expected a warm, toasted toaster strudel. If nosotros pose a bagel inwards our toaster, nosotros await a toasted bagel. Because nosotros know the component subdivision of a toaster (to toast things), nosotros tin predict the output based on the input.
What if this wasn't the case? What if I pose inwards a slice of bread, expecting toast, together with a toasted poptart comes out? What if I sometimes acquire toast together with sometimes acquire a poptart? Life would for sure survive exciting, but nobody inwards their right heed would purchase that toaster for anything to a greater extent than than novelty purposes. If I'm eating breakfast together with I similar poptarts, I mightiness non survive also broken-hearted if I pose inwards breadstuff together with got out a poptart. But, what if I'm wanting to brand a BLT on toasted bread? I definitely don't desire to consume my bacon, lettuce, together with tomato plant sandwiched betwixt ii strawberry poptarts amongst lots of icing. Yuck! (And, for the record, my BLT would survive made amongst these. I've never genuinely eaten actual bacon.)
After introducing the concept of a toaster equally a component subdivision machine, I ran across an amazing comment on Sarah Rubin's Everybody is a Genius blog. The comment was from Ms. Mac/Mrs. G. And, she suggested showing students that a toaster allows several inputs amongst the same output. But, it doesn't allow i input to receive got several outputs. For example, if I pose wheat breadstuff inwards my toaster, I locomote out toast. If I pose rye breadstuff inwards my toaster, I locomote out toast. If I pose pumpernickel breadstuff inwards my toaster, I soundless locomote out toast. (I've never genuinely eaten rye breadstuff or pumpernickel bread. I justice I demand to locomote out more!) But, if I pose inwards a bagel, at that topographic point is no way that I could ever acquire anything but a bagel out of the toaster. There is null possibility of getting scrambled eggs or a waffle. Every input has just i output. I honey this addition, together with I think it volition genuinely aid my students. I'm definitely adding this to my unit of measurement side past times side year!
I justice I made a pretty big bargain out of my toaster analogy before Christmas. After Christmas break, I asked the next inquiry without genuinely thinking nigh the whole toaster thing.
Me: "Guys, I brought some component subdivision machines from my occupation solid today. Don't you lot receive got component subdivision machines at your house?"
Student: "Did you lot conduct your toaster?"
Me: "No... I justice a toaster is a component subdivision machine, but I brought a dissimilar type of component subdivision machine today."
Now, holler upwardly the flip chutes I made this summer? They are made out of saltine boxes. Though, I'm pretty sure I was supposed to brand them out of empty juice cartons. They've been sitting inwards my occupation solid for months, just waiting to brand their debut inwards this lesson.
I'm edifice upwardly to the thought of component subdivision notation. Last year, my Algebra 1 students had a terrible fourth dimension wrapping their brains around component subdivision notation. The whole thought behind this lesson is to essay to avoid the frustration together with confusion that seemed to persist terminal twelvemonth amongst my students. Why is the component subdivision notation necessary? Because humans don't just similar predictable things. They similar things amongst names. Here's what I told my kids:
Me: "Humans honey to holler things. We receive got an obsession amongst naming the things around us. We holler our children. We holler our pets. We holler our cards."
Student: "We holler our guns." (That's word to me. Of course, I don't genuinely ain a gun.)
Another Student: "We holler our shoes." (Yeah, I don't do that either.)
Influenza A virus subtype H5N1 Third Student: "I give names to my favorite t-shirts." (That is yet some other thing I don't name.)
Influenza A virus subtype H5N1 Fourth Student: "I named my straightener."
So, apparently, humans similar to holler things fifty-fifty to a greater extent than than I realized! Mathematicians similar to holler things, too. After all, they are humans. Mathematicians are also fans of parsimony. If they holler something, they desire to holler it inwards the easiest together with shortest way possible. Therefore, commonly the holler of a component subdivision is notated past times a unmarried missive of the alphabet of the alphabet. The word component subdivision starts amongst f, thus a lot of times, mathematicians merely telephone telephone a component subdivision f.
I receive got to component subdivision machines, thus I named i f together with the other g. To farther emphasize that f and g are just the holler of the function, I gave them to a greater extent than traditional names, too. Francine. And George. After I gave them these names, I realized that these are both the names of characters on Arthur. I loved to sentry Arthur when I was growing up. I tin holler upwardly beingness thus scared of going to the 3rd grade. I mean, what if my instructor ended upwardly beingness equally hateful equally Mr. Ratburn? Okay, plenty amongst the reminiscing.
Once I realized that I was channeling Arthur amongst my names, I made a comment nigh it to my students.
Me: "I justice I'm channeling Arthur today."
Student 1: "Who?"
Me: "You know! Arthur, the aardvark."
Student 1: "What?"
Me: "Have you lot genuinely never seen that present on television?"
Student 2: "Oh, I know what present you're talking about! What did you lot state he was again?"
Me: "Arthur was an aardvark."
Student 2: "He was an aardvark?!? I e'er thought he was a bear!"
I didn't realize that giving my functions names was going to Pb to thus many issues. My sixth catamenia degree decided to rename George equally Georgeio. Another pupil came inwards at tiffin to give me her thought on the names. "The component subdivision machines should survive named George together with Francisco. Then, they tin survive gay, Italian lovers. Italians brand the best lovers. I know this." I didn't fifty-fifty respond to that comment. I did non desire whatsoever to a greater extent than details.
I gave the degree a component subdivision to graph inwards component subdivision notation. And, I also gave them their inputs, the integers from -5 to 5. Students had to write the output inwards component subdivision notation, the output equally a number, the ordered span inwards component subdivision notation, together with the ordered span using solely numbers. We did the starting fourth dimension ii lines of the table. Then, I gave students several minutes to work. We would banking concern check their work, together with they would maintain on. It was time-consuming, but I think it was a positive learning experience.
It was awesome to run across students start to brand connections. Some realized they could real easily create total inwards the component subdivision notation columns. Others noticed that the numerical output increased past times ii every time. I walked around the room, pointing at pupil work, congratulating them on finding patterns. We graphed the ordered pairs, together with they formed a line!
With nigh xv to xx minutes inwards class, nosotros i time once again tried graphing functions using the component subdivision machines, cards, stars, together with shower drape coordinate plane. When students looked confused, I reminded them of the component subdivision notation nosotros had just used before that solar daytime inwards our notebook. That seemed to aid some. But, I soundless had some real frustrated kiddos who could non figure out how to pick out which star to utilisation or where to house it on the coordinate plane.
I was starting to write off this whole exercise equally a failure. Maybe I am just terrible at teaching component subdivision notation. Maybe my kids are just doomed to never sympathise this. I wasn't nigh to give up, though. My students were going to sympathise component subdivision notation i way or another!
The side past times side day, I decided nosotros were going to pass our entire 50-minute catamenia graphing functions. If nosotros continually graphed, checked our work, discussed discrepancies, together with tried again, they were saltation to receive got to acquire it! I had students opened upwardly their notebooks to our pages on component subdivision machines together with component subdivision notation to utilisation equally a reference.
My students were rocking the procedure of evaluating the functions. They were checking their answers using the component subdivision machines. Life was good. Well, life was goodness until they got to the stars. I decided to just topographic point myself past times the stars together with aid students amongst the graphing process. They were going to acquire this!
After helping pupil subsequently pupil uncovering their right star, I had an epiphany. They already receive got the carte inwards their manus that has the output written inwards component subdivision notation on i side together with the output written inwards numerical notation on the other side. Why don't I receive got them utilisation this carte to graph their ordered pairs?!
My conversation amongst students went something similar this.
Student: "I can't figure out which star I need!"
Me: "Well, what was your input?"
Student: "My input was 2, together with my output was 1. But, I don't run across a star labeled (2,1)."
Me: "Do you lot holler upwardly how nosotros had ii dissimilar ways of writing the output inwards our notes?"
Student: "Yes."
Me: "Well, if your carte says f(2), you lot demand to uncovering a star that also has f(2) inwards the ordered pair."
Problem solved. Find the star that matches the card.
This was neat except for the fact that they soundless wanted to pose the (2, f(2)) star on (2, 2). Every. Single. Time.
It was fourth dimension for a 2nd epiphany.
Me: "So, your output was f(2)?"
Student: "Yes."
Me: "And, what was the numerical output that you lot got when you lot plugged 2 into the function?"
Student: "When I plugged inwards 2, I got 1."
Me: "So, f(2) is 1?"
Student: "Yes, I just said that."
Me: "So, you're proverb that f(2) is equal to 1?"
Student: "Yes. f(2) equals 1."
Me: "If f(2) equals 1, thus I tin supersede the f(2) amongst 1. And, that volition tell me where to graph the ordered pair."
At this point, I made a present of flipping the carte over several times. I would present them that (2, f(2)) was the exact same equally (2, 1). After all, they had just told me that both sides of the carte meant the same thing.
All of a sudden, my students were graphing points similar nobody's business. They didn't demand my aid to uncovering their stars. And, they for sure didn't demand my aid to graph their ordered pairs. It was thus fun to sentry them fit the cards amongst the stars together with thus lay the cards on transcend of the stars. They would flip the carte over just similar I had shown them inwards enterprise to graph their ordered pair. Success!
My students were able to successfully graph horizontal lines together with quadratic functions using the component subdivision machines equally well. We ended upwardly calling f(x) = iii the Spam equation. No affair what our input, iii (spam) e'er comes out. (If you're confused past times this, you lot should genuinely become dorsum together with sentry the Meat-A-Morphosis video. I promise; you'll survive much less confused!)
So, why do I honey this lesson thus much?
It's hands-on. It got my kids out of their seats. It was self-checking. It was a prophylactic surround to brand mistakes. When students bespeak out that they think (3, f(3)) is inwards the wrong place, nobody knows who graphed that bespeak unless the pupil announces that it was their star. Each pupil worked at their ain pace. It was discussion-provoking. Students were craving vocabulary words similar gradient together with y-intercept that nosotros had yet to learn. They were bespeak questions, making predictions. "I bet this i makes a line, too!" After, nosotros graphed a quadratic, i pupil wanted to know how he could tell from the equation if the graph would brand a directly delineate or a parabola. I also felt similar I could differentiate this activity for my students. If f(x) was a quadratic together with g(x) was linear, I could force my advanced students toward solving the f(x) cards, together with my students who needed to a greater extent than back upwardly could survive farther practicing linear functions amongst the g(x) cards.
Basically, I desire this to survive what my classroom is similar every unmarried day.
Download files here.
[Warning: This is in all probability the longest postal service I've ever written. It's also got a ridiculous amount of photos together with videos. I hope you lot volition accept the fourth dimension to read through it together with offering your feedback inwards the comments, though. If you lot were coming to my degree to uncovering me teach, this is the lesson I would desire you lot to see. I know at that topographic point are tons of things I could do improve that you lot tin run across that I can't. I'm also real interested inwards hearing your ideas on how I tin capture the spirit of this activity together with utilisation it to engage students throughout the year.]
Let me set the phase thus you lot tin run across where this falls inwards my Algebra 1 curriculum. At this bespeak inwards the year, my Algebra 1 kiddos receive got reviewed the pre-algebra basics of the existent number system, integer operations, together with the enterprise of operations. We learned exponent rules together with named polynomials. We're experts (hopefully) at distributing together with undistributing (factoring) expressions. Equations receive got been written together with solved. Inequalities receive got been solved together with graphed (in i variable.) Students receive got learned the definitions of relations together with functions. We've also discussed domain together with attain together with independent together with theme variables. We tin graph points on the coordinate plane. It's at nowadays fourth dimension to delve into component subdivision notation, expand on the thought of a component subdivision machine, together with graph functions past times making an input/output table. We haven't discussed gradient or looked at the divergence betwixt linear together with non-linear functions. Those volition come upwardly inwards the side past times side unit.
When nosotros beak over how to differentiate betwixt a relation that is a component subdivision together with a relation that is non a function, I introduced the concept of a component subdivision machine. Humans similar functional machines. We similar to survive able to predict what's going to happen. When nosotros pose breadstuff inwards our toasters, nosotros await toast. If nosotros pose a toaster strudel inwards our toaster, nosotros expected a warm, toasted toaster strudel. If nosotros pose a bagel inwards our toaster, nosotros await a toasted bagel. Because nosotros know the component subdivision of a toaster (to toast things), nosotros tin predict the output based on the input.
What if this wasn't the case? What if I pose inwards a slice of bread, expecting toast, together with a toasted poptart comes out? What if I sometimes acquire toast together with sometimes acquire a poptart? Life would for sure survive exciting, but nobody inwards their right heed would purchase that toaster for anything to a greater extent than than novelty purposes. If I'm eating breakfast together with I similar poptarts, I mightiness non survive also broken-hearted if I pose inwards breadstuff together with got out a poptart. But, what if I'm wanting to brand a BLT on toasted bread? I definitely don't desire to consume my bacon, lettuce, together with tomato plant sandwiched betwixt ii strawberry poptarts amongst lots of icing. Yuck! (And, for the record, my BLT would survive made amongst these. I've never genuinely eaten actual bacon.)
After introducing the concept of a toaster equally a component subdivision machine, I ran across an amazing comment on Sarah Rubin's Everybody is a Genius blog. The comment was from Ms. Mac/Mrs. G. And, she suggested showing students that a toaster allows several inputs amongst the same output. But, it doesn't allow i input to receive got several outputs. For example, if I pose wheat breadstuff inwards my toaster, I locomote out toast. If I pose rye breadstuff inwards my toaster, I locomote out toast. If I pose pumpernickel breadstuff inwards my toaster, I soundless locomote out toast. (I've never genuinely eaten rye breadstuff or pumpernickel bread. I justice I demand to locomote out more!) But, if I pose inwards a bagel, at that topographic point is no way that I could ever acquire anything but a bagel out of the toaster. There is null possibility of getting scrambled eggs or a waffle. Every input has just i output. I honey this addition, together with I think it volition genuinely aid my students. I'm definitely adding this to my unit of measurement side past times side year!
I justice I made a pretty big bargain out of my toaster analogy before Christmas. After Christmas break, I asked the next inquiry without genuinely thinking nigh the whole toaster thing.
Me: "Guys, I brought some component subdivision machines from my occupation solid today. Don't you lot receive got component subdivision machines at your house?"
Student: "Did you lot conduct your toaster?"
Me: "No... I justice a toaster is a component subdivision machine, but I brought a dissimilar type of component subdivision machine today."
Now, holler upwardly the flip chutes I made this summer? They are made out of saltine boxes. Though, I'm pretty sure I was supposed to brand them out of empty juice cartons. They've been sitting inwards my occupation solid for months, just waiting to brand their debut inwards this lesson.
 |
| My Flip Chute Function Machines |
I'm edifice upwardly to the thought of component subdivision notation. Last year, my Algebra 1 students had a terrible fourth dimension wrapping their brains around component subdivision notation. The whole thought behind this lesson is to essay to avoid the frustration together with confusion that seemed to persist terminal twelvemonth amongst my students. Why is the component subdivision notation necessary? Because humans don't just similar predictable things. They similar things amongst names. Here's what I told my kids:
Me: "Humans honey to holler things. We receive got an obsession amongst naming the things around us. We holler our children. We holler our pets. We holler our cards."
Student: "We holler our guns." (That's word to me. Of course, I don't genuinely ain a gun.)
Another Student: "We holler our shoes." (Yeah, I don't do that either.)
Influenza A virus subtype H5N1 Third Student: "I give names to my favorite t-shirts." (That is yet some other thing I don't name.)
Influenza A virus subtype H5N1 Fourth Student: "I named my straightener."
So, apparently, humans similar to holler things fifty-fifty to a greater extent than than I realized! Mathematicians similar to holler things, too. After all, they are humans. Mathematicians are also fans of parsimony. If they holler something, they desire to holler it inwards the easiest together with shortest way possible. Therefore, commonly the holler of a component subdivision is notated past times a unmarried missive of the alphabet of the alphabet. The word component subdivision starts amongst f, thus a lot of times, mathematicians merely telephone telephone a component subdivision f.
I receive got to component subdivision machines, thus I named i f together with the other g. To farther emphasize that f and g are just the holler of the function, I gave them to a greater extent than traditional names, too. Francine. And George. After I gave them these names, I realized that these are both the names of characters on Arthur. I loved to sentry Arthur when I was growing up. I tin holler upwardly beingness thus scared of going to the 3rd grade. I mean, what if my instructor ended upwardly beingness equally hateful equally Mr. Ratburn? Okay, plenty amongst the reminiscing.
Once I realized that I was channeling Arthur amongst my names, I made a comment nigh it to my students.
Me: "I justice I'm channeling Arthur today."
Student 1: "Who?"
Me: "You know! Arthur, the aardvark."
Student 1: "What?"
Me: "Have you lot genuinely never seen that present on television?"
Student 2: "Oh, I know what present you're talking about! What did you lot state he was again?"
Me: "Arthur was an aardvark."
Student 2: "He was an aardvark?!? I e'er thought he was a bear!"
I didn't realize that giving my functions names was going to Pb to thus many issues. My sixth catamenia degree decided to rename George equally Georgeio. Another pupil came inwards at tiffin to give me her thought on the names. "The component subdivision machines should survive named George together with Francisco. Then, they tin survive gay, Italian lovers. Italians brand the best lovers. I know this." I didn't fifty-fifty respond to that comment. I did non desire whatsoever to a greater extent than details.
And, let's non forget to advert the fact that Francine is taller than George. One of my classes assumed that George together with Francine were dating. And, Francine is taller than George. The daughter is taller than the guy. Oh, what outrage!
Next, it was fourth dimension to accept some notes over component subdivision notation. These notes were heavily influenced past times (read: stolen from) Everybody is a Genius. We labeled each component subdivision of a component subdivision inwards component subdivision notation. And, thus nosotros practiced evaluating functions using a component subdivision machine.
I'm going to survive genuinely honest. After going over the parts of a function, my Algebra 1 kids were VERY confused. But, subsequently nosotros worked through the ii examples amongst the component subdivision machines at the bottom, low-cal bulbs started going off. "Oh, that's what that means!"
Next, it was fourth dimension to accept some notes over component subdivision notation. These notes were heavily influenced past times (read: stolen from) Everybody is a Genius. We labeled each component subdivision of a component subdivision inwards component subdivision notation. And, thus nosotros practiced evaluating functions using a component subdivision machine.
I'm going to survive genuinely honest. After going over the parts of a function, my Algebra 1 kids were VERY confused. But, subsequently nosotros worked through the ii examples amongst the component subdivision machines at the bottom, low-cal bulbs started going off. "Oh, that's what that means!"
 |
| Function Machine / Function Notation Booklet Foldable - Outside |
Inside the foldable, students were given ii novel functions to evaluate for diverse inputs.
I included some problems where students added functions together with constitute the composition of functions. Normally, I would receive got skipped over these because they are non tested inwards Algebra 1, but I'm like shooting fish in a barrel starting to piece of work inwards increasing the rigor of my Algebra 1 degree inwards training for Common Core. I also included ii problems at the bottom where students had to evaluate the component subdivision for a dissimilar variable.
On the right side of the within of the page, I include iii sample EOI questions for students to solve. I desire my students to survive familiar amongst how the questions volition survive phrased on their end-of-instruction exams.
This took in all probability 2/3 of our 50-minute degree period. For the terminal 3rd of class, nosotros started a hands-on activity using my shower drape coordinate plane together with the component subdivision machines.
Before class, I prepared v pairs of functions. Each span of functions f together with g were written on a post-it note. Since I receive got a limited domain together with attain on my shower curtain, I prepared cards for each ordered span of the component subdivision that fit my restricted domain together with range.
Here's a close-up of the cards inwards i bag. The post-it banking concern complaint is a reminder for me. It tells me what functions to write on the SMART Board. One i side of the cards (cut out of card stock), students are told what to solve for. For example, if a pupil picks upwardly a g(2) card, they know that they are finding the output that results from putting 2 into g. The resulting output is written on the other side of the card.
To give students an chemical factor of choice, I set out all of the cards on the desk. Students were instructed to pick a carte upwardly off the desk together with provide to their seat. At their seats, they receive got calculators, dry out erase boards, markers, together with erasers. The students were to evaluate the right component subdivision for the given value of x.
If a pupil picked upwardly the f(3) card, they would provide to their seats together with substitute iii inwards for x inwards the component subdivision f(x) = 2x-3. They should acquire an output value of 2(3)-3 = 6-3 = iii if all their piece of work was done correctly. Once they receive got an answer, they must become dorsum to the component subdivision machine together with feed the work to the component subdivision machine. This is their input. They feed the input into Francine because they were asked to plug the iii into f.
This is where things acquire cool. Upon placing the f(3) carte inwards the component subdivision machine, the component subdivision machine volition flip the carte over thus that it comes out reading 3. To the students, it is MAGIC!
After checking their answer, students at nowadays had to graph their input together with output equally an ordered span on the coordinate plane. My giant coordinate airplane was pose inwards the flooring for this activity.
To plot the points, I created points out of some giant foam stars that my sis had given me. She had used component subdivision of them for an fine art project.
To farther emphasize component subdivision notation, I labeled the stars amongst generic ordered pairs. (1, f(1)) or (5, g(5)).
I was hoping that students would brand a connexion betwixt the fact that f(0) agency the output of f when the input is 0. I justice I didn't do the best task of explaining this on Day 1 because my students did genuinely good until nosotros got to the graphing part.
Instead of overnice linear functions, our attempts commonly ended upwardly looking a picayune something similar this.
After graphing each point, students picked upwardly a novel carte together with repeated the process. They placed the cards that had been graphed on transcend of the component subdivision machines to avoid farther confusion. Once all the ordered pairs had been constitute together with graphed, nosotros had a degree discussion.
What stars appear similar they are inwards the wrong house to you? Influenza A virus subtype H5N1 pupil would pick a star that they believed was incorrect. And, equally a class, nosotros would evaluate the component subdivision for that value together with determine the right house for it to go. Sometimes the star had been placed correctly. Most of the time, however, students made the exact same mistake. (3, f(3)) would survive placed on (3, 3). (-2, g(-2)) would survive placed at (-2, -2). Students were seeing the input of the component subdivision together with were confusing that amongst the output.
Here's an instance of some mis-placed stars.
Eventually, subsequently some difficult work, nosotros concluded that all the stars were at nowadays inwards their right places.
Students were shocked to uncovering that the points formed ii lines! Will this e'er happen? Students were convinced that sometimes they would brand lines. Other times they would brand X's. After graphing i span of functions, fourth dimension was up.
I felt similar nosotros had made some progress, but I knew nosotros had thus much to a greater extent than to go.
Enter Day 2. We started off degree past times watching the Meat-A-Morphosis: An Introduction to Functions video from youtube.
If you lot haven't seen this video before, accept 7 together with a one-half minutes together with just do it! I told my students that nosotros were going to sentry a movie, together with they were forthwith excited. I present lots of math songs together with small youtube clips, but I tell my students that I don't believe inwards movies. They think that nosotros should survive watching characteristic films before every unmarried holiday. Ummm....no. I receive got way also many exciting mathematical things to part amongst them. I don't receive got fourth dimension for us to sentry Finding Nemo.
They presently realize that nosotros are watching some other math video, thus their excitement fades. When the music starts, the complaints start upwardly again. "How onetime is this moving painting you're making us watch?!?" "I don't know. Just sentry together with see." The girls ooh together with aah over how cute the picayune chickens are. The degree collectively gasps when the (spoiler alert) chickens are chopped upwardly to brand chicken nuggets. "Why are you lot showing us this?" "Wait! You're a vegetarian. Why are you lot showing us a video nigh meat?" "Guys! I think she's trying to plough us into vegetarians!" No, I'm trying to plough you lot into mathematicians. At the destination of the video, i especially opinionated pupil said, "Some of the things I run across inwards hither scare me."
At this point, I'm soundless frustrated over my students' scrap amongst component subdivision notation from the previous day. How do I fix this? I know! We'll do an instance problem, together with I'll forcefulness them to utilisation component subdivision notation an obnoxious number of times!
 |
| Function Machine / Function Notation Booklet Foldable - Inside |
I included some problems where students added functions together with constitute the composition of functions. Normally, I would receive got skipped over these because they are non tested inwards Algebra 1, but I'm like shooting fish in a barrel starting to piece of work inwards increasing the rigor of my Algebra 1 degree inwards training for Common Core. I also included ii problems at the bottom where students had to evaluate the component subdivision for a dissimilar variable.
On the right side of the within of the page, I include iii sample EOI questions for students to solve. I desire my students to survive familiar amongst how the questions volition survive phrased on their end-of-instruction exams.
This took in all probability 2/3 of our 50-minute degree period. For the terminal 3rd of class, nosotros started a hands-on activity using my shower drape coordinate plane together with the component subdivision machines.
Before class, I prepared v pairs of functions. Each span of functions f together with g were written on a post-it note. Since I receive got a limited domain together with attain on my shower curtain, I prepared cards for each ordered span of the component subdivision that fit my restricted domain together with range.
 |
| Baggies of Function Cards |
Here's a close-up of the cards inwards i bag. The post-it banking concern complaint is a reminder for me. It tells me what functions to write on the SMART Board. One i side of the cards (cut out of card stock), students are told what to solve for. For example, if a pupil picks upwardly a g(2) card, they know that they are finding the output that results from putting 2 into g. The resulting output is written on the other side of the card.
 |
| Input/Output Cards for Function Machines |
To give students an chemical factor of choice, I set out all of the cards on the desk. Students were instructed to pick a carte upwardly off the desk together with provide to their seat. At their seats, they receive got calculators, dry out erase boards, markers, together with erasers. The students were to evaluate the right component subdivision for the given value of x.
 |
| Students select a carte to evaluate. |
If a pupil picked upwardly the f(3) card, they would provide to their seats together with substitute iii inwards for x inwards the component subdivision f(x) = 2x-3. They should acquire an output value of 2(3)-3 = 6-3 = iii if all their piece of work was done correctly. Once they receive got an answer, they must become dorsum to the component subdivision machine together with feed the work to the component subdivision machine. This is their input. They feed the input into Francine because they were asked to plug the iii into f.
 |
| Putting the carte into the component subdivision machine. |
This is where things acquire cool. Upon placing the f(3) carte inwards the component subdivision machine, the component subdivision machine volition flip the carte over thus that it comes out reading 3. To the students, it is MAGIC!
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| Function Machine Output |
I would LOVE to set upwardly a video photographic boob tube camera to capture my students' faces the starting fourth dimension time they pose a carte inwards the component subdivision machine. Their expressions are priceless! Of course, they presently realize that the reply is written on the dorsum of the card. And, all the component subdivision machine does is flip the carte over. Even subsequently discovering how it works, they soundless seem to honey it. I mightiness receive got seen i or ii kids all solar daytime forgo feeding their cards into the component subdivision machine together with just flip it over to banking concern check their reply instead.
I similar that this got my students checking their answer, together with I hope that it reinforced the thought of inputs together with outputs. If you've never seen a flip chute inwards action, I made a small video. I've never genuinely done anything similar this before thus I'm non sure it's that great, but I hope it volition give you lot a improve thought of the process. Don't worry. It's non that long of a video! (If you're reading this via a RSS reader, you lot may demand to click through to view the video file.)
After checking their answer, students at nowadays had to graph their input together with output equally an ordered span on the coordinate plane. My giant coordinate airplane was pose inwards the flooring for this activity.
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| Giant Coordinate Plane |
To plot the points, I created points out of some giant foam stars that my sis had given me. She had used component subdivision of them for an fine art project.
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| Ordered Pair Stars for Giant Coordinate Plane |
To farther emphasize component subdivision notation, I labeled the stars amongst generic ordered pairs. (1, f(1)) or (5, g(5)).
I was hoping that students would brand a connexion betwixt the fact that f(0) agency the output of f when the input is 0. I justice I didn't do the best task of explaining this on Day 1 because my students did genuinely good until nosotros got to the graphing part.
Instead of overnice linear functions, our attempts commonly ended upwardly looking a picayune something similar this.
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| Graphing Mishap |
After graphing each point, students picked upwardly a novel carte together with repeated the process. They placed the cards that had been graphed on transcend of the component subdivision machines to avoid farther confusion. Once all the ordered pairs had been constitute together with graphed, nosotros had a degree discussion.
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| Students discard cards subsequently evaluating together with graphing. |
What stars appear similar they are inwards the wrong house to you? Influenza A virus subtype H5N1 pupil would pick a star that they believed was incorrect. And, equally a class, nosotros would evaluate the component subdivision for that value together with determine the right house for it to go. Sometimes the star had been placed correctly. Most of the time, however, students made the exact same mistake. (3, f(3)) would survive placed on (3, 3). (-2, g(-2)) would survive placed at (-2, -2). Students were seeing the input of the component subdivision together with were confusing that amongst the output.
Here's an instance of some mis-placed stars.
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| Mis-placed Stars |
Eventually, subsequently some difficult work, nosotros concluded that all the stars were at nowadays inwards their right places.
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| Corrected Graph |
Students were shocked to uncovering that the points formed ii lines! Will this e'er happen? Students were convinced that sometimes they would brand lines. Other times they would brand X's. After graphing i span of functions, fourth dimension was up.
I felt similar nosotros had made some progress, but I knew nosotros had thus much to a greater extent than to go.
Enter Day 2. We started off degree past times watching the Meat-A-Morphosis: An Introduction to Functions video from youtube.
If you lot haven't seen this video before, accept 7 together with a one-half minutes together with just do it! I told my students that nosotros were going to sentry a movie, together with they were forthwith excited. I present lots of math songs together with small youtube clips, but I tell my students that I don't believe inwards movies. They think that nosotros should survive watching characteristic films before every unmarried holiday. Ummm....no. I receive got way also many exciting mathematical things to part amongst them. I don't receive got fourth dimension for us to sentry Finding Nemo.
They presently realize that nosotros are watching some other math video, thus their excitement fades. When the music starts, the complaints start upwardly again. "How onetime is this moving painting you're making us watch?!?" "I don't know. Just sentry together with see." The girls ooh together with aah over how cute the picayune chickens are. The degree collectively gasps when the (spoiler alert) chickens are chopped upwardly to brand chicken nuggets. "Why are you lot showing us this?" "Wait! You're a vegetarian. Why are you lot showing us a video nigh meat?" "Guys! I think she's trying to plough us into vegetarians!" No, I'm trying to plough you lot into mathematicians. At the destination of the video, i especially opinionated pupil said, "Some of the things I run across inwards hither scare me."
At this point, I'm soundless frustrated over my students' scrap amongst component subdivision notation from the previous day. How do I fix this? I know! We'll do an instance problem, together with I'll forcefulness them to utilisation component subdivision notation an obnoxious number of times!
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| Graphing Functions On The Coordinate Plane - INB Page |
I gave the degree a component subdivision to graph inwards component subdivision notation. And, I also gave them their inputs, the integers from -5 to 5. Students had to write the output inwards component subdivision notation, the output equally a number, the ordered span inwards component subdivision notation, together with the ordered span using solely numbers. We did the starting fourth dimension ii lines of the table. Then, I gave students several minutes to work. We would banking concern check their work, together with they would maintain on. It was time-consuming, but I think it was a positive learning experience.
It was awesome to run across students start to brand connections. Some realized they could real easily create total inwards the component subdivision notation columns. Others noticed that the numerical output increased past times ii every time. I walked around the room, pointing at pupil work, congratulating them on finding patterns. We graphed the ordered pairs, together with they formed a line!
With nigh xv to xx minutes inwards class, nosotros i time once again tried graphing functions using the component subdivision machines, cards, stars, together with shower drape coordinate plane. When students looked confused, I reminded them of the component subdivision notation nosotros had just used before that solar daytime inwards our notebook. That seemed to aid some. But, I soundless had some real frustrated kiddos who could non figure out how to pick out which star to utilisation or where to house it on the coordinate plane.
I was starting to write off this whole exercise equally a failure. Maybe I am just terrible at teaching component subdivision notation. Maybe my kids are just doomed to never sympathise this. I wasn't nigh to give up, though. My students were going to sympathise component subdivision notation i way or another!
The side past times side day, I decided nosotros were going to pass our entire 50-minute catamenia graphing functions. If nosotros continually graphed, checked our work, discussed discrepancies, together with tried again, they were saltation to receive got to acquire it! I had students opened upwardly their notebooks to our pages on component subdivision machines together with component subdivision notation to utilisation equally a reference.
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| Function Notation INB Pages |
My students were rocking the procedure of evaluating the functions. They were checking their answers using the component subdivision machines. Life was good. Well, life was goodness until they got to the stars. I decided to just topographic point myself past times the stars together with aid students amongst the graphing process. They were going to acquire this!
After helping pupil subsequently pupil uncovering their right star, I had an epiphany. They already receive got the carte inwards their manus that has the output written inwards component subdivision notation on i side together with the output written inwards numerical notation on the other side. Why don't I receive got them utilisation this carte to graph their ordered pairs?!
My conversation amongst students went something similar this.
Student: "I can't figure out which star I need!"
Me: "Well, what was your input?"
Student: "My input was 2, together with my output was 1. But, I don't run across a star labeled (2,1)."
Me: "Do you lot holler upwardly how nosotros had ii dissimilar ways of writing the output inwards our notes?"
Student: "Yes."
Me: "Well, if your carte says f(2), you lot demand to uncovering a star that also has f(2) inwards the ordered pair."
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| Finding the Correct Star |
Problem solved. Find the star that matches the card.
This was neat except for the fact that they soundless wanted to pose the (2, f(2)) star on (2, 2). Every. Single. Time.
It was fourth dimension for a 2nd epiphany.
Me: "So, your output was f(2)?"
Student: "Yes."
Me: "And, what was the numerical output that you lot got when you lot plugged 2 into the function?"
Student: "When I plugged inwards 2, I got 1."
Me: "So, f(2) is 1?"
Student: "Yes, I just said that."
Me: "So, you're proverb that f(2) is equal to 1?"
Student: "Yes. f(2) equals 1."
Me: "If f(2) equals 1, thus I tin supersede the f(2) amongst 1. And, that volition tell me where to graph the ordered pair."
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| Star With Ordered Pair In Function Notation |
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| Star With Ordered Pair inwards Numerical Notation |
All of a sudden, my students were graphing points similar nobody's business. They didn't demand my aid to uncovering their stars. And, they for sure didn't demand my aid to graph their ordered pairs. It was thus fun to sentry them fit the cards amongst the stars together with thus lay the cards on transcend of the stars. They would flip the carte over just similar I had shown them inwards enterprise to graph their ordered pair. Success!
My students were able to successfully graph horizontal lines together with quadratic functions using the component subdivision machines equally well. We ended upwardly calling f(x) = iii the Spam equation. No affair what our input, iii (spam) e'er comes out. (If you're confused past times this, you lot should genuinely become dorsum together with sentry the Meat-A-Morphosis video. I promise; you'll survive much less confused!)
So, why do I honey this lesson thus much?
It's hands-on. It got my kids out of their seats. It was self-checking. It was a prophylactic surround to brand mistakes. When students bespeak out that they think (3, f(3)) is inwards the wrong place, nobody knows who graphed that bespeak unless the pupil announces that it was their star. Each pupil worked at their ain pace. It was discussion-provoking. Students were craving vocabulary words similar gradient together with y-intercept that nosotros had yet to learn. They were bespeak questions, making predictions. "I bet this i makes a line, too!" After, nosotros graphed a quadratic, i pupil wanted to know how he could tell from the equation if the graph would brand a directly delineate or a parabola. I also felt similar I could differentiate this activity for my students. If f(x) was a quadratic together with g(x) was linear, I could force my advanced students toward solving the f(x) cards, together with my students who needed to a greater extent than back upwardly could survive farther practicing linear functions amongst the g(x) cards.
Basically, I desire this to survive what my classroom is similar every unmarried day.
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