When I was in college I studied Electrical Engineering. I got an MS. I spent 5 years in the field before getting laid off. Then I switched to teaching for reasons I usually make up during interviews to sound good.

I only introduce my past like that because when I was studying electrical engineering I was told a story that I would only later come to realize how it not only defined engineering – but also defined math education. And if I was really being bold – education.

It came from Professor Pedrotti at UC Santa Cruz, who was telling us about a time in his life when he was asked to fix a toaster. His grandmothers toaster had stopped working, and Prof Pedrotti was home for the holidays while getting his Ph.d. So his grandmother asked him – “Ken, maybe you could fix my toaster? Have you taken a class on toasters?”

There is no class on toasters.

When learning electrical engineering, you learn basic fundamentals about how electricity works, and how different circuit components behave. You learn about when assumptions about behavior are valid, and when they are not valid. You learn how things work individually, and how they should work when you put them together. And you are expected to apply these fundamentals to a wide variety of applications. For example – here is how such and such behaves. Here is how it is used to blah blah. Now use such and such to design a circuit that’s going to muck muck. Or something like that. You get the point.

That’s conceptual understanding. And that’s mathematics. And that’s education. What mathematical reasoning allows this process to work? What mathematics does this new application remind me of? And if there is some component missing – what might that component need to do?

The story’s conclusion fits in nicely with the idea of a thinking classroom. But on a more micro level probably – in our profession it’s more like “there is no lesson on toasters”. We can’t teach you how to do every specific thing. We can teach you to apply fundamentals so you have the ability to do whatever it is you have been asked to do.

“I must have been absent when we covered this?” Nope. You’re here now.

Here’s a post regarding the My Favorite I did at TMC16 on how I used the Mathalicious lesson “Licensed To Ill” to have a conversation that mattered. And since Karim inspired me to go this way with his blog post On Purpose – I’ll defer to him for the reminder about why it’s important:

@cmmteach@TheMillerMath As a species, we’ve had a rough last few weeks, huh? We need to be able to have these conversations more than ever.

The video of the presentation is here: (Thanks Glen!)

I’m going to highlight what changes I made from the suggested lesson delivery. Go to the Mathalicious website for the specifics of the lesson. I took the pulse of the room and a lot of students had opinions about health insurance and Obama care but did not understand what it was or know any of the details. Mathalicious provides a great video introduction to health care that allowed me to make sure every student had a basic idea of it before going into the scenarios outlined in the lesson.

The most effective thing I added to Mathalicious lesson was the physical simulation. There was a story telling benefit, a pedagogical benefit, and a mathematical benefit:

Allowed me to increase the drama around denying Daniel.

Anytime you put something in the students hands it is a win.

We got to use all the simulation data to better discuss it’s relationship to the expected value.

I think there could be a cool teacher move where the students come up with how to simulate these probabilities, but I choose to have that all setup from the start. I made sure the room was setup a bit different than normal, and each table already had a decks of cards, a coin, and dice.

Alonzo’s probability on the card draw isn’t 1%, so I talked to the students about whether or not we thought it was sufficiently improbable for our purposes. My student teacher said we could have them draw from a 50 card deck, and then if they draw correctly then they flip a coin. I thought that was cool – but the idea came after we had already started so I didn’t go with it.

I did have the students keep a record of all their simulations, which allowed us at the end to compare our expected values with the results of all our simulations. This is particularly helpful in the “set your price scenario” where they expect a profit of $0, but the only possible results with a single Daniel is +/-$20,000.

One edit I did which was subtle and only done for storytelling purposes was to make the students the insurance company. You can make the story more interesting to students by injecting them into it. Now instead of asking the students to calculate the expected profit of the insurance company, I would instead ask “calculate your expected profit”.

Although I did most the work at VNPS – I did do the reflection portion on the videos with the exact same handout Mathalicious gives.

Here’s the final grid. I started with just two columns and had the students add a column with every new situation. That way they didn’t know how many we were going to do and it kept the direction of the lesson more surprising.

Out of all the scenarios, the only one I actually added was the generalization at the end. I also changed the instruction on the second scenario – preferring to have them “set their own price” to try and fix the problem from the first scenario. Mathalicious is more explicit with their instruction.

I am asking students to figure out the profit for the best and worst case scenarios. I’m undecided if this actually provides them insight into the situation and I’m mostly doing it because I want to ask them to calculate the probability of the best and worst case scenario which will allow me to see if they are adding or multiplying probabilities.

In the moment of giving the lesson I would periodically put the mandate in effect. For instance, when you can deny Daniel I told the kids to have Alonzo buy. Sometimes the mandate was still in effect, other times it was more a better safe than sorry thing. Once the mandate is first introduced, the students wanted to know if it was in effect for each of the subsequent scenarios.

The questions I keep repeating over and over again are “what are the advantages and disadvantages?”, “who wins, who losses?”, “Is this fair to _______ “, “Do you agree with… “.

The Conversation That Matters:

I frame each scenario to the students pretty much like I did in the video. I had to skip a few elements of each scenario in my TMC presentation in order to make the 10 minute window.

The mathematics has to better inform the students of how heath care works, and how changing specific parameters effects the situation as a whole. I am not trying to tell students that the Mandate is fair or unfair for Alonso, or that it’s fair or unfair to deny Daniel coverage just because he is likely to have a heart attack. I am trying to give them a sense of how insurance works, and to look at how changing certain parameters effects other parameters – sometimes in positive ways, other times in negative ways. Then I want that understanding to then form the basis for an informed opinion of “fairness” and ultimately whether or not something is a good idea.

Here’s some more of the slides from my presentation.

And here’s the comic that a girl drew up on her whiteboard after we went through the scenario of denying Daniel health care for his pre-existing condition:

Keeping to my simple TMC15 definition of “Real World” as being math that someone needed to do at their job – I love this one. I believe these types of math problems deserve to be a spice in our curriculum. Especially in a geometry class! I used this problem at the end of Math 2 because it does a nice job of connecting angle measure and slope. I used it on an exam, so I will also show the two scaffolds I used.

Basically my wife is building a giant barn / man cave and they need to construct a retaining wall. Here is how my wife explained the problem, and how I gave it to students. I gave them the below diagram as well. Too see how she delivered the question to me take a look at the text message at the bottom of this post.

Here’s how my wife wrote it:

A General Contractor is building a retaining wall in Sonoma County. The General Contractor is required to have the portion of the retaining wall which is over 3’ tall designed by a licensed engineer. The base of the wall follows a driveway at a 6% slope. The top of the wall is level. How much of the wall needs to be engineered?

Here’s how I gave it to my students on their exam last year:

A General Contractor is building a retaining wall in Sonoma County. The General Contractor is required to have the portion of the retaining wall which is over 3’ tall designed by a licensed engineer. The base of the wall follows a driveway at a 6% slope. The top of the wall is level. The short side of the wall is 0.75 feet tall, and the tall side is 6.5 feet tall. How much of the wall needs to be designed by the licensed engineer? (Hint: When is the wall 3 feet tall?) (diagram not to scale)

I ended up loving this problem because it connected slope to angle measure, algebra to geometry. I got to tell them that in Math 1 we always talked about slope, but the angle measure was always there.

I used this problem as an exam question. In order to scaffold it I did the following two activities as openers. Both of them combined did a nice job of familiarizing the students with the language of the question. I also had gone over what a general contractor is before the exam.

Opener Scaffold #1

What would this road sign look like if we used slope or angle measures for gradients instead of grade? Draw all three. I didn’t run straight to this question – I developed it a bit first by asking questions like “what do you think this sign means?”.

Here’s an example of student work:

The point of this first activity was for students to become familiar with grades – the fact they are just the rise divided by the run in percent. Also I wanted to make the connection between slope and angle measurements, algebra and geometry.

For the question on the final I want them to have the ability to convert percent grade to an angle measure.

Opener Scaffold #2

Pick two of these streets and draw a diagram of them using the angle measure instead of the percent grade. If you finish early build an accurate physical model of the street using string and tape.

This opener went well – I didn’t take any pictures.

Results:

Here are some of the results of their work on the exam. I was surprised how many different ways the students went about solving it.

The Real Problem:

Here’s the actual text from my wife. Notice the dimensions she gave (0.8 and 6.56 ft) were different than the ones I used (0.75 and 6.5 ft). I changed the measurements because I didn’t feel I had done enough work with measurements for students to be able to deal with the actual dimensions. I still the question qualifies as real-world. Real enough world.

A Note On Construction Convention:

In construction – they would say “one over seven” to describe a slope of 1/7, and it would likely be written 1:7. A slope of 4/9 would be called “four over nine”. I talked about slope using that language during these activities.

My colleague Dave Casey designed the initial lesson.

I enjoy this modeling activity. It’s based on developing a relationship between the width and height of a car jack – which just happens to be a perfect shape to be modeled by a rhombus. I bring a car jack, but also like to show these videos because the jack opens a little slow for the students to get a great feel of the movement.

Or check it out in double time

Check out the cool graph a scissor jack makes (height as a function of width)

Here is one of the guiding questions I use – it’s also the big thinking moment that I make sure to stay very intentional about:

Is it more difficult to raise the car in the beginning or the end?

I phrase it that way as to make students have to define what it means to be difficult. Some may think the end is more difficult because it takes longer to lift the car, and others think the beginning is harder because it takes more force (since you are lifting the car more). Either way, they need to resolve what it means for it to be difficult. Here is another way I phrase the question:

Let’s say 16 of us each signed up to move the jack screw 1″. Would you rather be the 1st person, the last person, or does it not matter?

Here is an activity handout you can use. I didn’t use it – but it was nice to have for students who were absent that day.

I start by talking about the shape. I have the students draw the shape and we make conjectures about the key characteristics of the shape. The students have graph paper, so I recommend a uniform figure – I tell them to make it eight gridlines long, and +/- two gridlines up and down.

A critical point here is let the students know that the car jack is actually not a rhombus because the sides don’t actually touch. So we are modeling the real car jack as a rhombus even though that is not actually the shape. The rhombus is a mathematical model of an actual car jack. Even the word “modeling” itself implies that we are not working with the real thing – rather a model of the real thing.

To improve the visual I usually draw a representation on the board of the jack close to starting position and ending position. I add the length of the sides as 8″, and then I ask the these two questions:

What is the largest width we need to consider?

What is the shortest width we need to consider?

I then put them at whiteboards and tell them the whiteboard number they are at is the width of the car jack they need to calculate the height for. I also tell them to pick one other width. I do not tell them how to do it – just what to do. I’ve done this with classes that knew the pythagorean theorem and used it, and I’ve done it with classes that didn’t and we used this a vehicle to reteach the pythagorean theorem.

Common Mistake Alert: A few students will anticipate a linear relationship and just quickly make a table that goes 16-0, 15-1, 14-2 etc.

Once they have the table, I have them graph it. I show the Desmos graph, talk about how to write the function.

Once the graph is up, and the table is complete, I give them a couple minutes to consider when is it the most difficult to lift the car. I survey the entire room. Once each student is on record with an opinion – moving them into a talking point on the following prompt:

“It doesn’t matter. It will be the exact same difficulty to turn the crank at all times because the car’s weight doesn’t change.”

Once the talking circle is over I am generally satisfied that they have all had sufficient opportunity to think and discuss. I go around the room again and tally the results of the talking point and have a group discussion, calling on various students to give their reasoning.

I also like to bring up the slope of the lines, and ask them “What does slope represent in the context of this problem?”. Between the first and second person is the largest slope – what does that represent in this context?” (By the end most of us are thinking of the scenario where 16 of each are each going to turn the screw one inch)

My goal by the end is to get the students to see that the car moves more in the beginning than in the end, and understand how that validates arguments for the beginning or end being more difficult based on how the word “difficult” has been defined.

Here is a students work – but a lot of the great student thinking that happened here today was not caught on paper. I didn’t take any pictures of the whiteboards.

Regrets: I never had them write their arguments for when they thought it was the most difficult to use the Car Jack. At least not formally. I need to do a better job with the written communication piece of modeling.

Cheers!

-B

And even my textbook thought this was a good problem:

Here’s how I implemented Robert’s great Ms. Pac-Man lesson. Going into it I wanted to do the lesson as Robert outlined, but also make game of it, as well as have students do some work at whiteboards. All in all – I feel pretty happy with how it went:

Here’s a Pac-Man game board to use with Roberts Ms. Pac-Man lesson. It’s nice because it doesn’t use all the ink of the black background in the actual Ms. Pac Man. I asked my TA to draw a Pac-Man board and he responded “I’m on it”. I told him a bunch of other math teachers would use it – so don’t make me a liar!

I recommend cutting out a bunch of 1/2″ Ms. Pac Man’s to use with it. Here’s my desk of supplies!

The small Pac-Man’s were an excellent manipulative for the students – every student was using the one I gave them. There’s a big difference between imagining how Ms. Pac Man would be oriented when she rotates, and actually rotating her and seeing how she is oriented.

The larger Ms. Pac Man’s are because I am going to have the students go to whiteboards and discuss that first turn where Ms. Pac-Man both reflects and rotates. But I’m not showing my hand there yet – meaning I’m not going to bring up that first difficult corner. I’m going to wait until they find it.

To learn about the lesson just go to Robert’s site. But here is the extension that my colleague Alex Wilson offered up:

Each student gets 20 transformations to get the highest possible score. Each dot is worth 1 and the big dots are worth 10. Game on.

I thought about going right to playing the game – but I would have had to tell them the rules of Ms. Pac Man’s motion. By first starting with Roberts lesson they were able to discover her motion themselves.

Here’s my first question to them after showing the video:

“What does she do first?”

After collecting some ideas from the class – they seemed to all agree that this was not just about translations. I showed them the video where Ms. Pac Man only moves with translations and discussed some of those properties.

We collectively talked about her first two moves, and then I left them to go. We had not directly talked about rotations – so this activity was their introduction to describing rotations by degree and CW / CCW. I just let them get started and fielded questions as they arose. Then after a couple minutes I talked to them about how to describe the first rotation.

Definitely the part where she first flips and rotates is a big thinking moment and I was not going to miss it. One tell that students were engaged is that they did ask me about it when they first came to it . Once the question had been asked multiple times, I grouped them quickly and had them work out that first difficult corner on a whiteboard:

A couple groups hadn’t made it to that corner yet – so I first asked them something to the effect of “Why do you think that I picked this corner to bring you all up to whiteboards for collaboration?”. They manipulated the Pac Man and then told exactly why. So all the students can develop the question regardless if they are actually at that corner on their board.

Once they were up together I also asked them if we could make any generalizations about Ms. Pac Man’s movement. Does she always go up and down the same way?

The next day I brought Ms. Pac Man back to Whiteboards and this was the opener – in groups of 2 or 3 they had to make Ms. Pac-Man move around the obstacle and collect all the dots. Engagement was very high – helped largely by the fact that they all felt they could be successful.

This opener allowed me to go around an clean up some of their notation. There were definitely lingering misunderstandings about what line Ms. Pac Man should reflect over, and they were forgetting to say what point she rotated around.

I decided to assign the game as homework. 20 transformations, extra credit to the top three scores. I’m torn because part of thinks the game should be done in class rather than homework. I probably should have done the game right after the whiteboard opener. In my head I was thinking that I needed to move onto a different topic. Stupid pacing guide and it’s subconscious control over me! Here are some examples of the quality of their homework:

(I never got around to making those copies – so just trust me it was good shit)

The activity served to solidify their use of proper notation for translations and reflections, and also served as a great introduction to rotations. I think it would work perfectly fine as an introduction to – or review of – transformations. All in all – Thanks Robert!

p.s. It’s also super easy to turn this into a test question – just make a small course for Ms. Pac Man and it’s all good.

lastly – here’s a handy makeup assignment for absent students:

I was going through some old files on my computer and found this letter that I had received during my 3rd year of teaching:

It reminded me of the importance of the poem above. That poem was recited by by John Wooden – legendary UCLA basketball coach (and also high school english teacher) during a Ted Talk he gave. If you have 17 minutes somewhere in your life to spare – I would spare it by listening to this talk:

I know not all my students were born into great upbringings. One of my philosophies in teaching is that I believe students probably have adults in their lives who are emotionally unpredictable – happy one day, sad or stressed the next. So I will not be that for them – They will only ever see me in a good mood. I will not carry any negative energy into the classroom. And it’s kind of funny because on the other side it ends up being a self-fulfilling prophecy.

The year prior to receiving the letter above I had a 2nd period class that was particularly disruptive. And I had a 3rd period class was well behaved to a point that I felt like I probably didn’t deserve them. The problem was that the events from my 2nd period class often left me upset and stressed out, and I would carry that into my 3rd period. Near the end of the year I realized that too often 3rd period didn’t get the teacher they deserved, because he was always altered by the earlier interactions. That was never going to happen again. It was a learning moment for me. Don’t carry baggage class to class – Let it go and start each anew.

A retiring colleague of mine gave the graduation speech last year. He is someone who I looked up to a lot and always appreciated my interactions with him. One point he made during his speech that struck me was when he spoke of how thankful he was to have spent his life in such a “spiritually challenging profession”. He talked about how it made him a better person, and then a better teacher. I understood myself to know what he meant because much of my own classroom clarity came from a similar realization. There is a beautiful quote from a book I love by Robert Pirzig called Zen And The Art Of Motorcycle Maintenance which describes this spiritual challenge perfectly, albeit not from the perspective of a teacher:

“The application of this knife, the division of the world into parts and the building of this structure, is something everybody does. All the time we are aware of millions of things around us – these changing shapes, these burning hills, the sound of the engine, the feel of the throttle, each rock and weed and fence post and piece of debris beside the road – aware of these things but not really conscious of them unless there is something unusual or unless they reflect something we are predisposed to see. We could not possibly be conscious of these things and remember all of them because our mind would be so full of useless details we would be unable to think. From all this awareness we must select, and what we select and call consciousness is never the same as the awareness because the process of selection mutates it. We take a handful of sand from the endless landscape of awareness around us and call that handful of sand the world.”

It is so hard to know if you are really seeing your classroom for what it is – rather than what you want it to be or what bugs you about it. Or as simply a collection of small events that deviated from your predictions of how you thought it would go, or heard it would go, or read it would go. We share stories of our classrooms and those stories are a collection of things that stood out. Little pieces of awareness that we brought into consciousness. But who knows what we missed or how our consciousness mutated the actual events in the room. They are by definition incomplete stories and maybe at the end of the day the best teachers are the ones who can reconstruct the clearest and most accurate picture of their class as a whole.

How do we thrive in a spiritually challenging profession? These bullet points are my two cents:

Along the way I’ve realized that every time I felt out of control, or stressed, it could be alleviated by improved lessons, or new structures. No doubt about that. But somewhere below all of those great teaching strategies that we know work – there are still personal insecurities that need to be understood. Teaching will make you confront those. That’s the spiritual part. I thank my profession and my students for exposing me to my own. It made me a better person… who then became a better teacher.

This letter really is for me – I probably won’t even spell check it. I’m certainly not going to tweet it. I’m writing it because I feel I have to. Even if only discretely – I want to go on the record somewhere as a challenge to myself.

My main goal for second semester is to increase the amount of thinking students do.

I want to decrease the amount of time they spend reproducing and practicing any particular procedure.

I want more of their procedural fluency to come from interesting questions, engaging tasks, or a conceptual understanding.

I need to be more purposeful with my questions. When assigning tasks I want to have two good questions predetermined for each.

I want to keep track of each activity – specifically whether or not they had these three pillars of a great activity present in them:

open middle

low entry, high ceiling

solution method not immediately clear but solution seems attainable with effort.

Homework to be more review oriented. 1 review, 1 conceptual, 1 new. Something like that. Write it out a week in advance.

Continue to rely heavily on vertical non-permenaent surfaces and visible random groupings.

Give students time alone for quite contemplation before having them begin group tasks.

Focus on MP.4 and MP.3.

MP.4 Modeling: What does it mean to model with mathematics? Unanswerable Questions type openers.

MP.3 Viable Arguments. – How often do you get students to compare and contrast approaches? I want to maximize the effectiveness of this both at whiteboards and at their desks.

Meaningful discourse versus call and response, or low level transmitting of process. An example of the level of communication I want to avoid: I ask students to share with their partner I get this sort of dialogue:

Student 1: “I did this, and then this”

Student 2: “ok, I did this and this”

Student 1: “ok”

Me: “Did every one get a chance to share with their partners?”

Student 1 and 2: “yes”

Keep having fun and knowing that it’s impossible to reach every student.

Keep learning more and more and more about your students as people.

Read the standards you are being asked to teach.

Can we represent this differently?

I want to say CONVINCE ME a lot.

Focus on process versus answer getting.

Be more honest with myself when I am giving problems that have only one solution method.

Use Michael’s student teaching as an opportunity to study the effectiveness of lessons without having to facilitate the lesson.

Managing “flow” and allowing for the existence of “productive struggle”.

Allow the last 5 minutes of class for debrief / exit tickets.

Visit more classrooms.

Finish my presentation on modeling for TMC16.

Determine what a conceptual understanding of each topic would look like, and how would we use that to build procedural fluency?

This is a popular activity – here is a link to a post from Dan’s blog back in 2008 about the same lesson. I’m writing about it now because I love it and want to remind people of it if they had forgot.

Students get to question the assumptions its based on

Students get to see what effect those assumptions have on the model

Students get to validate the model through experimentation.

Allows students to estimate the solution. (Based on these results would we perceive the forest to have a high or low population of deer?)

Has a high ceiling as part B above is a fairly high level evaluation of the model.

The lesson has a few drawbacks

It doesn’t have an open middle. The only variation in methods comes from how they set up the proportion.

Brief overview of Capture / Recapture: It’s a technique used to estimate populations. For example, let’s assume we wanted to estimate the population of fish in a lake. We would first capture and tag a set of fish. After releasing them and giving them time to disperse back into the lake we would go back out and capture another set of fish and see how many of these recaptured fish had a tag from the first capture. We would then assume that the ratio of tagged fish in our second capture, to the size of our second capture, is exactly the same as the ratio of fish we tagged in our first capture to the total population of the lake.

Is that model valid? Turns out to be surprisingly accurate based on some interesting assumptions that the worksheet at the end will take the students through.

I use regular and pink goldfish to take students through a simulation. In groups of two, each student gets a lake with a lot of fish in it, a lake with not very many fish in it, and a bag of pink fish.

h

Then I have them capture 10 fish from each lake

(first capture)

Now they take each captured fish (by replacing them with a pink fish)

(tagging all the fish in the first capture)

Each fish goes back into the lake (bag) and are given sufficient time to reintegrate into fish society (shake the bag). Then go back out to the lake and again capture 10 fish. Record how many of the fish in this second capture have a tag from the first capture.

(take a look at how many of the fish from the first capture are in the second)

Now setup your proportion and solve. Math says that the proportion of tagged fish to non-tagged fish in the second capture is exactly the same as the proportion of fish captured in the first capture, to the total population. That those little piles of fish up there are a model of the entire fish population.

Of course we want to verify our results so we can test the accuracy of our mathematical model. After all we are trying to be the “least wrong” here, and ultimately accepting that our model doesn’t give the exact answer… but it is close enough to be acceptable to us? Let’s see, now I have them count to see exactly how many fish were in each lake:

(lake with lots of fish)

(lake with a few fish)

Then the next day give them these two slides (also from Dan)

“What if this were the results of the second capture. Would we predict the total fish population was large or small?” (red fish are tagged, blue are not tagged)

“A lot or a little?”

Ok that’s it. Other than that I just recommend doing some research on capture / recapture because there are a lot of interesting research articles and youtube videos on the topic.

The Goods

Here is the worksheet that I give the students after completing the lab. I’m pretty sure this originally came from Dan Meyer’s blog, I can’t find it on there now so… yeah:

This application isn’t real world but it’s real enough world. It’s not real world in the sense that nobody needed to figure out this exact question at their job. But it’s real enough world because a financial advisor at a well known wealth management firm told me he does calculations like this a lot. He simply invented the following scenario because he thought young kids could relate to it. I’m not sure if they can – but I’m damn sure they can learn from it. Since he’s the expert I’ll just leave it all in his words:

Additionally, here are some well- known abbreviations that I’ll reference in two scenarios:

PV = present value

FV = future value

N = years to goal

i = assumed annual growth rate

PMT = annual payment

Scenario 1 – When his son is age 18, a dad opens a Roth IRA for the boy with a $1000 investment (PV). The dad tells the boy “I’m giving you this money under one condition…and that is, you must contribute $600 per year (PMT) and leave it alone until you turn age 65, which is 47 years from now (N). We’re going to invest the money in an aggressive growth stock mutual fund that over time, I expect, should grow 9% per year on average (i). At age 65, I expect your account value will be in the neighborhood of $433,535 (FV).” Pretty amazing what time and compounding will do, huh?

Scenario 2 – A 13 year-old girl wants to purchase a used car at age 18, 5 years out (N). She expects the car to cost $8000 (FV). So far, she has saved $3000 (PV) and wants to know how much she must save annually (PMT) if her money is invested at a 4% annual rate (i). Solving for PMT:

PV = $3000

FV = $8000

i = 4%

N = 5 years

PMT = $803.13

I only ended up using scenario 1 and my teacher move was to block out the $433,535 and have students go through the estimation process about that account value after 47 years. What’s it going to be? Give me a couple dollar amounts it definitely won’t be because they are too high or low. Brave guesses only. (I heard Dan use the term “brave” after prompting for an estimation and it works well).