## Can Someone Tell Me What I’m Looking At Here?

January 6th, 2012 by Dan Meyer

I downloaded a clip of the game show *The Price Is Right* and analyzed the footage in Excel and AfterEffects.

It probably goes without saying I’m wondering, “Is it predictable?” What model underlies the showcase spinner? If you knew the initial position of the spinner and, say, the amount of time it took the spinner to complete one revolution, how close could you get to predicting its final position?

This one has me outclassed, though. Someone teach me something, okay?

**2012 Jan 7**: Here is the timecode data I gathered.

**2012 Jan 7**: And the spinner.

on 06 Jan 2012 at 6:51 am1 NoahHmm…if the wheel’s rotation slows at a constant rate, I’d expect those plots of beeps vs time to be parabolic. Plotting the beep rate vs time would be linear with the slope related to the angular acceleration and the intercept the initial beep rate. If we knew how many spokes are on the wheel we could convert the beep rates to radian per sec (or to revolutions per second or to degrees per second).

The angular acceleration should be independent of the initial rotation rate. One could check that be analyzing the data from the two spins. You might be able to accurately predict the time to stop and the number of rotations of a new third spin if you measured the initial rotation rate of that spin.

on 06 Jan 2012 at 7:28 am2 GregYou need to know more than what you posit in the post, I think. The impacting variables are:

1. The initial acceleration of the spin

2. The deceleration of the wheel

3. The initial position

The actual calculus and physics are a bit beyond me at the moment but intuitively it seems equivalent to dropping a ball from a bridge and finding out how long it will take till it hits the ground (with the differences being the wheel is initially in a state of rest and the ball is in a state having potential energy and the accel-/deceleration is angular with the wheel and linear with the ball).

But I’m sure it’s only moments before someone smarter than me finishes this off. :)

on 06 Jan 2012 at 7:35 am3 SamThe data looks similar to the speed of a falling object experiencing a drag force linearly dependent on speed, or the charge on a capacitor charging through a resistor. C(1 – e^-t/k) C would be the final number of beeps on a given spin, and maybe k would be constant for any spin because it’s based on the arrow-peg resistance and not on the initial push. k is called the time constant for RC circuits, and determines how long it takes to get up to a certain % of the number of final beeps (after 1k seconds, you are at (1 – e^-1k/k) or 63% of the final number of beeps, after 2k seconds, you are at (1-e^-2k/k) or 86% of the final number, etc). This doesn’t immediately give you a way of calculating the number of beeps as a function of spin speed, but it could be part of the process.

on 06 Jan 2012 at 7:38 am4 Mark BarnesAs a language arts guy, I must say, my head began to spin, like that wheel, considering your diagram and reading the comments of your readers.

Still, Dan, as always, you and your readers fascinate me.

Thanks for making my head spin in a delightful way.

on 06 Jan 2012 at 7:39 am5 Emily VAThinking out loud… it’s a case of rotational friction. The time for one revolution would give you the initial angular velocity.

Doing some sort of fit to the first curve (not sure what functional form would be most physically meaningful and 3 minutes of googling failed to tell me the answer) should give you some way of empirically measuring the angular deceleration friction from the ticker and whatever friction is between the bearing and the wheel.

Then the frictional factor applied to a new initial angular velocity might let you predict the curve for the second spinner and predict where it would stop… Unless the frictional factor is itself dependent on the angular velocity (which it well might be) in which case you’d need to determine that before you could make the prediction you’re looking for.

Overall this seems like a great problem for a college mechanics class, but maybe not so great for any high school math classes I know…

on 06 Jan 2012 at 7:44 am6 Jon in MaineI’m going to present this to my AP Physics Class today. We’ll try to find other clips of the wheel online to gather more data. The students have been using the program Audacity as a timer for event which make sound, so they should be able to get accurate beep vs. time data. It could be a few days before we have a result. On a side note: If you read this Dan, I look forward to meeting you this summer at the MSSM STEM educators camp in Maine.

on 06 Jan 2012 at 8:11 am7 Dan AndersonSidenote:

Awesome book that describes a group of people in the 80’s with custom toe-controlled computers that would predict where the ball will fall on a roulette simulation. Also chaos theory. Great book.

http://www.amazon.com/Eudaemonic-Pie-Thomas-Bass/dp/0595142362

on 06 Jan 2012 at 8:18 am8 Kelly John Rose“It seems equivalent to dropping a ball from a bridge and finding out how long it will take till it hits the ground”

Actually, to me it’s more equivalent to throwing a ball up in the air and seeing when it changes direction.

Basically, the wheel is constantly decelerating, and you have the initial speed v_i from the first 2 beeps (the distance of one block / t_1, t_i = the time between beeps i, and i+1). After a few calculations times through, you could sort out the average deceleration (a), in a similar fashion to how we can calculate g in first year physics. (we can sort out d by taking the number of beeps and multiplying them by the length of a single square.)

d = v_i*t + 1/2 a t^2

(d-v_i*t)*2/t^2 = a

Now, obviously I’m assuming constant deceleration, etc, but I think that’s a reasonable relaxation of the problem.

Now that you can calculation a and v_i. It’s just a matter of popping it in and solving for v_f = 0

v_f = v_i – a * t_f

t_f= (v_f – v_i)/-a

d_f = v_i*t_f+1/2 a t_f^2

and then divide d_f by the length of a square. You’d know how many squares it moves, and then you could easily figure out what square that is from the original square.

And you are done.

on 06 Jan 2012 at 8:23 am9 Kelly John RoseNote, all of my math is basically ignoring edge effects due to the tickers and other parts of the system. Viewing it as simply a dynamic system with velocity and deceleration.

on 06 Jan 2012 at 8:50 am10 Rhett Allain (@rjallain)This is where you need a student to collect data. If you get enough clips of this spinning wheel, you can gather the data you need to make a model. I would suggest the following:

– wheel starting position

– wheel ending postion

– starting angular speed

– time of spin

That would be a good place to start. I would guess the wheel would have a constant angular acceleration independent of the initial angular velocity.

on 06 Jan 2012 at 8:56 am11 Roy WrightUsing GeoGebra, each of the two graphs do seem to fit parabolas pretty well, and there does seem to be a constant deceleration of very close to the same magnitude for both spins (since each quadratic has approximately the same leading term).

on 06 Jan 2012 at 9:03 am12 Julie StrongI don’t think I can add much to the math of this one, but there is a great piece from This American Life (radio show) about a guy who did something similar with predictability. Also applies to the difficulty of generating “randomness”: http://www.thisamericanlife.org/radio-archives/episode/412/transcript. Scroll down to Act Four: Don’t Hate the Player.

on 06 Jan 2012 at 9:30 am13 Jim HardyAs an aside, I often use this clip – also from The Price is Right (the worst contestant ever):

http://www.youtube.com/watch?v=2QZpnBKZHVo

Ask: What maths comes from this video?

on 06 Jan 2012 at 10:11 am14 Mr. KYour beeps/second are proportional to the angular velocity of the wheel. You want to know when the wheel stops, i.e. dbeep/dt = 0. Your beeps are units, so your local approximation for the slope is 1/(time between beeps). I’d like to see a graph of that, to see if its really linear.

on 06 Jan 2012 at 10:32 am15 NicoSome side questions:

There are 3 contestants spinning the wheel.

a) Spinner#1-

Assuming that the wheel is random, what strategy should spinner #1 take? (In other words, at what number should spinner #1 spin again? Where do we set the line?)

b) Spinner #2-

If spinner #1 goes over (loses), where do we set the line for spinner #2?

I spent a few hours working out probabilities of different TPIR games back in my University days. I definitely had some fun with wheel spinning strategies. Also fun for strategy talk that leads to very common math themes are, The Dice Game, The Clock Game, and Plinko just to name 3.

TPIR is definitely full of mathematical treasures. Now if we could only find someone who can pull some video off the internet and create some graphics.

Someone should also call CBS…lots of fun to be had.

on 06 Jan 2012 at 10:32 am16 Scott ThomasFrom wikipedia, the mass of the big wheel is 2 short tons (3600 lbs -> ~1600 kg). Drew’s height is 5’10” so I’d guess the wheel is a radius of about 5 ft (~1.5 m). The inertia for rotation of a wheel is (1/2) m r^2.

The kinematics for rotation are similar to that of translation, so if you know the initial angular speed, the angular acceleration and the time elapsed, you can predict the final angular speed. Likewise, you can predict the total rotations it will take to stop.

I don’t have the time to analyze the video, but if you do, determine how fast the wheel is deceleration (formally called alpha{the lower case}). The deceleration is caused by the torque (symbol tau) due to friction. Just like F-ma, torque = I * alpha where is is the moment of inertia (for disk => (1/2) mr^2).

All kinematic equations for constant acceleration also similar:

v=vo + at

becomes

omega = omega-o + alpha * t

d=vo*t + (1/2)a t^2

becomes

theta=omega-o * t + (1/2) alpha * t^2

d=(v^2 – vo^2)/(2a)

becomes

theta =(omega^2 – omega-o^2)/(2 * alpha)

where v is final velocity, vo is initial velocity, a is acceleration, d is displacement, t is time, omega is angular speed, omega-o is initial angular speed, alpha is angular acceleration.

I’m guessing if you really want the full scoop, hit up Rhett Allain at wired.com!

on 06 Jan 2012 at 10:33 am17 Mark WatkinsIt’s a rotational kinematics problem in physics. It involves using something called the moment of inertia as the “mass” of the wheel and doing position, velocity and acceleration in a polar coordinate system. The angular acceleration is found using the rotational equivalent of Newton’s laws of motion; Torque = (moment of inertia) x (acceleration) which is just F=ma for rotation. The acceleration should be approximately constant until you get to the last little bit where the higher order drag terms take over.

Just off the top of my head, if I were going use this in my math class I would have the woodshop build a large-ish solid wooden wheel and hang weights off it. I would steer them towards a plot of weight vs time and curve fit to get the acceleration experimentally. Maybe throw in standard deviation or just range to get error and have them use it to predict a stopping point within a certain number of spaces as the error.

on 06 Jan 2012 at 10:46 am18 Steven PetersI think the important physics are related to the friction acting on the wheel as it spins. There are several different physical models for friction.

For a block sliding on a surface, a common model is Coulomb friction, which assumes that there is a roughly constant drag force proportional to the weight of the block that opposes it’s motion. With constant force, there’s constant deceleration and a parabolic type of trajectory.

For something moving slowly through a fluid (like water or air), the drag force is approximately proportional to the relative speed between the object and the fluid ( http://en.wikipedia.org/wiki/Drag_(physics) ). This leads to a differential equation, for which an exponential curve is a solution (as @Sam suggested). For a wheel spinning on lubricated bearings, there drag in the bearings is typically modeled like this.

For something moving at high speeds through a fluid, the fluid becomes turbulent and the drag force becomes proportional to the velocity squared. This differential equation is actually pretty tough to solve analytically. I don’t think that there’s much turbulence acting in this case, though.

For the wheel in this video, I’m guessing that the drag in the bearings plays a part, but that the needle hitting the edge of the wheel has an effect as well. That effect could be modeled as a series of impacts.

That’s my first take.

on 06 Jan 2012 at 11:08 am19 Frank NoscheseI’m going with Rhett: frictional torque is likely constant, thus it slows with constant angular deceleration.

The graph of the beeps have 2 distinct features: A “happy” parabola while the player is accelerating the wheel, and a “sad” parabola after the player lets go and the wheel is decelerating.

If you time-shifted the graph so t=0 is when the player lets go (thus just the “sad” parabola part) and did a curve fit, I bet the x^2 coeffient (which represents 1/2 the deceleration) would be nearly the same for both spins.

[goes to analyze graph]

Sure enough!

http://fnoschese.files.wordpress.com/2012/01/priceisrightanalysis.png

There’s percent difference of just 1.9% in the x^2 coeffiecients for the two spins!

So the wheel slows down a constant, consistent rate (like Roy said). Great. But to make the prediciton on where it stops, you would still need to know the angular velocity when the player lets go and the starting position of the wheel.

Better question though:

When you are the player, you really can’t control the starting position of the wheel. So, given the starting position, determine the angular velocity you must impart to the wheel so that it slows down to the position you want.

What a rich problem space!

[Note: I don’t know if the x^2 coeffiecient would be the same for spins on different days. I wonder if they lube the wheel from time to time, which would decrease the frictional torque and thus decrease the deceleration.]

Students could easily replicate this in the lab using a record player. Is the angular deceleration constant as the record player slows down? Is the angular deceleration the same when the record player slows down from 16 RPM, 33 RPM, 45 RPM, and 78 RPM?

on 06 Jan 2012 at 12:19 pm20 James C.Slightly different question here: Can we use a model here, ideally a logarithmic model, to estimate the number of beeps at a given moment in time? Does anyone potentially think this could be useful for teaching logarithmic growth (say as an introduction)? The idea of using sound to model functions as a learning technique is something I’m interested in.

on 06 Jan 2012 at 12:35 pm21 David GarciaDespite Frank’s analysis I’m skeptical that the acceleration will be constant as the wheel slows to a stop. I agree with Sam (comment #3) that a model resembling air resistance is probably more accurate. If the torque slowing down the wheel was solely due to friction in the axle, then the acceleration would be constant. However, I think a significant amount of the torque is caused by the pegs on the wheel hitting the arrow. When the wheel is spinning faster, these torques are applied with more frequency.

on 06 Jan 2012 at 12:50 pm22 Mark WatkinsThe growth isn’t logarithmic so much as an exponential decay. It does approach a final number of beeps and logarithmic functions grow without bound. You could use it to model decay, approximately, but a quadratic with a negative leading coefficient (negative acceleration) is definitely the best/easiest/simpliest model until second order friction takes over at slower speeds. (At that point the math becomes so messy that it was covered in my graduate level math methods course so it’s not worth getting into much detail with at high school levels)

on 06 Jan 2012 at 12:51 pm23 Frank Noschese@DavidGarcia: Very good point. But now we can debate about modeling and which model is “best.” Is the best model the one which is the most accurate portrayal of reality? Is the best model the simplest one that still yields accurate predictions? (Simpler models are cheaper and faster to use.)

We don’t include relativistic effects in our physics courses, but they are still there none-the-less. Same for ignoring air resistance at low speeds.

I wonder if the constant acceleration is accurate only when the initial velocity of the wheel is smaller than a certain value. If the initial velocity of the wheel is very large, then the repeated hits from the peg will slow the wheel down more rapidly than friction alone. Similarly, air resistance and relativity must be taken into account for some scenarios, but can be neglected for others.

More data please! I wonder if a gaming wheel found at carnivals (the local fire department or police department might actually have one) could be borrowed for data collection and modeling.

on 06 Jan 2012 at 12:54 pm24 Bob LochelIf somebody’s class does a full analysis of the wheel beeps, follow up with this question: what do you do when the beeper mechanism breaks?

http://www.youtube.com/watch?v=UbXwmnPhMNg

Along the same lines, students might be interested in knowing how the beeper mechanism works:

http://i8.photobucket.com/albums/a16/sideshowPA/beeper.jpg

Accoring to goldenroad.net (a site I frequent): “The truth about the big wheel sounds: The yellow device on the right is a light sensor that is mounted on the back of the wheel. A wire attached to the light sensor runs over the stage and is attached to the sound keyboard shown elsewhere in the gallery. Looking on the wheel, you will see repeating patters of black and white areas that are directly in the path of the sensor. Every time that the sensor picks up a change in this color, it sends a signal to the sound keyboard to make a “boop”. When looking at the wheel on tv, this is the left side of the wheel…the side that is never shown on tv”

Finally, some analysis of spinning strategy, and other TPIR probability discussions:

http://www.golden-road.net/index.php/topic,12104.msg326554.html#msg326554

on 06 Jan 2012 at 3:14 pm25 Mark WatkinsI thought about this on the way home for work. I’m certain that for a high school class Frank has the most accessible way to approach this. The trouble with real honest-to-goodness physics is that any measurement you make, no matter how accurate, is an approximation. You can always increase your accuracy by an order of magnitude or two but at the end of the day you have to pick the decimal place you stop worrying about accuracy for and move on. For my high school math classes that’s the hundreths place more often than not.

That wheel is big enough compared to the pivot to basically ignore complicated drag leaving a nice parabolic graph to discuss. Each beep corresponds to 1/20th of a trip around the big wheel leaving behind a solid position vs. time graph. To me the classroom question “where does it stop?” is estimated by either nailing down the rotational acceleration from that graph or using the first few points in the video to fit the rest of the quadratic. I think I’ve seen Dan put together something similar for predicting free-throw shots from the first half of the shot. In this case they would just need to use the curve fit to find the maximum number of beeps to know where it lands. The fact that the wheel is broken into nice neat 5% increments makes me a little excited to talk about measurement uncertainty without having to get extremely accurate in order for the students to feel like theirs is “right.

on 06 Jan 2012 at 4:35 pm26 CathyWow, I love the way you’ve shown the graph as the video is played. Amazing tech skills! As far as what math is taking place or how we can represent it, I have to admit that I really have no idea. I do, however, have a gut reaction thinking that the diameter (approximate since it isn’t a circle) of the spinner is also important. If we had a smaller spinner or a larger one, how would that affect the rate of the beeps if all of the other dimensions are kept proportional.

on 07 Jan 2012 at 8:14 am27 TimYou’re question of “is it predictable” is close but veered away from the question that was obvious in my mind: “Do they cheat?” As a kid, ever since I learned that the people they call down to play are chosen by their producers, I always wondered whether they cheated on the wheel too. If the class turned into more of a detective case, I think that’d be more engaging for the kids. Predicting where it should land doesn’t seem realworld-applicable. No one is ever going to be able to train to always get a dollar on the show. But…finding out whether or not the show is fair, that (to me at least) is really interesting.

Maybe you could have the students convert the graphs into speed vs. time for several different contestants. If they are all linear with the same slope, it’s fair. And even if it is fair, you could ask them what a graph might look like if someone backstage, or even a computer program, were rigging the process.

on 07 Jan 2012 at 10:27 am28 Roy WrightRelated to Tim’s comment, here’s a thought: Imagine how small a change in the deceleration constant it would take to completely change the outcome, given how randomly (so to speak) the cash amounts are placed on the wheel!

on 07 Jan 2012 at 1:01 pm29 Gilbert BernsteinIt would be interesting to ask in a statistics class, “How predictable of a model can you build?” So for instance, the suggestion that many people have had “fit a parabolic model.” turns into the questions: “What is the best possible parabolic model you can fit?” and “Is the uncertainty/variance in that model so high that it’s basically rubbish?” which could also be varried to “Suppose I let you guess which three consecutive numbers the wheel might stop on. Now how good of a prediction can you make?”

on 07 Jan 2012 at 5:36 pm30 Aaron F.@Frank Noschese: Nice analysis! But let me try to do you one better. ;) You saw when the contestant let go of the wheel by watching the video, but I think I can figure it out just by looking at the timecode data.

Here’s my analysis, in Excel and Open Document format.

The “count” variable in the timecode data gives the wheel’s angular position (according to Mark Watkins, each beep is 1/20 of a full revolution), and the “seconds” variable gives the time. From that, it’s easy to compute the wheel’s mean angular velocity from beep to beep.

Differentiating always adds noise, so the angular velocity curve isn’t quite as tidy as the position curve, but for both spins it shows a rather gorgeous linear decrease in angular velocity over time. So the wheel’s deceleration (once the player lets go of it) is very nearly constant, just like Roy Wright and Frank Noschese said.

Least squares regression says the angular deceleration of the wheel is about 0.45 counts / s^2, or 8.1 degrees / s^2, for both spins. Like Frank Noschese said, it would be interesting to take data from a bunch of different episodes and see how the deceleration changes over time!

(By the way, here’s a fun little data analysis puzzle. When the wheel is moving fast, why do the data points on my plots fall into neat horizontal rows?)

on 07 Jan 2012 at 6:12 pm31 Bowen KerinsHey Dan, quit stealing the material from my Mathematics of Game Shows talk! (Just kidding, I’ve got a bunch of other questions from TPIR to think about instead.)

I’ll agree with the majority opinion: you can ‘simplify’ the problem by calling it a constantly-decelerating wheel. In this case it works like something thrown in the air: constant negative acceleration. And since “beeps” is basically “vertical distance”, it should be like a parabola.

It’s not actually continuous: there is a small constant deceleration due to the wheel itself, but a larger discrete deceleration when the wheel hits one of the pegs. So it actually works more like the way some kids draw a quadratic: down 7, over 1, down 5, over 1, down 3, over 1. Either way a quadratic fit should be very accurate.

Frank and Aaron are pretty much right on the money here, and you’d need a more accurate reading to know more.

The show’s wheel has crazy variation in how fast or slow it is show-to-show. When it’s ‘greased up’ it spins like crazy, while at other times it can be difficult to get the thing all the way around.

Dumb but related: what’s the least likely thing that can happen in a 3-player round of spins, and how likely is it?

on 08 Jan 2012 at 11:07 am32 Aaron F.@Bowen Kerins: Cool! Is the talk posted online somewhere? How did you measure the deceleration due to hitting the pegs? Is your show-to-show data publicly available? :D

whatâ€™s the least likely thing that can happen in a 3-player round of spins, and how likely is it?I’m kind of confused by the question. Since every spin is different in terms of initial angular velocity and position, not to mention how greasy the wheel is, it seems hard to come up with a probability model based on pure thought. Is the wheel chaotic enough that the section it lands on is more or less uniformly distributed?

on 08 Jan 2012 at 12:38 pm33 Mark WatkinsChaos Mathematics doesn’t have anything to do with a single spin. Chaos would have to do with trying to predict the landing place on several consecutive spins with a specific initial condition (and the limit in that predictability does become uniform as the number of spins increases). If initial speed is known (the average of the first couple beeps isn’t a terrible approximation) as well as the starting location, then predicting the ending spot within a couple locations is completely doable.

on 08 Jan 2012 at 8:15 pm34 Bowen KerinsSorry, I meant the least likely outcome when 3 players compete, something like “Player 1 gets a total of 50 cents on 2 spins then …” I also need to qualify it by saying “without including bonus spins” because there is the potential for unlimited tying by players.

Click through to the Patterns in Practice blog and you’ll find the slides from last year’s NCTM game show talk. There will be another game show talk given this year with much different content. My data on the greased wheel is purely anecdotal, but there are easily visible changes in the ease or difficulty in spinning the wheel.

on 10 Jan 2012 at 3:46 am35 Robert Hansen“It probably goes without saying I’m wondering, “Is it predictable?” What model underlies the showcase spinner? If you knew the initial position of the spinner and, say, the amount of time it took the spinner to complete one revolution, how close could you get to predicting its final position?”

Given the large mass of the wheel, the system is almost entirely defined by the rotational inertia of the wheel and the opposing torque of the bearings (due to friction) which appears to be substantial and constant. Drag (air) is insignificant and the wheel is very predictable (as the smoothness in your data shows). This rotational problem (as some posters have already noted) is identical (mathematically) to a ball thrown up in the air (constant acceleration). I am sure that with practice a player could “land” the wheel on a given spot with fairly high probability in the same manner that a basketball player can sink a shot from different positions on a court.

on 10 Jan 2012 at 4:00 am36 Robert HansenOne other point. Forcing the player to make at least one revolution with the wheel is similar to making the basketball player shoot from further back on the court (say past the 3 point line). That doesn’t make the system chaotic, it only requires the player to be that much more accurate with their spin (or shot). The system is still linear because any error by the player results in a predictable (near) miss in the result. In a chaotic (non linear) system, there are no near misses because a small error causes an unpredictable result.

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on 10 Jan 2012 at 7:36 pm38 KalebIt makes me think of the book, the Quants. When they tried to predict the outcome of roulette. Everything else I was thinking was already said.

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on 14 Feb 2012 at 7:02 am41 KevinI think this discussion begs the question of why we think this problem needs a formula or exact physical properties in order to solve. The tough reality of applied mathematics is that if you want to actually get down to the “why it works” or “what’s the formula?” most of the time it’s way beyond middle or high school math which can totally destroy an otherwise successful lesson.

The real world is really messy and difficult to model exactly which is the beauty of data. The simple act of making that graph is really all students need to answer the question using the tools they know and feel comfortable with – algebra and some quadratics maybe. A few others alluded to the idea that simplicity is sometimes as good or better than reality when it comes to modeling. I think students need to appreciate the complexity, but not get bogged down in it.

I think looking for the perfect formula and the exact solution in situations like this actually does the opposite of the problem’s original intention (although fun for the teachers to explore!). I’ve run into a few situations where a problem looked really interesting and the class and I got really invested in it only to discover that it was rediculously detailed and required things that they weren’t able to do. Talk about your ultimate let down after a hard day’s work!

There is a definite skill needed in picking those perfect problems that are developmentally appropriate within the scope and sequence of the curriculum being taught, can be differentiated over various levels of complexity while still being able to provide some kind of real tangible answer at the end in a relatively neat and manageable way. Not an easy task, but so worthwhile when it happens!

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