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Re: Another Desmos feature

Richard Strausz
I get it now. You are looking for specific ideas for you, and not necessarily for general classroom uses for Desmos. I don't feel qualified to tell you what works in your special summer classroom. Actually, the one with the best chance of success might be something like the self-checking exercise(see the repost below). Depending on your goals something like this - at a proper level of content - could increase students' involvement time. What I am saying is that the students might start some activities with you and use Desmos to check and correct their work...

I hope your summer session is a success!

Richard


> > > >
> > > > He had his students solve a number of 'two
> > > equation, two unknown' problems
> > > > algebraically. The students' homework
> assignment
> > > was to check each problem
> > > > graphically using Desmos (or a graphing
> > > calculator). They had to correct
> > > > the ones with mistakes algebraically. He said
> he
> > > liked this for two
> > > > reasons. First, the students had to find their
> own
> > > mistakes. Second, for
> > > > the problems with no solutions or an infinite
> > > number of solutions, he liked
> > > > the graphical feedback that Desmos provided.
> "Mr.
> > > Jones, I thought I made a
> > > > mistake when I only saw one line, and then I
> > > realized that since there were
> > > > an infinite number of answers, they were
> actually
> > > two versions of the same
> > > > line.
> > > >
> > > > Richard
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Re: Another Desmos feature

Carleton Washburne
In reply to this post by Richard Strausz
https://xkcd.com/447/

I gather that "game" and "math" are mutually exclusive or nearly so in Trigg World. Glad I'm not in it. Glad my kids weren't. No pain, no gain? No games, just pain?

Why do I bump into mathematicians talking about mathematics precisely in terms of games: knowing the pieces, the moves, the rules, and the aim (which shockingly can be to find beautiful results)? These ideas pop up with some frequency in Herb Gross' lectures. And that's hardly the first time I've encountered such notions. In fact, it's implicit in Hilbert's mathematical philosophy. I suspect Kirby, our Wittgenstein expert, could contribute more than a little on what Ludwig might have thought about mathematics and games. I challenge the notice that anyone needs to "turn math into a game," Timotha: it already is one. Of course, one may choose to try to turn it into *drudgery," and we seem to have more than a few people on this list oddly eager to do so for the vast majority of children. Bless whatever benevolent forces of the universe have managed to keep them out of K-12 classrooms, and particularly out of K-5 classrooms. We have no shortage of drudges and drudgery as it is.

Timotha Trigg wrote:

> Richard, I notice that some of these examples seem to
> be trying to turn
> math into a game.  While I am pleased when students
> enjoy the puzzles that
> are an inherent part of mathematical practice, I
> don't try to turn math
> into a game.  I only have a very small amount of
> time, and the amount of
> ground I'd love to cover is pretty much unlimited so
> my top priority is
> always trying to maximize the amount of learning per
> hour. I have not found
> a math game that is a particularly efficient way of
> learning so I don't
> choose to use them. (I feed them snacks; I invite
> them to come early and
> stay late to socialize or play board games; I try to
> be cheerful and kind,
> but I don't play games during the math lessons.)
>
> As I think I mentioned, my students are volunteers
> who come to do math of
> their own free will over the summer; perhaps I would
> feel differently about
> trying to turn math into a game if I had a classroom
> with different
> students.  I doubt it, though, because I think the
> best way to help
> students feel better about math (and/or their ability
> to succeed in math)
> is to do the most effective job possible *teaching*
> them math. The more
> they fall behind, the more they hate math and the
> stupider they feel, even
> if the falling behind is through no fault of their
> own. If their math class
> is taught inefficiently, they fall behind.  Although
> they might enjoy it in
> the moment, I don't think it serves them well in the
> long run.
>
> I guess that is a super inefficient way of saying I
> think the games are
> inefficient.
>
> I don't understand what the student is supposed to do
> with the MathyCathy
> stuff, for example:
>
> https://www.desmos.com/calculator/e9kvhjcttk
>
> I think this is written for people who already
> understand how to use
> Desmos, but also I wonder if the verbiage is
> defective or is it just me?
> (e.g., "Use a vertex-form equation to plot a parabola
> through the three
> points.")
>
> My favorite of the three you kindly posted was the
> Riemann sum gizmo that I
> can't find right now, from the calculus guy's
> website.  But still, I like
> this one quite a bit better:
>
> http://community.plu.edu/~heathdj/java/calc2/Riemann.h
> tml
>
> It's a shame it isn't bigger, but it does a nice job
> of demonstrating
> families of Riemann sums.
>
> I will have more positive comments about your popcorn
> room when I have time
> to write more.  Thank you for trying to help me.
>
> -Timotha
>
>
>
> On Mon, May 11, 2015 at 6:37 PM, Richard Strausz <
> [hidden email]> wrote:
>
> > > Yes, thanks.  I hope to find the time to reply to
> > > that one and also the
> > > Mathy Cathy one soon.  Very busy here right now.
> > >
> > > -Timotha
> >
> > That's fine; I understand.
> >
> > Richard
> >
> > > On Thu, May 7, 2015 at 3:41 AM, Richard Strausz <
> > > [hidden email]> wrote:
> > >
> > > > Timotha, I made a total of 3 Desmos
> suggestions; I
> > > should have stuck with
> > > > one at a time. The nature of posts here is that
> > > something can get lost
> > > > quickly. I began the posts on 5/6 with one
> about
> > > using Desmos in an Algebra
> > > > class.
> > > >
> > > >
> > >
> http://mathforum.org/kb/message.jspa?messageID=9765447
> > > >
> > > > Here is what it said -
> > > >
> > > > Timotha, I haven't tried the Polygraph Parabola
> > > activity (below) which
> > > > sounds worthwhile, but I want to share the way
> that
> > > a friend used Desmos
> > > > this year.
> > > >
> > > > He had his students solve a number of 'two
> > > equation, two unknown' problems
> > > > algebraically. The students' homework
> assignment
> > > was to check each problem
> > > > graphically using Desmos (or a graphing
> > > calculator). They had to correct
> > > > the ones with mistakes algebraically. He said
> he
> > > liked this for two
> > > > reasons. First, the students had to find their
> own
> > > mistakes. Second, for
> > > > the problems with no solutions or an infinite
> > > number of solutions, he liked
> > > > the graphical feedback that Desmos provided.
> "Mr.
> > > Jones, I thought I made a
> > > > mistake when I only saw one line, and then I
> > > realized that since there were
> > > > an infinite number of answers, they were
> actually
> > > two versions of the same
> > > > line.
> > > >
> > > > Richard
> > > >
> > > > > Still *another* Desmos strength are classroom
> > > > > activities designed by teachers and
> constructed
> > > by
> > > > > Desmos.
> > > > >
> > > > > For a good example, check out the Polygraph
> > > Parabolas
> > > > > activity.
> > > > >
> > > > > https://teacher.desmos.com/
> > > >
> > > > Richard
> > > > =========================
> > > > > Well, I guess we have about the full range of
> > > > > reactions, assuming we're all
> > > > > talking about the same problems. I think
> these
> > > > > calculus problems are fine
> > > > > (don't seem dumbed down to me), but I don't
> see
> > > why
> > > > > anyone would want
> > > > > Desmos for them.  Maybe I just don't
> understand
> > > > > Desmos well enough? (The
> > > > > Smith teacher links to these:)
> > > > >
> > > > >
> > >
> https://docs.google.com/document/d/1q5jet6GMEWOPU6ykwC
> > > > > zP9lsu9qaeN_GGQOS7FgyfQvk/edit
> > > > >
> > > > > -Timotha
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Re: Another Desmos feature

Joe Niederberger
In reply to this post by Richard Strausz
Lou Talman says:
>It's gilding the lily to do so. Mathematics is already a game!
And reflecting on the time and energy that kids will put into games---sports in particular, but video games as well---may be a personal investment that will repay a teacher.

I suppose probability should be mentioned here - I'm too tired from yard and garden work to do so. (That's not clearly a "game" though we could find some affinity between math and gardening I think. It took me a long time to find gardening a good use of time.)

Going on, its certainly a fine line between puzzles and games, and mathematics is certainly about puzzles. I'm all on-board the game wagon as well.

Cheers,
Joe N

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Re: Another Desmos feature

Timotha Trigg
In reply to this post by Richard Strausz
Yes, sort of.  I think I will probably find a use for the paper popcorn holders (rolled the long way and the short way) and other clever ideas not directly related to calculus.  But yeah, the specific things I'm looking for are those that help lay the foundation for calculus.

-Timotha

On Tue, May 12, 2015 at 12:25 PM, Richard Strausz <[hidden email]> wrote:
I get it now. You are looking for specific ideas for you, and not necessarily for general classroom uses for Desmos. I don't feel qualified to tell you what works in your special summer classroom. Actually, the one with the best chance of success might be something like the self-checking exercise(see the repost below). Depending on your goals something like this - at a proper level of content - could increase students' involvement time. What I am saying is that the students might start some activities with you and use Desmos to check and correct their work...

I hope your summer session is a success!

Richard


> > > >
> > > > He had his students solve a number of 'two
> > > equation, two unknown' problems
> > > > algebraically. The students' homework
> assignment
> > > was to check each problem
> > > > graphically using Desmos (or a graphing
> > > calculator). They had to correct
> > > > the ones with mistakes algebraically. He said
> he
> > > liked this for two
> > > > reasons. First, the students had to find their
> own
> > > mistakes. Second, for
> > > > the problems with no solutions or an infinite
> > > number of solutions, he liked
> > > > the graphical feedback that Desmos provided.
> "Mr.
> > > Jones, I thought I made a
> > > > mistake when I only saw one line, and then I
> > > realized that since there were
> > > > an infinite number of answers, they were
> actually
> > > two versions of the same
> > > > line.
> > > >
> > > > Richard

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Re: Another Desmos feature

kirby urner-4
In reply to this post by Joe Niederberger

I'd not worry about some fixed universal true meaning of "game" in Plato's World of Ideas.

Being of "language game" (LW) and "World Game" (RBF) heritage, I'll obviously come with my own spin, but so does everyone "mean" (i.e. "spin") in some way.

Are maths "really" but language games? But what is "language" here? "Forms of life" said LW. So it just keeps on going. One spin after a next.

Follow what beaten tracks you will, or go off beat. Many games to play that way too. Be unusual. It's OK.

Kirby

On May 12, 2015 7:47 PM, "Joe Niederberger" <[hidden email]> wrote:
Lou Talman says:
>It's gilding the lily to do so. Mathematics is already a game!
And reflecting on the time and energy that kids will put into games---sports in particular, but video games as well---may be a personal investment that will repay a teacher.

I suppose probability should be mentioned here - I'm too tired from yard and garden work to do so. (That's not clearly a "game" though we could find some affinity between math and gardening I think. It took me a long time to find gardening a good use of time.)

Going on, its certainly a fine line between puzzles and games, and mathematics is certainly about puzzles. I'm all on-board the game wagon as well.

Cheers,
Joe N

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Re: Another Desmos feature

Bishop, Wayne
In reply to this post by Richard Strausz
I'm afraid that you do not get it; moreover, that you never
will.  Instead of being inspired by some particular demonstration
that appears to "nail it" better than other presentations you have
seen of some particularly difficult concept or procedure, your MO is
to look for the next presentation and present it as something
inspired and useful.  Nearly always, they are not.  In fact, they
usually turn out to be time-wasting math-avoidance in place of
inspired communication of mathematics.

Wayne

At 09:25 AM 5/12/2015, Richard Strausz wrote:

>I get it now. You are looking for specific ideas for you, and not
>necessarily for general classroom uses for Desmos. I don't feel
>qualified to tell you what works in your special summer classroom.
>Actually, the one with the best chance of success might be something
>like the self-checking exercise(see the repost below). Depending on
>your goals something like this - at a proper level of content -
>could increase students' involvement time. What I am saying is that
>the students might start some activities with you and use Desmos to
>check and correct their work...
>
>I hope your summer session is a success!
>
>Richard
>
>
> > > > >
> > > > > He had his students solve a number of 'two
> > > > equation, two unknown' problems
> > > > > algebraically. The students' homework
> > assignment
> > > > was to check each problem
> > > > > graphically using Desmos (or a graphing
> > > > calculator). They had to correct
> > > > > the ones with mistakes algebraically. He said
> > he
> > > > liked this for two
> > > > > reasons. First, the students had to find their
> > own
> > > > mistakes. Second, for
> > > > > the problems with no solutions or an infinite
> > > > number of solutions, he liked
> > > > > the graphical feedback that Desmos provided.
> > "Mr.
> > > > Jones, I thought I made a
> > > > > mistake when I only saw one line, and then I
> > > > realized that since there were
> > > > > an infinite number of answers, they were
> > actually
> > > > two versions of the same
> > > > > line.
> > > > >
> > > > > Richard
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Re: Another Desmos feature

Richard Strausz
In reply to this post by Richard Strausz
Wayne, please stay on-task and tell what you like and dislike about this particular activity. It seems quite effective to me.

Richard

> I'm afraid that you do not get it; moreover, that you
> never
> will.  Instead of being inspired by some particular
> demonstration
> that appears to "nail it" better than other
> presentations you have
> seen of some particularly difficult concept or
> procedure, your MO is
> to look for the next presentation and present it as
> something
> inspired and useful.  Nearly always, they are not.
>  In fact, they
> usually turn out to be time-wasting math-avoidance in
> place of
> inspired communication of mathematics.
>
> Wayne
>
> At 09:25 AM 5/12/2015, Richard Strausz wrote:
> >I get it now. You are looking for specific ideas for
> you, and not
> >necessarily for general classroom uses for Desmos. I
> don't feel
> >qualified to tell you what works in your special
> summer classroom.
> >Actually, the one with the best chance of success
> might be something
> >like the self-checking exercise(see the repost
> below). Depending on
> >your goals something like this - at a proper level
> of content -
> >could increase students' involvement time. What I am
> saying is that
> >the students might start some activities with you
> and use Desmos to
> >check and correct their work...
> >
> >I hope your summer session is a success!
> >
> >Richard
> >
> >
> > > > > >
> > > > > > He had his students solve a number of 'two
> > > > > equation, two unknown' problems
> > > > > > algebraically. The students' homework
> > > assignment
> > > > > was to check each problem
> > > > > > graphically using Desmos (or a graphing
> > > > > calculator). They had to correct
> > > > > > the ones with mistakes algebraically. He
> said
> > > he
> > > > > liked this for two
> > > > > > reasons. First, the students had to find
> their
> > > own
> > > > > mistakes. Second, for
> > > > > > the problems with no solutions or an
> infinite
> > > > > number of solutions, he liked
> > > > > > the graphical feedback that Desmos
> provided.
> > > "Mr.
> > > > > Jones, I thought I made a
> > > > > > mistake when I only saw one line, and then
> I
> > > > > realized that since there were
> > > > > > an infinite number of answers, they were
> > > actually
> > > > > two versions of the same
> > > > > > line.
> > > > > >
> > > > > > Richard
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Re: Another Desmos feature

Carleton Washburne
In reply to this post by Richard Strausz
Why should that bother you, Wayne? You haven't taught K-12 mathematics in decades and haven't observed anyone doing so in recent memory. You have nothing upon which to report. All you do here is recycle a handful of bilious comments, all negative, about anything outside your experience. You could just number them and tell us which one you wish us to think of when real teachers post ideas and practices from their classrooms or those of other real teachers. We know already that you're going to hate anything and everything that involves technology. You would have been fun to have around when earlier tech tools were invented: Wayne on the Abacus; Wayne on the Knotted String; Wayne on the Counting Board; Wayne on the Slide Rule; Wayne on Pebbles in the Sand; Wayne on Papyrus, etc. It would make a fabulous series of curmudgeonly screeds, always calling for the elimination of the new tech as part of systematic "math avoidance."

We're fortunate that the vast majority of mathematics teachers happily remain ignorant of your pronouncements and proscriptions or are indifferent to them. I shudder at the thought of losing Herb Gross' brilliant calculus revisited courses from 1970 because you determined that video tape was part of math avoidance.

What you don't discuss, of course, is teaching avoidance. Might cut a little too close to home for comfort. But then, to quote you, "I'm afraid that you do not get it; moreover, that you never will."

Wayne Bishop wrote:

> I'm afraid that you do not get it; moreover, that you
> never
> will.  Instead of being inspired by some particular
> demonstration
> that appears to "nail it" better than other
> presentations you have
> seen of some particularly difficult concept or
> procedure, your MO is
> to look for the next presentation and present it as
> something
> inspired and useful.  Nearly always, they are not.
>  In fact, they
> usually turn out to be time-wasting math-avoidance in
> place of
> inspired communication of mathematics.
>
> Wayne
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Re: Another Desmos feature

Bishop, Wayne
In reply to this post by Richard Strausz
Please reread the message.  Staying on-task is EXACTLY what I was
doing.  Moreover, that is what I was trying (albeit with little hope
of success) to encourage you to do for a pleasant change.

Wayne

At 05:33 AM 5/13/2015, Richard Strausz wrote:

>Wayne, please stay on-task and tell what you like and dislike about
>this particular activity. It seems quite effective to me.
>
>Richard
>
> > I'm afraid that you do not get it; moreover, that you
> > never
> > will.  Instead of being inspired by some particular
> > demonstration
> > that appears to "nail it" better than other
> > presentations you have
> > seen of some particularly difficult concept or
> > procedure, your MO is
> > to look for the next presentation and present it as
> > something
> > inspired and useful.  Nearly always, they are not.
> >  In fact, they
> > usually turn out to be time-wasting math-avoidance in
> > place of
> > inspired communication of mathematics.
> >
> > Wayne
> >
> > At 09:25 AM 5/12/2015, Richard Strausz wrote:
> > >I get it now. You are looking for specific ideas for
> > you, and not
> > >necessarily for general classroom uses for Desmos. I
> > don't feel
> > >qualified to tell you what works in your special
> > summer classroom.
> > >Actually, the one with the best chance of success
> > might be something
> > >like the self-checking exercise(see the repost
> > below). Depending on
> > >your goals something like this - at a proper level
> > of content -
> > >could increase students' involvement time. What I am
> > saying is that
> > >the students might start some activities with you
> > and use Desmos to
> > >check and correct their work...
> > >
> > >I hope your summer session is a success!
> > >
> > >Richard
> > >
> > >
> > > > > > >
> > > > > > > He had his students solve a number of 'two
> > > > > > equation, two unknown' problems
> > > > > > > algebraically. The students' homework
> > > > assignment
> > > > > > was to check each problem
> > > > > > > graphically using Desmos (or a graphing
> > > > > > calculator). They had to correct
> > > > > > > the ones with mistakes algebraically. He
> > said
> > > > he
> > > > > > liked this for two
> > > > > > > reasons. First, the students had to find
> > their
> > > > own
> > > > > > mistakes. Second, for
> > > > > > > the problems with no solutions or an
> > infinite
> > > > > > number of solutions, he liked
> > > > > > > the graphical feedback that Desmos
> > provided.
> > > > "Mr.
> > > > > > Jones, I thought I made a
> > > > > > > mistake when I only saw one line, and then
> > I
> > > > > > realized that since there were
> > > > > > > an infinite number of answers, they were
> > > > actually
> > > > > > two versions of the same
> > > > > > > line.
> > > > > > >
> > > > > > > Richard
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Re: Another Desmos feature

Richard Strausz
In reply to this post by Richard Strausz
Wayne, this thread is on Desmos and its uses. It would be helpful if you would critique the use I recommended below.

Thanks in advance, Richard

> Please reread the message.  Staying on-task is
> EXACTLY what I was
> doing.  Moreover, that is what I was trying (albeit
> with little hope
> of success) to encourage you to do for a pleasant
> change.
>
> Wayne
>
> At 05:33 AM 5/13/2015, Richard Strausz wrote:
> >Wayne, please stay on-task and tell what you like
> and dislike about
> >this particular activity. It seems quite effective
> to me.
> >
> >Richard
> >
> > > I'm afraid that you do not get it; moreover, that
> you
> > > never
> > > will.  Instead of being inspired by some
> particular
> > > demonstration
> > > that appears to "nail it" better than other
> > > presentations you have
> > > seen of some particularly difficult concept or
> > > procedure, your MO is
> > > to look for the next presentation and present it
> as
> > > something
> > > inspired and useful.  Nearly always, they are
> not.
> > >  In fact, they
> > > usually turn out to be time-wasting
> math-avoidance in
> > > place of
> > > inspired communication of mathematics.
> > >
> > > Wayne
> > >
> > > At 09:25 AM 5/12/2015, Richard Strausz wrote:
> > > >I get it now. You are looking for specific ideas
> for
> > > you, and not
> > > >necessarily for general classroom uses for
> Desmos. I
> > > don't feel
> > > >qualified to tell you what works in your special
> > > summer classroom.
> > > >Actually, the one with the best chance of
> success
> > > might be something
> > > >like the self-checking exercise(see the repost
> > > below). Depending on
> > > >your goals something like this - at a proper
> level
> > > of content -
> > > >could increase students' involvement time. What
> I am
> > > saying is that
> > > >the students might start some activities with
> you
> > > and use Desmos to
> > > >check and correct their work...
> > > >
> > > >I hope your summer session is a success!
> > > >
> > > >Richard
> > > >
> > > >
> > > > > > > >
> > > > > > > > He had his students solve a number of
> 'two
> > > > > > > equation, two unknown' problems
> > > > > > > > algebraically. The students' homework
> > > > > assignment
> > > > > > > was to check each problem
> > > > > > > > graphically using Desmos (or a graphing
> > > > > > > calculator). They had to correct
> > > > > > > > the ones with mistakes algebraically.
> He
> > > said
> > > > > he
> > > > > > > liked this for two
> > > > > > > > reasons. First, the students had to
> find
> > > their
> > > > > own
> > > > > > > mistakes. Second, for
> > > > > > > > the problems with no solutions or an
> > > infinite
> > > > > > > number of solutions, he liked
> > > > > > > > the graphical feedback that Desmos
> > > provided.
> > > > > "Mr.
> > > > > > > Jones, I thought I made a
> > > > > > > > mistake when I only saw one line, and
> then
> > > I
> > > > > > > realized that since there were
> > > > > > > > an infinite number of answers, they
> were
> > > > > actually
> > > > > > > two versions of the same
> > > > > > > > line.
> > > > > > > >
> > > > > > > > Richard

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Re: Another Desmos feature

Richard Strausz
In reply to this post by Richard Strausz
Wayne, it has been a week since you criticized my thinking but refused to comment on the message I had made. I presume that means that you still don't like my thinking BUT you have nothing to say about the message (see below).

Richard

> Wayne, this thread is on Desmos and its uses. It
> would be helpful if you would critique the use I
> recommended below.
>
> Thanks in advance, Richard
>
> > Please reread the message.  Staying on-task is
> > EXACTLY what I was
> > doing.  Moreover, that is what I was trying (albeit
> > with little hope
> > of success) to encourage you to do for a pleasant
> > change.
> >
> > Wayne
> >
> > At 05:33 AM 5/13/2015, Richard Strausz wrote:
> > >Wayne, please stay on-task and tell what you like
> > and dislike about
> > >this particular activity. It seems quite effective
> > to me.
> > >
> > >Richard
> > >
> > > > I'm afraid that you do not get it; moreover,
> that
> > you
> > > > never
> > > > will.  Instead of being inspired by some
> > particular
> > > > demonstration
> > > > that appears to "nail it" better than other
> > > > presentations you have
> > > > seen of some particularly difficult concept or
> > > > procedure, your MO is
> > > > to look for the next presentation and present
> it
> > as
> > > > something
> > > > inspired and useful.  Nearly always, they are
> > not.
> > > >  In fact, they
> > > > usually turn out to be time-wasting
> > math-avoidance in
> > > > place of
> > > > inspired communication of mathematics.
> > > >
> > > > Wayne
> > > >
> > > > At 09:25 AM 5/12/2015, Richard Strausz wrote:
> > > > >I get it now. You are looking for specific
> ideas
> > for
> > > > you, and not
> > > > >necessarily for general classroom uses for
> > Desmos. I
> > > > don't feel
> > > > >qualified to tell you what works in your
> special
> > > > summer classroom.
> > > > >Actually, the one with the best chance of
> > success
> > > > might be something
> > > > >like the self-checking exercise(see the repost
> > > > below). Depending on
> > > > >your goals something like this - at a proper
> > level
> > > > of content -
> > > > >could increase students' involvement time.
> What
> > I am
> > > > saying is that
> > > > >the students might start some activities with
> > you
> > > > and use Desmos to
> > > > >check and correct their work...
> > > > >
> > > > >I hope your summer session is a success!
> > > > >
> > > > >Richard
> > > > >
> > > > >
> > > > > > > > >
> > > > > > > > > He had his students solve a number of
> > 'two
> > > > > > > > equation, two unknown' problems
> > > > > > > > > algebraically. The students' homework
> > > > > > assignment
> > > > > > > > was to check each problem
> > > > > > > > > graphically using Desmos (or a
> graphing
> > > > > > > > calculator). They had to correct
> > > > > > > > > the ones with mistakes algebraically.
> > He
> > > > said
> > > > > > he
> > > > > > > > liked this for two
> > > > > > > > > reasons. First, the students had to
> > find
> > > > their
> > > > > > own
> > > > > > > > mistakes. Second, for
> > > > > > > > > the problems with no solutions or an
> > > > infinite
> > > > > > > > number of solutions, he liked
> > > > > > > > > the graphical feedback that Desmos
> > > > provided.
> > > > > > "Mr.
> > > > > > > > Jones, I thought I made a
> > > > > > > > > mistake when I only saw one line, and
> > then
> > > > I
> > > > > > > > realized that since there were
> > > > > > > > > an infinite number of answers, they
> > were
> > > > > > actually
> > > > > > > > two versions of the same
> > > > > > > > > line.
> > > > > > > > >
> > > > > > > > > Richard

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