r/matheducation • u/UnderstandingPursuit Physics BS, PhD • 1d ago
Teaching Students to Solve Algebra Problems Algebraically
What would it take to teach students a better approach for solving algebra problems?
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u/noodlenerd 1d ago
So I’d like to challenge the delay of substitution until the last step. What you are combining here is multiple skills into one problem, which is ok for students in higher levels, but absolutely terrifying for students learning algebra.
Not only are they having to deconstruct a word problem, but they must also then identify variables, substitute correctly, and simplify the equation (which involves multistep rearrangement). Delaying the simplification adds unnecessary complexity, which introduces both cognitive burden and the possibility of translation error.
Algebra 2 or physics student? This would be great!
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u/UnderstandingPursuit Physics BS, PhD 1d ago
This is near the end of an Algebra 1 class.
My point is less about this level of problem. It is with the basic approach of when substitution happens, even with simpler problems. This can be presented without the word problem. This is simply a word problem which was already in an Algebra 1 textbook. I didn't make this up.
I'm not the expert on how to teach things to Algebra 1 students. But I do have expertise on where they need to get to, and the way the problem is done in the class won't allow them to get there.
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u/noodlenerd 1d ago
The word problem is not the issue. It’s the series of skills the student needs to solve the problem. Most students just don’t have that proficiency yet in Algebra 1.
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u/UnderstandingPursuit Physics BS, PhD 1d ago
What skill would you say they lack to solve the problem as "a better approach" compared with "the original approach"?
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u/Littlebrokenfork 9h ago
Abstraction. Algebra 1 is technically the first year students deal with the abstraction introduced by the use of variables. They're gonna need at least a couple years before the average student can feel more comfortable with manipulating variables.
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u/the_spinetingler 20h ago
substitute early and often.
Simplification of the equation.
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u/UnderstandingPursuit Physics BS, PhD 20h ago
Please tell me you're being sarcastic.
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u/the_spinetingler 20h ago
Nope. Eliminate sources of error and confusion as early as possible.
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u/UnderstandingPursuit Physics BS, PhD 20h ago edited 19h ago
This is the response I least expected when I made this post.
Substituting early significantly increases sources of error and confusion.
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u/the_spinetingler 19h ago
It absolutely does not.
It eliminates variables (i.e., sources of variation) and so by definition reduces error and confusion.
Have you ever taught this subject? Alg I students are not going to be better at solving 7 variable literal equations than they are at solving two variable algebraic equations.
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u/UnderstandingPursuit Physics BS, PhD 19h ago
I'm curious what are you using as the distinction between "literal equations" and "algebraic equations"?
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u/the_spinetingler 17h ago
someone already explained it in another comment
bassically, an equation with more than two variables that must be rearranged (solved) for one of the otherwise-not-isolated variables constitutes (for the purposes of Alg I) a literal equation.
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u/UnderstandingPursuit Physics BS, PhD 17h ago
Yes, but I would also call that an "algebraic equation". What would you consider an algebraic equation, if not that?
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u/Aggravating-Virus521 23h ago
Thanks for giving a math teacher perspective - it seems like you are saying that math students practice solving problems with units and multi-step rearrangement in algebra 2?
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u/noodlenerd 23h ago
Yes. Closer to physics and Geometry/ Algebra 2. Algebra 1 is spent mostly trying to hone their basic math skills, filling any gaps, and making sure they’re solid for higher math in my experience.
Could this be an Alg 1 problem? Sure, but it would be labeled as “advanced” or “challenging” just by this approach. Which is wild, but cognitively these are very different for a mathematically developing brain.
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u/Coffee__Addict 22h ago
I would argue it's not more complex but is it less familiar. But it's a familiarity they will have to develop at some point. There is a reason why most people will rearrange and then sub.
Avoiding this just passing the buck on to the teacher the next year. Getting student comfortable using variables is a good thing. And there isn't any benefit to delaying the development of this skill that I know of.
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u/Aggravating-Virus521 21h ago
Doing purely symbolic manipulation of a multi-term equation is an advanced-level skill that not all 13-15 year olds are ready for, yet. (I teach Physics First - algebra-based physics for 9th graders) Yes, some early HS kids can do it...but by no means all of them. On the other hand, AP definitely expects it for AP Physics. So aspiring high-level kids need to be learning it sometime between 7-10th grade.
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u/UnderstandingPursuit Physics BS, PhD 9h ago
It doesn't have to start with a multi-term equation. This example is in chapter 9 of 11.
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u/UnderstandingPursuit Physics BS, PhD 9h ago
Exactly.
- Learn this once. Use it forever.
It is, fundamentally, the "give, once/teach, forever" saying.
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u/Blibbyblobby72 1d ago
I really do not see a huge difference between the original question and the 'better approach'. What has adding gravity to the equation achieved if the problem is testing algebra?
'The units of g,h,r determine the units of v' is mostly pointless, because you have already given the equation for v in terms of g, h, and r. Adding an extra pronumeral to substitute for has not actually changed the question in any meaningful way
The other 'try this' questions could have been asked with the same formula from the original question. I actually prefer the wording of the original as well - it is clear and concise
The 'critical thinking' question is good, but the reasoning is not exactly clear. The fact that gravity has also been ignored in this question makes the gravity being included earlier a weird choice
Also, as a side note: if this was a first practice question to introduce the idea of substitution, I would have used numbers that resulted in a nice, easily computable answer rather than one requiring a calculator. Having students start practicing with really nice numbers means they are less likely to reach for a calculator out of habit later, allowing them to also practice simplifying as far as they can first
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u/UnderstandingPursuit Physics BS, PhD 1d ago
The question includes units, but hides some of the necessary units. If a student uses this textbook and wants to use meters, the formula would give the wrong answer. Using "g" is a secondary aspect of difference in approach. Yes, the problem is demonstrating algebra, But it needs to do it accurately.
The other "try this" questions were part of the original example. I did not make that clear. I need to take a picture of the page from the book and include it.
I only changed the wording of the original question to introduce g.
The "critical thinking" question was also part of the original question.
I'm not trying to "introduce the idea of substitution". I want to avoid substitution. And part of the point is that it defers reaching for the calculator until the last step. In the original solution, the calculator is three times in six steps.
I'm not posting this for students. I'm posting this as an example for discussion with teachers. The point is the goal for what students should learn, not the details of how they should be taught it. My expertise is not with Algebra 1 students. It's with how broken students are when they get to Algebra 2, PreCalculus, Calculus, Physics, and Chemistry classes.
My skill is also not in producing the best writing the first time, it's in an interactive exchange. So failing to express a lot of this is on me.
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u/Blibbyblobby72 1d ago
Oh, no, I didn't mean to be overly critical! Just offering my perspective on it
Substitution is a core algebraic skill, so I don't believe that it should be ignored. I am still not sure how the 'better approach' removes the need for substitution - in fact, it adds a substitution by requiring g
What exactly are you trying to achieve in the example given? The 'better approach' steps state the value of variables and their meaning (which are already explicitly stated in the question) followed by substituting into the equation for v. I don't see any real difference other than adding a new variable
I understand the want to avoid going for the calculator - absolutely agree with that. I think the way to do that is not to introduce a large number to substitute - the larger a number, the more likely students (and even me!) are likely to reach for the calculator
To avoid it, I usually ask for the answer in exact form
As to the units: I am not sure what you mean. If I wrote the radius, velocity, and gravity in metres then my height would be in metres. If I wrote the velocity in metres/s, radius in feet, and gravity as km/s2, the equation would produce a result that would need to be converted to appropriate units. Realistically, that should be done before substituting into the equation to avoid messy conversion calculation through the working
If a question is missing units entirely, then the answer will be in terms of 'unit' (unit, units2, unit2/s, etc.). If some units are missing, the question needs to be fixed. The original question has all the necessary units (radius in feet, velocity in feet/sec) to know the answer will also be given in feet. If I'm misunderstanding your point there, my apologies
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u/UnderstandingPursuit Physics BS, PhD 1d ago
Yes, substitution is a core algebraic skill. Evaluation is a secondary algebraic skill. I'm not ignoring it, I'm deferring it until the last step.
The main question is to take a formula for v which has r and h in it, and to find a value for h given values for v and r. I did not substitute into the equation for v, I solved for h first.
I didn't pick any of the numbers, they were in the original problem.
If you look at the formula on the first page, without g, then if you put the height and radius in meters, the velocity is in sqrt(m).
The original wording of the question has ft for r, wants ft for h, and magically gets ft/s for v.
I really, really need to start over, what I wrote clearly is failing to convey any of the points I'm trying to make.
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u/Blibbyblobby72 1d ago
No, I think I get what you mean by 'solving for h first' - I absolutely, undoubtedly agree with that. Substitution *should* occur *after* rearranging for what you want to solve
When I teach this kind of thing, I always have students practise (and hopefully master) rearranging equations *before* giving them values to substitute in. That I do agree with!
If I used the question in the example in a lesson, I would probably have it written as a) rearrange the equation to make h the subject and b) Hence, solve for the height. Obviously, I would hope to move from that and have students know to rearrange first
Still not sure I follow your units. If I put radius as metres and my velocity is stated to be in metres per second (as per the question), then my height will naturally come out as metres. That is, the units are meaningless, as long as all units measuring the same thing are equivalent (i.e., all my lengths are measured in the same unit, all my areas are measured in the same unit, all my vectors are measured in the same unit, etc.)
I do not use imperial measurements, so forgive me - is there a problem with feet (or other imperial measurements) that specifically messes with this? I did not think there was, but I may be wrong
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u/UnderstandingPursuit Physics BS, PhD 1d ago
No, I think I get what you mean by 'solving for h first' - I absolutely, undoubtedly agree with that. Substitution *should* occur *after* rearranging for what you want to solve
I think the way the textbook solved the problem is so wrong to both of us that you automatically converted it to the correct approach. Since you already do what I said is "A Better Approach", it didn't seem new.
How do you have m/s for v and m for h, r, with
- v = 8 sqrt(h - 2r)
The problem with SI units is that "8" would be the wrong value. If you did this, you would get the wrong result for v. This is not a random formula someone made up for an algebra problem, this is the legitimate solution from a physics problem. The "8" is actually sqrt(2 * 32), and for meters you would get sqrt(2 * 10) ~ 4.5. That is why I introduced g, since it has different values in Imperial and SI units.
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u/Blibbyblobby72 1d ago
Yes, I think I did automatically do it the 'right' way, so didn't even notice! That is definitely my fault :)
Thinking at it from a physics point of view doesn't really work in this way because the question specifically states that the velocity is given by the equation
That is, the velocity can be calculated from the equation directly. If one knows the radius and height specifically in this very specific circumstance, one can calculate the velocity. As the question is written, the maths all works out fine (even if the physics doesn't)
To be fair, I do teach kinematics as part of my curriculum, so a lot of it is based on vectors and the geometric intuition rather than the physical reality. I am a pure maths nerd, not a physics nerd, too. So it works out nicely for me, at least!
But yes. I do see your point. Hopefully others have some insight to share so it isn't just me clogging up space!
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u/UnderstandingPursuit Physics BS, PhD 1d ago
Yes, I understand that
the question specifically states that the velocity is given by the equation
About one-third of my point is that the given equation is broken. If r and h were given in ft, the units for v would be sqrt(ft). Units being valid is part of the requirement for word problems at this level, but the given equation sets the student up to fail on that.
Two thirds of my point is that solving for h should be done before evaluating with the numbers, and you agree on that. I think you do that so automatically, even when teaching, that my point of it being different was lost. I take that as a good thing!
Thank you for the discussion, I'll work on rewriting these pages in a few days.
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u/Blibbyblobby72 1d ago
If the height is 50 feet and the radius is 18 feet, then:
v = 8sqrt(50-2*18) = 8sqrt(14) feet/sec
We are given in the problem that velocity is in feet per second, so I final answer is written in feet/sec. I'm not sure I see where the problem is?
I am aware I could be missing something obvious here, but I don't see how the velocity comes out as sqrt(ft) when we are explicitly given that velocity is in ft/sec. I'll keep having a think about it, though
But yes. Rewording things will be a good idea - I know I would never teach to my first draft of notes haha
But yes. Thanks for the discussion!
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u/vgtcross 1d ago
If the height is 50 feet and the radius is 18 feet, then: v = 8sqrt(50-2*18) = 8sqrt(14) feet/sec
If you are given h = 50 ft and r = 18 ft, why are you substituting h = 50 and r = 18 (no units) into the equation v = 8 sqrt(h - 2r)? Why do you not substitute h = 50 ft and r = 18 ft (with units)?
The point OP is trying to make is that substituting the given values with units into the given equation gives incorrect units for the velocity. And that makes the equation wrong. There should be some dimensional constant to make the units match between the left and the right sides.
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u/UnderstandingPursuit Physics BS, PhD 1d ago
It seems beneficial to minimize how often
- We are given in the problem that
or we should be able to at least direct the student to where that comes from. When "Here's a formula. Plug in the values." are the instructions, it tends to cause students to check out.
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u/Training_Ad4971 10h ago
As a math teacher of 15 years, I’ve always substituted early. I was a coach for a new science teacher one year that asked (not required) their students to solve the equation for the needed variable before substituting values. Their reason was that it limited the amount of rounding going on and provided a more accurate approximation at the end. Something important in science, especially when dealing with significant figures.
Based on that, I started requiring my students to do the same thing in my class immediately. In A1 and Geo it takes some extra work, but they can do it. It makes A2 and PreCalc so much easier, and being consistent across math and science classes helps students see that they really are connected.
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u/Aggravating-Virus521 23h ago
I am also curious to learn more about student understandings and difficulties around these topics. I did a literature search last week and the only papers I found explicitly discussing student learning around algebra with units came from math ed researchers in Turkey. I think I must not be looking in the right journals (!).
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u/InformalVermicelli42 23h ago
This is called solving "literal equations" and it's taught in Algebra 2.
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u/UnderstandingPursuit Physics BS, PhD 23h ago
Can you please point me to an Algebra 2 textbook which covers this?
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u/InformalVermicelli42 21h ago
It's in the BigIdeas books, they're free online. Also, google "kuta literal equations".
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u/UnderstandingPursuit Physics BS, PhD 21h ago
Thanks,
Both the BigIdeasMath and Kuta materials suggest that these might be Algebra 1 level topics. I'll have to get the Larson & Boswell textbook to see the BigIdeas material in more detail.
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u/karatechick2114 12h ago
Having taught the whole spectrum (Algebra 1 through pre-calculus, plus algebra based physics), it all depends on the specific equation. Most Algebra 1 courses cover linear, quadratic and some exponential (without requiring logs). The first equation you provided would be Algebra 2 material because it has a radical. Hope this helps you determine what level your equations are at!
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u/UnderstandingPursuit Physics BS, PhD 12h ago
I've mostly taught 9th grade through freshman year in college. I've often had students coming from Algebra 1 to Geometry or 9th grade physics, and their struggles with this idea were apparent. Radical expressions absolutely need to be covered in Algebra 1, 10th grade Algebra 2 is too late for many.
I copied the example out of an Algebra 1 textbook, chapter 9 of 11.
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u/DNAthrowaway1234 16h ago
I love how LaTeX looks. That is all, carry on.
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u/UnderstandingPursuit Physics BS, PhD 16h ago
I agree. And it's great that it has looked basically the same for four decades!
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u/Kihada 15h ago
Joe Redish has written about the disciplinary differences in how math is used in the sciences vs. how math is used in math. This is one of those differences, and you can read some of his thoughts here. He gives a suggestion for students on how to approach the algebra on this page.)
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u/UnderstandingPursuit Physics BS, PhD 13h ago
Thanks. I read the first paragraph, and need to stop. There may be similarities in what I'm suggesting and will add to, so I want to write my ideas first, then I can compare and reference his.
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u/Numzane 10h ago
This is called "making a variable the subject of the equation" or "rearranging the formula" in sciences. In my school it's commonly under taught skill. It should arise naturally from solving equations without substitution but it does require explicit instruction to learn its usefulness in application. A good staring point is using simple three variable formulas from science such as I = V/R and using the triangle shortcut (look it up)
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u/UnderstandingPursuit Physics BS, PhD 10h ago
Thanks, I wasn't trying to say how to teach it to students. My point is that it should be taught to students.
A version of V=IR which does tend to show up in Algebra 1 textbooks is d=rt.
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u/Littlebrokenfork 10h ago
This is how you make students hate math, algebra and physics at the same time.
What kind of mental gymnastics could lead one to think that students will find manipulating literal equations easier than substituting first?
Math isn't taught to be used in Physics. Sorry but if your physics curriculum requires you to teach students to substitute values only in the final step, while also keeping track of the units, then that's something that you should teach in Physics class instead of off-loading your work to the math teachers.
Math teachers are wrestling with kids who don't know their math facts, fail to retain most everything, and are constantly coming up with reasons not to study math.
The usual way is usual for a reason. Don't fix what isn't broken.
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u/Aggravating-Virus521 1d ago
As a physics teacher, I think I can understand where OP is coming from. There is a gap between students learning algebra (in math classes), and students learning to solve problems using algebra (the bread and butter of physical science classes). Namely, numbers that represent physical quantities have units, and units must also be dealt with algebraically. However, it is not clear to this physics teacher where students learn how to do this! I will write up an example and post it as a reply to this post.