Aops intro to algebra

[Lilaclady]

We have finally reached the famous farmer Fred optimization problem. My dd took all her math time yesterday on it and was able to get 2 of the questions but could not figure out the last one. I am of no help with it and could not find the other topics here on it. I wonder if anyone has any ideas of how to get more problems like this that she can work through. Thanks

[wapiti]

This topic is covered in the 1989 Dolciani algebra 1 that I recently bought (it wasn't cheap on Amazon but I got a better deal at Follet).  Unfortunately I don't have it where I am today, so I can't look, though that text tends to have quite number of good word problems.  I may have seen this topic in other algebra texts that I have though I can't recall and I'm not at home to check.  In the Dolciani, the topic has a different name.

 

Are you saying that she's not done with the lesson problem yet?  Maybe she'll get it after she sleeps on it.  (If she doesn't get it in her next attempt, personally, I'd probably just look at the solution at this point and think of some leading questions to help her along, though I know everyone has a different perspective on that.)

 

Eta, Farmer Fred is the lesson-too-far for my ds.  We are reviewing earlier chapters before we go back to take a stab at that.

[Lilaclady]

Thank you wapiti. Yes, she had to look at the solution after spending so much time on it she was able to figure out what she was doing wrong. I was just wondering if there are other problems like this out there that she can work on. I will look at the Dolciani book

[wapiti]

Has she done the exercises already?  What about Alcumus?  And I should double-check the Dolciani when I'm home tomorrow.

 

Eta, in the other books I looked at, I believe the optimization topic is called linear programming, a term which is also referenced in AoPS.

[Lilaclady]

Yes she has done the exercises up to 9.5 where the farmer Fred question is. I have tried to google questions in optimization for algebra but coming up empty. I will look into linear programming. Thanks again.

[wapiti]

A while back, Kathy in Richmond posted that she found it helpful to first visualize the graph for the Farmer Fred problem before solving it.

 

FWIW, it seems that the topic may not be typically included in standard algebra 1 courses.  I looked at the various algebra books I have.  In Dolciani's Modern Algebra Book 1 (1975), Algebra Structure and Method Book 1 (1988) and Algebra and Trigonometry Book 2 (1986), linear programming is an extra challenge sort of section.  The only texts in which linear programming was a regular topic were Dolciani's Algebra 1 (1989) and Algebra 2 and Trig (1989), though there the graphs look a little different from the other books - the solution graph was not always bound on all sides (eek - I haven't been through the lesson myself yet so that may not even be an accurate description).  Linear programming is also a lesson in Frank Allen's Modern Algebra, A Logical Approach (1964) though it's an extra there as well, an example of math-in-the-real-world.  I did not see anything like this topic in my other collection of algebra 1 books (Jacobs, Foerster, Prentice Hall) though it's possible that I missed it.  I'd like to read the different texts on this particular topic when I have a chance - it's actually interesting.

 

Question for anyone who happens to read this - is linear programming, a.k.a. optimization, a standard Alg 2 topic?  (Just wondering.  I see that there's more optimization in Intermediate Algebra.)

[raptor_dad]

In the mid-60s Dolciani Structure Method Algebra books it is in an "Extras for Experts" section after CH13 Functions and Variation in Algebra 1 and in the Extras section after CH3 Systems of Linear Open Sentences in Algebra 2. These are probably the same sections as Wapiti's 80s copies. However, I thought I would chime in since some people ascribe magical properties to the mid-60s editions.

 

The second book has farmer problem that involves corn/wheat production where each requires different amounts of land, capital, labor in August, and labor in Sept. Based on the returns per crop and these constraints you optimize. It follows up with 3 problems merely based on the bound convex polygons and two more that involve TV manufacturers and vitamin pills. I don't own AoPS Algebra but this looks the most similar.

[kiana]

I don't think it's standard. The only place we taught it at college was in finite math courses, which were applied math courses post-college-algebra for business majors. There was also a linear programming for majors course but that would be considerably more in-depth.

[avilma]

I think I saw some linear programming topics in an Algebra 2 book used in an advanced algebra 2 class in a local high school. When I was in high school in another country I also had a short chapter in linear programming. That chapter somehow left me with the idea that I had learned how to do linear programming since I was able to do all the assigned problems. After a few years I saw some more linear programming problems and realized how little I had learned in high school. 

Based on my own personal experience I would suggest not to try to go beyond what AOPS provides. I think that the farmer Fred problem itself is one of the few problems in Intro to Algebra that can be safely ignored as it doesn't lead to other problems or other useful knowledge inside the AOPS curriculum.

Just for the curious I should mention that linear programming is a problem very suitable for computers. There exist lots of computer programs where one can enter their linear programming problem and retrieve the solution if it exists.