[Azyg] math in the garden-forward

[Roberta Gibson]

>Below is a newsletter that contains some ideas for incorporating math in
the garden. The author requested that it be transmitted in its entirety,
although some of the material is off topic for this list.

-Roberta Gibson
rlgibson@ix.netcom.com

>X-Sender: mathkits@mail.cyber-trail.net
>Date: Thu, 26 Apr 2001 10:16:14 -0600
>To: algebra@mystery.com
>From: josh rappaport <josh@algebrawizard.com>
>Subject: Algebra Times - April 2001 (II)
>Sender: owner-algebra@angus.mystery.com
>Reply-To: josh rappaport <josh@algebrawizard.com>
>X-Loop: Majordomo @ NSTS
>
>"QUICK-AND-EASY" LESSON PLAN #28
>
>Spring: allegies, runny eyes ... and then, finally, ah ...
>gardening.
>
>I love to garden, myself, but it wasn't until this year,
>working with a homeschooler who wanted to start his
>own garden that I realized how much math can be involved
>in the pursuit.
>
>Sure, you can always just turn some dirt over,
>sprinkle some seeds, and hope that something
>sprouts. But if you want to do it in a more
>disciplined way, the possibilities are enormous
>for math/gardening problems.
>
>What I'd like to do in this month's lesson plan is
>simply show how as a teacher or homeschooler,
>we educators can use gardening as a springboard
>to get children interested in some neat math problems.
>
>The best way for me to do this is to describe what
>I've been doing with this one student, and then you'll
>see how you can take the same kinds of situations
>to develop questions and lesssons for the children
>with whom you work.
>
>This boy, Chris, is an 18 year old Santa Fean, who has worked
>a bit with a landscaper, and who wanted to see what he can
>accomplish himself in terms of growing vegetables.
>
>(By the way, this boy is in the Clonlara School program,
>whereby he gets credit for everything he does during
>this project. If you'd like to check out the
>Clonlara School, which I personally find wonderful,
>visit: http://www.clonlara.org)
>
>Anyhow, here are some problems that have come up as he
>has been trying to plan out his garden:
>
>PROBLEM 1:  BEST DIMENSIONS OF THE GARDEN -
>Chris was on a budget, and we determined that, based
>on his budget (lots of arithmetic here, as you can imagine),
>he could afford, at most, 48 linear feet of chicken wire
>for fencing. With that in mind, he started to think about the
dimensions for
>his garden. Then he said, and I do quote:
>"You can use the same amount of fencing and still have
>the same area for diferent gardens. Can't you?"
>
>Oh boy! I couldn't have scripted that question any
>better. This question launched us into an exploration into
>the relationship between perimeter and area of rectangles. I asked
>Chris to test out several rectangular plots with perimeter
>of 48, and to see if their areas all come out the same.
>
>Chris tested three rectangular plots with perimeter of 48
>(20 x 4; 10 x 14; 12 x 12), Chris started to see the
>light: that area increases as the rectangle more and more
>approximates a square. As a follow-up problem, I asked
>Chris to calculate the area of his garden if he took his fencing
>and used it as the circumference of a circular garden
>(he was amazed when he came up with the answer -
>see this month's POTM below for the challenge stated
>formally.)
>
>Classroom teachers:  here is a problem that is worthy in
>its own right. As your students explore such a question,
>you might also ask them why they think a square maximizes
>the area of all rectangles with the same perimeter.
>
>Homeschoolers:  whether or not you are planning a garden,
>this would be an interesting problem for any student who
>has studied area and perimeter of rectangles. What's great
>about this problem is that the answer defies children's
>"common sense," and thereby engages their sense of
>wonder.
>
>PROBLEM 2: TESTING THE SOIL -
>  Once Chris decided on the shape of his garden
>(10 x 14, only because it fits better with the contour
>of his land), we decided that we had better test the
>soil. To do this, we ordered a Garden Soil Test Kit
>out of the Edmund Scientific catalog (800/728.6999
>or www.scientificsonline.com)
>[Maybe I should start charging for all these free
>ads, huh?]
>
>The kit is very cool, and it allows you to test your
>soil for pH, and then for concentrations of
>nitrogen, phosphorous and potash. Doing the tests
>involves both chemistry and a lot of arithmetic.
>We found out that Chris' soil has good pH
>for vegetables (6.5), but that it is extremely weak
>in nitrogen, phosphorous and potassium.
>
>Classroom teachers: there's no reason why,
>if your school allows for interdisciplinary projects,
>you couldn't have your children use such a soil
>test kit to evaluate the soil for a school garden.
>
>Homeschoolers: here's a fun way to blend the
>studies of horticulture, chemistry and math.
>Not only that, you'll get better tasting vegetables
>and prettier flowers!
>
>PROBLEM 3:  HOW TO ENRICH THE SOIL
>& CALCULATING UNIT PRICES FOR SOIL
>ENHANCERS -
>
>Armed with our soil-test information, we visited
>the local greenhouse. We were all set to purchase
>separate bags of nitrogen, phosphorous and
>potash to make our own customized blend of
>fertilizer for Chris' soil. But the salesman who
>waited on us warned us that this would be
>something of a waste, since Chris' soil had never
>been cultivated. Turns out that when dealing
>with virgin soil, you must first use mulch and compost
>to give it fiber, and peat moss to help your plants grow.
>Once your soil has a good fiber content, then you can
>also add fertilizer in the customized manner we'd been planning.
>
>The salesman at this first greenhouse recommended that
>we use equal parts of mulch, compost and peat moss.
>
>We did some comparison shopping to find the best prices
>on these three enhancers. To do this, Chris called up
>five different nurseries and made a chart with headings like the one
>below:
>
>Store     Product        Cost      Cubic Feet     Cost / Cubic Foot
>
>He found  these best deals:
>
>mushroom mulch:  $1.67/1.25 cubic feet = $1.33/ cubic foot
>cotton compost:  $2.90/2 cubic feet = $1.45 cubic foot
>peat moss:  $9.97/5.5 cubic feet = $1.81/ cubic foot
>
>Classroom teachers:  you could have your own students
>do research on this in the same way that we did our
>research. Have your students present their information
>in a spreadsheet (a computer spreadsheet if you have
>access to computers).
>
>Homeschoolers:  You can do this sort of project just
>as we did it.
>
>PROBLEM #4:  FIGURING OUT WHAT WE
>COULD AFFORD -
>
>Using the information we had, we set out to find
>out if we could afford to purchase equal quantities
>of the three soil enhancers.
>
>Our budget for soil enhancers was $100, and
>we had previously calculated that we needed a total
>of 70 cubic feet of enhancers for Chris' 10 x 14 foot garden
>(70/140 = .5 cubic feet of enhancer for every
>square foot of garden space, a previous calculation
>we had made).
>
>I asked Chris how to figure out whether or not we could
>afford a 1:1:1 ratio of enhancers, recommended by the salesman
>at the first greenhouse. To do this, Chris divided the number
>of total cubic feet needed by 3 (70/3 = 23.33 cubic feet)
>and then he did this calculation:
>
>23.33(unit cost of mulch) + 23.33 (unit cost compost)
>+ 23.33 (unit cost peat moss).
>
>Factoring out the 23.33 and plugging in values, he got:
>23.33(1.33 + 1.45 + 1.81) = $107.08. Pretty close to
>our $100 budget, but budgets are budgets. This led
>to the next question: could we still use 70 cubic feet
>of enrichers and stay within budget?
>
>We spoke to more greenhouse people to find
>out where we should cut back, and most of those
>we spoke to said that when dealing with virgin soil,
>the least important ingredient is peat moss, so we decided
>to cut back on that. No problem, since that was the
>most expensive enhancer anyhow.
>
>The math question then became:  if we keep the
>two quantities of compost the same and decrease
>the peat moss, how much do we have to decrease
>it to hit our target of $100.
>
>Chris set up an algebraic equation as follows:
>
>let x = the amount of mushroom mulch in cubic feet
>
>Since we want to use the same amounts of compost
>as mushroom mulch, x is also equal to the amount of compost.
>
>Since all three enhancers must add up to 70 cubic feet,
>the amount of peat moss may be expressed as 70 - 2x
>
>Using our unit prices, we can then come up with
>an equation to give us the information we need,
>namely:
>
>(cost mushroom mulch) + (cost cotton compost)
>+ (cost peat moss) = $100
>
>Plugging in our values, we got:
>
>$1.33x + $1.45x + $1.81(70 - 2x) = $100
>
>Solving this, Chris got:
>x = 31.78, which we rounded to 32,
>meaning that to get the budget down to $100,
>we must use 32 cubic feet of the mushroom mulch
>and compost. That leaves us room for just
>6 cubic feet of the peat moss. (70 - 32x2 = 6)
>
>Classroom teachers:
>You can give your students a problem like this, or
>let them set a budget, set a cubic foot amount
>of enhancers, and have them use their actual data
>to get the answer.
>
>Homeschoolers: Once again, this sort of problem
>gives you a chance to use a real-life math. And it's more
>motivating, of course, if you actually use the information
>to create a garden.
>
>That's all for now.
>Happy gardening!
>
>XYXYXYXYXYXYXYXYXYXYXYXYXYX
>
>PROBLEM OF THE MONTH
>
>SOLUTION TO MARCH'S
>PROBLEM OF THE MONTH:
>
>Here once again is the problem -
>
>Imagine that you are playing darts on an unusual
>dartboard. This dartboard has two zones only:
>1) an outer zone, and
>2) the inner bull's-eye zone.
>Any dart landing in the outer zone earns you 5 points.
>Any dart landing in the inner zone earns you 8 points.
>
>The question is this:
>If you can throw any number of darts whatsoever
>what is the highest score that you CANNOT possibly
>earn using this dartboard.
>
>A little explanation:  It's easy to see that when using this
>dartboard, you cannot score 1 point. Nor can
>you score 2, 3, or 4 points. You can score 5
>points (by hitting the 5-zone), but you can't score
>6 points or 7 points. You can score 8 points
>(by hitting in the 8-zone). So you can see that
>there are certain scores you cannot attain. For all numbers
>beyond a certain number, you can - through combinations
>of darts - attain any score. This problem asks
>you to figure out the largest number that you
>CANNOT attain using this dartboard and an unlimited
>supply of darts and throws.
>
>The first subscriber who sent in the right answer was ...
>Kim O'Hara, 44-year-old homeschool mom, with 2 kids
>(mostly self-taught by now) still at home in Olympia,
>WA. Kim is also the Articles Editor for Home
>Education Magazine, and she has been a regular contributor
>to the Problem of the Month for many years now.
>
>So congratulations, Kim. You just won yourself a free
>copy of the Algebra Survival Guide.
>
>Kim used an interesting solution that involved
>analyzing the possible last digits in any given score.
>Here it is, in her own words:
>
>One way to solve this is to look at the last digit of the number you
want to
>reach. For example, once you can reach 5, you can also reach every
number
>that ends in 5 by adding enough pairs of 5 to reach it. So with 5's
alone,
>every number ending in 5 is reachable:
>       5+5x (where x is some even number of 5's)
>
>Here are the formulae for each ending digit:
>
>0 --> 5x
>1 --> 8+8+5+5x (start from 21, the first reachable number ending in 1)
>2 --> 8+8+8+8+5x (start from 32, the first reachable number ending in
2)
>3 --> 8+5+5x (start from 13, the first reachable number ending in 3)
>4 --> 8+8+8+5x (start from 24, the first reachable number ending in 4)
>5 --> 5+5x (start from 5)
>6 --> 8+8+5x (start from 16)
>7 --> 8+8+8+8+5+5x (start from 37)
>8 --> 8+5x (start from 8)
>9 --> 8+8+8+5+5x (start from 29)
>
>In each case, the last non-reachable number is:
>
>0 --> 0
>1 --> 11
>2 --> 22
>3 --> 3
>4 --> 14
>5 --> none
>6 --> 6
>7 --> 27
>8 --> none
>9 --> 19
>
>So the highest non-reachable number is 27.
>
>This month we had a total of 22 respondents who got
>this problem right. And now here are the first nine people who
>sent in their correct answers quickly enough
>to be included in the March "Winner's Circle":
>
>- Randy Lomas, 37, of Pleasanton, CA,
>a teacher at Harvest Park Middle School
>
>- Shelly & Skona Brittain, 11 & 46,
>of Santa Barbara, CA, student and teacher/Mom
>at SB Family School
>
>- Joyce Fetteroll, 44, of Medfield, MA, (although
>originally from Pittsburgh, PA), who is homeschooling her
>daughter Kathryn, age 9. Joyce adds that she has a fantastic
>husband, a dog, 2 cats and not enough time in the day!
>
>-  James Shapiro, high school math teacher at Santa Cruz
>High School in Eloy, AZ.
>
>-  Meridan Pickett, 13, of Olympia, WA,
>a homeschooled student who really likes math. y=mx+b YEAH!
>
>-  Catherine Mokede, 46, of Reading, PA,
>who is homeschooilng two children and says:
>"I actually enjoy doing stuff like this and am trying to convince my
kids
>that it's fun.  I had my 13 yo do the problem too to be sure I was
right!"
>
>-  Jim Poston
>
>-  Sarah Gopher-Stevens, a mother of 4 from Lafayette, CA
>
>--  Marna, 36, of Kirkland, Wa, homeschooling mother of two,
>and former nuclear engineer
>
>
>APRIL'S PROBLEM OF THE MONTH -
>
>First person to email me the answer receives a free copy
>of my book, the Algebra Survival Guide. All you have to pay
>is the cost of shipping.
>
>In the next newsletter I'll name the first 9 others who got this
>right. Due to overwhelming response, I can no longer give
>subscribers second chances. But you'll find out your status
>in the next newsletter.
>
>DON'T FORGET: when you submit answers for the Problem
>of the Month, please write
>POTM
>on the Subject line of your email's header and also tell me:
>
>-- your age
>-- your hometown
>-- status:  student, teacher, tutor, etc.
>-- whether in school or homeschooled
>-- anything else about you of interest
>Here's the problem -
>You have a set amount of fencing for enclosing a garden.
>You are pondering whether to make the shape of your
>garden square or circlular.
>
>a)  Which kind of garden - square or circlular - would
>maximize the garden's area?
>
>b)  If you compare the areas of the two gardens,
>what is the ratio of the area of the larger garden
>to the area of the smaller garden.
>
>Express your answer first as a ratio. Then calculate
>and show by what percent the larger garden's area exceeds
>the smaller garden's area.
>
>ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
>
>TRICK OR TREAT: ANSWERS TO PRACTICE PROBLEMS -
>
>a)  32/39
>7 apart; 7 doesn't divide in, so it's reduced
>
>b)  40/51
>11 apart; 11 doesn't divide in, so it's reduced
>
>c)  25/38
>13 apart; 13 doesn't divide in, so it's reduced
>
>d)  28/33
>5 apart; 5 doesn't divide in, so it's reduced
>
>e)  52/65
>13 apart; 13 does divide in; reduces to 4/5
>
>f)  27/44
>17 apart; 17 doesn't divide in, so it's reduced
>
>g)  88/95
>7 apart; 7 doesn't divide in, so it's reduced
>
>h)  69/92
>23 apart; 23 does divide in; reduces to 3/4
>
>j)  22/25
>3 apart; 3 doesn't divide in, so it's reduced
>
>k)  51/80
>29 apart; 29 doesn't divide in, so it's reduced
>
>Practice makes perfect!
>
>YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
>Copyright 2001, by Josh Rappaport. All rights reserved.
>May be redistributed if the entire newsletter, including signature,
>is used.
>
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>--
>Josh Rappaport
>Publisher, Singing Turtle Press
>#770, 3530 Zafarano Drive #6
>Santa Fe, New Mexico 87505
>
>Voice:  1-505/438-3418
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