Sunday, October 26, 2014

Leafy Math

This post is in honor of my friend who is studying math – algebra, to be specific.

This afternoon Younger Daughter, Son, and I raked leaves, because it is the most pressing thing on my list, because it was a beautiful day, and because the Common Household labor force nearly doubled with the arrival of my son.
How much does that yard bin hold, anyway?
After about 30 minutes of raking, I dismissed the kids, partly to reward Son and Younger Daughter for doing it without complaint (! it does happen !), and partly to promote sibling camaraderie.  I steered them to two other projects: putting up some Halloween decorations, and composing word problems having to do with leaves and raking. 

When I rake leaves I am usually doing informal science or math in my head.   Just the other day I did an experiment to see which was a faster way to get the leaves to the top of the hill. 

Method 1: rake the leaves into a pile, lift the pile into the yard waste bin, drag the yard waste bin to the top of the hill, and then dump it out at the edge of the street. 

Method 2:  rake the leaves onto the tarp and drag the tarp to the top of the hill, then dump it out at the edge of the street.  

Method 1 takes umpteen years and Method 2 takes an eon.  The end result of my experiment was blisters on both my hands.

Or I do some math musing in my head – How many leaves are still in the trees?  What is the probability that the leaves will stay in the piles at the top of the hill?

There are a few more weeks' worth of raking up there.

How long before they fall to the ground?

In recognition of the courage and determination of my friend who is studying algebra, I asked my kids to come up with their own leaf algebra problem.

The Common Household Mom pondering some leaf math.
Just in case you would like extra math practice, or you wish to punish your children with word problems, I included the problems below.  I will admit I needed help to set up the equations for these problems.  Algebra forces you to think in a certain way (which is why it should be required in most curricula), and I am out of practice. 

Perhaps more interesting than the solution to the word problems is to gauge how realistic they are.  One of the tricky parts to these problems is that the leaves continue to fall while the raking is occurring.  That’s realistic, all right.

The algebra word problem lab

Younger Daughter’s word problem
Leaves are falling at a rate of 160 leaves per hour.  You can rake leaves at a rate of 3,000 per hour.  The leaves have been falling all week since you last got a chance to rake them, and you decided that you can’t stand it any more.  However, you only have 1.5 hours before you have to go to your church meeting.  How long would it take you to clear the yard, and can you do it before you go to church?  A calculator will be helpful.

Son’s word problem
Son, Older Daughter, and Younger Daughter are raking leaves.  Son can rake 1,000 leaves per hour, Older Daughter can rake 500 leaves per hour, and Younger Daughter can rake 1,100 leaves per hour.  If the total number of leaves in the yard is 5,000 and leaves are falling at a rate of 100 leaves per hour, how long will it take to clear the yard if
a) everyone starts at the same time, and
b) Younger Daughter starts 1 hour after Son and Older Daughter?

Answers in the comments.

If you have gotten all the way through this post, I am amazed and reward you with these questions:  Do you like algebra?  What is the most pressing thing on your list?

A fine afternoon's work.


Common Household Mom said...

Here are the answers.

YD's problem:
9.5 hours. No way can you get them all raked before your meeting. This is very realistic.

Son's problem:
a) 2 hours
b) 2.44 hours
This problem is less realistic - why is O.D. portrayed as such a slacker? There are way more than 5,000 leaves in the yard.

Common Household Mom said...

Details on solutions:

YD's problem:
One week = 7 days x 24 hours = 168 hours.
Number of leaves in yard before raking = 168 x 160 = 26,880 leaves.
26880 + 160x = 3000x

Son's problem:
a) Equation:
1000x + 500x + 1100x = 5000 +100x

b) Equation:
1000x +500x + 1100(x-1) = 5000 + 100x

Green Girl in Wisconsin said...

I have to say figuring out a story problem while doing manual labor (like raking leaves) sounds like the perfect blend of body/mind exercise.
You and those leaves. Every year. I like it.

Angie said...

I never did like math. My most pressing task is finishing a quilt hanging that my mother had started and couldn't complete prior to being diagnosed with Alzheimer's. I have to finish it before we come home for Thanksgiving.

Cheri @ Blog This Mom!® said...

"A calculator will be helpful"

Uh huh.


Cassi Renee said...

Isn't is amazing the way a whole paragraph can be condensed to simple equations? It's no wonder someone had to invent that shorthand.

When I was a kid, we had to rake beautiful sugar maple leaves. But our yard was MUCH smaller. (Now, with our very large yard, we just use the mower to mulch them once they've dried up.) But I always enjoyed the raking --it's such a gorgeous time of year, and those piles were always so tempting. Often we'd need to rake those leaves back into piles several times.

I love your photos up into the leaves. So pretty.

The Crislers said...

The only thing I liked about algebra was the word problems, especially if they made them somewhat realistic, with phrases like "and you decided you can't stand it anymore."

I admire your family's endurance, because holy cow, that's a lot of (pretty) yard, and a lot of (pretty) leaves.