Mathematics and scheduling of buses


Image courtesy of welshmackem


I found this article via Math Teachers at Play carnival... it talks about a new method of scheduling buses, which uses mathematics, namely Markov Chains.

I'm not personally familiar with Markov chains, but whenever mathematics solves a problem in real life, I find it fascinating!

The mathematical method "evens out" the headway (the spacing between buses) by holding each bus at the start of the route a specified amount of time. That avoids "bus bunching", and it also makes it where the bus drivers don't need to adhere to any schedule--reducing their stress.

The researcher John Bartholdi says

The trick is to hold the bus for an adjustable amount of time at one stop. We simply control how long they wait at the end of the route, and then we tell them, ‘drive comfortable with the traffic to the other end. Don’t worry about where you are. Just flow with the traffic.’

Read more: Markov Chains avoid buses arriving in threes
Sounds cool!

Read it Again

My summer reading theme this year is "Re-Reads."  Last year I caught up with various sub-cultures a bit as I read through some of the most influential books from the past 10 years.  This year, I'm catching up with my former self and standing her back-to-back with my present self, measuring the distance between.  I'm re-reading those books that had significant influence on me, maybe left me breathless, and asking the question, "Does this book still hold its place?"

These books are attached to moments and experiences.  When I read them I endured a hard/interesting season of life, pondered a question, sought out a perspective.  As I re-read the books, I'm looking back to where I was the first time, seeing where I've changed, how life took turns I wasn't expecting.

The list is significant.  But the moments re-lived may be more overwhelming.

East of Eden
East of Eden
To begin I've picked up the book that began my love of Steinbeck, East of Eden.  I took it along on a roadtrip with my husband up California's Highway 101 through Steinbeck country. Three years into marriage.  One church down.  Seminary educations in full swing. Childless and idealistic.  This story gave me a dose of reality I couldn't relate to and yet couldn't shake.  Have I seen and understood more of this reality in the time in between?  And just when I had my Steinbeck collection all in matching orange spines, why did Penguin go and change the design?
The Pursuit of God
The Pursuit of God

Secondarily, I've re-opened The Pursuit of God by A.W. Tozer on my Kindle.  This is one that I read during our impossible journey years ago.  When we were packing up everything to put in storage for an undetermined amount of time, I kept this book out knowing I needed what it had for me.  While waiting, praying, crying out to be heard, hoping, sighing, living without a home, without a job, or a purpose  I'd put our two kids to bed in our borrowed rooms at my parents house and go out on the arbor swing and just let Tozer teach and calm my anxieties.  Our current season is the opposite of that one.  Will Tozer still give clarity toward a fuzzy future?


The rest of the list:
Walking on Water by Madeleine L'Engle
Ordering Your Private World by Gordon MacDonald
The Count of Monte Cristo by Alexandre Dumas
Surprised by Hope by N.T. Wright
Men and Women in the Church by Sarah Sumner
Working the Angles by Eugene Peterson
Gift from the Sea by Ann Morrow Lindbergh
Night by Elie Wiesel
Entering His Presence by Don McMinn
The Skilled Helper by Gerard Egan
Native Son by Richard Wright
Sacred Companions by David Benner
Bittersweet by Shauna Niequist




More about Area - free worksheet for grade 3

Here's a little sample of what is keeping me so busy right now - third grade math, aligning things to the Common Core standards.

This is a complete lesson (4 pages) about area for grade 3 math. It's not the initial lesson - it assumes the student already knows how to find the area of a rectangle. (Sorry - there is no answer key at this time; it hasn't been made yet!)



The lesson (worksheets) is specifically about these two standards:

3.MD

7.c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

 d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

The Effort of Growing


God in the Yard: Week Two
"Spiritual life and growth is complicated by the question of whose job it is to keep our lives from falling into nothing." -- L.L. Barkat
This week, the Salvia were engorged with blooms, the Columbine were coming to their grand finale, the Jack Frost was saying a sweet good-bye with his dainty blue flowers, the Hostas were just emerging with force and fullness.  And this week we had a hailstorm.

There was utter helplessness in my chest that night. All the beauty that I had partnered with the plants to create was decimated.  Cleared away.  I could do nothing to save them.  I could only hope they would endure into the next season.  As I deadheaded the beds last night I found myself encouraging them onward.  You don't have to fall into nothing.  There's still life left in you.  Keep trying to grow.   I'll help.

Around me, lives are falling into nothing.  Friends who once had strength are currently powerless.  Those in a place of influence are finding that fewer and fewer are paying attention anymore.  Among them is a sense of general ineffect.  And though their great griefs tug at me, in some sense it reassures me that everything is as it should be.

It's the ebb and flow.  One cannot be hot all the time, though we fool ourselves into thinking it's possible.  My experience says that the ebbs are what's real and the flows are really a sort of appearance.  What's for certain is the rhythm.  When we are full the thing that assuredly comes next is a great emptying.

If that's true, then is there anything that can be done to keep my life from falling into nothing?  And if there is, then does that effort fall to me or to God?

Years ago, the depths of my Calvinism would have pronounced, "Whatever will be, will be."  Through the miracle of wandering homelessness I saw that in the end there's nothing I can really do.  No one is truly self-made.  In recent years,however I felt more of that effort falling to me, a sort of gift from God who says, "In this season you can take some of the reigns, but be wary of snakes in the way of the horse."  I recognize the partnership we keep with God much more readily, and as a result my part has risen from nothing to actual service.  And still when we fly he keeps us aloft; when we fall he inclines to rescue us.

I believe it's my job to preserve my soul because there are not six degrees of separation that will do it on my behalf.  Not my identity. Not my husband. Not my reputation -- my previous story.  What grows me is my service, my following a prompting, my disciplines, my open heart, my seeking... and my falling.  And yet, I also believe it is God's job to preserve my soul because the Spirit of God has been given that role to move, to prompt, to focus.  It is a partnership of sacrifice and submission, not primo and secundo, but as in most things of love, it is both/and.

I do not know how my garden will fare this summer.  Once the leaves are gone, and they are not all gone, how can the plant produce its food?  But I believe that the earth will replenish; that it is far more enduring than I.  Everything is as it should be.  And everything will be changed.