If our atoms were expanded to the size of BBs,

a simple molecule would be the size of a marble,

a chain molecule might be the size of a cat, a molecular structure would be the size of a

trailer tractor, a cell would be as big as an ocean liner, anda human being would be about the size of

North America.

Many molecules

If you could mark all of the water molecules in a single glass of water and then pour them and mix them thoroughly into the seven seas, and then you took a glass of water anywhere from the out of the ocean, you would find in the glass about a hundred of your marked molecules.

Life runs on sugarSun, CO2, H2O -> Glucose, O2

• Each year terrestrial and marine plants make enough glucose to fill a freight train 30 million miles long.

Who are you?

• Each day about 7% of your molecules are "turned over." So ~100% of your molecules will turn over in about 2 weeks.

The Way Life WorksBy Mahlon B. Hoagland, and Bert DodsonTimes Books, a division of Random House, Inc. (October 1995)

From the 4-7th grade mathematics framework statement

While engaged in meaningful mathematical experiences, students define and solve problems, reason logically, and communicate theirunderstanding and knowledge to parents, peers, teachers, and themselves. Appropriate problem-solving experiences provide students with a real world context in which to learn the concepts and mechanics of mathematics. Students are encouraged not only to think, but also to reflect on their thinking. They communicate their understanding through words, pictures, or numbers; develop an appreciation for the fact that there are many ways to solve problems; and develop an understanding of the role mathematics plays in their lives.

While engaged in meaningful mathematical experiences, students define and solve problems, reason logically, and communicate theirunderstanding and knowledge to parents, peers, teachers, and themselves. Appropriate problem-solving experiences provide students with a real world context in which to learn the concepts and mechanics of mathematics. Students are encouraged not only to think, but also to reflect on their thinking. They communicate their understanding through words, pictures, or numbers; develop an appreciation for the fact that there are many ways to solve problems; and develop an understanding of the role mathematics plays in their lives.

From the 4-7th grade mathematics framework statement

While engaged in meaningful mathematical experiences, students define and solve problems, reason logically, and communicate theirunderstanding and knowledge to parents, peers, teachers, and themselves. Appropriate problem-solving experiences provide students with a real world context in which to learn the concepts and mechanics of mathematics. Students are encouraged not only to think, but also to reflect on their thinking. They communicate their understanding through words, pictures, or numbers; develop an appreciation for the fact that there are many ways to solve problems; and develop an understanding of the role mathematics plays in their lives.

From the 4-7th grade mathematics framework statement

While engaged in meaningful mathematical experiences, students define and solve problems, reason logically, and communicate theirunderstanding and knowledge to parents, peers, teachers, and themselves. Appropriate problem-solving experiences provide students with a real world context in which to learn the concepts and mechanics of mathematics. Students are encouraged not only to think, but also to reflect on their thinking. They communicate their understanding through words, pictures, or numbers; develop an appreciation for the fact that there are many ways to solve problems; and develop an understanding of the role mathematics plays in their lives.

From the 4-7th grade mathematics framework statement

How we learned mathematics...

Sketch each function between ±2

6.1 Core Content: Multiplication and division of fractions and decimals

• 6.1.D Fluently and accurately multiply and divide non-negative fractions….

In the CCSS this is 6.NS.1: “Apply and extend previous understandings of multiplication anddivision to divide fractions by fractions.”

We can do better

What are algorithms?

“An algorithm is a precise, systematic method for solving a class of problems. An algorithm takes input, follows a determinate set of rules, and in a finite number of steps gives output that provides a conclusive answer.” Stephen B. Maurer

Maurer, S.B. (1998). What is an algorithm? What is an answer? In Morrow & Kenney (eds.), The Teaching and Learning of Algorithms in School Mathematics. Reston, VA.: National Council of Teachers of Mathematics.

• 54 x 17 = ?• Algorithms are, by their nature, abstractions and

abbreviations. • Students may develop “transitional algorithms”

on their way to procedural fluency with the standard algorithms.

• All algorithms should be efficient, mathematically valid, and generalizable.

What are algorithms?

http://www.wsmc.net/pubs/WaMath/fall_2008/Algorithms.pdf

Bornemann, et. al. (2008). Algorithms. Washington Mathematics. WSMC