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What Algebra Is Traditional and What Is Reality?. Johnny W. Lott Past President, NCTM [email protected] What does “algebra” mean to you?. Why the switch to algebraic thinking versus algebra?. What is algebraic thinking? Do I ever engage in it? Do my students ever engage in it?

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What algebra is traditional and what is reality

What Algebra Is Traditional and What Is Reality?

Johnny W. Lott

Past President, NCTM

[email protected]



Why the switch to algebraic thinking versus algebra
Why the switch to algebraic thinking versus algebra?

  • What is algebraic thinking?

  • Do I ever engage in it?

  • Do my students ever engage in it?

  • Does anyone think that it is important?

  • What do parents think about algebra for the very young?


What are nctm s thoughts
What are NCTM’s Thoughts?

  • All students need algebraic thinking to learn analytical and reasoning skills.

  • With algebraic thinking as a strand, students see interconnections in math.


Algebraic thinking as a strand requires re-evaluating your entire mathematics curriculum and re-thinking teaching methods used.


What are some guiding themes for an algebraic thinking strand
What are some guiding themes for an algebraic thinking strand?

  • Mathematical Language

  • Patterns

  • Relations and Functions

  • Multiple Representations

  • Modeling

  • Structure


Contributing themes
Contributing Themes strand?

  • Problem Solving

  • Communication

  • Connections

  • Reasoning


Principles
Principles strand?

  • All students need algebraic thinking.

  • Algebraic reasoning is as fundamental as algebraic manipulation.

  • Appropriate tools should be used to teach algebraic thinking.

    Do you accept these principles?


Now i get to pose problems you get to work
Now, I get to pose problems; you get to work. strand?

  • Would you work for me?


On the first day, I will pay you $1000 early in the morning., At the end of the day, you must pay me a commission of $100. At the end of the day we will both determine your next days’ salary and my commission. I will double what you have at the end of the day, but you must double the amount that you pay me. Will you work for me for a month?


What is a variable
What is a variable? morning., At the end of the day, you must pay me a commission of $100. At the end of the day we will both determine your next days’ salary and my commission. I will double what you have at the end of the day, but you must double the amount that you pay me. Will you work for me for a month?


Mathematical language
Mathematical Language morning., At the end of the day, you must pay me a commission of $100. At the end of the day we will both determine your next days’ salary and my commission. I will double what you have at the end of the day, but you must double the amount that you pay me. Will you work for me for a month?

  • Pick any number; keep it a secret.

  • Add 15.

  • Multiply the sum by 4.

  • Subtract 8

  • Divide the difference by 4.

  • Subtract 12

  • Tell me your answer and I know the original number.


I can find your number. morning., At the end of the day, you must pay me a commission of $100. At the end of the day we will both determine your next days’ salary and my commission. I will double what you have at the end of the day, but you must double the amount that you pay me. Will you work for me for a month?

Let n be the number.

n + 15

4(n + 15)

4(n + 15) – 8

[4(n + 15) – 8]/4

[4(n + 15) – 8]/4 – 12

What is the result?


Patterns relations functions
Patterns, Relations, Functions morning., At the end of the day, you must pay me a commission of $100. At the end of the day we will both determine your next days’ salary and my commission. I will double what you have at the end of the day, but you must double the amount that you pay me. Will you work for me for a month?

  • Use pattern blocks.

  • Use Illuminations from www.nctm.org/illuminations/

  • They can be fun!

    • O, T, T, F, F, S, S, E, N, T, …

    • 61, 52, 63, ____, 46, 18, 001

    • They can drive you nuts!


Let s try common patterns and push them
Let’s try common patterns and push them. morning., At the end of the day, you must pay me a commission of $100. At the end of the day we will both determine your next days’ salary and my commission. I will double what you have at the end of the day, but you must double the amount that you pay me. Will you work for me for a month?

  • Think of all the sum families that add to 12.

  • Think of all the sum families that add to 15.

  • Think of all the sum families that add to 20.

  • And so on.

  • Graph the pairs in the sum families of each one.

  • What do you see?


Explore over 1 and down 2 on a hundreds chart
Explore “over 1 and down 2” on a hundreds chart. morning., At the end of the day, you must pay me a commission of $100. At the end of the day we will both determine your next days’ salary and my commission. I will double what you have at the end of the day, but you must double the amount that you pay me. Will you work for me for a month?

  • 1 2 3 4 5 6 7 8 9 10

  • 11 12 13 14 15 16 17 18 19 20

  • 21 22 23 24 25 26 27 28 29 30

  • 31 32 33 34 35 36 37 38 39 40

  • 41 42 43 44 45 46 47 48 49 50

  • 51 52 53 54 55 56 57 58 59 60

  • 61 62 63 64 65 66 67 68 69 70

  • 71 72 73 74 75 76 77 78 79 80

  • 81 82 83 84 85 86 87 88 89 90


Patterns relations and functions
Patterns, Relations and Functions morning., At the end of the day, you must pay me a commission of $100. At the end of the day we will both determine your next days’ salary and my commission. I will double what you have at the end of the day, but you must double the amount that you pay me. Will you work for me for a month?

  • Sally is hosting a party. Every time the doorbell rings, three more people come to join the party. Identify the pattern and write a mathematical sentence for this pattern.


Consider the pattern that follows
Consider the pattern that follows: morning., At the end of the day, you must pay me a commission of $100. At the end of the day we will both determine your next days’ salary and my commission. I will double what you have at the end of the day, but you must double the amount that you pay me. Will you work for me for a month?

  • 4, 7, 10, 13, 16, 19, …

  • Let’s use a spreadsheet.



Where did we move from lower grades to high school
Where did we move from lower grades to high school? overhead.

  • Can the same type of problem be used?


What does any linear function look like
What does any linear function look like? overhead.

  • f(n) = an + b

  • What does almost every linear pattern (sequence) look like?

  • Retreat to the overhead!


Patterns relations and functions1
Patterns, Relations, and Functions overhead.

  • With technology, use a pendulum to measure a 5-second interval accurately.

  • How?

  • Use a motion detector to collect data that measures the motion of a pendulum. Examine the graph; determine the function.


Patterns relations and functions for most
Patterns, Relations, and Functions--for Most overhead.

  • Spreadsheet yields recursive formula

  • Algebra yields explicit formula

  • Which is better?

  • What are we learning?


How do you prove that the pattern is what you say it is
How do you prove that the pattern is what you say it is? overhead.

  • At what level?

  • How would you do this in high school?


Revisit a twisted old problem
Revisit a Twisted Old Problem overhead.

  • Consider all the pairs whose product is 12

  • Consider all the pairs whose product is 24.

  • Consider all the pairs whose product is 36.

  • Consider all the pairs whose product is 72.

  • What do these graphs look like?


Back to early grades teddy bear mathematics
Back to Early Grades overhead.Teddy Bear Mathematics

  • A set of teddy bears are on a shelf in a child’s classroom. Count the number. Represent a relation between the number of bears and the total number of ears.

  • Try eyes.

  • Try paws.

  • Try noses


Move to figure this
Move to Figure This! overhead.

  • Examples abound where algebraic thinking exists and equations can be solved.



Old but good modeling problems
Old BUT Good Modeling Problems prices?

  • How do you model operations with integers?

  • Postman Stories

  • Elevator

  • Number Lines

  • Can this be extended to algebraic thinking?


For many
For many prices?

  • A “wave” is often seen at sporting events. How long would it take the wave to travel around this room?

  • How many times would you have to simulate this to be reasonably sure that you could write a formula for the process?


Modeling for algebra
Modeling for Algebra prices?

  • According to The Missoulian, the trees in a certain land area are being cut at a rate of 15% per year. The lumber company claims that it replants 2000 trees every year in this area. Discuss the future tree production of this land area if this plan continues.


Old but good modeling problems1
Old BUT Good Modeling Problems prices?

  • How do you model operations with integers?

  • Postman Stories

  • Elevator

  • Number Lines

  • Can this be extended to algebraic thinking?


For many1
For many prices?

  • A “wave” is often seen at sporting events. How long would it take the wave to travel around this room?

  • How many times would you have to simulate this to be reasonably sure that you could write a formula for the process?


Modeling for 9 12
Modeling for 9-12 prices?

  • According to The Missoulian, the trees in a certain land area are being cut at a rate of 15% per year. The lumber company claims that it replants 2000 trees every year in this area. Discuss the future tree production of this land area if this plan continues.


Structure
Structure prices?

  • The stained glass window consists of two regular hexagons. How many ways can you represent the green as a part of the whole?


Two examples more
Two examples--more? prices?

What is the structure that is being presented here? What are other ways to show this structure?


More structure
More Structure prices?

  • The set of all square matrices

    • Operations of addition and multiplication

      What properties of real numbers still hold?



Selected references
Selected References of your mathematics program?

  • Montana Council of Teachers of Mathematics. Algebra: A Strand PreK-12. Missoula, MT: MCTM, 1998.

  • www.nctm.org

  • www.figurethis.org

  • www.nctm.org and search for illuminations


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