belleville hs math ed stuff
"Technology and/or Developmental Math"
© 2017, Agnes Azzolino

Schedule for the Day
Work Session:9:00-10:30
Break: 10:30-10:45
Work Session:10:45-11:30
Work Session:1:00-2:30
Work Session:2:45-3:30
Faculty Meeting:   4:00-5:00
Morning Work Session Topics
"Visit My Classroom"
My cart comes with a laptop, a projector, and a "bamboo and stylus" with which I write notes which I post daily to the web. Mostly I lecture a little and keep quiet a lot so students can do their work then share work with those around them. I have spreadsheets, a Geometry Sketch pad page or two, manipulatives/models, and games to teach the "point" better than a lecture.
"Math Class Languages and Pedagogy"
The 20 or so "languages" with which I present material has four major language families: VERBAL/Auditory, WRITTEN/Symbolic, PICTORIAL/Visual, and CONCRETE/Kinesthetic. These languages are the motivation for the pedagogy.
"Forty+ Years of Math Teaching Experience and Techniques in Four+ Hours"
From whole number, decimal, fraction representation and computation to percents, area formulas, teacher education topics, prealgebra, algebras, and statistics, to precalc and calc, I've collected a few trick of the trade. Examine them. Maybe even have a little fun.
Afternoon Work Session Topics
"Forty+ Years of Math Teaching Experience and Techniques in Four+ Hours, Continued"
More of the same.

Visit My Classroom
Hi. Welcome to my class.
We will be doing things with an asterisk. We will be skipping things with no asterisk.
Feel free to examine them if you wish.
The workshop is mostly a lecture with detours in which participants can work on the material presented.
math home

  • Please read this letter.
  • *"Everything with the tablet was really sick."
  • *How Sick

The Languages of the Math Classroom
'98, '08, '09 Agnes Azzolino

Most Sophisticated and also the Most Basic

Most Sophisticated, Most Basic

VERBAL / Auditory WRITTEN / Symbolic     PICTORIAL / Visual     CONCRETE / Kinesthetic    
  formal spoken mathematics   written word   DIGITAL MANIPULATIVE     object
  informal spoken mathematics       written symbol   moving picture   model
  spoken symbol   semisymbolic   static picture   manipulative/token
  symbol speak   calculator symbol   numeral
  calculatoreze/computereze   graph
  web speak   nonverbal body language

Manipulative and Languages
  • *Features of the Languages of the Math Classroom
  • *manipulative use

Choose to Improve Instruction.
  • Choose a modality first.
            Visual, Auditory, Symbolic, or Kinesthetic
  • Introduce in the concrete.
  • Review in the abstract.
  • Sometimes use multiple modalities at the same time.
  • Strive for comfort in all modalities, not just your favorite.
  • Repeatition improves retention,
            especially in different modalities.

Vary the technique to achieve the above.
  • Use color rather than black and white.
  • Use active rather than passive.
  • Use pictures rather than just text.
  • Use spoken word rather than just text.
  • Use white space to improve clarity.
  • Larger is usually clearer than smaller.
  • Sometimes start with the big picture rather than the details.
  • Reference to earlier work improves comprehension.
  • Showmanship -- a performance -- improves retention.

Courses or Focus
Operations With Whole Numbers
Operations With Fractions
Operations With Decimals and Percents
Algebra Basics
First Degree Equations and Inequalities in One Variable
Systems of Linear Equations in Two Variables
Exploring Functions Through Manipulatives
Teacher Ed
One Last Use of Technology

Operations With Whole Numbers

  • *The Hundreds Board -- page and images
  • *The Hundreds Board -- Digital Manipulative and master page for multiples
  • page 40, 50 - Whole Number Addition and Subtraction and Multiplication and Division with multiples masters
  • *A Game for Two Players
  • *A Game for Two Players Digital Manipulative as 42.xls
  • *Prime Numbers Presented Logically as prime.xls
  • Recognition of Prime Numbers Exercise for Students in a Lab
  • slide rule
  • *slide.xls - slide rule digital manipulative
  • Someone's Powers of 10

Operations With Fractions

  • So, What's A Fraction?
  • *fract.xls -- Movable fraction bars from 1/1 to 1/15 on multiple sheets
  • Why "Invert and Multiply" When Dividing FRACTIONS
  • Multiple Strips & Fraction Bars
  • *strips.xls --DIGITAL MANIPULATIVE on a spreadsheet with multiple strips & fraction bars
  • Fraction Computation Stated in Symbolic Form (so you do not have to bother writing it out)
  • *Lessons in Dictionary
  • *Dictionary

Operations With Decimals and Percents

  • *compute.xls
  • *compute areas and volumes with area.xls

Algebra Basics and
First Degree Equations and Inequalities in One Variable

  • *Invisible Ones
  • *Term Tiles & Tokens
  • *A Number, Zero, the Opposite, "Zero Out" to Simplify and Solve
  • Tough Topics to Teach
  • *Digital Manipulatives Spread Sheets:
        (for students, basic tiles only) tiles.xls
        (for students, basic + hot cells) create.xls
        (for teacher and to supplement students', all tiles, tokens, hot cells) hot.xls
  • Unit 17: Algebraic Multiplication
    *student work on a test

  • The Square of the Difference of Two Numbers.
    *** On the spreadsheet, use term tiles to compute the square of the difference of two numbers.
    *** With manipulatives try to assemble the cube of the difference of two numbers.
  • Source page
  • Appendix and Masters and Digital Manipulatives
  • "Who Has" Linear Expressions Game
  • Code, Semi-equations/semi-expressions

Systems of Linear Equations in Two Variables

  • Solve A Systems of Linear Equations by Term Tiles
  • *cramers.xls
  • Solve Systems of Equations Graphically answers

Exploring Functions Through Manipulatives

  • Entire Book
  • Lecture Handout
  • *Please wait to do Graph by Placement
  • more graphing
  • *solving
  • *raw masters
  • absolute value dilator in hand
  • *Slopemeter w/movement
  • *"Math Exercises."
  • Math Exercises web page

Teacher Ed

  • nomogrf.xls -- Nomograph for whole, signed, fraction, decimal computation webpage
  • The Decimal Bakery
  • Napier's Bones spreadsheet webpage
  • *Area Formulas spreadsheet
  • Assessment Inventory -- A Journey with Self-Assessment as a Compass
  • *Math Tokens


  • Hypothesis Tests
  • *Spreadsheet to write Hypothesis Tests
  • Other Stuff


  • Name it with a title to make it more concrete and for better recall
    *2 mile race (visual with auditory and kinesthetic)
    *basic Veronica's bow tie
  • internet trig reasources at c33pg00.htm
  • *central angle, clockwise/counter-clockwise, coterminal, negative/positive angle measure
  • *Choose paper or spreadsheet
  • The Unit Circle
    *web, static, but with color
    *web, active
          The Animated Unit Circle. - animation w/text.
  • The Usual Way vs. The Active Way
    *sine/cosine only, web active
    *quadrantals notes the usual way
    *sine and cosine graphed
    *write on similar triangles
    tangent actively
  • Proofs
    *Pythagorean Identities -- graphically and by hand symbolically
    *Examine odd/even identities
  • Blank paper
        3 planes per page, gif
        3 planes per page, web
        *Other graph papers
  • *Review with Explorer
  • *Library of connect-the-dots puzzles/worksheets
  • *Assign students to build a cheat sheet and continue
  • *Curve shift with sine sine on a spreadsheet
  • *poly.xls
  • *Other spreadsheets


  • *Take a derivative with a pencil
  • *Casa Omar answers
  • Other Connect-the-Dots

One Last Use of Technology

  • Review with Windows Explorer.

[MC,i. Home, site home page] [Good Stuff -- free & valuable resources] math games toc [tutorials & resource material arranged by topic
[MSH! Home -- arithmetic & algebra dictionary home page] [master list of all MSH! dictionary words] math tokens
© 4/2017, A2