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Digital Manipulatives

© 12/2017, 11/2018, Agnes Azzolino  



"Tell me and I forget, teach me and I may remember, involve me and I learn."
- Benjamin Franklin

S152:   SATURDAY, 2:15-3:05 PM, CORONADO EF
      Hi. The session begins at 2:15.
      Feel free to work/play before that.
      Copies of the 5 spreadsheets are available on thumb drives in the room.

First We'll "Play," Then We'll Work

    Please introduce yourself to those around you.

    Manipulative theory says: "Begin with free play & unstructured time before setting the stage for work" so we will do something like that.

    Open the spread sheet just below and using the notes pictured on the spread sheet, see what you can do. Then we can better judge how to continue.

    Don't worry about messing up. You can always download another spread sheet.

Digital Manipulative Spread Sheet 1

Session Purpose and Objectives

    This session and page is built for both those who know how to use concrete manipulatives and those who do not.  If you do not know how to use a concrete manipulative, a digital manipulative is only a play thing. So we will consider a bit of theory first. This page is also written for those who have experience in cutting/selecting/pasteing and those who do not.

    The time it takes to complete this work depends on one's skill set.


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

Digital Manipulatives Purpose

    Digital manipulatives were created to:

  • Introduce in the concrete and debrief in the abstract.
  • Increase the number of graphics used in class - speak pictorial more often.
  • Increase visualization as a form of presentation, summary, debriefing.
  • Create math experts, those who have enough facility in multiple languages to choose the language in which to work and communicate.
  • Restate a mathematical idea in as many math class languages as possible for increased understanding on the part of both the speaker and the listener.
  • Bring a cart load of manipulatives to class without having to bring a cart load of manipulatives to class.

More Work and Play

Solve, & Create a Cartouche, & Add Vectors, & Write Digital Algebra Problems
Digital Manipulative Spread Sheet 2

Use multiple strips & fraction bars, & Add & Subtract on a Nomograph, & Assemble a jig saw puzzle, & Use Napier's Bones to Multiply, & Demonstrate that 2/9 + 1/3 is 5/9.
Digital Manipulative Spread Sheet 3

Compute & Play on the 100s Boards
Digital Manipulative Spread Sheet 4

Calculi and Boards, & Egyptian Multiplication, & Slide Rules, & The Sine Law
  • Chinese suan pan: 2/5, 2 over 5 (biquinary)
  • Japanese soroban: 1/4, 1 over 4 (decimal)
Digital Manipulative Spread Sheet 5
Multiply 25 x 12 by Doubling and Halving

Material Used Today

Digital Manipulative Spread Sheet 1


select copy paste Intro to Digit Manipulatives
This work from "Term Tiles & Tokens"
@ and
problems 2 - 4 Using Term Tiles to complete manipulative work.
This work from "Term Tiles & Tokens"
game for two players     The best middle school mental computation and vocabulary game ever.
This work from "A Game for Two Players"
@ and
extra tiles & tokens Storage. Because sometimes you need extra pieces.
answers Answers to some problems.

Digital Manipulative Spread Sheet 2


solve Solve 3 + 2x = 4x - 1
This material is found
@ and
hyro Create your own cartouche or write ancient Egyptian numbers or glyphs.
This is page:
vectors Add 4 @ 45° and 6 @ 135° by building the resultant vector on the rectangular plane.
This stuff is found
@ and
write problems Spread sheet completes polynomial computation and solves linear and quadratic equations.
This material is found

Digital Manipulative Spread Sheet 3


multiple strips/fraction bars     Write equivalent fractions. Add or subtract fractions.
nomograph To add & subtract wholes, integers, fractions, decimals.
See and for more info.
unit circle Assemble a unit circle jig saw puzzle.
Unit circle spread sheet found
Animated unit circle found at
Manipulative master found at
napier's bones Napier's Bones for multiplication, division, and taking roots. Problem 4: Multiply 2467 by 8.
Stored @
Multiplication with Napier's Bones is @
Division with Napier's Bones is @
Root Extraction with Napier's Bones is @
thirds, sixths, ninths Problem 5: Demonstrate that 2/9 + 1/3 is 5/9.
Found @
with many other fraction bars.

Digital Manipulative Spread Sheet 4


100s regular 100s board
negs positive, negative board
multiples 12 copies of the 100s board colored with multiples of 2 to 13
uncolored multiples 12 copies of the 100s board

Digital Manipulative Spread Sheet 5


calculiAbaxes (Sand Tables), Reckoning Boards, Banks (Using Calculi)
See: for history & use of abacus.
See: The "units" worksheet of
      for a table which does not have powers of the same base.
Egyptian Mult       Egyptian Multiplication
      for Egyptian Computation.
      Multiplication by Doubling and Halving (Mediation & Duplation) w/Calculi

Additional Material Not Used Today

Additional Material Not On This Site

The International Slide Rule Museum
  • Slide Rule History
  • Simulated Pickett N909-ES Slide Rule, © 2005, Derek Ross

  • [MC,i. Home, site home page] spread sheet notes spread sheet notes [master list of all MSH! dictionary words] [tutorials & resource material arranged by topic Good Stuff -- free & valuable resources math games toc   © 12/2017, 11/2018, 9/2023, A2