number sense & computational calisthenics

Judah L Schwartz, Apr 13, 2016

[b]This is a series of applets drawn from the website mathMINDhabits [[url]sites.google.com/site/mathmindhabits[/url]] that is designed for teachers of mathematics who want to deepen their understanding of the mathematics they teach and that their students are expected to learn.[/b] The applets in this collection all focus on the skills students need to develop with arithmetic computation and algebraic manipulation. In addition, the reader is urged to look at the program "What Do You Do With A Broken Calculator?" [url]http://www.teachscape.com/content/he012a/03_s3/07_online/he012a_calculator.htm[/url] and to read the essay on the "Broken Calculator" [url]https://sites.google.com/site/mathmindhabits/on-th-teaching-of-mathematics/what-do-you-do-with-a-broken-calculator[/url] that is on the mathMINDhabits website.

Table of Contents

  1. ordering
    1. Mystery Fraction Search
    2. Exploring the Ordering of Fractions
    3. Assessing Understanding of Fraction Ordering
    4. Common Denominator Finder
    5. Common Factor Finder
    6. Locate a mystery point!
    7. Binary Search on a plane & in a cube
    8. By all means! - averaging 2 numbers
    9. p R I m E !
    10. Inverse Prime Factorization
  2. estimating
    1. Power + Drill - Estimating Sums
    2. Power + Drill - Estimating Products
  3. place value
    1. Place Value - Decomposing for addition & subtraction
    2. Place Value - Decomposing Multiplication
    3. Power to the Base !
  4. computing
    1. The Array Model of Multiplication
    2. Many-answer addition
    3. Many-answer subtraction
    4. 'between-ness' in multiplication/division
    5. 'between-ness' in addition/subtraction
    6. The Fraction Workbench - working w/ UNcommon Denominators
    7. the game of arrows - adding signed numbers
    8. SeeSaw - multiplying signed numbers
    9. Common Factor Finder
  5. computing with measurements
    1. adding & subtracting uncertain measurements
    2. multiplying & dividing uncertain measurements
  6. transforming
    1. Exploring Factoring & 'FOIL' - I
    2. Exploring Factoring & 'FOIL' - II

1.

ordering

  • 1. Mystery Fraction Search
  • 2. Exploring the Ordering of Fractions
  • 3. Assessing Understanding of Fraction Ordering
  • 4. Common Denominator Finder
  • 5. Common Factor Finder
  • 6. Locate a mystery point!
  • 7. Binary Search on a plane & in a cube
  • 8. By all means! - averaging 2 numbers
  • 9. p R I m E !
  • 10. Inverse Prime Factorization

1.1.

Mystery Fraction Search

Find the mystery fraction by varying the numerator and denominator sliders. [the RED slider sets the maximum value of the denominator]
– Judah L Schwartz

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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1.8.

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1.9.

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1.10.

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2.

estimating

  • 1. Power + Drill - Estimating Sums
  • 2. Power + Drill - Estimating Products

2.1.

Power + Drill - Estimating Sums
– Judah L Schwartz

2.2.

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3.

place value

  • 1. Place Value - Decomposing for addition & subtraction
  • 2. Place Value - Decomposing Multiplication
  • 3. Power to the Base !

3.1.

Place Value - Decomposing for addition & subtraction

This applet is designed for teachers to use as a challenge environment with their students.[br][br]Some of the issues that can be explored with this applet include -[br][br][b]place value[br]estimation[br]"borrowing" and "carrying"[br][br][color=#ff0000][i]What questions could/would you ask your students based on this applet?[/i][/color][/b]
– Judah L Schwartz

3.2.

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3.3.

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4.

computing

  • 1. The Array Model of Multiplication
  • 2. Many-answer addition
  • 3. Many-answer subtraction
  • 4. 'between-ness' in multiplication/division
  • 5. 'between-ness' in addition/subtraction
  • 6. The Fraction Workbench - working w/ UNcommon Denominators
  • 7. the game of arrows - adding signed numbers
  • 8. SeeSaw - multiplying signed numbers
  • 9. Common Factor Finder

4.1.

The Array Model of Multiplication

IT IS CUSTOMARY TO TEACH MULTIPLICATION TABLES IN THE USA UP TO 12 x 12 BECAUSE THE COUNTRY WAS ONCE PART OF THE BRITISH EMPIRE If you have an array of cells that is 3 [math]cm[/math] on one side and 4 [math]cm[/math] on the perpendicular side then the product is an area of 12 [math]cm^2[/math]. If you have an array of cells with 3 rows and 4 columns and you multiply 3 rows by 4 columns the product is 12 row*columns. Use this applet to understand what multiplication in the array model really means.
– Judah L Schwartz

4.2.

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4.3.

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4.4.

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4.5.

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4.6.

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4.7.

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4.8.

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4.9.

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5.

computing with measurements

  • 1. adding & subtracting uncertain measurements
  • 2. multiplying & dividing uncertain measurements

5.1.

adding & subtracting uncertain measurements

All measurements are of necessity uncertain to some degree.[br][br]This applet allows you to study how this uncertainty is compounded[br]when measurements are added or subtracted.[br][br][color=#ff0000][i][b]What questions could/would you ask your students based on this applet?[/b][/i][/color]
– Judah L Schwartz

5.2.

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6.

transforming

  • 1. Exploring Factoring & 'FOIL' - I
  • 2. Exploring Factoring & 'FOIL' - II

6.1.

Exploring Factoring & 'FOIL' - I

Adjust the A, B and C sliders so that the BLUE g(x) = [math]Ax^2 + Bx + C[/math] matches the RED function f(x). How are the factors of f(x) related to the roots of g(x)? What can you learn form the graph of f(x) - g(x)?
– Judah L Schwartz

6.2.

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