Concrete-Pictorial-Abstract and the Teaching for Mastery

The Central Maths Hub, Jun 5, 2017

The GeoGebra Resource Work Group, part of the Central Maths Hub, are a small group of teachers, both primary and secondary, who have come together to design, create and trial resources to support using "Concrete-Pictorial-Abstract" approaches in their teaching. Using GeoGebra to act as the pictorial phase of the approach, this book is a collection of those resources. Thanks to Tom Carpenter, Michael Borcherds and the GeoGebra community for the creation of these files.

Table of Contents

  1. Counters and Number Lines
    1. Coin Counter
    2. Dot Counter
    3. Numbered Counters
    4. Coloured counters
    5. Number Line
    6. Number line (mental addition/subtraction)
  2. Bar Models for fdp
    1. Fraction of an amount
    2. Fraction of a given amount (probability)
    3. Dividing an amount into a given ratio
    4. Part of the ratio given
    5. Percentage of an amount (10%)
    6. Percentage of an Amount (1%)
    7. Percentage Change
    8. Percentage Changed
    9. Reverse Percentages
  3. Other Primary
    1. Place Value Blocks
    2. Number Bond Grids
    3. Virtual Numicon Shapes
    4. Fractions of Shapes
    5. Part-Part-Whole Model using Bar and Circles
    6. Part-Part-Whole Estimation
    7. Super Clock
  4. Other Cross-Phase
    1. Multiplicative Commutativity
    2. Adding Double Digits
    3. Double digit subtraction
    4. 100 Grid for Equivalent Fractions, Decimals and Percentages
    5. Fraction and Decimal Hundredths Grids
    6. Multiplying Fractions
    7. Multiplying Fractions 2
    8. Equivalent fractions
    9. Simplifying Fractions
    10. Comparing Two Fractions
    11. Add and Subtract Fractions
    12. Multiplying & Dividing by Powers of 10
    13. Line Graph of Simpson's characters
  5. Other Secondary
    1. Improper to Mixed Fractions
    2. Mixed to Improper Fractions
    3. Area of a triangle
    4. Area of a parallelogram
    5. Area of a trapezium
    6. Area of a Circle
    7. Expanding a Single Bracket
    8. Double Bracket Expansion
    9. Histogram: different widths
    10. Histogram: Equal Widths
    11. Sketch a Cumulative Frequency Curve
    12. Bounds and Errors in Measurement

1.

Counters and Number Lines

  • 1. Coin Counter
  • 2. Dot Counter
  • 3. Numbered Counters
  • 4. Coloured counters
  • 5. Number Line
  • 6. Number line (mental addition/subtraction)

1.1.

Coin Counter

Instructions
1. If you want to use a given question, adjust the slider to the maximum amount of money, in pounds, to be collated.[br]2. Click on the "Question" button for a given question.[br]3. Click and drag a choice of coins into the boxed area.[br]4. Click on the "Show Total" box to reveal the total value of the money inside the boxed area.[br]5. Click on the "Reset" button to move the coins back to their starting positions.
– Tom Carpenter

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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2.

Bar Models for fdp

  • 1. Fraction of an amount
  • 2. Fraction of a given amount (probability)
  • 3. Dividing an amount into a given ratio
  • 4. Part of the ratio given
  • 5. Percentage of an amount (10%)
  • 6. Percentage of an Amount (1%)
  • 7. Percentage Change
  • 8. Percentage Changed
  • 9. Reverse Percentages

2.1.

Fraction of an amount

Instructions
1. Click on the "Question" box if you wish use a provided question. [br]2. Type in the amount representing the whole in the "Total" box.[br]3. Adjust the slider to select the appropriate number of parts in total (denominator).[br]4. Click on the "Part value" box to show what each part is worth.[br]5. Click on "Fractional value" box.[br]6. Adjust the slider to select the appropriate number of red parts (numerator) in question.[br]7. Check the solution if you used the provided question by clicking on "Solution" box.[br]8. Click on the "Reset" button to start again and get a new question.
– Tom Carpenter

2.2.

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2.3.

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2.4.

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2.5.

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2.6.

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2.7.

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2.8.

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2.9.

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3.

Other Primary

  • 1. Place Value Blocks
  • 2. Number Bond Grids
  • 3. Virtual Numicon Shapes
  • 4. Fractions of Shapes
  • 5. Part-Part-Whole Model using Bar and Circles
  • 6. Part-Part-Whole Estimation
  • 7. Super Clock

3.1.

Place Value Blocks

Instructions
1. Drag the ones, tens and hundreds into their relevant columns to create a 1/2/3-digit number.[br]2. If 10 ones end up in the ones column then they will turn red. Click on the checkbox that appears in order to convert the 10 ones into a ten. This ten can then be dragged into the tens column as a carry. The values are adjusted accordingly.[br]3. If 10 tens end up in the tens column, then a similar process can be followed.[br]4. The Reset button will restart the process.
– Tom Carpenter

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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4.

Other Cross-Phase

  • 1. Multiplicative Commutativity
  • 2. Adding Double Digits
  • 3. Double digit subtraction
  • 4. 100 Grid for Equivalent Fractions, Decimals and Percentages
  • 5. Fraction and Decimal Hundredths Grids
  • 6. Multiplying Fractions
  • 7. Multiplying Fractions 2
  • 8. Equivalent fractions
  • 9. Simplifying Fractions
  • 10. Comparing Two Fractions
  • 11. Add and Subtract Fractions
  • 12. Multiplying & Dividing by Powers of 10
  • 13. Line Graph of Simpson's characters

4.1.

Multiplicative Commutativity

Instructions
1. Select your multiplication by adjusting the sliders[br]2. Click on the "switch" box to alternate between collective rows and columns in the array of coloured dots. The total number of dots doesn't change whichever grouping is selected.[br]3. Click on the "What do you see?" box to give the groups of counters.
– Tom Carpenter

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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4.10.

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4.11.

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4.12.

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4.13.

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5.

Other Secondary

  • 1. Improper to Mixed Fractions
  • 2. Mixed to Improper Fractions
  • 3. Area of a triangle
  • 4. Area of a parallelogram
  • 5. Area of a trapezium
  • 6. Area of a Circle
  • 7. Expanding a Single Bracket
  • 8. Double Bracket Expansion
  • 9. Histogram: different widths
  • 10. Histogram: Equal Widths
  • 11. Sketch a Cumulative Frequency Curve
  • 12. Bounds and Errors in Measurement

5.1.

Improper to Mixed Fractions

1. Adjust the red and blue sliders to create fraction you wish to convert.[br]2. Adjust the green slider so that the sectors come together to show how many wholes and spare parts there are. The abstract calculation will also be shown.[br]3. The number can be hidden if you wish be clicking on the 'Count Sectors box'.[br]
Adapted from [url=https://www.geogebra.org/m/vxUuhPe6]https://www.geogebra.org/m/vxUuhPe6[/url] by Andrew Creer
– The Central Maths Hub

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5.3.

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5.4.

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5.5.

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5.6.

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5.7.

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5.8.

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5.9.

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5.10.

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5.11.

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5.12.

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Information