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Modeling Products of Mixed Numbers

Modeling Products of Mixed Numbers
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2022-11-18 23:46
Geometric ConstructionSlider Animation
This example demonstrates how to use an area model and interactive tools to show the multiplication of mixed numbers. The key mathematical concepts include multiplying mixed numbers, combining fractions and whole numbers, and the geometric meaning of area models. Construction involves GeoGebra commands such as Shuffle, Polygon, PointIn, Denominator, If, etc., to build mixed number pairs and their products. Suitable for students in grades 4-5 to understand the visual representation and calculation process of mixed number multiplication.

GeoGebra Commands & Parameters

  1. Shuffle({{1 / 2, 3 / 4}, {3 / 4, 1 / 2}, {1 / 2, 2 / 3}, {2 / 3, 1 / 2}, {1 / 3, 1 / 2}, {1 / 2, 1 / 3}, {1 / 4, 3 / 5}, {3 / 5, 1 / 4}, {2 / 5, 1 / 3}, {1 / 3, 2 / 5}, {2 / 3, 3 / 4}, {3 / 4, 2 / 3}, {1 / 2, 3 / 5}, {3 / 5, 1 / 2}, {2 / 5, 1 / 4}, {1 / 4, 2 / 5}}) → fracPairs, {{0.6, 0.25}, {0.6666666666666666, 0.5}, {0.6, 0.5}, {0.5, 0.75}, {0.25, 0.6}, {0.75, 0.5}, {0.6666666666666666, 0.75}, {0.4, 0.25}, {0.5, 0.6666666666666666}, {0.25, 0.4}, {0.4, 0.3333333333333333}, {0.3333333333333333, 0.5}, {0.75, 0.6666666666666666}, {0.3333333333333333, 0.4}, {0.5, 0.6}, {0.5, 0.3333333333333333}}
  2. Shuffle({{1, 2}, {2, 1}, {1, 1}, {1, 2}, {3, 1}, {2, 3}, {1, 3}, {1, 1}, {2, 1}, {2, 2}, {1, 2}, {2, 1}, {1, 3}, {3, 1}, {2, 1}, {2, 2}}) → wholePairs, {{1, 2}, {3, 1}, {2, 2}, {1, 1}, {1, 2}, {1, 3}, {1, 3}, {1, 2}, {2, 1}, {3, 1}, {1, 1}, {2, 3}, {2, 1}, {2, 1}, {2, 1}, {2, 2}}
  3. Polygon({PAnchor + (1, -1), PAnchor + (5, -1), PAnchor + (5, -5), PAnchor + (1, -5)}) → moveArea
  4. PointIn(moveArea) → PMove
  5. Denominator(fracB) → denomB
  6. Denominator(fracA) → denomA
  7. If(showsolution, (numA, (-numB)), PMove) → PReference
  8. Numerator(fracA) → numerA
  9. Numerator(fracB) → numerB
  10. RemoveUndefined(Flatten[Sequence[Sequence[If[i ≥ (wholeA * denomA) ∨ j ≥ (wholeB * denomB), (i / denomA, ((-j)) / denomB) + PAnchor], i, 0, numCols - 1], j, 0, numRows - 1]]) → startPositions
  11. Flatten(Sequence[Sequence[(i / denomA, ((-j)) / denomB) + PAnchor2 + (((wholeA * wholeB) * (1 + spacer)), 0), i, 0, denomA - 1], j, 0, denomB - 1]) → endPositionTemplate
  12. Sequence(If[A ≤ numCompositeWholes, (denomA * denomB), ((fracAB * denomA) * denomB)], A, 1, numCompositeWholes + 1) → numBlocks
  13. Flatten(Sequence[Zip[Translate[A, ((K * (spacer + 1)), 0)], A, endPositionTemplate], K, 0, Length[numBlocks] - 1]) → endPositionsWithExtras
  14. Sequence(endPositionsWithExtras(A), A, 1, Sum[numBlocks]) → allEndPositions
  15. Polygon({PAnchor, (x(PAnchor), y(PReference)), PReference, (x(PReference), y(PAnchor))}) → studentArea
  16. Polygon({(0, 0), (1 / denomA, 0), (1 / denomA, -1 / denomB), (0, -1 / denomB)}) → smallBoxTemplate
  17. Zip(Translate[smallBoxTemplate, A], A, startPositions) → allSmallBoxes
  18. Flatten(Sequence[Sequence[(i, (-j)), i, 0, wholeA - 1], j, 0, wholeB - 1]) → startUnitPositions
  19. Polygon({(0, 0), (1, 0), (1, -1), (0, -1)}) → unitBoxTemplate
  20. Sequence(PAnchor2 + ((i * (1 + spacer)), 0), i, 0, (wholeA * wholeB) - 1) → unitEndPositions
  21. Zip(Translate[unitBoxTemplate, Vector[PAnchor, A]], A, startUnitPositions) → allUnitBoxes
  22. Sequence(Segment[(x(PAnchor) + A, y(PAnchor)), (x(PAnchor) + A, y(PCorner))], A, 0, x(PCorner), 1) → gridWholeVertical
  23. Sequence(Segment[(x(PCorner) - 1 + A, y(PCorner)), (x(PCorner) + A - 1, y(PAnchor) + If[x(PMove) < 1, -1, 0])], A, 0, 1, 1 / Denominator[fracA]) → gridPartsVertical
  24. Sequence(Segment[(x(PAnchor) + If[y(PMove) < -1, 0, 1], y(PCorner)) + (0, A), (x(PCorner), y(PCorner)) + (0, A)], A, 0, 1, 1 / Denominator[fracB]) → gridPartsHorizontal
  25. Zip(Translate[A, Vector[(sliderSoln * Vector[B, C])]], A, allSmallBoxes, B, startPositions, C, allEndPositions) → smallBoxes
  26. Zip(Translate[A, Vector[(sliderSoln * Vector[B, C])]], A, allUnitBoxes, B, startUnitPositions, C, unitEndPositions) → unitBoxes
  27. Sequence(Segment[(x(PAnchor), y(PCorner) + A), (x(PCorner), y(PCorner) + A)], A, y(PAnchor) - y(PCorner), 0, -1) → gridWholeHorizontal
  28. Midpoint(PAnchor, (x(PAnchor), y(PReference)) + (-1, 0)) → AnchorNum2
  29. Circle(SolnPoint, 0.2) → solnOutline
  30. circlePixels(PMove, 7 + (7 * pulsing), Corner[1], Corner[3], Corner[5]) → pulsingPoint

AI Drawing Prompt Reference

Draw an area model to demonstrate the multiplication of two mixed numbers. First, randomly select two mixed numbers, such as 1/2 and 3/4, and draw a rectangular area to represent their product. Then, inside the rectangle, draw smaller squares to represent the components of each mixed number. Finally, adjust the positions of these smaller squares based on the selected mixed numbers to show the geometric meaning of the product.
The prompt above was compiled from the original drawing commands and is for reference only.