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Models of algebraic conceptsModels of algebraic concepts

Y2

XY

XX2

Y

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PowerPoint PresentationDeveloped by

Harvey Almarode

Title Slide

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The Quadrant Mat

The Quadrant Mat consists of 4 sections

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-Two positive sections And two negative sections. The channels are like the x and y axis

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The Area Model-

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-Areas of rectangles are used to model products.Length and width of the rectangles are in the channels.

LENGTH

WIDTH

Area

These represent the factorsThe area of the rectangle is in the quadrantsAnd this represents the product.The next series of slides illustrates this area model.

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The quadrant mat is used to model integer multiplicationPlace a +3 on one of the axis

1 1 1

Place a +2 on the other axis

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Make a 3 by 2 rectangle in the first quadrant

1 1 11 1 1

Remove the tray and read the resultsAnswer: +6

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Place a +3 on one of the axis

1 1 1

Place a -2 on the other axis

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Make a 3 by 2 rectangle in the fourth quadrant

1 1 11 1 1

Remove the tray and read the resultsAnswer: -6

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Place a -3 on one of the axis

1 1 1

Place a -2 on the other axis

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Make a 3 by 2 rectangle in the third quadrant

1 1 11 1 1

Read the results of the components in the quadrantsAnswer: +6

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Place a -3 on one of the axis

1 1 1

Place a +2 on the other axis

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Make a 3 by 2 rectangle in the second quadrant

1 1 11 1 1

Remove the tray and read the resultsAnswer: -6

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Multiplying a constant by a variable: 4 times XModel 4 on one of the axis

1 1 1 1

Model X on the other axis.

X

Place components in the first quadrant to model X times 1.

X X X X

Remove the tray and read the resultsAnswer: 4X

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Multiplying a negative number by a variable: -3 times XModel -3 on one of the axisModel X on the other axis.

X

Place components in the quadrant to model X times 1.

XXX

Remove the tray and read the resultsAnswer: -3X

1 1 1

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+1X 1

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Multiplying a monomial by a binomial: 3(X+2)Model X + 2 on one axisModel 3 on the other axisPlace 3 of the X rectangles in the first quadrant

XXX

Place 6 of the 1’s in the first quadrant

1 11 11 1

Answer: 3X + 6Remove the tray and read the results

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+1X

X2

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X

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The Quadrant Mat will also model the product of binomials Model X + 1 on one of the axis

X

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Model X + 2 on the other axisPlace an X2 representing X times X in the first quadrant. Place X’s representing 1 times X in the appropriate places.

Place 1’s representing 1 times 1 in the appropriate placesRemove the tray and read the resultsAnswer: X2 + 3X + 2

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+1 X

X2

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X

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The product of binomials involving negativesModel X - 1 on one of the axis

X

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Model X + 2 on the other axisPlace an X2 representing X times X in the first quadrant. Place X’s representing 1 times X in the appropriate places.

Place 1’s representing 1 times 1 in the appropriate placesRemove the tray and read the resultsBe sure to remove a zero pair of X’sAnswer: X2 + X - 2

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+1 X

X2

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X

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The product of binomials involving negativesModel X - 1 on one of the axis

X

1

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Model -X + 2 on the other axisPlace an X2 representing X times X in the fourth quadrant. Place X’s representing 1 times X in the appropriate places.

Place 1’s representing 1 times 1 in the appropriate placesRemove the tray and read the resultsAnswer: X2 + 3X - 2

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Modeling the product using two variables. Y (X + 2)Model Y on one of the axis

X

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Model X + 2 on the other axisPlace an XY representing X times Y in the first quadrant. Place Y’s representing 1 times X in the appropriate places.

Remove the tray and read the resultsAnswer: XY + 2Y

Y

XY

YY

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Model 2Y(-2X) Place 2 Y’s on one of the axis.

Y Y

Place 2 X’s on the other axis in the negative direction.

X

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Complete the rectangles for the product.

XY XY

XY XY

Remove the tray and read the results Answer: -4XY

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Model (2X - 1)(X - 2)Place 2X - 1 on one of the axis.

1 X X

Place X - 2 on the other axis.

X

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X2 X2

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X

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Place rectangles in the appropriate places on the quadrant.Remove the tray and read the resultsAnswer: 2X2 - 5X + 2