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Algebra Practice
with a Geometry Connection
Bob Battinich
Pacent Learning Solutions
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The Problem
Students entering Geometry are not retaining Geometry concepts taught in General Math courses from grades 3rd – 7th.
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Equity
Students who repeat Algebra 1, either in 9th or 10th grade, are 2+ years removed from Geometry concepts taught in General Math curriculum.
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Focus Geometry Concepts
Basic Angle Relationships
1) Supplementary Angles
2) Complementary Angles
3) Vertical Angles
4) Triangle Sum Theorem
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Basic Angle Relationships
3MG 2.4 Identify right angles in geometric figures or in appropriate objects and determine whether other angles are greater or less than a right angle.
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Basic Angle Relationships
4MG 3.5 Know the definitions of a right angle, an acute angle, and an obtuse angle. Understand that 90°, 180°, 270°, and 360° are associated, respectively with ¼, ½, ¾, and full turns.
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Basic Angle Relationships
5MG 2.1* Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler, compass, protractor, drawing software).
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Basic Angle Relationships
5MG 2.2* Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems.
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Basic Angle Relationships
6MG 2.1 Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms.
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Basic Angle Relationships
6MG 2.2* Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.
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Basic Angle Relationships
7th Grade Standards
Nothing
8th Grade Standards
Nothing
Grade Level Students2 years without any Angle Relationships
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Question
Where can we embed the practice of Algebra concepts in a Geometric context to keep basic angle relationships fresh in students minds.
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Algebra Concepts
Algebra Units/Strands1) One Variable Equations
2) Linear Equations
3) Systems of Equations
4) Operations with Polynomials
5) Quadratic Equations
6) Rational Expressions
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Supplementary Angles
One Variable Equations
(3x)°(4x + 5)° 12x + 20x2
Quadratic Equations
(4x + 5) + (3x) = 180A1 4.0*, A1 5.0*
x2 + 12x + 20 = 180A1 14.0*
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Complementary Angles
(4x)°
(7x + 2)°
One Variable Equations
x2
4x + 30
Quadratic Equations
(4x) + (7x + 2) = 90A1 4.0*, A1 5.0*
(x2) + (4x + 30) = 90A1 14.0*
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Vertical Angles
Systems of Equations
60°3x – y
120°3x + 3y
(5x – 6)°(3x + 8)°
One Variable Equations
5x – 6 = 3x + 8 A1 4.0*, A1 5.0*
3x + 3y = 1203x – y = 60
A1 9.0*
Quadratic Equations
(x2)° (2x + 24)°
x2 = 2x + 24A1 14.0*
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Triangle Sum Theorem
Quadratic Equations
503x
x2
(3x – 5)°(2x)°
(4x – 4)°
One Variable Equations
(2x) + (3x – 5) + (4x – 4) = 180A1 4.0*, A1 5.0*
(x2) + (3x) + (50) = 180A1 14.0*
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1 Variable Equations
Linear Equations
Systems PolynomialsQuadratic Equations
Rational Expressions
Supp x xComp x xVert x x xTST x x
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How about Area and Perimeter?
Where can we embed the practice of Algebra concepts in a Geometric context to keep basic area and perimeter concepts fresh in students mind.
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Area and Perimeter
3MG 1.2* Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them.
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Area and Perimeter
3MG 1.3* Find the perimeter of a polygon with integer sides.
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Area and Perimeter
4MG 1.4 Understand and use formulas to solve problems involving perimeters and areas of rectangles and squares. Use those formulas to find the areas of more complex figures by dividing the figures into basic shapes.
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Area and Perimeter
5MG 1.1* Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle.
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Area and Perimeter
6AF 3.1 Use variables in expressions describing geometric quantities (e.g., P=2w + 2I, A = ½ bh, C = πd — the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively).
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Area and Perimeter
6AF3.2 Express in symbolic form simple relationships arising from geometry.
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Area and Perimeter
7MG 2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
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Area and Perimeter
7MG 2.2 Estimate and compute the area of more complex or irregular two- and three- dimensional figures by breaking the figures down into more basic geometric objects.
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Area and Perimeter
Simplifying Algebraic Expressions
3x2x
15
5x8
Combining Like Terms
42x
3x + 5Distributive Property
8
2x + 1
23x – 7
2x + 15 + 3x + 8 + 5x 4(3x + 5)8(2x + 1) – 2(3x – 7)
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Area and Perimeter
One Variable Equations
4
2x + 3
3xP = 45m 6
x + 4
A = 30m2
4 + (2x + 3) + (3x) = 45 = 306(x 4)2
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Area and Perimeter
Linear Equations
Find the area of the polygon created by the given linear equations.
a. y = 2x + 3
b. y = -3
c. 3x + 2y = 6
Area = 7 62
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Area and Perimeter
Operations with Polynomials
2x2 – 9
3x2 – 2x + 4
x2 + 5x
Addition
2x – 5
3x – 4
Multiplication
2x + 5
3x – 4
x2x – 3
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Area and Perimeter
Quadratic Equations
x2 + 2x + 8 = 23 x(x – 4) = 45
x
x + 4
A = 45m28
x2
P = 23m2x
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Area and Perimeter
Rational Expressions
2
3x 12x 25
2
2
x x 20x 16
2
2 2
3x 12 x x 20Area
x 25 x 16
45x
2
1x
2
35x 2 2
1 4 3Perimeter
x 5x 5x
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1 Variable Equations
Linear Equations
Systems PolynomialsQuadratic Equations
Rational Expressions
Supp x xComp x x
Vertical x x xTST x xArea x x x x x
Perimeter x x x x
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