Download One Variable Equations - Pacent Learning Solutions

Algebra Practice
with a Geometry Connection
Bob Battinich
Pacent Learning Solutions
•Supporting Teachers. Serving Students
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
Grade
LevelStandards
Students
7th Grade
8
2 years without any
Angle Nothing
Relationships
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Basic Angle Relationships
Geometry
5th Grade
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Basic Angle Relationships
Geometry
6th Grade
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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/Strands
1) 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
Quadratic
VariableEquations
Equations
(4x +x25)° 12x
(3x)°
+ 20
(4x
x2 ++ 12x
5) + +(3x)
20 = 180
A1 4.0*,
A1 14.0*
A1 5.0*
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Complementary Angles
Quadratic
Equations
One
Variable
Equations
(7x + 2)° x2
4x(4x)°
+ 30
2) ++(4x
(x
(4x)
(7x++30)
2) = 90
A1 A1
4.0*,
14.0*
A1 5.0*
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Vertical Angles
One
Systems
Quadratic
Variable
of Equations
Equations
Equations
(2x
3x + 3y 3x
(3x–++y24)°
8)°
2)°
(x
60° 120°
(5x – 6)°
3x + 3y = 120
5x x–2 6= =2x3x+ +248
3x – y = 60 A1 4.0*,
A1 A1
14.0*
5.0*
A1 9.0*
•Supporting Teachers. Serving Students
Triangle Sum Theorem
Quadratic
One
VariableEquations
Equations
(4x – 4)°
3x
(2x)°
x2
(3x – 5)°
50
2) +–(3x)
(2x) +(x(3x
5) ++(4x
(50)
–=
4)180
= 180
A1 4.0*,
A1 A1
14.0*
5.0*
•Supporting Teachers. Serving Students
1 Variable
Equations
Linear
Equations
Systems
Polynomials
Quadratic
Equations
Supp
x
x
Comp
x
x
Vert
x
TST
x
x
Rational
Expressions
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.
•Supporting Teachers. Serving Students
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.
•Supporting Teachers. Serving Students
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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Basic Angle Relationships
Geometry
7th Grade
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Area and Perimeter
Simplifying Algebraic Expressions
Combining Like Terms2x + 1 Distributive Property
15
3x + 5
3x – 7 2x
3x
2x
4
8
2
8
5x
2x + 15 + 3x + 8 8(2x
+ 5x + 1) – 2(3x – 4(3x
7) + 5)
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Area and Perimeter
One Variable Equations
2x + 3
4
P = 45m
A = 30m2
3x
6
x+4
4 + (2x + 3) + (3x) = 45
6(x  4)
= 30
2
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Area and Perimeter
Linear Equations
Find the area of the polygon created by the
76
given linear equations.
Area =
2
a. y = 2x + 3
b. y = -3
c. 3x + 2y = 6
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Area and Perimeter
Systems of Equations
•Supporting Teachers. Serving Students
Area and Perimeter
Operations with Polynomials
3x – 4 Multiplication
Addition
2x2 – 9
x2
2x + 5 + 5xx
2x – 3
3x – 4
2x – 5
3x2 – 2x + 4
•Supporting Teachers. Serving Students
Area and Perimeter
Quadratic Equations
x+4
2x
8
P = 23m
x
A = 45m2
x2
x2 + 2x + 8 = 23
x(x – 4) = 45
•Supporting Teachers. Serving Students
Area and Perimeter
Rational Expressions
x 2  x  20
2
1
x  16 3
2
2
5x
x
3x  12
x2  25
4
5x
3x  12 x2  x  20
Area  2
 2
x  251 x4  163
Perimeter  2   2
x 5x 5x
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1 Variable
Equations
Supp
Comp
Vertical
TST
Area
Perimeter
x
x
x
x
x
x
Linear
Equations
Systems
Polynomials
x
x
x
x
Quadratic
Equations
Rational
Expressions
x
x
x
x
x
x
x
x
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Curriculum & Presentations
2010 CMC-South
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Contacts

Bob Battinich
–
–
(916) 296-3958
[email protected]
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