Download Angle and Triangle Relationships involving Real

:
Campus: Clark Jr High
Author(s):Montgomery, Donaldson, Luelf,
Anderson, Lincoln
Date Created / Revised: 1/2/17
Six Weeks Period: 4
Grade Level & Course: 8th Math
Timeline: 13 days
Unit Title: Unit 08: Angle and Triangle Relationships
involving Real Numbers
Stated Objectives:
TEK # and SE
Lesson #
1 of 1
8.1 Mathematical process standards. The student uses mathematical processes to acquire
and demonstrate mathematical understanding.
8.2: The student applies mathematical process standards to represent and use real numbers
in a variety of forms. The student is expected to:
8.2A: Extend previous knowledge of sets and subsets using a visual representation to
describe relationships between sets of real numbers.
8.2B:Approximate the value of an irrational number, including π and square roots of numbers
less than 225, and locate that rational number approximation on a number line.
8.2D: Order a set of real numbers arising from mathematical and real-world contexts
8.6: The student applies mathematical process standards to develop mathematical
relationships and make connections to geometric formulas. The student is expected to:
8.6C: Use models and diagrams to explain the Pythagorean theorem
8.10B: Differentiate between transformations that preserve congruence and those that do not.
8.10C: Explain the effect of translations, reflections over the x- or
y­axis, and rotations limited to 90°, 180°, 270°,
and 360° as applied to two­dimensional shapes on a coordinate plane using an algebraic repr
esentation.
8.10D: Model the effect on linear and area measurements of dilated two-dimensional shapes.
See Instructional Focus Document (IFD) for TEK Specificity
Key
Understandings
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Visual representations can be used to represent relationships between sets and subsets of
numbers.
The value of an irrational number can be approximated using the relationship between
perfect squares of consecutive integers.
A number line is composed of an infinite series of points that are labeled according to a
specified unit length and its distance from zero.
The sum of the area of the squares of the legs of a right triangle is equivalent to the area of
the square of the hypotenuse.
A right triangle can be formed from any two points on a non-horizontal, non-vertical line by
drawing a vertical line from one point and a horizontal line from the other point until the lines
intersect.
A special relationship exists between the measures of the interior angles of a triangle and
their related exterior angles.
Misconceptions
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Students may mislabel the hypotenuse as a or b rather than labeling it as c.
Some students may think the Pythagorean relationship can be used on all triangles instead
of only right triangles.
Key Vocabulary
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Adjacent angles – angles that share a common vertex and side
Angle – two rays with a common end point (the vertex)
Angle-angle criterion for triangles – if two angles in one triangle are congruent to two
angles in another triangle, then the measure of the third angle in both triangles are congruent
Axes – the vertical and horizontal lines that act as a reference when plotting points on a
coordinate plane
Complementary angles – two angles whose sum of angle measures equals 90 degrees
Congruent angles – angles whose angle measurements are equal
Coordinate plane – a two-dimensional plane on which to plot points, lines, and curves
Counting (natural) numbers – the set of positive numbers that begins at one and increases
by increments of one each time {1, 2, 3, ..., n}
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Degree – the measure of an angle where each degree represents
of a circle
Exterior angles of a triangle – angles that are outside of a triangle between one side of a
triangle and the extension of the adjacent side
Hypotenuse – the longest side of a right triangle, the side opposite the right angle
Integers – the set of counting (natural numbers), their opposites, and zero {-n, …, -3, -2, -1,
0, 1, 2, 3, ..., n}. The set of integers is denoted by the symbol Z.
Interior angles of a triangle – angles that are inside of a triangle, formed by two sides of
the triangle
Intersecting lines – lines that meet or cross at a point
Irrational numbers – the set of numbers that cannot be expressed as a fraction ,
where a and b are integers and b ≠ 0
Legs – the two shortest sides of a right triangle
Order numbers – to arrange a set of numbers based on their numerical value
Origin – the starting point in locating points on a coordinate plane
Parallel lines – lines that lie in the same plane, never intersect, and are always the same
distance apart
Place value – the value of a digit as determined by its location in a number such as ones,
tens, hundreds, one thousands, ten thousands, etc.
Quadrants – any of the four areas created by dividing a plane with an x-axis and y-axis
Rational numbers – the set of numbers that can be expressed as a fraction ,
where a and b are integers and b ≠ 0, which includes the subsets of integers, whole
numbers, and counting (natural) numbers (e.g., -3, 0, 2, - ,
, etc.). The set of
rational numbers is denoted by the symbol Q.
Real numbers – the set of rational and irrational numbers. The set of real numbers is
denoted by the symbol R.
Right triangle – a triangle with one right angle (exactly 90 degrees) and two acute angles
Square root – a factor of a number that, when squared, equals the original number
Supplementary angles – two angles whose sum of angle measures equals 180 degrees
Transversal – a line that intersects two or more lines
Triangle – a polygon with three sides and three vertices
Vertical angles – a pair of non-overlapping angles that are opposite and congruent to each
other when two lines intersect
Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
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Ascending
Consecutive
Converse
Corresponding angles
Descending
Formula
Informal argument
Interval
Number line
Open number line
Ordered pair
Perfect square
Pythagorean theorem
Radical symbol
Repeating decimal
Square
Terminating decimal
x-axis
x-coordinate
y-axis
y-coordinate
Suggested Day
5E Model
Instructional Procedures
(Engage, Explore, Explain, Extend/Elaborate, Evaluate)
Materials, Resources,
Notes
Day 1
Warm Up: Relating Rational Numbers #1-3
Relating Rational
Numbers
Real Numbers
Examining Real Numbers
Exploring Real Numbers
Day 2
Warm Up: Rational Numbers Application
Irrational Numbers
Irrational Numbers Interactive Bundle – Notes
Irrational Number Interactive Bundle – Task Cards
Examining Real Numbers
Exploring Real Numbers
Rational Numbers
Application
Irrational Numbers
Interactive Bundle – Notes
Irrational Number
Interactive Bundle – Task
Cards
Day 3
Irrational Numbers
Irrational Numbers Interactive Bundle – Word Problems (as a class)
Irrational Numbers Interactive Bundle – Independent Practice –
Approximating Square Roots
Irrational Numbers
Interactive Bundle – Word
Problems (as a class)
Irrational Numbers
Interactive Bundle –
Independent Practice –
Approximating Square
Roots
Day 4
Real Number Review
Day 5
Real Number Quiz
Ready to go on – Go Math pg 27
Ready to go on – Go Math
pg 27
Day 6
Pythagorean Theorem
Go Math Differentiated Instruction Reading Strategies – pg 150
Understanding Pythagorean Theorem Notes – ppt
Examples and Non Examples of Pythagorean Theorem
Differentiated Instruction
Reading Strategies
Understanding
Pythagorean Theorem
Notes – ppt
Examples and Non
Examples of Pythagorean
Theorem
Day 7
Pythagorean Theorem in Three Dimensions - 8.1 pg 223
Go Math Example 2 – pg 223
Go Math Your Turn #6 - pg 224
Go Math Differentiated Instruction Practice D – pg 148
Go Math Differentiated
Instruction Practice D
Day 8
Converse of the Pythagorean Theorem – 8.2 pg 227
Identifying a Right Triangle
Go Math Example 1 – pg 228
Go Math Your Turn #2-5 – pg 228
Go Math Differentiated
Instruction Practice C
Using the Converse of the Pythagorean Theorem – pg 229
Go Math Example 2 – pg 229
Go Math Your Turn #6-8 – pg 229
Go Math Differentiated Instruction Practice C
Day 9
Distance between Two Points – 8.3 pg 233
Go Math Example 1 – pg 233
Go Math Your Turn #1 – pg 233
Go Math Explore Activity – pg 234
Distance Formula –
Coloring Activity
Finding the Distance Between Two Points
Go Math Example 2 – pg 235
Go Math Your Turn – pg 236
Distance Formula – Coloring Activity
Day 10
Pythagorean Theorem Quiz – Ready to Go On? Pg 239
Day 11
Unit Review
Day 12
Unit Test
Day 13
Review Unit Test
Accommodations
for Special
Populations
Accommodations for instruction will be provided as stated on each student’s (IEP)
Individual Education Plan for special education, 504, at risk, and ESL/Bilingual.
Ready to Go On? Module
Quiz