Last updated: 6/9/2015

Tuckahoe Common Logo.jpg

Math- 8 Core- Quarter 3

Squares and Square Roots

Pythagorean Theorem



Quarter 3

Squares and Square Roots

Square Root

Principal Square Root

Perfect Squares

Estimating Square Roots

Ordering Square Roots

Rational and Irrational Roots

Expressions wlith Square Roots

Real and Imaginary Roots

Graphing Irrational Numbers


Pythagorean Theorem

Determine parts of right triangle

They hypotenuse of a right triangle is always the longest side

The right angle of a right triangle is always included between the legs

Regect negative root, because its a length

Using Pythagorean Theorem to determine a missing side

Determining if a triangle is a right triangle (converse of Pythagorean Theorem)

Pythagorean Triples




Isometric Figures




Composition of Transformations



Proportionality of sides




Squares and Square Roots

How can you find the side length of a square when you are given the area of the square?

Why can't we find the root of a negative number?

How do the roots of numbers help us classify numbers as rational and irrational numbers?

If we add a constant to an irrational number is the sum rational or irrational?

How can we use knowledge of perfect squares to estimate values of square roots?

How many real roots do square roots of positive real numbers have?

Are both negative and positive roots always reasonable solutions?


Pythagorean Theorem

How are the lengths of the right triangle connected?

Is there a common rule to determine the length of the diagnol of a triangle?

Is there on side of a right triangle whose length is always the longest?

What is a theorem?

How can we demonstrate Pythagorean Theorem on a coordinate plane?

Can you prove that a radicand is a group?

Is the hyponetuse of a right triangle always opposite the right angle?

How can we determine the legs of a right triangle?

Can you use Pythagorean Theorem in geometric applications of area and perimeter?

How does Pythagorean Theorem help us with indirect measurement?

Why is it useful to know the Pythagorean Triples?



What does congruence mean?

Do rigid movements change size or shape of figures on a plane?

Does sliding a figure from one position to another maintain congguence?

How can we describe a slide (translation)?

Can we create a general rule to apply to coordinates for translations?

How do descriptive words of a translation affect the coordinates of the original image?

How can we distinguish the new image from the original on a graph?

What is a reflection?  Can you describe a real life reflection?

Why do you think a reflection is described as a flip?

Can we reflect images over the x-or y-axes?

What affect do reflections over the axes have on the coordinates?

Can a figure that is drawn over two or more quadrants be reflected?

What is significant about the distances from reflected points to the original?

How can you determine if a figure is reflected or translated if the new image appears to be either reflected or translated?

Can we create a general rule to apply to coordinates for reflections?

Can you describe what a rotation means to you in real life?

How do we turn an object on a plane?

What are the two directions objects can be rotated?

Why is it important to know the center point of rotation, how does it affect the movement?

What are the quarter turn degrees of a coordinate plane?

What is rotational symmetry?

Can we describe the same position multiple ways?

Can we create general rules for rotations and how the affect the coordinates?

Can we do multiple transformations and maintain congruence?

How can we determine which image is first, second, last?

Why do you think multiple movements are called compositions?  How does that word applyinyour life?

What conclusions can you draw about corresponding angles and sides when translations, reflections and rotations are performed on a figure?

What does it mean to dilate?  Can you think of any real life situations where you have heard the word dilation before?

How does a scale factor affect the size of a figure?

What are the parameters of the scale factor that make a figure reduce in size?

What are the parameters of the scale factor that make a figure enlarge?

How is the distance from the center of the dilation to each point relate when compared to the dilated new images' points?

Are dilated figures congruent or similar?

What is the affect on corresponding sides and angles?

How do corresponding sides look on a graph?  What is the relationship you see?

Can we develop a general statement in regard to corresponding sides and parallelism?

Can we develop a general rule that applies to the coordinates when doing dilations?

What value must the scale factor be to be similar and congruent?

Can we use proportions to determine corresponding segment length?

How does similarity help us measure distances that cannot be measured easily with rulers?




(1) 8.G.1 Verify experimentally the properties of rotations, reflections, and translations:
(1) 8.G.1.a Lines are taken to lines, and line segments to line segments of the same length.
(1) 8.G.1.b Angles are taken to angles of the same measure.
(1) 8.G.1.c Parallel lines are taken to parallel lines.
(1) 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
(1) 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
(1) 8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
(1) 8.G.6 Explain a proof of the Pythagorean Theorem and its converse.
(1) 8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
(1) 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Squares and Square Roots

Determine the parts of a radical (index, radicand, radical)

Determine the square roots of of whole numbers

Explain why square roots less than zero are imaginary

Determine when to use the principle root, the negative root or both

Simplify expressions with radicals

Estimating roots, using knowledge of perfect squares

Determine if square roots are rational or irrational

Place square roots on the number line

Solve equations with square roots


Pythagorean Theorem

Recognize if a triangle is a right triangle

Determine the parts of the right triangle (hypotenuse, legs)

Draw conclusions about the length of the hypotenuse in relation to the legs

Demonstrate the understanding of the sum of the squares of the legs is equal to the square of the hypotenuse on the coordinate grid

Use Pythagorean Theorem to find a missing side of a right triangle

Find the perimeter and area of a right triangle or other geometric figures, after using Pythagorean Theorem (with and without square root sides)

Use the converse of the Pythagorean Theorem to determine if a triangle is in fact a right triangle

Discover real life applications where Pythagorean Theorem would be helpful to know



Demonstrate understanding of congruence

State corresponding sides of figures

Write statements of congruence using symbol and proper labeling of figures

Perform translation from words, the T symbol, rules to coordinates

Determine the translation rule used for given points and apply rule to another point

Graph and translate figures with proper notation for pre-image and new-image

Perform a reflection through the x-axis and y-axis, label images properly

Perform a reflection through vertical and horizontal lines (not axes)

Establish the distance from points to the line of reflection are congruent

Determine the reflection based on the changes described to the coordinates

Determine degrees and direction of quarter turns on coordinate plane using origin as center

Graph rotations of quarter and 180 degree rotations by rotating plane

Graph rotations of quarter and 180 degree rotations by using protractor and ruler

Graph rotations of quarter and 180 degree rotations using coordinate rules

Determine the applied rotation when given pre-image and image points

Graph compositions of rigid transformations, with proper image labeling

Verbally explain why figures maintain size and shape (congruence)

Identify combined transformations

Graph dilations and label new image

Identify scale factors that result in reductions and those that result in enlargement

Write rules of dilations based on scale factor

Write similarity statements between original and new images

Identify the relationship between corresponding sides of similar figures

Finding missing sides of similar figures

Demonstrate understanding that angles of corresponding sides are congruent in similar figures

Determine similar figures from markings of angles and sides

Write similar statements in corresponding form

Use similarity to measure indirectly

Examine slope triangles and connect to parallelism

Establish that corresponding sides of similar figures are parallel








Squares and Square Roots

Square Root






Principle Root

Negative Root

Perfect Squares








right angle


included angle




pythagorean triples

angle of rotation

center of dilation

center of rotation



scale factor


line of reflection




rotational symmetry



composition of transfomations

corresponding angles








Math Chat

Writing prompt

Classroom assessments

Pass paper

Judy Dodge exits

Exit tickets

Daily and weekly spirals



Glencoe Course 3 Text

Big Ideas Text-Grade 8

Holt-McDouglal Mathematics Grade 8


New York State Modules

Data is Loading...