Unit 1 - Square Roots and Applications

In this unit, students will be engaging in finding perfect squares and square rooting various numbers. Students will be estimating non-perfect squares and seeing what value they can obtain. They will be understanding and connecting to the Pythagorean Relationship and applying the Pythagorean Relationship to real world situations.

Lesson # Lesson Notes1 Perfect Squares and Cubes2 Square Roots3 Cube Roots4 Estimating Square Roots5 Pythagorean Relationship Square and Cube Roots Quiz6 Using the Pythagorean Relationship7 Pythagorean Relationship Project8 Review Pythagorean Quiz9 Unit Test

Curricular Competencies Reasoning and Analyzing

o Students will be reasoning and exploring square roots and how to estimate reasonably when it is not a perfect square.

Understanding and Solvingo Students will be developing and demonstrating their mathematical

understanding through inquiring about how the Pythagorean relationship was developed. Students will be using multiple strategies to solve problems

Communicating and Representingo Students will be communicating mathematical thinking in many ways including

written and visuals. Students will be explaining and justifying their decisions Connecting and Reflecting

o Students will be conducting a Pythagorean relationship project to see how it connects to the real world

Lesson #1 - Perfect Squares and Cubes

Big Idea - Perfect squares and cubes are perfect because they are rational number side lengths of geometric shapes.

What is prime factorization?

Example 1. What are the prime factors of 4, 9, 16, and 20?

Example 2. What are the prime factors of 24, 36, and 81?

How does this relate to perfect squares?

How does perfect squares relate to geometry?

What are all the perfect squares?

Example 3. Is 484 a perfect square?

Example 4. Determine the side length of this picture frame that has an area of 400cm2

Lesson #2 - Square Roots

Big Idea - When you square root a value, you are finding the side length of a square that has that value as an area.

How do you find square roots using a calculator?

Example 1. Square root the following using your calculator - 60, 100, 144, 200

What happens if a number is not a perfect square?

Example 2. What is the square root of 2, 3, 4?

What about numbers that are decimals? are they perfect squares too?

What makes 0.09 a perfect square, but 0.9 not a perfect square?

Example 3. What is the square root of 0.64?

Is 6 the only solution to the square root of 36?

Example 4. What are the square roots of 64 and 100?

Can you square root a negative number?

Lesson #3 - Cube Roots

Big Idea - When you cube root a value, you are finding the side length of a cube that has that value as a volume.

How does a cube root relate to a square root?

Example 1. What is 3 x 3 x 3? Is there another way to represent it?

How can you apply prime factorization to solve cube roots?

Example 2. What is the prime factorization of 1000?

How do you use a calculator to solve?

What are all the perfect cubes from 0-1000?

Example 3. What are the cube roots of 8, 64, 729

Can you cube root negative numbers?

Example 4. What is the cube root of -125?

Can you also cube root decimals?

Example 5. Which of the following is not a perfect cube? 0.08 or 0.008?

Example 6. The Avengers needs to contain Ultron in a cube box. Ultron will take up a total volume of 512ft2. What must the side length of the cube box be?

Lesson #4 - Estimating Square Roots

Big Idea - You can estimate square roots by using fractions and reasoning.

How do we estimate square roots that are not perfect squares?

What is the approximation method?

Exaxmple 1. What is the square root of the following numbers?

√56

√70

√139

√7

√90

What is the fraction method?

Example 2. Estimate the following using the fraction method.

√8

√70

√90

√135

√56

How do you find square roots of large numbers?

Example 3. Estimate the following

√6400

√7000

√9000

Lesson #5 - Pythagorean Relationship

Big Idea - Pythagorean Relationship is a relationship where you can find the side length of any unknown side of a right triangle as long as you have the other 2 sides provided.

What conditions are needed for a right triangle to be formed?

What happens if you form squares around each side of the right triangle?

What formula can be created to help solve for all Pythagorean relationships?

Example 1. Solve the following missing area

How can you solve the Pythagorean Theorem?

Example 2. Solve the following triangles

What are Pythagorean Triples?

Example 3. Find the missing side

Lesson #6 - Using the Pythagorean Relationship

Big Idea - Pythagorean Relationship is everywhere in the real world and is used to solve various problems.

How does the Pythagorean Relationship help us in real world situations?

Example 1. Mr Chio is walking home from work and has to walk across the park. He wants to know how many meters he will save if he cuts diagonally around the park as compared to walking around the park on the sidewalk. The park's dimensions are 1500m by 650m.

Example 2. Mr. Chio is working on his home and wants to buy a ladder. He knows that in order for the ladder to be safely anchored to the side of his home, he needs to have it angled away from the house. If the ladder he wants to buy is 10m long and he needs to angle it 5m away from the bottom, will he reach the top of his house which is 7m tall?

Example 3. Mr. Chio is captaining a cruise ship, yes he is that cool. He travels from Port Vancouver south at a speed of 35km/h for 4 h. Then turns 90° and travels west at 40km/h for 5 h. When he reaches the middle of the Pacific Ocean, how far is the ship from Port Vancouver?