Square Root Solutions (Part 1 of 2)

To begin the lesson , hand each student the SQUARE ROOT WORKSHEET.docx and ask them to do Activity 1. Students should complete this activity in pairs. Before they begin it is a good idea to ask:

• What do you know about the sides of a square?
• How do you find the area of a square?
• How do you find the area of a rectangle

I will also ask my students if they recall how to find the area of a right triangle. Each of these answers should be written on the board, so that the students can reference them while completing the activity. 

Do not provide more information than what is already in the diagram, even if asked to do so. Most students will quickly find that the area of the big square is 49 sq.cm. and that BU must be 3.

The activity assumes no knowledge of the Pythagorean Theorem. If some students know the theorem...

• Don't allow them to provide answers to other classmates in this manner 
• Don't overestimate these student's knowledge of the theorem. Many times they don't know more than the equation a² + b² = c² 
•   Gear these students as well to finding the triangle areas with everyone else.   

All students should fine the triangle areas and and arrive at the answer. Some may make a common mistake and state that the answer is 25, instead of 5.  In an case....I state:

"You have found that the area of the inner square, GUMS is 25 sq cm. If the area of a square is bxh and all sides of a square are equal, then what should each side measure?"

Write on the board:  S² = 25     S x S = 25      S = 5

Ask the same pairs of students to do Activity 2 on SQUARE ROOT WORKSHEET.docx.

They soon find that the area of GUMS is 5 square units, but may get stumped when finding the length of GU. Some students may say 2.5. Encourage these students to square 2.5. They should see that they are wrong. Ask them to find a number that when squared gives 5. They may soon claim it is impossible. 

When students appear ready, tell the class that the length of a side of a square whose area is x is called the square root of x. Use the current task as an example. The length of GU is the square root of 5, √5 . Ask them to find the square root of 5 with their calculators and they will see.

It is often the case that the calculation of √5 sparks another discussion about irrational numbers. State that numbers like 9, whose square root is a whole number are called "perfect squares", and that the rest of the numbers' square roots are irrational numbers, just like √5.  

I like to make a digression in our conversation at this point. I ask students to identify the perfect squares from 1 to 100. For those that don't really know perfect squares, this may sound over-whelming. But they get the pattern after the first few numbers.

I say...."Ok, is 1 a perfect square?...1 times 1 is?....1....ok....next....(if I feel a pause, I say...come on 2 times 2)" They say 4.... After this point they usually get going, but may say 8 or 10....and I quickly ask....what times what give me 8? So, I like to do this until the class identifies all of the perfect squares from 1 to 100.

Do this drill two or three times. Afterward, I end the digression by stating that in Activity 1, the square root of 9 is 3, because a square with area 9 has side 3. I show them that the answer to the square root of 5 is written as √5.

Finally,  I write a definition of Square Root on the board:

If A = s², then s is a square root of A, (√A)

Ask the students to individually do Activity 3 on SQUARE ROOT WORKSHEET.docx

For Part 1b Ask students to write the answer with a radical symbol and as a decimal with 4 places after the decimal point. Tell them they can write dot, dot, dot (...) to mean that it continues. 

I threw in this Part 3 because I've found that students can be very careless with using the terms square root and squaring correctly. I encourage students to use math language appropriately because if not, they often carry carelessness over to problem solving. For example: When solving √ x = 4 they take the square root of both sides.

This extension problem targets the area interpretation of the Pythagorean Theorem. 

A hypotenuse whose length is not a whole number was chosen purposely. 

• An option could be to ask the student(s) to sketch the squares on their own graph paper.
• Ask them to check their answers by counting the number of grid squares inside the squares and estimating.

Students who already know the Pythagorean Theorem but know it as being merely about
lengths of sides in a right triangle will be impressed by this task.

See: Extension square root solutions 2.docx