Jim Rahn www.jamesrahn.com James.rahn@verizon.net Activities that Build Understanding for Working with Radicals and Radical Equations Jim Rahn www.jamesrahn.com James.rahn@verizon.net

Use the geoboard to show a square that has an area equal to one of the numbers from the Do Now questions. How long is the side of the square?

Use the geoboard to show a square that has an area equal to another number from the Do Now questions. How long is the side of the square?

Is it possible to build a square for each of the numbers on the Do Now list if we had a larger geoboard? What would be the length of the side of each square?

Suppose the square contained 25 squares. How long would each side be? What would be the length of the side of this square? Notice that given the area of a square we can find the length of the side of the square.

We’ll introduce a symbol to connect these two thoughts. Since the square with an area of 25 has a side that measures 5, we’ll say that the

How can we visualize ? Therefore, since the square has a side that measures 4.

Build a square that holds 4 square units. Use a second rubber band to connect the midpoints of each side of the original square. What is this shape called?

What is the area of this inscribed square? How long is the side of this inscribed square? It must be the

How long is ? Place your thumb or index finger between to pegs on the geoboard. Can you feel both pegs on either side of your finger?

How long is ? Now place the same finger on the side that equals . What does this tell you about ?

How long is ? Take an edge of a piece of paper and make a unit ruler.

Line the unit ruler up along the side of the square that holds two square units. What does this tell you about .

Make a record of the various area and lengths from this first activity on “These Squares are Really Radical” Chart. 2 4 2 1.4

Build a square that holds 16 square units. Use a second rubber band to connect the midpoints of each side of the original square.

What is the area of this inscribed square? How long is the side of this inscribed square? It must be the

How long is ? So Is the made up of two segments? How long are those segments?

Line the unit ruler up along the side of the square that holds two square units. Complete row 2 of the chart. What does this tell you about .

Build a square that holds 36 square units. Use a second rubber band to connect the midpoints of each side of the original square.

What is the area of this inscribed square? How long is the side of this inscribed square? It must be the

How long is ? So Is the made up of two segments? How long are those segments?

Line the unit ruler up along the side of the square that holds two square units. Complete row 3 of the chart. What does this tell you about .

Build a square that holds 64 square units. Use a second rubber band to connect the midpoints of each side of the original square.

What is the area of this inscribed square? How long is the side of this inscribed square? It must be the

How long is ? So Is the made up of two segments? How long are those segments?

Line the unit ruler up along the side of the square that holds two square units. Complete row 4 of the chart. What does this tell you about .

Can you use a rubber band and show ? About how long is ? So Fill in row 5 on your chart. What is the area of the square that has this side length? What is another name for ?

Let’s study the chart. What patterns do you see? What other predictions can you make from the chart? Study your chart to see how you would change . 2 4 2 1.4 4 16 8 2.8 6 36 18 4.2 8 64 32 5.6 10 100 50 7.1 20 484 98

Can we see with same relationship in other squares?

Build a square that holds 9 square units. Is this shape still a square? Use a second rubber band to connect the points that are 1 unit from the corner of each side of the original square.

What is the area of this inscribed square? How long is the side of this inscribed square? It must be the

Line the unit ruler up along the side of the square that holds five square units. What does this tell you about .

Make a record of the various area and lengths from this first activity on “These Squares are Really Radical” Chart. 3 9 5 2.2

Build a square that holds 36 square units. Is this shape still a square? Use a second rubber band to connect the points that are 2 units from the corner of each side of the original square.

What is the area of this inscribed square? How long is the side of this inscribed square? It must be the

Do you see another name for ?

Line the unit ruler up along the side of the square that holds two square units. Complete row 2 in your chart. What does this tell you about .

Build a square that holds 81 square units. Is this shape still a square? Use a second rubber band to connect the points that are 3 units from the corner of each side of the original square.

What is the area of this inscribed square? How long is the side of this inscribed square? It must be the

Do you see another name for ?

Line the unit ruler up along the side of the square that holds two square units. Complete row 3 in your chart. What does this tell you about .

Can you use a rubber band and show ? About how long is ? So Fill in row 4 on your chart. What is the area of the square that has this side length? What is another name for ?

Can you use a rubber band and show ? About how long is ? So Fill in row 5 on your chart. What is the area of the square that has this side length? What is another name for ?

Let’s study the chart. What patterns do you see? What other predictions can you make from the chart? Study your chart to see how you would change . 3 9 5 2.2 6 36 20 4.4 9 81 45 6.6 12 144 80 8.8 15 225 125 11.1 21 289 2000

What is your understanding about square root from this activity? Do you have more understanding for square root because you now have a visual understanding?

Place the Drawing Squares template in your communicator. With a partner find all the other radicals you can visualize on this 5 x 5 square.

What is another name for ? If a square had a side that measured , what would the area of the square be? If a square had an area of 147, how long would the side the square be?

Can the Area Model Help Me Solve Radical Equations?

x Recall that if a square has an area of 4, then its side equals or 2. Similarly, if a square has an area of 3, then its side has a length of . Label the side of this square if the area of the square is x. x

Label the area of this square if the length of the side is . X-1

The Radical Equation Template contains two sets of equations that show either two squares are equal. What does that mean when the expressions are equal?

Use the Radical Equation Template to represent the following equation: Check your answer. Does it make sense? x 81 If the sides of the squares are represented by these expressions, how much area is in each square? What do you know about the areas?

Use the Radical Equation Template to visualize what each equation is describing. Then use the model to help you solve for the value of x. Check your answer when you are done.

Use the Radical Equation Template to visualize what each equation is describing. Then use the model to help you solve for the value of x. Check your answer when you are done.

Use the Radical Equation Template to visualize what each equation is describing. Then use the model to help you solve for the value of x. Check your answer when you are done. Would you find the area of the square whose side is

Use the Radical Equation Template to visualize what each equation is describing. Then use the model to help you solve for the value of x. Check your answer when you are done.

Use the Radical Equation Template to visualize what each equation is describing. Then use the model to help you solve for the value of x. Check your answer when you are done. -2 X -2 X + 1 -2x 1x

Jim Rahn www.jamesrahn.com James.rahn@verizon.net Activities that Build Understanding for Working with Radicals and Radical Equations Jim Rahn www.jamesrahn.com James.rahn@verizon.net