activities that build understanding for working with radicals and radical equations jim rahn ...
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Activities that Build Activities that Build Understanding for Understanding for
Working with RadicalsWorking with Radicalsand Radical Equationsand Radical Equations
Jim Rahnwww.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.
length of the side of the squarearea
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 25 5
How can we visualize ? 16
Therefore, since the square has a side that measures 4.
16 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 2area
How long is ? 2
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 ? 2
Now place the same finger on the side that equals .2
What does this tell you about ?2
How long is ? 2
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 .2
Make a record of the various area and lengths from this first activity on “These Squares are Really Radical” Chart.
2 4 2 1.42
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 8area
How long is ? 8
Is the made up of two segments? 8
How long are those segments?
So 8 2 2
Line the unit ruler up along the side of the square that holds two square units.
What does this tell you about .8
Complete row 2 of the chart.
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 18area
How long is ? 18
Is the made up of two segments? 18
How long are those segments?
So 18 3 2
Line the unit ruler up along the side of the square that holds two square units.
What does this tell you about .18
Complete row 3 of the chart.
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 32area
How long is ? 32
Is the made up of two segments? 32
How long are those segments?
So 32 4 2
Line the unit ruler up along the side of the square that holds two square units.
What does this tell you about .32
Complete row 4 of the chart.
Can you use a rubber band and show ? 5 2
What is the area of the square that has this side length?
So 50 5 2
What is another name for ?5 2
About how long is ? 5 2
Fill in row 5 on your chart.
Let’s study the chart. What patterns do you see?
2 4 2 1.42
4 16 8 2.88 2 2 6 36 18 4.218 3 2 8 64 32 5.632 4 2
What other predictions can you make from the chart?
48420
98
10 100 50 7.150 5 2
12 2100
Study your chart to see how you would change . 72
Can we see with same relationship in other squares?
Build a square that holds 9 square units.
Use a second rubber band to connect the points that are 1 unit from the corner of each side of the original square.
Is this shape still a square?
What is the area of this inscribed square?
How long is the side of this inscribed square?
It must be the 5area
Line the unit ruler up along the side of the square that holds five square units.
What does this tell you about .5
Make a record of the various area and lengths from this first activity on “These Squares are Really Radical” Chart.
3 9 5 2.25
Build a square that holds 36 square units.
Use a second rubber band to connect the points that are 2 units from the corner of each side of the original square.
Is this shape still a square?
What is the area of this inscribed square?
How long is the side of this inscribed square?
It must be the 20area
Do you see another name for ? 20
Line the unit ruler up along the side of the square that holds two square units.
What does this tell you about .20
Complete row 2 in your chart.
Build a square that holds 81 square units.
Use a second rubber band to connect the points that are 3 units from the corner of each side of the original square.
Is this shape still a square?
What is the area of this inscribed square?
How long is the side of this inscribed square?
It must be the 45area
Do you see another name for ? 45
Line the unit ruler up along the side of the square that holds two square units.
What does this tell you about .45
Complete row 3 in your chart.
4 5Can you use a rubber band and show ?
What is the area of the square that has this side length?
So 80 4 5
What is another name for ?4 5
About how long is ? 4 5
Fill in row 4 on your chart.
5 5Can you use a rubber band and show ?
What is the area of the square that has this side length?
So 125 5 5
What is another name for ?5 5
About how long is ? 5 5
Fill in row 5 on your chart.
Let’s study the chart. What patterns do you see?
What other predictions can you make from the chart?
28921
200011 5320
Study your chart to see how you would change . 72
3 9 5 2.25
6 36 20 4.42 5
9 81 45 6.63 5
12 144 80 8.84 5
15 225 125 11.15 5
• 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 ? 27
What is another name for ? 2 7
If a square had a side that measured , what would the area of the square be?
27
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?
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.
3
x x
Label the area of this square if the length of the side is .1x
1x
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: 9x
9x
If the sides of the squares are represented by these expressions, how much area is in each square?
x 81
What do you know about the areas?
Check your answer.
Does it make sense?
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.
7 7x 7x 7
7x 49
56x
56 7 7
49 7
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.
6 13x 23
Would you find the area of the square whose side is
6 13 23x
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.
6 13 23x
6 13x 23
6x 36
6x 1296
216x
6 216 13 23
1296 13 23
36 13 23
23 23
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.
2x x
2x x
2x 2x
2
2
2
2 0
2 1 0
2 0 1 0
2 1
,
,
x x
x x
x x
x x
x x
2x
-2
X -2
X
+
1
-2x
1x
Activities that Build Activities that Build Understanding for Understanding for
Working with RadicalsWorking with Radicalsand Radical Equationsand Radical Equations
Jim Rahnwww.jamesrahn.com
James.rahn@verizon.net
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