Simplifying Non-perfect Square Roots
Arlena Miller
Sullivan County
9/Algebra 1
Lesson Title: Simplifying Non-perfect Square Roots
Grade: 9(Algebra I)
Alignment with state standards:
CLE 3102.2.1 Understand computational results and operations involving real numbers in
multiple representations.
SPI 3102.2.1 Operate (add, subtract, multiply, divide, simplify, powers) with
radicals and radical expressions including radicands involving rational numbers and
algebraic expressions.
3102.2.3 Operate with and simplify radicals (index 2, 3, n) and radical
expressions including rational numbers and variables in the radicand.
Mathematics Goals: Students will be able to simplify square radicals that are not
perfect squares but have perfect square factors.
Students’ Needs: Students will have a prior knowledge of square numbers and their
geometric representation as a square. (ex.
because 9 can be arrange as a square of
3-sides
)
Materials:
for each student:
handout
square guide in page protector
dry-erase marker & cleaning cloth
100 – 1cm squares
for class:
Roots Matching Cards
Roots Bingo Game
document camera
Lesson Plan:
Before:
As students come in give them a note card with one of the following symbols:
On the smartboard have the instructions to find the table with their matching
symbol. (This is just to randomly put them in groups. The activity is individual; however,
working in a group will allow them to easily help each other.) Give students a page
protector with the handout inside of 1-cm grid paper on one side and the 4, 9, 16, 25 grid
on the other side, a dry-erase marker, and a cleaning cloth. Remind them that in the last
lesson we found the perfect squares by arranging the square numbers into an actual
square like the one labeling their tables. Instruct them to take out 27 1-cm squares and
arrange them on the grid paper to make a square. (Use document camera to
demonstrate.) After letting them struggle with this task for a few minutes ask, “Who
has found a perfect square?” (No one will.) Then ask, “What is wrong?” Then tell them,
“We need to find a way to write these numbers that are not squares.”
During:
Have them turn over the page protector to show the side with the 4, 9, 16, &
25 layout. Now, arrange the 27 squares in the 4-grid row and color the spaces. (Will
need to instruct them to start on the left side and color in solid, not just randomly if
teacher notices this happening while observing.) Continue this onto the 9-grid row, 16grid row, & 25-grid row. Ask, “In which row do the 27 squares take up whole squares with
no leftovers”?
Results should look like:
4
9
16
25
The answer is the 9-square. The 27 squares take up exactly 3 of the nine squares.
Pass out handout and go through how to record this result in the first column.
Erase those results with your cleaning cloth.
Repeat with 8 squares.
with 50 squares.
with 32 squares.
Guide as needed and go through these four examples on their handout. (When they get
to 32, you need to ask, “What happened differently with 32?”. There will be whole boxes
filled in on 4 and 16. Question them to guide to using 16. If they are not getting to that
conclusion, have them record both and look at it during next step.)
“Look at these four results. Now in the „rewrite‟ column, we want to rewrite the number
of squares by how many perfect squares we can make.”
“What type of square could we fill with 27?” 9
“Rewrite
as
.”
“What do we know about
? It equals 3.
“We now can simplify to
“What type of square could we fill with 8?” 4
“Rewrite
as
? It equals 2.
“How many were there?” 2
“We now can simplify to
“What type of square could we fill with 50?” 25
as
.”
.”
“What do we know about
“Rewrite
“How many were there?” 3
.”
“How many were there?” 2
.”
“What do we know about
? It equals 5. “We now can simplify to
.”
“What type of square could we fill with 32?” 4 & 16
“We want to use the largest squares possible.
So which will we use to rewrite?”
16
“How many were there?” 2
“Rewrite
as
.”
“What do we know about
? It equals 4. “We now can simplify to
.”
MY IDEA IS THAT THIS WILL HELP STUDENTS VISUALIZE WHY WE BREAK DOWN
THE RADICALS BEFORE SIMPLIFYING. STUDENTS OFTEN SEEM TO NOT
UNDERSTAND WHY WE ARE PULLING OUT THESE NUMBERS. I HAVE NOT TRIED
THIS YET IN CLASS. I GOT THIS IDEA WHEN WE WERE WORKING ON THE SQUARE
NUMBERS IN MATHLETES. IF WE HAVE ALREADY DISCOVERED IN CLASS THE
SQUARE NUMBERS, WHY THEY ARE SQUARE ROOTS, AND WHAT THEY LOOK LIKE AS
SQUARES I THINK THIS WILL HELP THEM MAKE THE CONNECTION.
After:
“Work together in your group to simplify the rest of the given numbers.”
“Do not use calculators.”
Give students time to complete front of handout. Observe and offer guidance as needed.
“How can we work backwards to check our answers?”
the conclusion that, for example,
Go over results.
Want to get students to come to
.
Activity to Move a Little
Give each student a note card from „ROOTS MATCHING CARDS‟. Play music and have
them walk around the class until the music stops. Then have them pair up by high-fiving.
See if the pair is a match. If not, discuss what would be a match with them. Do this until
all students are paired. Can switch around cards and play again as time permits.
Return to groups.
“What other squares could we use as our numbers get larger?”
36, 49, 64, 81, 100, etc
“Using these as well as the 4, 9, 16, & 25 squares, try to simplify the ones on the back.
You may use your calculator to help with these as needed.”
This may be finished as homework if not finished in class.
Homework challenge:
See how many more you can simplify less than 1000.
Bonus points given for each extra one you find and have full explanation for.
Full explanation means:
and not just an „answer‟.
Assessment:
Next class, we will reinforce these by playing “ROOTS” Bingo. Students may use 4, 9, 16,
25-grid paper as a help for this. (This game is attached.) Teacher calls out the root such
as
and students mark
on their bingo card.
Daily “bell ringer” questions will assess individuals‟ ability to correctly simplify nonperfect square roots.
Listen to students as the play the Matching Game to observe which students can simplify
on their own.
Individuals asked to simplify roots as these come up in the next several lessons of adding,
subtracting, multiplying, and dividing radicals. (This is randomly done by having the
students‟ names written on a craft stick and pulling one out of a jar to choose the next
student asked.)
Essay question on test to explain how to simplify a non-perfect square root without the
use of decimals.
Accommodations:
Students who need „extra time‟ or „less work‟ as indicated in IEPs will be told to work out
every other one initially and then fill in extras that they do not get to as we go over
them.
When playing “ROOTS” Bingo or working with other lessons, a sheet of simplified radicals
in order can be given to students to help them when they need to work more quickly.
Working in their group will help all students who benefit from being able to talk to those
around them to think through the processes.
Sources:
graph paper printed from http://mathbits.com
Roots Bingo game template made from http://print-bingo.com .
Cards to assign groups.
Copy on card stock – as many pages as needed for class. Cut apart.
Label each table with a different card. Give one to each student as
they come in to class.
Simplifying Radicals Activity
Name ______________
RESULTS SHEET
Materials needed:
4, 9, 16, & 25 grid layout in page protector
100 1-cm squares
dry-erase marker
cleaning cloth
Examples:
if there are left-overs, mark with an X
#
square
units
whole
squares
used
whole
squares
used
whole
squares
used
whole
squares
used
27
4 _____
9_____
16_____
25_____
8
4 _____
9_____
16_____
25_____
50
4 _____
9_____
16_____
25_____
32
4 _____
9_____
16_____
25_____
# square
units
whole
squares
used
whole
squares
used
whole
squares used
whole
squares used
12
4 _____
9_____
16_____
25_____
20
4 _____
9_____
16_____
25_____
24
4 _____
9_____
16_____
25_____
40
4 _____
9_____
16_____
25_____
45
4 _____
9_____
16_____
25_____
48
4 _____
9_____
16_____
25_____
54
4 _____
9_____
16_____
25_____
63
4 _____
9_____
16_____
25_____
75
4 _____
9_____
16_____
25_____
80
4 _____
9_____
16_____
25_____
90
4 _____
9_____
16_____
25_____
rewrite
with
radical
simplify
rewrite with
radical
simplify
Continue:
Simplify using perfect squares.
18
28
32
44
72
80
98
108
125
147
150
175
180
200
242
250
300
363
500
Bonus roots:
ROOTS Bingo Game
Copy the cards onto card stock, cut apart, and laminate.
Copy the Call Sheet onto card stock (2 copies). Cut out the numbers you will
call out (8, 12, etc.) from one of the sheets. Then use the one not cut out to
place the pieces on as you call them.
Cut up squares for playing pieces or use any available counters.
Give each student a bingo card & several small squares.
Call out the roots to simplify (such as 8) and they find the simplified version
on their card (2√2).
Regular BINGO rules apply.
If this lesson plan is printed a copy of the bingo cards
will be printed. If you are receiving by email it will
be a separate attached pdf file.
ROOTS – Bingo Call Sheet
8
2√2
80
4√5
12
2√3
98
7√2
18
3√2
108
6√3
20
2√5
125
5√5
24
2√6
147
7√3
27
3√3
150
5√6
28
2√7
175
5√7
32
4√2
180
6√5
40
2√10
200
10√2
44
2√11
242
11√2
45
3√5
250
5√10
48
4√3
300
10√3
50
5√2
363
11√3
63
3√7
500
10√5
72
6√2
75
5√3
Roots Matching Cards for Activity in Class