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, 16- grid row, & 25-grid row. Ask, In which row do the 27 squares take up 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 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 How many were there? 3 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 How many were there? 2 Rewrite as. What do we know about? It equals 2. We now can simplify to. What type of square could we fill with 50? 25 How many were there? 2 Rewrite as. 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? Want to get students to come to the conclusion that, for example,. Go over results. 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: Examples: 4, 9, 16, & 25 grid layout in page protector 100 1-cm squares dry-erase marker cleaning cloth # square units if there are left-overs, mark with an X squares used squares used squares used squares used 27 4 9 16 25 8 4 9 16 25 50 4 9 16 25 32 4 9 16 25 Continue: # square units squares used squares used squares used 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 rewrite with radical simplify simplify
Simplify using perfect squares. Bonus roots: 18 28 32 44 72 80 98 108 125 147 150 175 180 200 242 250 300 363 500
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