The other day I was playing bowling in the kitchen with my three year old. As always, we were being creative with the game. I had set up the pins and asked him to look from above and tell me what shape the pins were making. He correctly said a triangle and then proceeded to ask me to make them into a circle. I did and we knocked them down. Next he asked for a square. In setting up the square formation, I quickly realized I wouldn't be able to use all ten pins. The most I could use would be three sets of three because the square's sides must be the same length. We knocked them down and he asked for another square. My mind just immediately started working out how many more pins I'd need to make the next bigger square. At first I thought twelve but quickly realized that would make a rectangle size three by four. I'd need a four by four which required sixteen pins. And that's when I found the WOW. I realized that the length of the side was the square root of the total number of pins. I love how the language reflects the underlying principle. It's what I mean by any two subjects are integratable.
While I was excited by my discovery, I was also a little confused as to why I was never taught what a square root IS. We were taught how to memorize them. We were taught that the opposite of finding a square root is squaring or multiplying a number by itself. But nobody ever illustrated WHY this is a universal truth and HOW it is rooted (pun intended) in our natural environment.
Here is a lesson plan that my brain hatched because of my discovery. I hope someone uses it and tells me about how their students react.
Day 1: Dance Integration: Choreography and Square Roots
The Big Idea: Choreography can be used to give students a personal, visceral understanding of what a square root is and how to find it without looking it up.
Teaching Objective: At the end of this lesson, students should be able to:
* answer the question: What is the relationship between a number and its square root?
* figure out the square root of a number by drawing a square.
* use the formation of different sized squares to make a choreographed dance.
* understand the difference between movement that does and does not move through space.
> sound system
> song of teacher's choice
> whiteboard/blackboard/easel and paper to write on during math explanations
> paper and writing implements for students
Introduction to the art form:
To warm up students bodies and get their brains ready to choreograph, lead a physical dance warm up to any song. The warm up should include simple patterns such as marching for 8 counts, then jumping jacks for 8 counts, repeat a few times. Some patterns should require that the students retain their formation within the group or their personal space and some patterns should move through space, such as step out, step together or run, run, run, leap. Ideally you should be teaching at least one pattern that moves through space and one pattern that does not move through space that will be used later. ( 5 minutes)
Break students up into groups of four and have each group come up with one simple pattern ( no more than two movements) that moves through space and one pattern that does not move through space. ( 5 minutes)
Have each group quickly demonstrate their patterns for each other. (10 minutes)
Have each group put themselves in a square formation. Ask them what the definition of a square is. A qudrilateral with four equal sides and four right angles. Ask them how you find the area of a square. By multiplying the length and the width. But since a square's sides are the same size you are also squaring the length of one side. ( 2 minutes)
Have all of the groups do their pattern that does not move through space in their square formation at the same time. Ask them how many people are on each side of their square. Ask them how many people total are in their square. ( 2 minutes)
Have the students figure out how many other groups they would need to combine with to make the next biggest square possible. They may not exclude anyone. (Hint: they will not be able to make a 3x3 square. They will need to combine with 3 other groups to make a 4x4 with 16 total students. The square must be solid not an outline of a square but rows of 2x2, 3x3, 4x4, etc.) ( 5 minutes)
Have them figure out how to position their groups so that they can can do their pattern that does not move through space twice through or for 8 counts and then use their patterns that do move through space to become one big square within 8 counts. Then the big square should do each of the patterns that do not move through space once through from each of the groups. Have each set of groups demonstrate for each other. Ask them how many students are on each side of the square. Ask them how many students they needed to complete a solid square. Ask them how they figured that out. Point out that the number of students on the side of the square is the square root of the total number. That's why they couldn't just add one more group or one more line to get a bigger square. ( 5 minutes)
Have the students figure out of there are enough students to combine the big squares into a gigantic square. If yes, have them combine their squares using one of the patterns that move through space that you taught, followed by one of the patterns that do not move through space that you taught them earlier. Have them begin in their groups of four and run through the dance combining into larger groups and then one big group. Your students have just choreographed a dance using square roots. (5 minutes)
If you have access to a video camera, video your class's dance so they can watch it.
Closing: Have your students write down the relationship between the total number of people and the number of people on one side of each square they were in. Explain the relationship between squaring to find the area and finding the square root. ( 2 minutes)
Formative Assessments: Teacher's Observation- Were all of the groups able to put themselves into square formations? Were all of the groups able to figure out how many students would be needed to complete the rows for the next biggest square? Were all if the groups able to fistinguish between movement that moves through space and movement that doesn't?
Summative Assessment: Written- collect the papers for the closing to see if eveyone understood the math connection
Day 2: Visual Art and Game Integration: sudoku, mosaics and square roots
The Big Idea: Sudoku puzzle boards can illustrate the relationship between a number and its square root. Mosaics can give students the ability to use squares to demonstrate square roots while expressing themselves through abstract art.
> one blank sudoku puzzle board for each student
> one set of different colored crayons, pens or pencils
> one piece of cardboard at least 12x14
> box of assorted mosaic tiles in only square shapes
> graph paper
By the end of class, students should be able to:
* Find squares within squares and write the mathematical equation for the sqaure root relationship illustrated.
* Create abstract art using mosaic tiles that illustrates the relationship between a number and its square root.
Warm up activity:
Give each student a blank sudoku puzzle board and a set of crayons, colored pencils, markers or colored pens. Ask the students to use each color to draw an outline of a square of a different size. Then ask them to use each color to write a mathematical expression for the square outlined in that color. The expression should be the square root of the total number of "tiles" = the number of tiles on one side of the square.
Introduction to the art:
Provide several examples of abstract art. Ask the students what qualities the art has. ( Examples: They all use different colors and shapes. None of them are trying to be a representation of a person, place or thing in our natural environement. All of them have shapes that appear to move in some direction. All of them have "white space", space that is not taken up by any shape or color.)
Give the students pieces of cardboard and access to large tubs of different colored square mosaic tiles. Give the students 15 minutes to explore with the tiles making different size squares on their cardboard. Hint: the sqaures can overlap if they wish but do not have to. Give them a blank piece of graph paper and ask them to use their colored pencils to " sketch a design" that they will then create using the mosaic tiles, glue and the cardboard.
Every mosaic must use only squares of different sizes. There must be at least three different sized squares in each finished mosiac. Have the students write the mathematical expressions for each square they created on their sketch sheet.
Once their sketches are finished, give out the glue and allow the students to create their mosiacs. Make sure they sign their work and give their mosaic a title.
Formative Assessments: Teacher observation: Has the student found different sized squares in each step of the activity? Has the student written accurate mathematical expressions to represent the square root/ area relationship for each square? Has the student used more than one color in their mosiac plan? Does the design plan for the mosaic clearly contain at least three different sized squares?
Summative Asessment: End result of the mosaic- Did the student make a mosaic using only squares as shapes? Does the mosaic include at least three different sized squares? Did the student accurately write mathematical expressions that describe each square in their mosaic?
Follow Up Activity for Review:
Give each student or group of students a bag or box with a number of squares in it that results in a whole number when the square root is taken. The squares can be leftover mosiac tiles or just pieces of paper cut with a paper cutter. Have them figure out how many tiles they have and the square root of that number by creating a filled in square with the tiles. Have them write the mathematical expression that corresponds. You can make it a small group activity and a competition or keep it individual. Either way every box/bag should have a different number of squares in it.
I hope this is helpful. Please try it and let me know how it turns out.