Authors Aleta Kandle, with helpful feedback from Dodi Marvell – Special Education
Sandy High School
Title: Making Music with Pythagoras and the Ratios
Grades: 3-10
Pythagoras and the Ratios
Written by Julie Ellis
Published by Phyllis Hornung Peacock
Charlesbridge, Watertown, MA
ISBN: 978-1-57091-775-2
Supplies: Children’s book, handout, rulers, straws, scissors, tape
Do you have straws? During this time of continued budget concerns all teachers are looking for ways to continue teaching effectively with limited supplies. Perhaps one solution comes from a time before the school’s purchase of even the oldest piece of technology still in use. In “Pythagoras and the Ratios,” (a children’s story) Julie Ellis writes a story about children, one of whom is named Pythagoras, who discover the ratios that allow pan flutes to be played together. The narrative walks the reader through the discovery of ratios with illustrations that clearly explain the math. Given those ratios students can calculate and build their own pan flutes with straws. (Thankfully, the straw pan flutes are a very quiet instrument, so even if students “cut the wrong note” the resulting sounds should not disturb neighboring classes.)
Below are listed ideas for different ages/strands.
Measurement for early elementary: Students can use centimeters to measure 6 straws lengths 6, 7.5, 8, 9, 10, and 12 cm. Have students cut each length, arrange them in order from smallest to largest, aligning them at one end, and tape them together so that the instrument lays flat. If some students us the length 3 3/4, 4, 4 1/2, 5, and 6 in inches or 3.8, 4, 4.5, 5, and 6 centimeters the resulting tones should, according to the story, fill your room with beautifully, blending notes.
Number and Operations for elementary: Students may complete the handout applying their multiplication and addition skills to make pan flutes.
Ratios for middle/high school: Use the handout provided or simply write the ratios on the board, complete an example, and then have student choose a number less than ½ the length of the straw (in centimeters or inches). To discourage copying, I requested that students choose a number different than the number selected by their neighbors. Students will each calculate the pipe lengths following the example ratio problem. I quickly looked over each student’s work to determine that their pipe lengths increased from shortest to longest and that the last number was double the first. (A student could also be asked to check their classmate’s papers.) Once their paper passed my inspection, students picked up supplies, shared tape and began to play.
Dodi Marvell our special education teacher extraordinaire tried the most basic lesson in her high-school-age, self-contained classroom. Student abilities range from Kindergarten level to early middle school. She found the hands on lesson to be effective. Conversations during the lesson lead to discussing how real people come up with rules (formulas) for math. She like the way the lesson incorporated different manipulative, a different activity for the day, and multi-step problems that were not too hard. At first students were overwhelmed with the appearance of fractions on their page, but they enjoyed the lesson once they caught onto the process. Best of all, students cheerfully played the whistles when they had finished.
So, grab your straws and scissors and prepare for some musical fun!
Pythagoras and the Ratios Name: __________________
Use the ratios from the story, “Pythagoras and the Ratios” to make your own pan flute.
1) The ratios are 5/4, 4/3, 3/2, 5/3, 2/1. Write the numerators (top numbers) on the left. Multiply each by 6 and write the answer on the right.
Numerators * 6 = answer
____5____ * 6 = ____30___
_________ * 6 = _________
_________ * 6 = _________
_________ * 6 = _________
_________ * 6 = _________
2) Write your answers from above on the left. Divide by the denominator and write your answer on the right.
____30___ ÷ 4 = _________
_________ ÷ 3 = _________
_________ ÷ 2 = _________
_________ ÷ 3 = _________
_________ ÷ 1 = _________
3) (Optional) Reduce the fractions below.
30/4 = ____7.5___
24/3 = _________
18/2 = _________
30/3 = _________
12/1 = _________
4) (Optional) Do you get the same answers as above? Why or why not. ________________________
_______________________________________________________________________
Cut straws for each of the lengths in number 2 or 3 above. Use inches. Tape the straws together aligning them along the top. Play away!
Solutions:
1) Numerators * 6 = answer
____4___ * 6 = ____24____
____3___ * 6 = ____18____
____5___ * 6 = ____30___
____2___ * 6 = ____12___
2) Write your answers from above on the left. Divide by the denominator and write your answer on the right.
____30___ ÷ 4 = ___7.5___
____24___ ÷ 3 = ___8_____
____18___ ÷ 2 = ___9_____
____30___ ÷ 3 = ___10____
____12___ ÷ 1 = ___12____
3) Same answers as number 2.
4) Reducing fraction is the same as dividing by the denominator.
Pythagoras and the Ratios Name: __________________
Use the ratios from the story, “Pythagoras and the Ratios” to make your own pan flute.
The ratios are 5/4, 4/3, 3/2, 5/3, 2/1. Solve each ratio equation to find a new set of pipe lengths that will play in tune with the story’s characters.
Pipe 1: 20
Pipe two is given as an example. Write an equation with the ratio and the length of pipe 1.
5/4=x/20
5*20=4x cross multiply
100/4=x Divide by 4 on both sides of the equation.
Pipe 2: 25
1) Find the error in the Pipe 3 or 4 calculations.
Pipe 3: Pipe 4: Describe error: _______________
4/3=x/20 3/2=x/20
80=3x 3x=40 ____________________________
80/3=x x=40/3
26.7=x x=13.3 ____________________________
Pipe 3: 26.7
Pipe 4: 13.3
2) Calculate the lengths of pipes 5 and 6.
Pipe 5: ________
Pipe 6: ________
3) In number one the longest straw length was 40 and the shortest (our starting number) 20. What is the largest starting number you can choose given the length of a straw? __________inches or __________ centimeters
4) Choose a number less than your answer to number 2. Use ratio equations to calculate the lengths of your pipes.
5) Cut each length, arrange them in order from smallest to largest, aligning them at one end, and tape them together so that the instrument lays flat. Play away!
Solutions:
1) Error found in 4 calculations. Numerators and denominators were multiplied straight across rather than cross-multiplied.
2) Calculate the lengths of pipes 5 and 6.
Pipe 5: ___33.3__
Pipe 6: ___40___
3) ____8____inches or ____21____ centimeters