Course: Foundations & Pre-Calculus 10, Outcome 2
Stage 1 – Desired Results
Outcome: FP10.2
Demonstrate understanding of irrational numbers in both radical (including mixed radical) and exponent
forms through representing, identifying, simplifying, ordering, relating to rational numbers, applying
exponent laws.
Indicators:
a, e, f, h, i, j, k, m, n
Enduring Understandings:
Essential Questions:
Any number that can be written as a fraction m/n,
n≠0, m, n Є I (terminating or repeating decimal)
is rational; any number that cannot be written as a
fraction is an irrational number.
 Like rational numbers, irrational numbers can be
simplified.
 Exponents can be used to represent irrational
numbers (roots) and reciprocals of rational
numbers.
 The exponent laws can be extended to include
powers with rational and variable bases, and
rational exponents
Students will know:
 The subsets of the set of real numbers.
 The laws of exponents and be able to apply them
to rational numbers
 The difference between a mixed radical and an
entire radical
 That powers and radicals are two ways of
expressing real numbers
Vocabulary
 Rational and Irrational Numbers
 Radical, Index (Indices), Radicand, Radical
Symbol
 Mixed Radical
 Entire Radical
 Real Numbers
Integer
Rational
Irrational
 Power, Exponent, Base
 Rational Exponents
Rational and Variable Bases
Integral and Rational Exponents
 How are rational and irrational numbers
different? The same?
 How do you express entire radicals as mixed
radicals and mixed radicals as entire radicals?
 How are the exponent laws related to rational
and irrational numbers?
Students will be able to:
 Sort real numbers into rational and irrational
numbers.
 Express radicals as mixed radicals in simplest
form, and mixed radicals as entire radicals.
 Represent the relationships between natural
numbers, whole numbers, integers, rational
numbers and irrational numbers.
 Apply exponent laws to powers (rational and
variable bases and integral and rational
exponents).
 Express powers with rational exponents as
radicals and vice versa.
 Create a representation to illustrate the
relationship between powers, rational numbers,
and irrational numbers.
 Analyze patterns to generalize why a-n=1/an, a ≠
0
 Analyze patterns to generalize why a1/n = n a ,
n ≠ 0, n є I, and a > 0 when n is an even integer
Extension:
 Approximate the value of a given irrational
number and explain the strategy used.
Stage 2 – Assessment Evidence
Formative (Pre-assessment, AFL, …) :
Performance Task:
Tuning a Piano Adapted from:
http://www.bced.gov.bc.ca/careers/aa/lessons/ao
1.
Pre-assessment/ Diagnostic:

Assess your understanding sections
m15.htm
Note: the above website has a mistake; the

correction ( 12 2 ) has been made in the Activity
Sheets below. Rubric 2.1

Summative
1. Quiz 1 that includes questions similar to the
exercise questions on pages 204-221 or
appropriate sections/questions of Unit Review
and Practice Test on pages 246-249, Pearson:
Foundations and Pre-Calculus 10, and for
which students:
 identify and differentiate numbers as either
rational or irrational.
 apply strategies for writing radicals in
simplest form, writing entire radicals as
mixed radicals, and writing mixed radicals
as entire radicals.
 represent the relationships between natural
numbers, whole numbers, integers, rational
numbers and irrational numbers with
examples using a Venn diagram, as
illustrated on page 220, Pearson:
Foundations and Pre-Calculus 10.
2.
3.
2.
from text (Pearson: Foundations and
Pre-Calculus 10)
Venn diagram of set of Real
Numbers
Diagnostic assessment regarding the
laws of exponents learned in grade
9.
Assessment For Learning
 Assess your Understanding Activities in
text
 Teacher observation, questioning, and
verbal &/or written feedback
 Name that Song Game
 Line up Cards Activity
 Homework Activities with feedback, self
and peer evaluation and peer coaching
 Exit Slips
 Journal Activity: At the end of each class
neatly write/draw/diagram what you
learned that day and what you find
challenging.
 Concept Map as through a number of
refinements
 Reviews and Practice Test in text
Quiz 2 that includes questions similar to the
exercise questions on pages 222-243 or
appropriate sections/questions of Unit
Review and Practice Test on pages 246-249,
Pearson: Foundations and Pre-Calculus 10,
and for which students:
 apply exponent laws to powers (rational
and variable bases and integral and
rational exponents).
 express powers with rational exponents
as radicals and vice versa.
Unit 2 Concept Map Rubric 2.2
Stage 3 – Learning Plan
Learning Activities:
Period 1___________________________________________________________________________
1. Diagnostic Assessments. (10 minutes)
Have students to the best of their ability individually complete the following without a calculator:
 Make a Venn diagram of the set of Real numbers
 Read page 204, Pearson: Foundations and Pre-Calculus 10
 Complete “Assess your Understanding”: page 206, Pearson: Foundations and PreCalculus 10
As students are doing the above, circulate and observe to get an idea of where individuals and the
class’ understanding is at, so that appropriate learning activities from below may be selected.
2.
Irrationals: Motivational set. (5 minutes)
Songs/Music of Numbers.
Prior to this activity the teacher will need to download and install a free copy of Wolfram
Mathematica player from http://www.wolfram.com/products/player/. Then download the two
demonstrations: Math Songs (Sounds of Irrational Numbers); and, Music from the Rationals,
which can be found at:
http://demonstrations.wolfram.com/search.html?query=math+songs. Finally, before doing this
activity in class, play around with the two demonstrations so you are familiar with them.
Have students in small groups. Ask the class if they can detect any patterns in the songs of
numbers that will be played that would allow them to place the numbers into alike groups. In
class, open up the two demonstrations. Without students seeing the computer screen, choose one
of the demonstrations (either “Math Songs” or “Music from the Rationals”) and have student
groups listen to the “song” of the number selected. Be sure the repetition slider is set to the left. (I
found setting both the “duration per note” and “digit” sliders at about one third is a good place to
start when I did this myself. And, because of my Scottish heritage I chose the bagpipe as the
instrument!). Play the songs for a number of rational and irrational numbers. Allow the class to
offer their ideas to the question, but don’t give them any answers. Instead just say that today we
will investigate some different types of numbers.
3.
Irrationals: Construct Understanding. (10 minutes)
Cutting and Herding Numbers:
Prepare index cards, each with a number from the two “Construct Understanding” activities on
page 205 (section D) and on page 207(Try This section), Pearson: Foundations and Pre-Calculus
10. Include the symbol π as well on one index card. Display the numbers randomly on the white
board. Students, working in small groups of 3-4 students, will take the numbers and place them
into categories of the students’ choosing based on criteria that the students themselves develop.
Have each group report their categories and criteria to the class.
4.
Irrationals: Construct Understanding. (15 - 20 minutes)
Get groups working cooperatively to place the numbers used in the previous activity into a table
(like the table on page 205, Pearson: Foundations and Pre-Calculus 10), complete the table, order
the numbers on a number line, and answer Questions E and F on page 205, Pearson: Foundations
and Pre-Calculus 10.
Then have groups do part A of the “Construct Understanding” activity on page 207, Pearson:
Foundations and Pre-Calculus 10. Each group to report.
Next, have groups generate criteria/definitions regarding Rational and Irrational numbers, and
have them compare their definition of Irrational Numbers to the one in the text on page 208:
Pearson: Foundations and Pre-Calculus 10. Have each group briefly report how their definition
compared to the text definition.
Have groups do part B of the “Construct Understanding” activity on page 208, Pearson:
Foundations and Pre-Calculus 10. Discuss as a class.
Exit slip: Parts C and D of the “Construct Understanding” activity on page 208, Pearson:
Foundations and Pre-Calculus 10.
Journal Activity: Students are to include terms from page 204. Use Rubric 1.4 from Unit 1 –
discuss Journal prompts and rubric with students prior to starting this activity. Note that journals
may be viewed periodically for formative assessment purposes.
Period 2___________________________________________________________________________
5. Irrationals: Name that “Song” Game Formative Assessment. (10 -15 minutes)
Students detect the type of number (Rational or irrational) by listening to its “song”.
As noted for activity 2 above, prior to this activity the teacher will need to download and install a
free copy of Wolfram Mathematica player from http://www.wolfram.com/products/player/. Then
download the two demonstrations: Math Songs (Sounds of Irrational Numbers); and, Music from
the Rationals, which can be found at:
http://demonstrations.wolfram.com/search.html?query=math+songs Finally, before doing this
activity in class, play around with the two demonstrations so you are familiar with them.
In class, open up the two demonstrations. Without students seeing the computer screen, choose
one of the demonstrations (either “Math Songs” or “Music from the Rationals”) and have student
groups listen to the “song” of the number selected. Each student in the group is to determine if the
“song” is a rational or irrational number and why. Have students discuss this briefly in their group
and come to a consensus. Then have a group with the correct answer report their reason for
choosing their answer. Discuss as appropriate. Choose another number (either rational or
irrational) and repeat. Now introduce the game rules.
Game rules: Each student is to record their own answer on their score card. Each student writes
down whether they think the song is from a rational and irrational – introduce the symbols for
these two sets at this time. This game is like golf – everyone plays on their honour, but members
of their group help to monitor this! Oh, yeah, no Mulligans with this game. If you want to throw
in a little fun competition, then the student or group with the highest score (or highest average
score if there are different numbers of students in groups) will be declared the winner.
Now, let the game begin! The teacher selects a number and plays it’s song. After the song has
been played and students have recorded their answer, state the correct answer and ask students to
raise their hand if they got the correct answer. This will give you a quick assessment of the
students’ understanding of the difference between the two types of numbers. Continue to play
songs from different numbers. When most/all students are getting answers correct, reduce the
“duration” slider (the length of each note), and/or reduce the “digits” slider (ie. .length of the
song), and/or change the instrument, and have fun!
6.
Irrationals: Example Exercise Questions. (10 -15 minutes)
Place example questions 1 and 2 from page 209 up on the board and have individual students
attempt them. Then have students work within small groups to compare their solution to the
text’s solutions, and help each other out (peer tutor) as necessary. Teacher monitors and helps out
groups as necessary.
7.
Irrationals: Line-up Cards Formative Assessment Activity. (10 minutes)
Use the prepared index cards from activity 2. Give an index card to each student. Do the activity
as described on the website: http://regentsprep.org/REgents/math/ALGEBRA/AOP1/Tcards.htm.
Then orally discuss as a class questions #1 & 2, Page 211, Pearson: Foundations and Pre- Calculus
8.
Irrationals: In Class Individual Assignment. (15 minutes)
Assign #3-20, page 211-212, Pearson: Foundations and Pre- Calculus.
Homework: Complete #3-20 above and “Assess your Understanding” on page 206, Pearson:
Foundations and Pre-Calculus 10
Journal Activity: Rubric 1.4
Period 3___________________________________________________________________________
9. Irrationals: Peer tutoring. (15 minutes)
Arrange peer coaching for students who need support with challenges they had in the assignment
on page 211. Any students not paired up with a partner who needs tutoring can do questions #2124, page 212, and/or Part A and/or B of page 251, Pearson: Foundations and Pre- Calculus.
10. Irrationals: Sponge Activity. (10 minutes)
(Click the hyperlink to be directed to the activity, which is near the end of this document)
Exit Slip: Represent sets of numbers using a Venn Diagram with examples. Note: Students
should produce something like the Venn diagram on page 220, but do not direct them to this page.
Instead use the exit slips to determine students that need more support, which might be given in or
out of class depending upon the number of students.
Homework: Start making a concept map of content from section 4.1 to 4.3 on pages 202 – 221,
Pearson: Foundations and Pre- Calculus). Discuss Rubric 2.2 with students and how their final
Unit Concept Map will be used as a summative assessment.
The following article has a detailed description about how to make concept maps if students don’t
know how to do this: Novak, J.D., & Cañas, A.J. (2008). The theory underlying concept maps and
how to construct and use them, Technical Report IHMC CmapTools 2006-01 Rev 01-2008,
Florida: Institute for Human and Machine Cognition. Retrieved from
http://cmap.ihmc.us/Publications/ResearchPapers/TheoryUnderlyingConceptMaps.pdf. And, here
is the link to the free Cmap software site referred to in the article (note that it needs to be installed
before it can be used (division techies may need to tinker to get it to work on the network), and it
takes some time to use if not familiar with concept mapping technology):
http://cmap.ihmc.us/conceptmap.html
11. Mixed and Entire Radicals: Construct Understanding. (20-25 minutes)
Students work with a partner to do the “Construct Understanding” on page 213-214, Pearson:
Foundations and Pre- Calculus).
Journal Activity: Rubric 1.4
Period 4___________________________________________________________________________
12. Mixed and Entire Radicals: Direct Teaching. (10 minutes)
Show the students how to write radicals as mixed radicals in simplest form, and how to write
mixed radicals as entire radicals as per pages 214-217, Pearson: Foundations and Pre- Calculus).
13. Mixed and Entire Radicals: In Class Individual Assignment: (30 - 40 minutes)
Have the students work on a variety of questions from pages 218-219, Pearson: Foundations and
Pre-Calculus 10: Section 4.3, while the teacher circulates and assists as needed.
Homework: “Assess your Understanding” page 221: Pearson: Foundations and Pre- Calculus as a
formative assessment.
Journal Activity: Rubric 1.4
Period 5___________________________________________________________________________
14. Mixed and Entire Radicals: Peer tutoring. (15 - 20 minutes)
Arrange peer coaching for students who need support with challenges they had in the assignment
on page 218-219 or page 221.
Any students not paired up with a partner who needs tutoring can do either:
 questions Part A and/or B of page 251, Pearson: Foundations and Pre- Calculus. Jigsaw
Activity (Extension Activity).
Exit Slip: oral report to class

Or, these students might do the following jigsaw activity. Have half of these students go
to: http://www.solving-math-problems.com/calculate-square-root.html, which explains
Guess and Check, and the other half of the students go to: http://www.solving-mathproblems.com/calculate-square-root.html, which explains Newton’s Method. Have the
students read the website and use the method they were given to determine the square of
a number to “x” decimal places until they understand and are fully comfortable in using
that method. Once all students have an understanding of the strategy they will pair up
and explain the strategy they learned to the rest of their group/class.
Exit Slip: Find the square root of 41 to two decimal places using Guess and Check and
Newton’s Method.

Or, they might try the following enrichment Multiple Choice Questions:
http://regentsprep.org/REgents/math/ALGEBRA/AO1/pracRad.htm
15. Study for Quiz (next class): (15 - 20 minutes)
Students work in pairs or a small group to do sections 4.1 – 4.3 on the Review on pages 246 –
247. Peer coach as necessary.
Student individually do #1 – 3 on the Practice Test on page 249, Foundations and Pre-Calculus 10.
16. Mixed and Entire Radicals: Word Problems (15 - 20 minutes)
Assign a couple of word problems involving radicals for students to solve working in groups.
Have each group report, explaining their solution(s) after each problem attempted. The text,
Pearson: Foundations and Pre- Calculus, does not provide word problems at this point, so use
other sources for appropriate word problems.
Journal Activity: Rubric 1.4
Homework: Study for quiz and revise concept map of content from section 4.1 to 4.3 on pages 202
– 221, Pearson: Foundations and Pre- Calculus). Remind students of Rubric 2.2
Period 6___________________________________________________________________________
17. Quiz 1 (as per description in Stage 2 above) (20 - 25 minutes)
18. Laws of Exponents: Brief Diagnostic Assessment using positive bases and natural number
exponents. (5 - 10 minutes)
19. Laws of Exponents: Review the Laws of Exponents (from grade 9), as/if necessary. (10 minutes)
Journal Activity: Rubric 1.4
Period 7___________________________________________________________________________
20. Rational Exponents & Radicals: Constructing Understanding. (10 -15 minutes)

Give the student groups a list of equations such as 4 1/2 = 2, 251/2 = 5, etc., and try to get the

students to generalize that x1/2 = x , or use activity on page 222, Pearson: Foundations and
Pre- Calculus.
Give the student groups a list of equations such as 27 1/3 = 3, etc., and try to get the students to

generalize that x1/3 = 3 x , or use page 222, Pearson: Foundations and Pre- Calculus.
Discuss terms index, radical, and radicand.
21. Rational Exponents & Radicals: Direct Teaching. (10-15 minutes)
Demonstrate and have the students evaluate powers, with rational exponents, rewrite powers in
Radical and Exponential Form, evaluate powers with Rational Exponents and Rational bases. See
pages 223-226, Examples 1-4, Pearson: Foundations and Pre-Calculus 10.
22. Rational Exponents & Radicals: Pair Work. (15 - 20 minutes)
Have students work in pairs on questions #1-16, page 227,Pearson: Foundations and Pre-Calculus
10, Section 4.4, with peer coaching as required. Complete for homework if not done this period.
Homework: Revise concept map started previously to include content from Sections 4.4 to 4.6 on
pages 222 - 245. Remind student of Rubric 2.2
Journal Activity: Rubric 1.4
Period 8___________________________________________________________________________
23. Rational Exponents & Radicals: Problem Solving. (15 - 20 minutes)
Have student work cooperatively in small groups to do #17, 20, 21, 22 on page 228, Pearson:
Foundations and Pre-Calculus 10.
Homework: #18 & 19, page 228, Pearson: Foundations and Pre-Calculus 10.
24. Negative Exponents & Reciprocals: Construct Understanding. (15 minutes)
The students will work with a partner to read page 229 and complete the activity on page 230,
Pearson: Foundations and Pre-Calculus 10. The students will try to determine a generalization for
negative exponents (ie. x-n = 1/n, and 1/x-n = xn, x ≠ 0)
25. Negative Exponents & Reciprocals: Direct Teaching. (10 - 15 minutes)
Demonstrate and have the students evaluate Powers with Negative Integer Exponents, evaluate
Powers with Negative Rational Exponents, and apply Negative Exponents. See pages 231-232,
Examples 1-3, Pearson: Foundations and Pre-Calculus 10.
26. Performance Assessment: Start to do for homework. This is due the day of Quiz 2.
Discuss rubric with students.
Rubric 2.1
Homework: Questions as appropriate from page 233, Pearson: Foundations and Pre-Calculus 10.
Journal Activity: Rubric 1.4
Period 9___________________________________________________________________________
27. Rational Exponents & Radicals and Negative Exponents & Reciprocals: Peer Tutoring (20 - 25
minutes)
Arrange peer coaching for students who need support with challenges they had in the assignment
on page 218-219 or page 221. Teacher circulates among pairs.
Any students not paired up with a partner who needs tutoring can do “Assess Your
Understanding” on page 236, Pearson: Foundations and Pre-Calculus 10. This is homework for
these students and students who are struggling, but it is not required of student tutors.
28. Applying Exponent Laws: Construct Understanding. (10 minutes)
Starting on their own, students are to read “Make Connections” and do “Construct Understanding”
on page 237, Pearson: Foundations and Pre-Calculus 10. They end up working in pairs.
29. Applying Exponent Laws: Direct Teaching. (10 minutes)
Demonstrate and have the students simplify numerical expressions with rational number bases,
and simplify algebraic expressions with Integer exponents. See pages 238-239, Examples 1& 2,
Pearson: Foundations and Pre-Calculus 10.
Homework: #1-8 from page 241-242, Pearson: Foundations and Pre-Calculus 10.
Journal Activity: Rubric 1.4
Period 10___________________________________________________________________________
30. Applying Exponent Laws: Direct Teaching. (10 minutes)
Demonstrate and have the students simplify algebraic expressions with rational exponents, and
solving problems using the exponent laws. See pages 240-241, Examples 3& 4, Pearson:
Foundations and Pre-Calculus 10.
31. Applying Exponent Laws: Pairwork. (25 - 30 minutes)
Assign #9-21 from page 242-243, Pearson: Foundations and Pre-Calculus 10.
Homework: Read pages 244-245, Pearson: Foundations and Pre-Calculus 10 and revise concept
map started previously to include content from Sections 4.4 to 4.6 on pages 222 - 245. Remind
students of Rubric 2.2
Journal Activity: Rubric 1.4
Period 11___________________________________________________________________________
32. Study for Quiz 2. (40 – 50 minutes)
Students work in pairs or small group to do sections 4.4 -4.6 on the Review on pages 247 – 248.
Student individually do #4 – 8 on the Practice Test on page 249, Foundations and Pre-Calculus 10.
Or alternatively, the teacher could prepare a Review Activity, such as:
MillionairesRadicals:http://regentsprep.org/REgents/math/ALGEBRA/AO1/Lmillion.htm
Period 12___________________________________________________________________________
33. Quiz 2 (as per description in Stage 2, above) (20 - 25 minutes)
34. Hand in Performance Task.
Rubric 2.1
35. Final revision & proof of Unit 2 Concept Map due next class.
Rubric 2.2
Supporting Information for Learning Activities:
I. Sponge Activity:
Hand students a half-sheet of graph paper. The overhead projector should be on showing
the following directions:




Please follow the directions as best you can! Record your results on the graph
paper.
Draw 2.
Draw a rectangle that shows 2.
Draw a square that shows 2.
The teacher’s mindset should be that there is no one right answer. Your interest is in
having the students respond, and you hope to get a variety of ways of representing 2.
Processing:
Select 4 students to go to the board and draw a shape that shows 2. (If, as you have
walked around, you have not found 4 students with different drawings, after the students
have put their representations of 2 on the board ask if there are other ways.)
Select another 4 students to go to the board and:
 Draw a rectangle that shows 2.
Ask the students: (a)What is a rectangle? (b) What is a square?
 Direct students to draw a square that shows 2.
If a student is able to draw a square that shows 2, discuss the length of the side.
Is it a whole number?
Repeat the entire process using other numbers, such as 4, 5, 8, 9.
Discuss as a class. Hopefully they make a connection back to unit where there was an
activity where numbers that are perfect squares were easily drawn as squares.
II. Tuning In Performance Task
Piano Repair Technician
Adapted from BC: Applications of Mathematics 10 http://www.bced.gov.bc.ca/careers/aa/lessons/aom15.htm
Lesson Idea by: David Ward, Rutland Senior Secondary School,
Central Okanagan School District
Some people think piano tuners have something called "perfect pitch," says Paul Brown, a
registered piano repair technician in British Columbia. But it's just a catch phrase someone
must have dreamed up years ago.
If someone had perfect pitch they could correctly tune the first note tuned on a piano, A 440,
without using a tuning fork or an electronic device. This is simply impossible. (The 440
relates to the number of cycles per second that the piano string completes while vibrating.)
In a band or orchestra setting, many instruments may need to "tune in" with the A 440 note
before practising scales, and certainly prior to playing together. Whether played on a piano
or an oboe, it is vital that the A 440 note be perfectly tuned to ensure the concert is made up
of an enjoyable, harmonious series of sounds.
Since proper tuning is so vital for good music, one can't rely on the myth of perfect pitch.
Therefore, piano repair technicians set the first note properly with a tuning fork or an
electronic device. The rest of the piano is tuned by comparing the notes to that first one.
The goal is to equally space all the other notes so that the sounds made relate to one another
in a harmonious way. In other words, the sounds must relate with harmonicity. Without this
harmonicity, the sounds made would be harsh to the ears.
Make a visit to the music room and have someone play the A 440 note on a piano. Listen to
it on its own, and as part of a scale.
Observe that for the more standard scales in the middle of a piano, playing most keys
activates a hammer that strikes three wires to form a single note! Is this a surprise?
Consider the numbered (#) set of questions as you work through the activities below.
Record your answers on the Student Activity Sheet.
Question #1: What is of concern to a tuner here? How about a listener?
If possible, have someone play some scales on a guitar, a clarinet, a trombone, or on other
instruments. What do you hear?
Question #2: How do you think mathematics relates to a range of notes played on a piano or
any other instrument?
Ask if it is possible to see and hear a tuning fork. If you can, observe the way an electronic
device may be used for tuning a piano or guitar.
The first job of the tuner is to set the A 440, described in terms of the cycles or "vibrations
per second." Then the tuner must equally space all the other notes in relation to A 440 and to
each other.
One of our early mathematical "stars," named Pythagoras (6th century BC), discovered
some of the relationships between the length of strings in musical instruments and
harmonious intervals. As methods of measuring the frequency of vibrations were developed,
Galileo (1564-1642) and Mersenne (1558-1648) established some important relationship
rules.
Today, several quite complex "Laws of Strings" govern the many special factors that affect
control of musical notes in a piano. However, all notes are essentially separated by one
special formula:
The frequency of any higher note is calculated by multiplying the frequency of the previous
note by 12 2 or 2 (1/12). Dividing the frequency of a note by 12 2 or 2 (1/12) establishes the
number of vibrations per second of a lower note.
To accomplish the task above on most calculators, you would press 2, followed by the 2nd
layer function button, then, the "x sq. root sign y" button, followed by 12. Try it. The result
should be 1.059463094.
Next, multiply 440 by
12
2 (the twelfth root of 2) or 2 (1/12).
Did you get 466.1637615? This is the frequency of the note, A#.
Now, answer these questions:
Question #3: Are the multiple decimal places necessary?
Question #4: Is the answer a Rational or Irrational number?
(Remember: All rational numbers are made up of the set of natural numbers, whole
numbers and integers. They can be written as decimals that are either terminating or
repeating. Irrational numbers are those numbers with decimal values where there is not a
clear pattern, that is, neither terminating nor repeating decimals. The value "pi" is an
example of an irrational number.)
Question #5: Is it wise to consider the recording of a "Rational explanation" to describe an
Irrational number? Would "rounding off" to a certain decimal place provide an accurate
answer? Explain.
For the purpose of the exercise on the Student Activity Sheet, simply record the first two
decimal values reached in your calculations. You do not need to round off or change the
answers on the calculator before each successive operation of multiplication or division.
(See instructions that follow.)
Apply the simple calculation described above in order to determine the number of string
vibrations per second for the notes of a piano as set out in the Student Activity Sheet. You
may start at the C (261.63) position and move upward to determine the frequency of the next
note on the piano keyboard on the Student Activity Sheet. Remember to simply record the
first two decimal values reached in your calculations, but don’t round off or change the
answers on the calculator.
To check your work, start at the A 440 value, and divide by 12 2 each time you arrive at an
answer to get values in a descending order. Again, don’t round off or change the answers on
the calculator.
Question #6: Do you notice any differences? Discuss your observations and why there are
any differences. Record this explanation on the Student Activity Sheet.
Now determine the exact values in radical form: Start at the A 440 value, and divide by 12 2
or 2 (1/12) each time you arrive at an answer to get values in a descending order. And, for the
two notes above A 440, multiply by 12 2 or 2 (1/12) each time you arrive at an answer to get
the exact frequency of A# and B. Be sure to place all the radicals in simplest form. Write
your exact answers to the left of each note’s name.
Additional Activity
Paul Brown, the piano tuner consulted in researching the application of rational and
irrational numbers in a specific setting, notes: "If an electronic device were used for all
tuning, the spacing of notes would be very rigid in terms of separation. Fortunately, the
human brain is such a marvelous creation that it can be used to help space notes apart
equally, at the same time, taking into account slight imperfections in the piano wire!" After
special training and lots of practice, it is possible for piano repair technicians to listen for
certain "intervals" or "beat rates," that distinguish notes from one another. For example, a
tuner has to be able to count from one to 10 beats per second on occasion. With plenty of
experience, piano tuners truly gain a sense of "feeling" when notes are in proper tune.
While many of the standard octaves can be tuned "aurally," an electronic device may be
used for the higher octaves, because of the high-level frequencies involved. The length,
thickness and tension of good quality steel wire are changed to produce notes of a higher
pitch.
The "hertz" or cycles per second can be as high as 1568.0 for G in the A octave directly
above A 440; 2793.8 for F in the A-7th octave, and 4186.0 for the highest C note of an 88
key piano (the last key on the right).
Question #7: How do you think changes in temperature could create problems for
musicians?
Perhaps the next time you hear music, you will consider for a moment the clever and
special mathematical way that instruments work together in order to produce the series of
sounds that we so often take for granted! And, any time that someone tells you they have
"perfect pitch," and can sing any note without a comparison first, you'll know that "perfect
pitch" is a myth!
"Tuning - In" Student Activity Sheet
Name: _______________________________________________
Course: _________________________ Block: _______________
Teacher: ______________________________________________
"Tuning - In" Student Activity Sheet…continuted
Questions
Question #1: What is of concern to a tuner here? How about a listener?
Question #2: How do you think mathematics relates to a range of notes played on a piano
or any other instrument?
Question #3: Are the multiple decimal places necessary? Why or why not?
Question #4: Is the answer a Rational or Irrational number?
Question #5: Is it wise to consider the recording of a "Rational explanation" to describe an
Irrational number? Would "rounding off" to a certain decimal place provide an accurate
answer? Explain.
Question #6: Do you notice any differences? Discuss your observations and why there are
any differences.
Question #7: Discuss: How do you think changes in temperature could create problems for
musicians?
Solution for Student Activity Sheet
The frequencies for the notes are:
C# = 277.18
D = 293.66
D# = 311.13
E = 329.63
F = 349.23
F# = 370.00
G = 392.00
G# = 415.31
A = 440.00
A#= 466.17
B = 493.89
C = 523.25
These 12 notes, along with C (261.63), comprise the
chromatic scale of C.
III. Math Journal Prompts and Rubric 1.4 (adapted from:
Science Journal Prompts and Rubric found at
http://uteach.utexas.edu/~gdickinson/pbi/Tony07Project
s/ecosystem/Content/Science_journal_promts.pdf )
Math Journal:
Journal Activity: Act the end of each class briefly and neatly write/draw/diagram what
you learned that day, what you find challenging, as well as thoughts or epiphanies you
have regarding the Essential Questions.
Journal Prompts:
This is a great way to avoid hearing the response “I don’t know what to write!” Here are
six Math journal-writing prompts to get the ball rolling:
• Today I discovered that……… I also learned that………The most interesting part of
the activity was……..I am still wondering………..
• Today I inquired about……..My hypothesis was……..I concluded that……..My next
inquiry will be about…………..
• Today I observed……..I predict that………I also measured…………I concluded
that………….
• Today I learned about (vocabulary word). I discovered that (vocabulary
word)………………
• Today I observed (topic). I now know what happened to ………….I am still unsure
about………………
• Today I performance task on (topic). I predicted that ………….I analyzed my results
and concluded that…………Another question that I have is……………………
FP 10.1 Rubric1.4 is from Unit 1 and is used for formative assessment/ Assessment for
Learning with Student Journals. Journal prompts for the students are also provided
above.
Criteria
4
3
2
1
All journal
All journal
Most journal
Journal entries
Completed
entries are done entries are done entries are
Challenges are
journal and
and
and
done, or all are not completed,
clearly
communicated
done but not all but some are
communicated communicated
clearly in detail, clearly.
were
attempted.
(wrote, drew,
illustrated as
communicated
and/or
necessary.
clearly
illustrated)
information &
thoughts
Knowledge
Knowledge
Knowledge
Clearly showed Knowledge
learned is
learned is
learned is
learned is
knowledge
always shown
consistently
usually shown. sometimes
learned
clearly.
shown.
shown.
Challenges are
Challenges are
Challenges are
Challenges are
Clearly
always
consistently
usually shown. sometimes
indicated
shown.
shown.
challenges with indicated
clearly.
topic or noted
that there were
on challenges
with this topic
Provided
detailed
descriptions
using
mathematical
terms, as
appropriate
Posed
questions for
further thought
and study, and
answered them
– answering
occurs after
more learning
has occurred.
Recorded
thoughts,
epiphanies, and
answers
regarding the
Essential
Questions.
IV.
Descriptions are
always detailed
and use
mathematical
terms, as
appropriate.
Descriptions are
consistently
detailed and use
mathematical
terms, as
appropriate.
Descriptions are
usually detailed
and usually use
mathematical
terms, as
appropriate.
Questions for
thought are
consistently
posed, and
usually.
Questions for
thought are
often posed,
and often
answered.
Questions for
thought are
sometimes
posed, and
sometimes.
Thoughts and
epiphanies are
recorded fairly
often, and the
answers are
clear and
correct.
A few thoughts
and epiphanies
may be
recorded, and
the answers are
correct.
Thoughts and
epiphanies are
not recorded,
but most
answers are
correct.
Descriptions
are not
detailed, or do
not use
mathematical
terms, as
appropriate.
Questions for
thought are
rarely posed,
and answered.
Thoughts,
epiphanies,
are not
recorded, and
few answers
are correct.
Rubric 2.1 – Tuning In Performance Task FP10.2
Criteria
Approximate
(Decimal)
frequency of
notes on piano
keyboard on
Student Activity
Sheet.
Exact (radical in
simplest form)
frequency of
notes on piano
keyboard on
Student Activity
Sheet.
4
All the notes’
frequencies
were calculated
correctly as
directed.
3
Most of the
notes’
frequencies
were calculated
correctly as
directed.
2
Some of the
notes’
frequencies
were calculated
correctly as
directed.
1
None or few of
the notes’
frequencies
were calculated
correctly as
directed.
All the notes’
frequencies
were calculated
correctly as
directed.
Most of the
notes’
frequencies
were calculated
correctly as
directed.
Some of the
notes’
frequencies
were calculated
correctly as
directed.
None or few of
the notes’
frequencies
were calculated
correctly as
directed.
Answering
Questions on
the Student
Activity Sheet
Answers to
Questions
All questions
were carefully
considered
All questions
were
considered.
Most questions
were
considered.
Few questions
were
considered.
All answers
were clear,
detailed, and
correct.
All answers
were clear and
correct.
Most answers
were clear and
correct.
Few answers
were clear or
correct.
V. Rubric 2.2 – Unit Concept Map
Criteria
4
3
2
1
Concepts
All required
concepts from the
unit, described
using proper
mathematical
terminology, are on
the concept map in
boxes or ovals.
Some extra relevant
concepts might be
included.
Concepts are
arranged
hierarchically from
most general at the
top to most specific
at the bottom of
each vertical strand
of the concept map.
Almost all
required
concepts from
the unit,
described using
proper
mathematical
terminology, are
on the concept
map in boxes or
ovals.
Concepts are
usually arranged
hierarchically
from most
general at the top
to most specific
at the bottom of
each vertical
strand.
Few required
concepts from
the unit,
described using
proper
mathematical
terminology, are
on the concept
map in boxes or
ovals.
Links
Concepts in vertical
threads are all
linked by lines with
concise, descriptive
labels.
Spatial
Arrangement
of Concepts
All Concepts that
are related to each
other are positioned
next to or near each
other.
Cross-links
Concepts from
Concepts in
vertical threads
are nearly all
linked by lines
with descriptive
labels.
Most concepts
that are related to
each other are
positioned next
to or near each
other.
Concepts from
Some of the
required
concepts from
the unit,
described using
proper
mathematical
terminology, are
on the concept
map in boxes or
ovals.
Concepts are
sometimes
arranged
hierarchically
from most
general at the top
to most specific
at the bottom of
each vertical
strand.
Concepts in
vertical threads
are sometimes
linked by lines
with descriptive
labels.
Some concepts
that are related to
each other are
positioned next
to or near each
other.
Concepts from
Hierarchical
Arrangement
of Concepts
Concepts are
rarely arranged
hierarchically
from most
general at the top
to most specific
at the bottom of
each vertical
strand.
Concepts in
vertical threads
are not linked by
lines with
descriptive
labels.
Few concepts
that are related to
each other are
positioned next
to or near each
other.
Concepts from
different vertical
threads that are
related to each
other are cross
linked with lines
and concise,
descriptive labels to
show a
sophisticated
understanding.
different vertical
threads that are
related to each
other are cross
linked with lines
and descriptive
labels to show a
healthy
understanding.
different vertical
threads that are
related to each
other are cross
linked to show a
limited
understanding.
different vertical
threads that are
related to each
other are not
cross linked to
show any
understanding.