Notation
Notation
Objectives
In this unit, students will explore a variety of notations and become proactive in discovering origins and accepted notations for mathematical ideas. Students will investigate and explore notation via:
- sets and functions.
- history of numbers and evolution of symbols.
- patterns and sequences.
Essential Questions
- What notations are generally accepted throughout mathematics? What role does notation play in the realm of mathematical understanding?
- How are mathematical notations used as part of the problem-solving process?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
- http://illuminations.nctm.org/ActivityDetail.aspx?ID=34
- http://nlvm.usu.edu/en/nav/frames_asid_191_g_4_t_2.html?from=category_g_4_t_2.html
- http://mathworld.wolfram.com/Set.html
- http://mathworld.wolfram.com/Function.html
- http://illuminations.nctm.org/ActivityDetail.aspx?ID=161
- http://illuminations.nctm.org/ActivityDetail.aspx?ID=8
- http://nlvm.usu.edu/en/nav/frames_asid_137_g_4_t_3.html?open=instructions&from=category_g_4_t_3.html
- Golden Rectangle http://nlvm.usu.edu/en/nav/frames_asid_133_g_4_t_3.html?open=instructions&from=category_g_4_t_3.html
- National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html
- The First Thousand Digits of Pit
http://www.factmonster.com/ipka/A0876705.html
- Number: The Language of Science by T. Danzig. Pearson, 2005.
- The Thirteen Books of Euclid’s Elements, 2nd ed. by Sir Thomas L. Heath. Dover Publications, 1956.
- The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number by M. Livio. Broadway Books, 2002.
- The Rhind Mathematical Papyrus: Free Translation Commentary and Selected Photographs, Transcription, Transliterations, Literal Translations: Classics in Mathematics Education, 8, by A. B. Chace. (Ed.). National Council of Teachers of Mathematics, 1979.
- A Number for Your Thoughts by S. P. Richards. S. P. Richards, 1982.
http://www.ebook3000.com/A-Number-for-Your-Thoughts_84216.html
Formative Assessment
-
View
Multiple-Choice Items:
1. Which of the following correctly uses set-builder notation to describe the set of all integers less than −2, denoted Set A?
2. Suppose we have the following sets: Set A = {−8, 1, 4} and Set B = {2, 0, 1, 3}. After performing an operation on the sets, we get the set {1}. Which relationship was examined?
A
B
C
A´
D
B´
3. Which relation is also a function?
A
B
0
8
4
−1
−1
6
1
3
4
2
C
D
4. Which of the following is a perfect number?
A
16
B
28
C
36
D
41
5. How many prime numbers are there in the range from 0 to 25?
A
8
B
9
C
10
D
11
6. Which of the following is a representation of divergence?
A
Π
B
C
.898989…
D
7. Which of the following is the correct rule for describing the triangular numbers?
8. Which is not a correct description/representation of the Golden Ratio?
A
B
Irrational
C
D
1.8180339887…
9. Which of the following represents an arithmetic sequence?
A
B
C
1
4
2
8
3
16
4
32
D
10. When converting the repeating decimal to a fraction, which would be a possible equation involved in the solution process?
- 10a = 18.75555
- a = 1.875555
- 100a = 1875.555
- 1000a = 1875.555
Multiple-Choice Answer Key:
1. C
2. A
3. A
4. B
5. B
6. D
7. B
8. D
9. D
10. D
Short-Answer Items:
11. Write the set of “whole numbers less than 9” using three other representations.
12. Write the fraction equivalent for the infinite decimal
. Show your work and tell whether this is an example of convergence or divergence.13. Describe three patterns found within Pascal’s Triangle.
Short-Answer Key and Scoring Rubrics:
11. Write the set of “whole numbers less than 9” using three other representations.
Points
Description
3
The student correctly writes three representations. These might include:
2
The student correctly writes two of the three answers.
1
The student correctly writes one of the three answers.
0
The student does not provide an answer or incorrectly answers all three parts.
12. Write the fraction equivalent for the infinite decimal . Show your work and tell whether this is an example of convergence or divergence.
Points
Description
3
The student writes the following:
- 35/99
- 100a = 35.3535
- This is an example of convergence.
2
The student correctly writes two of the three answers.
1
The student correctly writes one of the three answers.
0
The student does not provide an answer or incorrectly answers all three parts.
13. Describe three patterns found within Pascal’s Triangle.
Points
Description
3
The student describes three patterns. These might include:
- The natural number set can be found within the second diagonal.
- The sum of the numbers in each row is a power of 2.
- The sum of the diagonals equals the number in the next row.
2
The student correctly writes two of the three answers.
1
The student correctly writes one of the three answers.
0
The student does not provide an answer or incorrectly answers all three parts.
Performance Assessment:
Of all the material presented in this unit, which component or piece stands out as the most influential or key in your mind regarding the importance of notation in mathematics? Present your idea using a visual representation of your choice. Include the following in your visual representation:
- statement of the component or piece you are describing.
- three illustrations with supporting explanations.
- description of the connection between your component/piece and the topic of notation.
Performance Assessment Scoring Rubric:
Points
Description
5
The assessment includes the following requirements, all of which are mathematically correct:
- statement of the component or piece the student is describing.
- three illustrations with supporting explanations.
- description of the connection between the student’s component/piece and the topic of notation.
4
The assessment includes four of the five components, all of which are mathematically correct.
3
The assessment includes three of the five components, all of which are mathematically correct.
2
The assessment includes two of the five components, both of which are mathematically correct.
1
The assessment includes one of the five components, which is mathematically correct.
0
The student does not present any parts of the assessment or provides incorrect information for all five parts.
DRAFT 10/11/2011
