Constructing Rational Number and Operation

Tuesdays: 5-8 PM, Room 321 PKH

For the course syllabus, click here
and here.

For problems worked or assigned in class, click here.

For the readings packet, click here.

For links to other online readings, see below.

For student interviewing tips, click here.

For the sample interview write-up, click here.

For tips on writing papers on math problems,
click here.
For a math ed web resources page, click here.

**NOTE:** Technical information such as prerequisites, text materials,
course format and assignments, and other details can be found in the syllabus,
a copy of which is provided in a link at the top of this page.

*integers*: the whole numbers and their (additive) opposites: 0, +1, -1, +2, -2, ...*iterate*: to repeat (or make copies of). Iterate 1/3 five times to make 5/3.*natural number*: the counting numbers, beginning with 1 and going up.*[fraction as] operator*: a set of instructions or action to take, with regard to another object. A fraction as operator typically implies partitioning followed by iterating: for example, 2/3 as operator ("2/3 of 5", "2/3 of the votes cast", etc.) implies first partitioning the object into three equal parts and then making two copies of (iterating) one of those parts. This corresponds to multiplication by the fraction; the word "of" (in "2/3 of") evokes set-model multiplication.*partition*: to split into a given (usually whole) number of equal parts. This action corresponds to partitive division.*rational number*: any number that can be written as the ratio of two integers (the second one nonzero).*unitize*: to conceptualize or treat a quantity as an iterable unit: for example, to count in terms of thirds, "two-thirdses", or other composite quantities.*whole numbers*: zero and the natural numbers.

*Session 7:*Algorithms for dividing fractions --- Write full justifications in the simplest possible terms for the common-denominator and invert-and-multiply algorithms.*Session 8:*Place Value in Other Bases --- Respond to the last question on the worksheet using the earlier questions to support your answer (in other words, your paper should read as a single, organized narrative, rather than as a homework assignment with five numbered answers).*Session 9:*Fractions to Decimals --- Give a complete, justified answer to the question of when a given fraction's decimal representation will be terminating, and when repeating, including: (1) an explanation of what causes termination, (2) a*general*explanation of why nonterminating decimal representations for fractions must eventually repeat, and (3) an upper bound on the length of the repeating digit string. Organize your response around questions 2(b), 3(c) and 3(d) from the worksheet, and use examples (those provided or your own) to illustrate (but not prove) your points.*Session 9:*Decimals to Fractions --- Give a complete, justified*general*explanation of how to convert terminating or repeating decimals into common fractions. Your paper should address all possible cases (see examples on the worksheet).*Session 10:*Fractions and Decimals in Other Bases --- Use the first two questions to motivate and illustrate an explanation of the last one.*Session 12*: Going to Extremes --- A single cohesive narrative that addresses the questions in this activity. (Note all the questions are yes/no but require some explanation.)*Session 12*: Countable and uncountable infinities --- A single cohesive narrative that addresses the questions in this activity (the countability, or not, of the whole numbers, integers, rationals, and reals). If you choose to write this up, you might focus on the process.*Session 13*: Divisibility tests --- Group the common tests into families (as will have been discussed in class), explain the justification for each family, and address the last 3 questions on the page.*Session 13*: Odd and Even in Base Five --- a complete, justified answer to the question.*Sessions 14, 15*: How many factors? --- Explain and justify a general procedure for determining how many factors a given (arbitrary) natural number has.- Feel free to argue for including others.

- an online version of the FDRP casebook from the ERIC database
- learner.org's online courses "Learning Math" include Number and Operation.
- John Carroll University's Mathematical Vignettes provide short, informal essays on several math topics.

*Session 1*- Cuisenaire rod applet at learner.org 8B
- old class notes
- the factor table used in the factor problems

*Session 2*- Factor tree applet from USU which includes Venn diagrams of prime factorizations to find GCF and LCM
- Online activities for virtual Session 2 (from 2009)

*Session 3**Journal 2 (mini-interview) suggested prompts.*For most students, you can use the following problems: (1) 1/2+1/4, (2) 1/2+1/3, (3) 3/4+2/3. If necessary for student understanding, you may situate the problems in simple story problem contexts (refer to the CGI problem type table). For younger students who cannot respond to any of the problems, ask (after trying) a problem with like denominators. You may also allow the student to provide concrete or semiconcrete responses (ask if the answer has a number name, but don't report purely pictorial responses as ``the student was unable to answer''). Report the student's response precisely, and analyze the understanding the student shows of representing and adding fractions. If the student ``just knows'' (for example, that 1/2+1/4=3/4), probe for justification (including of the algorithm), especially for older students.

*Session 4**Journal 3 activity:*first page, second page (you'll need to copy this diagram from the first page)

*Session 5*- Cross-hatching applet at learner.org 9A

*Session 6*- An article that follows up on MMO Case 28

*Session 8*- Babylonian decimals video at learner.org 7B

*Session 12*- learner.org's Number and Operations course, Session 1C on extending the number line
- learner.org's Number and Operations course, Session 2A on number sets
- learner.org's Number and Operations course, Session 2B on infinities
- learner.org's Number and Operations course, Session 3B on exponents, logarithms, and scientific notation
- JCU's vignettes on infinities and irrationals (readings for this session)

*Session 14*- tiling/area model for GCF and LCM at learner.org 6A
- Sieve of Eratoshtenes at learner.org 6B