Math 97 Pedagogy & Assessment Plans€¦ · Web viewWeekly Topic (& relevant sections of the text) Math Knowledge Outcomes

MATH 180 PEDAGOGY & ASSESSMENT PLANS Rev: Winter 2008

Weekly Topic (& relevant sections of the text) Math Knowledge Outcomes

1. The role of the educator; the concept of “numbers” (2.1, 2.2, 2.4) ~1, ~3, ~112. Prop. of number systems (2.2, 2.3); Our # system – purpose/represn’ns (6/7/8/9.1,9.2) ~3, 4, ~7

3. (Short) Structure of numbers (3.3, 5.1, 5.2) 3

4. Concept of the operations (+ and – with wholes, dec, frac, int; 3.1, 6.2, 7.2, 8.1) ~5, ~7

5. (Gone) Algorithms (+ and – with wholes, dec, frac, int?; 4.2, 4.3, 6.2, 7.2) ~6, ~8

6. Concept of the operations ( and with wholes, dec, frac, int; 3.2, 6.3, 7.2, 8.2) 5, 7

7. (Short) Algorithms ( and with wholes, dec, frac, int?; 4.2, 4.3, 6.3, 7.2) 6, ~8

8. Other algorithms (4.1, 6.2/3, 7.1/2, 8.1/2,9.1/2); Ratios, Percents, Prop. (6.3,7.2,7.3,7.4) ~2, 8, ~9, ~10

9. Ratios, Percents, Proportions and Problem Solving (7.3, 7.4, 1.1, 1.2) ~2, 9, 10

10. Problem solving (1.1, 1.2); The Real Number System 2, 3+

11. (1 class day – student evals?; Finals) 1, 11

Learning Objectives

Thorough Knowledge of Specific Mathematical ContentAt the end of this course, you should be able to:1. Identify sources of mathematics standards and summarize major strands of the NCTM Standards

or Washington math EALRs/GLEs.2. Describe multiple problem solving strategies and use them to solve a variety of problems.3. Describe the concept of a number, along with the reasons for/use of different types of numbers in

the real number system.4. Describe features of our number system and use that information to analyze other number

systems or obstacles to children learning our number system.5. Describe the purpose of and concepts underlying basic operations on whole numbers.6. Use multiple algorithms for arithmetic computation, list the steps for executing each one, and

analyze several algorithms’ advantages and disadvantages.7. Represent decimals, fractions, and integers in multiple ways; compare and contrast them with

whole numbers and each other.8. List the steps for computing with decimals, fractions, and integers; compare and contrast the

standard algorithms with those for whole numbers.9. Define, compare, and contrast ratios (rates), percentages, and proportions; represent them in

multiple ways.10. Describe techniques for identifying and solving problems involving ratios (rates), percentages,

and proportions, then solve problems involving them.11. Communicate effectively orally and in writing about the concepts and techniques of the course.

Procedural Proficiency at the Elementary Student Level Able to accurately and efficiently perform computations and tasks expected of students in grades

K-8 without the use of a calculator or notes.Procedural Proficiency at the Elementary TEACHER Level

Able to accurately explain and use alternative strategies to perform computations or other tasks expected of students in grades K-8. (Some strategies will make use of a calculator.)

Robust Understanding of Core ConceptsDepth of understanding at highest levels of Bloom’s Taxonomy:

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Able to analyze mathematical techniques or student work to describe core assumptions and ideas being used.

Able to synthesize a variety of ideas into concept maps showing the relationships between mathematical concepts and techniques.

Able to evaluate and articulate one’s own understanding and uncertainty of mathematical ideas.

Increased Awareness of Own Attitudes and Behaviors About Mathematics and Learning Able to describe one’s own attitudes and behaviors, along with how they change and their

consequences for one’s future students.

Preliminary Schedule – Subject to change

Monday Tuesday Wednesday Thursday Friday-HW Due(2.1, 2.2, 2.4) Jan. 7Educators/Numbers

8 9 10 Wk1 11SPT 1 – Decimals

(2.2, 2.3) 14Properties of # sys.

15 16 (6/7/8/9.1, 9.2) 17Structure of # sys

Wk2 18TPT 1 –Numeration

No class 21(MLK Jr. Day)

(3.3, 5.1, 5.2) 22Structure of #sys, #s

23 24 Wk3 25SPT 2 – Fractions

(3.1,6.2,7.2,8.1) 28Concept of +, –

29 30 31 Wk4 Feb. 1TPT 2 – # structure

(4.2,4.3,6.2,7.2) 4Algorithms for +,–

5Erik in Atlanta

6Erik in Atlanta

7Erik in Atlanta

SPT 1/2 – Retake 8Erik in Atlanta

(3.2,6.3,7.2,8.2) 11Concept of , ÷

12 13 14 Wk6 15TPT 3 – Op concpt?

No class 18President’s Day

(4.2,4.3,6.3,7.2) 19Algorithms for , ÷

20 21 Wk7 22SPT 3 – Equations?

(4.1, 8.1/2, 9.1/2) 25Other algorithms

26 27 (7.3, 7.4) 28Ratio, %, Proportion

Wk8 29TPT 4 – Nonst alg.?

(7.3, 7.4) Mar. 3Ratio, %, Proportion

4 5 (1.1, 1.2) 6Problem Solving

Wk9 7SPT 4 – RPP?

(1.1, 1.2) 10Problem Solving

11 (9.1?, 9.2) 12Extending # system

13 Final is next Tue 14SPT 3/4 – Retake

Class Components Lecture In-Class Exploration Weekly Assignments and Presentations Concept Maps Self-Reflective Writing Student Proficiency Tests Teacher Proficiency Tests

Approach to Formal Mathematics Students must know the formal mathematics to read and translate instructor materials Procedure for addressing theorems or general statements:

o [Formal] Read the formal statemento [Example] Create an example to make the statement concreteo [Justify] If possible, use objects or several examples to explain why the statement

should always be trueo [Paraphrase] Rewrite the formal statement in informal language that captures the

major idea

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Week Topics, Outcomes,& Sections of Text Daily Plan

11/7

Sched

IntroductionsEstablishing process

The Role of an EducatorThe Concept of “Number”(2.1, 2.2, 2.4)

Math Knowledge Outcomes:o [MK~1] Identify sources of

mathematics standards and summarize major strands of the NCTM Standards or Washington math EALRs/GLEs.

o [MK~3] Describe the concept of a number, along with the reasons for/use of different types of numbers in the real number system.

o [MK ~11] Communicate effectively orally and in writing about the concepts and techniques of the course.

Procedural Proficiency (Student)Student Proficiency Test #1<Decimal Arithmetic – 5 points> Add/Subtr decimals Mult/Div decimals Round

Robust Understanding: [MK11] Writing task on

analyzing the concepts of “standard” and “number.”

Question(s) of the Week: What is a “number”?

Puzzle of the Week: P. 35 2(a) (Sec. 1.2 B)

Standard(s) of the Week: GLE 1.1.1, Grade 2:

“Represent a number to at least 1000 in different ways, including numerals, words, pictures, and physical models;

Monday [Introduction, Questions] Have them read cover page of syllabus, write into info as I

take roll. Write introductory info to share with groups (& hand in?):

o Name, where you are from.o [Why do you want to teach?]o What level do you want to teach, and why?o What’s your experience with children (age, setting)?o What’s your definition of a very smart (or “gifted”)

child?Assign: Give out HW 1, ask them to write about the following for in-class discussion (#1-3 on HW), plus questions about learning objectives/syllabus. Career – What are the differences between a teacher and a

day care provider? Teaching – What is the purpose of a “standard”?

(Compare w/ syllabus learning objectives & course components)

Math – What is a “number”? For example, “eight” is more than just a word or symbol.

Tuesday [Groups, Standards, 2.1, 2.2] – Bring copy of GLE Group creation. [10 min] Teachers vs. day-care providers? Share w/ grp [20 min] Discuss meaning/purpose of a standard

o Features: Essential knowledge/skill; Achievable (Determined how? What fraction of people? – Military analogy); Consequences if not met; Implications?

o Discuss syllabus in light of the above.o Key notes: List main standards’ names and source

organizations. [20 min] Discuss concept of a number w/ group.

o What’s challenging about answering this question?o Not just a symbol, sound, or a specific collection – it’s

an essential feature that links many different things. [Read first part of Ch.2 of Russell’s Introduction to Mathematical Philosophy]

o Compare/contrast with letterso Key notes: A number is a representation of

quantity or amount.

Wednesday [2.1, 2.2, 2.4] – Bring chips! [20 min] Key notes: HOW do quantities get

represented by numbers – what’s the process?o Activity: Distribute chips, place 4 dots on board, ask

students to show me how many dots there are.o How do you know what to show, or how do you

decide your answer is right? [Matching idea – sound,

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translate between representations.”

finger, chip for each dot; symbols drawn in air?]o Key notes: What associations are important to a

usable concept of number? [Sounds – English,…; Symbols – Arabic, …; Other objects – fingers, people, ???]

o Show rapid growth of complexity due to linking each idea with the others.

o Key notes: Introduce formal terminology for precise descriptions and generalization: Groups -> Sets Members -> Elements Association/Representation -> one-to-

one correspondence Revisit complexity of counting using this language – they

provide examples of how errors occur (over/undercounts). Note that counting is independent of the order of name assignments! [Functions/relations?]

[20 min] Alternate source of confusion: using chips in front of you, show me “two.” (3 ways – two chips, second chip, creating shape of symbol 2)o Last of these is narrow focus on word-symbol

associationo Second of these is confusing the set of two with the

“two” from the number chant. We try to eliminate this confusion by using different words – “second” vs. “two,” but it is a recognition that there are different meanings.

o Key notes: terminology for types of numbers Ordinal numbers – for describing order

or position Cardinal numbers – for describing the

amount in a set Identification numbers – for labeling

objectso Key notes: Counting chant is about internalizing a

set with a specific order that can be used as a “measuring stick” for other objects.

o Key notes: How does counting use both ordinal and cardinal numbers – how are ordinal numbers turned into a cardinal number? [Last ordinal number stated IS the cardinal number.]

Thursday [2.1, 2.2] – Bring chips! First “Student Proficiency Test” tomorrow @ start –

decimal arithmetic WITHOUT a calculator. Recall yesterday’s example of showing one chip as “two”

and how I validated the child’s reasoning. This reflects an important epistemology (theory of learning) called “constructivism”: it is the idea that humans build learning on what they already “know” (right or wrong). For me that translates into “rational people are always trying to draw sensible conclusions about their world, so when a person makes an error, seek and validate

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ways he/she reasoned correctly, and identify the assumptions that are erroneous.”

[10 min] Activity: Using formal set theory to describe the physical world. (Key idea)o Show me a set of chips. [Pile of objects]o Show me a subset of chips. [Some objects – all

white? Basis of classification.]o Show me the complement of that set. [Everything

BUT those – opposite or leftovers]o Show me an element of the complement.

[Identifying individuals of a group – individuality ]o Show me a second set equivalent (numerically) to

the first. [Basic concept of “sameness”]o Show me two separate sets. Show me their union.

[Basis of what operation? Boolean “or” in search engines]

o Form two groups of students – all those with (naturally) dark hair on one side of the room, those wearing jewelry on the other side. [Why confusion? Those with both are the intersection of the sets. Boolean “and” in search engines.]

[15 min] HW questions from 2.1? [They present. I answer theory. 2b/g, 4/5, 6, 8b/d, 10b]

[10 min] HW questions from 2.2? [Do #2, 4; discuss 5 – ways are circle subset, equiv to subset, position in counting chant]

[15 min] Babylonian number system

Friday [2.2] [10 min] SPT #1 – Decimals Distribute new assignment; highlight number system

creation task. Think/pair/share: What must be understood about our

number system to be able to correctly interpret the meaning of the symbol “4017?” [Create list of properties]

Discuss idiosyncracies of column names and decimal point placement for our number system.

Why look at other number systems? [Helps highlight features of our own, and reveals the idiosyncrasies – what features are truly arbitrary, or have reasons lost in history?]

Examine features of the different number systems, paying particular attention to the handling of bases in the Babylonian and Mayan systems.

Discuss their questions about ideas in Section 2.2.

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21/14

Summ

Features of Number Systems; Structure of our number system(2.2/3, 6/7/8/9.1, 9.2)

Math Knowledge Outcomes:o [MK~3] Describe the concept

of a number, along with the reasons for/use of different types of numbers in the real number system.

o [MK 4] Describe features of our number system and use that information to analyze other number systems or obstacles to children learning our number system.

o [MK~7] Represent decimals, fractions, and integers in multiple ways; compare and contrast them with whole numbers and each other.

Procedural Proficiency (Teacher)Teacher Proficiency Test #1<Standards, concept of #, properties of # systems> Sources of standards Identifying and classifying

error types in counting Analyze a child’s number

system for recognizable features.

Robust Understanding: [MK4] Construction/

articulation of own number system (Synth)

Question(s) of the Week: If a child memorizes the

alphabet in reverse order (z y x w …), does it have a significant effect on his/her ability with language? What if a child memorizes a list of numbers in reverse order (100, 99, 98, …)?

Puzzle of the Week: Page 19, prob. 18 in Sec. 1.1 B

Monday [2.2, 2.3] – Bring chips! [15 min] Each pair scatters a bunch of chips (no

organization). Ask them to “tidy up”/organize their chips. Have them describe how they chose to organize, then ask

them to count the chips efficiently.o Highlight role of bundles for efficiency in

organization AND counting.o Compare and contrast numbers of bundles with the

digits they tell me to write. [26 vs. 5x5+1]o Reorganize chips to explicitly reveal our numerals.o How do you justify that 12 and 21 are different?

[15 min] Key notes: The impact of bundling/grouping o Reduces the number of symbols required.o Creates a multiplicative system: the value of one

symbol is multiplied by the bundle size (say, 10).o Reusing symbols with different meanings forces

the order to matter: creates a positional system.o Physically creating and counting bundles is a

sorting and counting task, limits abstraction. [15 min] Chip abacus concept – next level of abstraction

o Want to use fewer objects – have each person create 3-column abacus on a sheet of paper: columns (R -> L) are “units,” “first grouping/bundle,” and “second grouping/bundle.”

o Start with 24 chips in the “units” column. Go through exchange process. They try with 36.

o Discuss benefit of using a second color chip for the second column (precursor to money exchange, too: pennies, dimes, dollars; poker chip color)

o Explore Babylonian system – describe thirty six chips, then 142, then 471. Compare abaci with base-10, and the symbols being used. (Can use “red” = 10 and “white” = 1 for the Babylonian numerals.)

[5 min] Key notes: Chip abacio Each column represents a bundle of the objects

before it (to the right).o To describe an amount efficiently, we continually

figure out the largest bundle size that fits inside, then use that to reduce the amount down to units.

o Slightly more abstract – have to hold the bundles in the mind.

Tuesday [Finish 2.2, 2.3] – Bring chips! [20 min] ID HW questions from 2.2, 2.3.

o Preferred: 2.2A #11 cdeil; 2.2B #15 – explain pattern, 52,603; 2.3A# 4 – 9

o ID one question and one group to present. [2 if time]o I answer theory questions

[15 min] Resume with chip abacus

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Standard(s) of the Week: GLE 1.1.1, Grade 3 evidence of

learning: “Represent and discuss place values of digits of whole numbers [up to 10,000] using words, pictures, or numbers.”

o Repeat with base-3 -> emphasize naming like “two-one-one” not “two hundred eleven”

o “Base n” – number of numerals, size of groupingo Discuss how Mayan chip abacus would look.

[15 min] Key notes: The features/cognitive issues children must grasp to handle our number system [Keep these findable for concept map later!]o Order matterso One symbol represents a bundle of a certain sizeo Bundle size is implicito Can make bundles of bundleso Each symbol is a specific quantityo Both adding and multiplying taking placeo Absence of 0 as placeholder allows ambiguityo Other? [Issues related to “number” or

numeration] Key notes: Definitions of classification terms:

o Positional – the physical location of a symbol affects the value of the overall number

o Place valued – each location in the number system changes the meaning of all symbols placed there in a consistent way

o Additive – the overall value of a set of symbols comes from adding the values associated with each symbol

o Multiplicative – the value of a symbol sometimes represents a fixed multiple of that symbol (can be determined by place value or accents)

o Subtractive – the overall value of a set of symbols comes from subtracting the values associated with each symbol

o Has zero – There is a symbol which represents “none” of a particular amount.

Wednesday [6.1, 7.1, 8.1] – “Why” day: # types & purpose List topics for TPT #1; remind them of online sample –

this test will probably be a bit different. [15 min] Guided discussion – brainstorm on:

o Why do we have so many numbers, and not just stop at describing quantity with “none,” “one,” “two,” “some,” and “many?”

o Examining above questions leads to understanding what motivates a need to learn the concept – at what stage/what view of the world would cause children to find numbers meet a need they have? Precision, standardization, communicating

with someone elsewhere (can’t just show ‘em) Response to cultural values: individuality,

ownership, competitiveness, “more is better” (no end to this, unlike with “many”)

[15 min] Guided discussion – brainstorm on:o Why does it make sense to call 1, 2, 3, … “natural #s”

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o Why does adding 0 to this list make it “whole?” (Provides starting point; measuring distance w/ ruler.)

o Why is it reasonable to give zero “number” status and not just leave it as a word like “none?” Represents quantity (absence of – “showing”

emptiness with empty circle); Benefit of placeholder symbol with other “number” symbols; completeness of symbolic calculations (5 – 5 = seems incomplete); noting a transition or reference point (temperature, time, money)

[15 min] Guided discussion – brainstorm on:o Key notes: Why do we need fractions or decimals,

and is there any benefit to having both? Representing subdivision/pieces; Dec –

standardized, easy comparison, linear models; Frac – compact symbols, highlights relationship betw two quantities, set models

Frac complexity – showing two quantities with their own meaning, plus a new meaning together (c.f. analogies)

Frac representations – region model prevents confusion of part & whole; similarity of part & whole in linear model can cause confusion

Response to cultural values: are “pieces” considered of value (herders, Nat. Amer., children – 2 quarters is better than $1)

Thursday [8.1, 9.1, 9.2] – # types & purpose, part 2 [15 min] Guided discussion – brainstorm on:

o Why do we have negative numbers? Can you physically show a negative amount of something? Negative numbers represent the opposite of

something else – must have a “partner” that does have meaning

o Brainstorm quantities, determine if there is a “negative” of it and what conveys direction. Height above ground, temp above freezing,

possession vs. debtedness Does red have a negative? (Same meaning?) Take away too much and want to record that

amount – words (debt, loss) vs. symbol; Notion of opposites; only exist in opposition to something else – need a reference direction

o Value of opposition/annihilation concept Chip games (Papy minicomputer) Significant understanding of world – application

to electricity; weather (pressure – “nature abhors a vacuum,” negative-pressure clean rooms)

[15 min] Guided discussion – brainstorm on:o What is a rational number? Do we actually use them?

These are all the numbers you can get by starting with natural numbers and doing arithmetic (all basic operations are “closed”)

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Contains every fraction, and every decimal that stops, or repeats forever -> basically every number we think of.

o Are there any numbers we use that AREN’T rational numbers? Do we really need them? Irrationals – infinite, nonrepeating decimals

(can’t write as a fraction); primarily found as fundamental relationships in nature

Complex – Complete number system: contains solutions to every polynomial equation that can be expressed using these numbers; primarily used to describe the invisible (electrical current, energy fields, subatomic particles)

http://mathforum.org/library/drmath/ sets/select/dm_imaginary.html

[15 mins] – HW Q’s? [If needed] Sample number system: “Celeste”

o Up to 4 ticks (s, st, sta, star) at compass points, then cap for crosshairs with concatenation of name

o 8 is a ring (mun), add a new tick for each 8 (muna, muni, muno, munae) up to 40. Two of these is a double-ring, called “sol.”

o Analyze this system, then ask students to present their own.

Friday [6.1, 7.1, 8.1] – Representing numbers [20 min] TPT #1 – Standards, Concept of number, and

Properties of number systemso Sources of standardso Explain aspects of the concept of number or analyze a

child’s learning of the number concepto Analyze a familiar or unfamiliar number system using

the terminology of properties of number systems. [5 min] Distribute new assignment; Summarize concept

map idea – refer back to GLE.o Brainstorm key terms, skills, conceptso Group similar ones, create networks of ideas

[20 min] Key notes: How can we represent the different types of numbers?o Whole numbers – individual objectso Fractions – MUST represent the whole in some

manner, as well as the part “of interest” since it’s the relationship between the two that’s described (pieces of a # line, of a pie, etc.)

o Decimals – Typically shown in linear or rectangular contexts; relies on knowledge of reference points (0.5 = half; 0.25 = quarter) and seeing subdivision by 10s.

o Integers – Arrows (vectors) or “annihilator” chipso Rational/Irrational numbers – Either represented

by a decimal approximation, a geometric object constructed to have the precise size, or a symbol.

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http://mathforum.org/library/drmath/%20sets/select/dm_imaginary.html

http://mathforum.org/library/drmath/%20sets/select/dm_imaginary.html


o Complex numbers – points or vectors on a 2-D grid (plane)

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31/21

Short

Summ

Structure of Numbers(3.3, 5.1, 5.2)

Math Knowledge Outcomes:o [MK3] Describe the concept of

a number, along with the reasons for/use of different types of numbers in the real number system.

Procedural Proficiency (Student)Student Proficiency Test #2<Fractions – 5 points> Each of the operations, some

mixed numbers; Reducing

Robust Understanding: [MK1] Written explanation of

concept of number and features of numeration (Anal) Concept Map (Synth)

Question(s) of the Week: What are the advantages of

fractions (especially when compared to decimals) that make them worth the frustration of trying to comprehend?

Puzzle of the Week: p.246 #26 in Sec. 6.1B


“Understand the concept and symbolic representations of integers as the set of natural numbers, their additive inverses, and 0.” Grade 7: “Understand the concept and … of fractions, decimals, and integers.”

Monday [] – MLK Jr. Day; No classesTuesday [3.3, 8.2 pp. 357, 358] <Return/discuss TPT #1 if ready – importance of clarity in

explanation, knowledge of vocabulary>

[25 min] Ordering of whole numbers – brainstorm ways to prove which of two amounts is “bigger.”o Key notes: Whole numbers can be ordered via

Counting chant (earlier is smaller) Number line (left is smaller) Subsets/Correspondence (fits inside->smaller) Essence of formal definition: if I can add an

amount to a number and get another number, then the first number is smaller.

o Properties – Transitivity is “linking together” reasoning (similar to cognition for: if stoves are hot and hot things hurt, then stoves will hurt)

o Try to understand proof on top of p. 137! [25 min] Ordering of integers – brainstorm: How can you

justify that –5 < –2 and –4 < 1? Which methods still work?o Key notes: the formal definition of “less than”

extends order and properties to integers (esp. negative numbers) Counting chant & subsets break down when

comparing negatives to positives OR neg. Negatives lead to competing notions of “less

than,” and the absolute value exists to help us clarify our meaning: size vs. position – in what sense is a debt of $5 less than a debt of $2, and in what sense is it greater?

Wednesday [3.3, 5.1] – Bring chips [15 min] HW Questions? [25 min] Exponents – brainstorm on what are they for,

why do they exist, and are they a new operation?o Key notes: Exponents are essentially a shorthand

for writing repeated multiplication. They “become” an operation when you begin thinking about numbers acting on each other through this process.

o How does multiplication itself serve a related purpose? [Increasing shorthand/compression of +]

o Key notes: Exponent rules come from counting or

extending patterns. The order of operations simply reflects the

increasing “compression” of a number through other operations.

o Is ? Why is this a common error?

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Euclidean Algorithm rests on the principle, “if x|a and x|b, thenx| (a – b).”

o What phrasing do you use to describe what an exponent means? [310 means multiply 3 times itself 10 times] Why is this problematic?

[10 min] Exploring the “innards” of numberso Form several groups of six chips. Make several

patterns using six chips. What patterns emerge, and what do they reveal about the structure of 6? Arrays: Shows divisibility by two & three;

Dimensions reveal factorization even for 1 x 6 Triangular number: 1 + 2 + 3 <Not a perfect square>

o Try with 7, 8, and 9.Thursday [5.1, 5.2] – Bring chips SPT #2 tomorrow – fraction arithmetic (includes mixed

numbers), reducing [10 min] Questions about concept maps? [20 min] Patterns in numbers – divisibility

o What does it mean for something to be divisible by something else? Can’t everything be divided?

o Why is divisibility important or worth investigating? Review the divisibility rules in the book, but

memorization is not the goal – purpose is to see how one reasons out the divisibility rules (like pp. 204 – 205, or top of p. 202).

Work through formal -> example -> informal process of interpretation.

[20 min] Key notes: Primeso Why are primes so important?

“Atoms” of numbers for multiplication Unique factorization – everyone gets the same

answer (compare the number of factor pairs or factor trees for 48 with the list of primes obtained from the factor trees)

o What is the value of factoring?Friday [5.2] – HW Due [10 min] SPT #2 – Fraction arithmetic Key notes: GCF, LCM

o Purpose Fraction manipulation – GCFs for reducing;

LCMs for common denominators Conceptually – tools for finding or creating

numbers that link two other numberso Language confusion – which is bigger?

“A number that two other numbers fit into” [LCM, bigger] vs. “A number that fits into two other #s” [GCF, smaller]

o Methods GCF – List all factors, find match; smallest

exponent of each prime; repeated division of previous divisor by remainder

LCM – List all multiples, find match; largest exponent of each prime; “what’s missing?”

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41/28

Summ

Concept of the Operations:+, – with whole, dec, frac, int(3.1, 6.2, 7.2, 8.1)

Math Knowledge Outcomes:o [MK~5] Describe the purpose

of and concepts underlying basic operations on whole numbers.

o [MK~7] Represent decimals, fractions, and integers in multiple ways; compare and contrast them with whole numbers and each other.

Procedural Proficiency (Teacher)Teacher Proficiency Test #2<Structure of numbers and concepts of + and –> Significance and use of primes Purpose, meanings, and

methods for finding GCFs, LCMs

Approaches and models for + and –

How fractions and integers change interpretations of + and –

Robust Understanding: <In a later concept map.>

Question(s) of the Week: What is the difference between

the concept of an operation and the skill for using it?

Puzzle of the Week: p. 257 #26 in Sec. 6.2 A


“Understand the meaning of addition and subtraction and how they relate to one another”

GLE 1.1.5, Grade 2 evidence of learning: “Use joining, separating, part-part-whole, and comparison situations to add and subtract.” “Illustrate

Monday [3.1, 6.2, 8.1] – Bring chips Key notes: Addition concept

o What is the essential idea behind addition? Put together/join, increase, union, “and”

o How does this help us recognize English words or phrases that reflect addition?

o Why is addition the first operation taught after counting? How are they related?

o What are the main perspectives (“approaches”)? Put together approach – count each set,

smash together, count result; only physical relationship between addends and sum

“Counting on” approach – shows continuity of sum through counting; challenge is duality of tracking “four more places” in the number chant

o What are the main representations and how is the meaning of addition revealed? Set model – smash idea Measurement model – counting on

o How does addition change for other numbers? Fractions – not every object is a “one”:

challenge of changing unit size [compare w/ 1 cm + 1 in is not 2 “cm in”]

Integers – objects have different attributes, so they’re not counted the same way and certain models are less “natural”

Tuesday [3.1, 6.2, 8.1] – Bring chips [15 min] ID HW questions

o Preferred: o ID one question and one group to present. [2 if time]o I answer theory questions

[35 min] Basic properties of addition & their justificationo Associativity/Commutativity [first clarify difference:

changing order of operations vs. changing order of symbols]

o Closure property for addition is a consequence of counting on idea

o Identity (root word is “identical”)

Wednesday [3.1] – Bring chips [15 min] ID HW questions


[35 min] Key notes: Subtraction concepto What is the essential idea behind subtraction?

Take away/remove, decrease

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addition and subtraction using words, pictures, and/or numbers.”

o How does this help us recognize English words or phrases that reflect subtraction (rather than division)?

o What are the main perspectives (“approaches”)? Take away – Start with a set, remove a

designated portion, amt left over is answer Missing addend – What must be added to the

“subtracting” number (subtrahend) to make the other number (minuend)?

Comparison – Matching elements of two sets one-to-one and identifying the amount that is different or unmatched

o Additional perspective: Directed difference – “a – b” means “To get

from b to a, I must travel __ units to the ___.” OR “Going from b to a is a change of ___.” [Important for extension to integers]

Other observations:o The missing addend approach reveals how + and – are

inverses; that – “undoes” +o It explains how some students avoid learning

subtraction as a separate operation.o It sets the stage for equation-solving in algebra.o It is the first example of a “backwards” question or

reasoning process.

Thursday [3.1, 6.2, 8.1] – TPT #2 tomorrow [15 min] ID HW questions


[35 min] Key notes: Subtraction concepto What are the main representations and how is the

meaning of subtraction revealed? Set – Removal (take-away); “What’s

missing?” (missing addend); Match-up and count leftovers (comparison)

Measurement – Cut and re-measure (take-away); “How much farther?” (missing addend); Start at b, measure to a (directed difference)

o Essential features to examine: What roles are played by the minuend and

subtrahend in the approaches? [Basis for why subtraction is not commutative and fails to have closure property]

Fractions introduce no new problems beyond those for addition [changing unit size]

How does subtraction naturally lead to the integers, and how can we make sense of problems like ? [Obstacle is different attribute – cannot take away/compare red with white; “adding zero”

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is a patch for sets; “add a negative” is a patch for measurement; directed difference??]

Friday [3.1, 6.2, 8.1] – HW Due Describe concept map part of HW #5 [20 min] TPT #2 – Structure of numbers; Concepts of

+ and – [30 min] Basic properties of subtraction & their

justification.o Associativity/Commutativity/Identity – do they hold?o Closure property & children’s error with computing

2-5. (Absence of such numbers)o How do we know 2-5 is not 3?o How to guide a child toward discovery of negative

numbers

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52/4

Gone

Summ

Algorithms for Computation: Estimation, + and – with wholes(4.1 – 4.3, 6.2, 7.2)

Math Knowledge Outcomes:o [MK~6] Use multiple

algorithms for arithmetic computation, list the steps for executing each one, and analyze several algorithms’ advantages and disadvantages.

o [MK~8] List the steps for computing with decimals, fractions, and integers; compare and contrast the standard algorithms with those for whole numbers.

Procedural Proficiency (Student)Student Proficiency Test Retake

Robust Understanding: [MK5,7] Concept map on

ideas of +, – (Anal, Synth)

Question(s) of the Week: What are the advantages and

disadvantages of the standard algorithm for the addition of whole numbers?

Puzzle of the Week: p. 185 #28 in Sec. 4.2 B

Standard(s) of the Week: GLE 1.1.6, Grade 3: “Use

computational procedures for addition and subtraction of whole numbers”

GLE 1.1.6, Grade 3 evidence of learning: “Explain and apply strategies or use procedures to add three 2-digit or two 3-digit numbers, and/or subtract numbers with 1, 2, or 3 digits.”

Monday [4.1, 4.2] – Bring chips [10 min] Reminder of schedule for week:

o Sub Tues, Wed, Fri; recommend meeting Thursday in class to discuss/work on HW together;

o retakes of SPT #1 (Dec) on Fri for those who haven’t passed it; otherwise retake SPT #2

o I MAY be able to communicate via email – don’t know how realistic that is right now.

Return concept maps, discuss strengths & weaknesses. [25 min] Key notes: Tools for rapid computation

o Properties of + (Associative, Commutative, Distributive) allow for grouping and adding numbers from a list in any order that’s convenient

o Compatible numbers – those which are easy to compute mentally (like #s that add to 10) Typically the #s we try to add first in long

columns of data, reorganizing via assoc./comm.o Compensation –

Transferring small amounts between addends before completing sum (additive compens.)

Increasing/decreasing both minuend and subtrahend before subtracting (equal-additions method) Physically justified via missing-addend

approach on measurement model: length is translation-independent, so move addend to “nice” spot on # line “ruler.”

Also reflects “add a special 0” concept.o Why “exchanging” is preferred to “borrowing.”

[15 min] Standard algorithm for whole number additiono Describe in words the process for adding any two 2-

to 3-digit numbers. (Highlight language precision) What’s the largest amount that will be carried?

o How does a chip abacus physically demonstrate the procedure, and how can we arrange the chip abacus to make the “carrying” process for digits obvious? Which approach to + does this represent?

Tuesday [4.2] [5 min] Key notes: Standard alg. for whole # addition

o Strengths: uses very little space and few symbolso Weaknesses: separation of digits, abstraction

[Consider motor skills, visual perception] [30 min] Alternative algorithms

o Intermediate algorithm 1 Intermediate between what? How is it more concrete, how does it reflect

expanded form of #s?o Intermediate algorithm 2

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How is this different, what are its advantages and disadvantages, how can it help someone struggling with the standard algorithm?

o Lattice algorithm How is this different, what are its advantages

and disadvantages, how can it help someone struggling with the standard algorithm?

[15 min] Standard algorithm for whole # subtractiono Strengths: uses very little space and few symbolso Weaknesses: “Magic” transformation of numberso Advantage of chip abacus for revealing idea.

Which approach to subtraction is demonstrated? [Comparison or take-away]

o Alternative: “subtract from the base” Always take from a 10 – provides consistency

Wednesday [4.3] [10 min] Similarities, differences between whole number

and polynomial arithmetic (and integer arithmetic?) [40 min] + and – algorithms in other bases

o Key notes: Practice with standard + algorithm in base 7, then try intermediate algorithms What makes you anxious – what parts of our

“familiar” algorithm become confusing or overwhelming?

What provides security, and in what way? (Addition facts table, chip abacus, interm. algs?)

At what steps of the calculations do we experience significant slowdown, and what does this reveal is an essential building block for learning the base-10 algorithm?

o Tom Lehrer’s “New Math” songo [Practice with lattice algorithm?]o Practice with standard subtraction algorithm in base 7.

<Same questions as above>

Thursday [Ch. 4] – SPT #1 or 2 retake tomorrow HW questions

o Preferred: 4.1A# 1a, 2a, 14d; 4.2A# 2, 4a, 5a, 6, 10, 12b; 4.3A# 1d, 2, 4b, 5, 6bc

o Several groups presento I answer theory questions

Friday [6.2] – Bring LEGOs? [10 min] SPT #1 or 2 retake [40 min] Key notes: Algorithms for adding fractions

o Why can’t we add them straight across? Visually represent 1/2 + 1/3. Identify visual

meaning/role of each symbol. Reminder: region model distinguishes part and

whole by geometric dissimilarity; linear model

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does not Denominators essentially aren’t “numbers” –

they’re just sizes. Sizes/units of measurement don’t add (recall 1 cm + 1 in = ??)

Result of a fraction addition problem answers TWO questions: “How many pieces, and of what size?”

o Abusing notation – when MIGHT you add across? (Never mathematically “equal,” but used when comparing distinct subsets of a total: 3/4 accuracy + 5.5/6 completeness = 8.5/10 HW points)

o Have students describe the addition algorithm in words for themselves. Students pair up, sit back-to-back. One creates a

fraction + problem, the other describes how to do it without ever seeing the numbers or getting any feedback from the one who is computing.

Algorithm: Find LCD. Convert each fraction to equivalent with LCD. Add numerators, write answer over LCD. Reduce.

o How does the algorithm change for subtraction? Other key ideas to consider:

o Mixed numbers (esp. subtraction with “borrowing”) Benefit of improper fractions

o How do we add three or more fractions? Finding LCD for all at once vs. iterative

methodo What visual representations can help demonstrate

the concept of/need for the LCD? (And what cognitive development is required before this will be meaningful for the child?) Fraction strips (and then rulers) Easily divisible (“cuttable”) geometric shapes LEGOs

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62/11

Summ

Concept of the operations of and ÷ with whole numbers, decimals, fractions, and integers(3.2, 6.3, 7.2, 8.2)

Math Knowledge Outcomes:o [MK5] Describe the purpose of

and concepts underlying basic operations on whole numbers.

o [MK7] Represent decimals, fractions, and integers in multiple ways; compare and contrast them with whole numbers and each other.

Procedural Proficiency (Student)Student Proficiency Test #3<Equations and Graphs> Graph linear and nonlinear

equations by point-plotting Solve a linear equation

involving parentheses

Robust Understanding: [MK4] Analysis of student

work showing idea behind an operation (Anal); Concept Map (Synth)

Question(s) of the Week: How are the basic operations

related to each other? What are two different ways to pair up the operations, and what is the logical basis for those groupings?

Puzzle of the Week: p. 233 #25 in Sec. 3.2 A


“Understand the meaning of multiplication and division of whole numbers”

GLE 1.1.5, Grade 3 evidence of learning: “Show and explain the relationship between multiplication (division) and repeated addition (subtraction)”

Monday [3.2] – Bring chips Key notes: Multiplication concept

o What is the essential idea behind multiplication? Repeated addition

o What English words or phrases reflect multiplication, and how do they express the repeated addition idea?

o What are the main perspectives (approaches), and how is 3 2 interpreted? The words in [brackets] indicate which model applies. Repeated addition [set or measurement] –

first number (“factor”) describes the number of repetitions, second factor describes the amount in each repetition

Rectangular array [measurement] – each number describes a dimension of the rectangle

Cartesian product [set] – form all ordered pairs where the first number is from one set and the second number is from a second set

Tree diagram [set or none] – list each object from the first set as a “branch” of the tree, then off EACH of these branches, do the same thing with each object from the second set

o What is significant about these approaches? Repeated add. – shows essence of

multiplication as merely a shorthand for addition; establishes different meanings for the different factors

Rect. array – establishes link between dimensions (“2 by 4”), area, and multiplication; makes commutativity of multiplication believable

Cartesian prod. – links counting of combinations of options [car colors and features] with multiplication; applies to probabilities of independent events (e.g. number of ways two dice can be rolled)

Tree diagram – links the number of results of successive decisions [which pants, which shirt, etc.] with multiplication; basic tool for conditional probabilities (where the number of new choices depend on previous decisions)

Basic properties of multiplication & their justificationo Commutative – demonstrate through rotation of arrayo Associative – demonstrate through 3-D volume

calculationso Distributive – Add first & duplicate is the same as

duplicate, then add; provides alternative to order of operations requirement that parentheses are done before multiplication

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“Illustrate multiplication and division using words, pictures, models, and/or numbers”

Puzzles for later p.257 #26; Fibonacci; Pasc Tr.

Tuesday [3.2, 6.3, 8.2] – Bring chips Key notes: Multiplication concept

o How does multiplicat’n change for other numbers? Integers – Repeated addition approach

doesn’t make sense if first factor is negative (e.g. “negative 3 groups of …”); this case can be addressed by extending the pattern 22=4, 12=2, 02=0, –12=??; difficulty stems from “–” truly meaning “additive inverse.”

Fractions – Repeated addition approach doesn’t make sense if first factor is a fraction; this is more easily addressed using an array approach wherein the fractions are exhibited as dimensions, showing the portion of a 11 square; note subtle change of unit sizes to still count “wholes” – is a side chopped into thirds showing “thirds” or “three?”

Key notes: Division concepto What is the essential idea behind division?

Subdividing/fracturing/partitioning/grouping (note roots of first two words)

o What English words or phrases reflect division, and how do they express one or more of these ideas?

o What are the main perspectives (approaches), and how is 6 ÷ 2 interpreted? The words in [brackets] indicate which model applies. Partitive division [set] – the number of

desired partitions (groups) IS known, but the size of each is NOT; 32 students in class, if I want 8 groups, how many need to be in each group (how should the students count off?)

Measurement division [set or measurement] – the size of each group (its measurement) is known, but the number of groups is not; 32 students in class, I have them count off by 4s to get 4 in a group, how many groups will I have?

Missing factor [neither] – What must multiply the “dividing” number (divisor) to get the other number (dividend)?

Repeated subtraction [set or measurement] – The divisor is the number of objects to take away from the dividend in each step; the answer is the number of removal steps

o What is significant about these approaches? Partitive/Measurement – reflects our view of

multiplication as describing the number of groups times the number of items in each group

Missing factor – reveals that division is the opposite of multiplication

Repeated subtraction – reveals that the relationship between division and subtraction

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is parallel to that between multiplication and addition

Other key significant features of divisiono Role of 0 – the missing-factor approach is used to

establish the impossibility of defining # ÷ 0. Language precision is important here.

o (Long) Division algorithm – simply applies the repeated subtraction approach with varying levels of shortcuts

o Note that addition and multiplication have one name for the components – addends or factors – while subtraction and division have two [reflects (non)commutativity]

Wednesday [3.2, 6.3, 8.2] Key notes: Division concept

o How does division change for other numbers? Integers – we rely on the missing-factor

approach to see how signs affect division Fractions – [this is the mind-bending one!]

Examine 7/8 ÷ 3/8, 5/6 ÷ 2/3, or 3/5 ÷ ½. Measurement approach – how many

groups of size 3/8, 2/3, or ½ fit inside …? Notice relative ease when “chunk sizes” are the same – matches idea for whole #s. What if the sizes aren’t? Draw a 1×1 square and shade 5/6. On same diagram, cut the other direction to show thirds. *This is finding a common denominator! 18 “chunks” in whole square (it also multiplies each numerator by the other denominator!!)* Now our focus is entirely on the region described by the divisor – the 2/3. Fill it with the shaded pieces and describe the portion which is filled. **Note the additional confusion of changing unit size – 5/6 and 2/3 are initially represented as portions of 1, but the measurement approach means we’re interested in “chunks” (units) of size 2/3 – the divisor essentially becomes the whole!

Missing-factor approach – very similar to the above: since fraction multiplication is shown in an array model via subdivision of the length and the width, we subdivide one side using the divisor, then look for the split of the second side which will produce the dividend’s number of pieces and subdivisions. This will be easier if we convert the dividend to an equivalent fraction with the LCD of the two fractions. Now we proceed as in the measurement approach.

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Thursday [3.2, 6.3, 7.2, 8.2] HW presentations Interpreting operations with decimals

Friday [7.2] – HW Due [10 min] SPT #3 – Equations and graphing Discussion of questions to ponder Alternative: Discuss the full spectrum of job duties of a

faculty membero Teaching loado Official office hours + unofficial hourso Prep time + Gradingo Phone + Emailo Committees – hiring, textbook (& publishing cycle),

curriculum devel, advising, letters of rec., student awards, grants/special projects, scheduling, tutor hiring/oversight, PT faculty hiring/oversight, student complaints

o Amount of sleep, tradeoffs, enjoyment, compensationo Don’t expect much different in first couple of years of

teaching (high attrition rate; student teachers taking TP with them)

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72/18

Short

Summ

Algorithms for Computation: and ÷ with whole #s, fractions(4.2, 4.3, 6.3, 7.2)

Math Knowledge Outcomes:o [MK6] Use multiple algorithms

for arithmetic computation, list the steps for executing each one, and analyze several algorithms’ advantages and disadvantages.

o [MK~8] List the steps for computing with decimals, fractions, and integers; compare and contrast the standard algorithms with those for whole numbers.

o [PPTL] … use alternative strategies to perform computations …

Procedural Proficiency (Teacher)Teacher Proficiency Test #3< Computational algorithms for all operations and concepts of × and ÷> Describe the approaches and

representations of × and ÷ Describe the steps of, or

compute with an alternative algorithm for each operation

Describe the strengths and weaknesses of algorithms for each operation

Analyze a student’s attempt to use a common algorithm and identify the error pattern as well as suggest an alternative

Robust Understanding: [MK4] Classroom presenta’ns

on algorithms for +, – (Anal)

Question(s) of the Week: What are the advantages and

disadvantages of the standard algorithm for the multiplication of whole numbers?

Puzzle of the Week: p. 185 #3 in Sec. 4.2 “Problems

Monday [] – President’s Day (no class)

Tuesday [4.2] – Bring chips Key notes: Standard alg. for whole # multiplication

o Strengths: uses very little space, each calculation is a single-digit multiplication

o Weaknesses: results of multiplications are “broken apart” (carrying step); multiplication and addition steps are mixed together

Key notes: Alternative multiplication algorithmso Intermediate algorithm 1

Process: Show each single-digit multiplication on its own line; recognize place value in multiplication (20 3, not 2 3 written over one place)

Strengths: results of each multiplication are recorded individually; all multiplications are done before additions

Weakness: takes more space **Bridge to algebra: this algorithm is FOIL

when multiplying two two-digit numbers** **Visualizable: use rectangular-array approach,

and can break whole array into sections reflecting each sub-multiplication

o Intermediate algorithm 2 Process: Multiply all of the upper number by

each digit of lower number, one at a time Strengths: uses less space; all multiplications

are done before additions; results of multiplications are written intact

Weakness: may require large multiplicationso Lattice method

Strengths: compact, visually organized, all multiplications before additions

Weakness: requires more writing to show lattice

Wednesday [4.2] [30 min] Groups discuss HW/post solution attempts on 4.1A #15acd; 20b; 28 4.2A #17 (latt); 18b; 22; 31 B#15c;314.3A #9, 10, or 12 Presentation consists of:

o State the tasko Describe the steps or your reasoningo Explain why the process works or is reasonableo Answer questions

[20 min] Key notes: Standard alg. for whole # divisiono Strengths: Uses the least space; everyone has the

same middle steps

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for Writing”

Standard(s) of the Week: GLE 1.1.7, Grade 4: “Apply

strategies and use tools appropriate to tasks involving multiplication and division of whole numbers.”

GLE 1.1.7, Grade 4 evidence of learning: “Select and use appropriate tools from among mental computation, estimation, calculators, manipulatives, and paper and pencil to compute in a given situation

o Weaknesses: lots of estimating; multiplications can be complicated; alignment is VERY important; unclear what process has to do with concept of division

o Process for developing intuition with standard alg: Start with single-digit divisors to facilitate use of

manipulatives; extend rules to multiple-digit divisors when trust of symbolic steps is established

Use chip abacus or base-10 blocks to show the grouping within each column (single-digit division steps), along with the exchange to the next-lower column for the “bring down the …” step to continue dividing

Key notes: Alternative division algorithmso Most basic: repeated subtraction

Strengths: Clear connection to division – using repeated subtraction to address measurement division question (“How many groups of size … are in …?”); simple procedure that can be demonstrated with objects

Weaknesses: SLOW!!o Next: scaffold method

Process: use multiplication to subtract more groups at once

Strengths: maintains concrete connection to repeated subtraction; shows why multiplication is helpful in the algorithm; allows use of multiples that “make sense” or are easy to compute

Weaknesses: allows variation in the choice of multiples, doesn’t provide guidance about the “right” multiples to use; slow

o Intermediate algorithm Process: Break the dividend into chunks of

digits – being mindful of place value – and only divide by as much as you need for the divisor

Strengths: Reduces size of multiplications by introducing “chunking” idea of standard algorithm; encourages finding largest allowable multiple at each step, leading to uniformity in approach

Weakness: “Chunking” idea weakens connection with repeated subtraction, so more abstract

Thursday [4.2, 4.3] [5 min] Other sources of confusion in the division

algorithmo Order of symbols – using a calculator vs. long divis.o Zero columns and shifting

[20 min] Multiplication and division in base seven

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o Which steps of the different algorithms are particularly slow or confusing? What does that tell you about the ideas that must be learned well prior to this process?

[25 min] Groups discuss HW/post solution attempts on 4.1A #15acd; 20b; 28 4.2A #17 (latt); 18b; 22; 31 B#15c;314.3A #9, 10, or 12 Presentation consists of:

o State the tasko Describe the steps or your reasoningo Explain why the process works or is reasonableo Answer questions

Friday [6.3, 7.2] – HW Due [20 min] TPT #3 – Concepts of and algorithms for

operations [30 min] Lead-in to next week’s topics

o Algorithms for decimals Rationale for manipulating decimal point

o Algorithms for fractions Complexity of mixed-number algorithms

o Estimation algorithms Range; one/two-column front-end; with

adjustment Rounding vs. truncating & rationale Significant digits; calculator issues

Page 25 of 35



82/25

Summ

Other Algorithms(4.1, 6.1/2/3, 7.1/2, 8.1/2, 9.1/2)Ratios, Percentages, Proportions(6.3, 7.2 – 7.4)

Math Knowledge Outcomes:o [MK~2] Describe multiple

problem solving strategies and use them to solve a variety of problems.

o [MK8] List the steps for computing with decimals, fractions, and integers; compare and contrast the standard algorithms with those for whole numbers.

o [MK~9] Define, compare, and contrast ratios (rates), percentages, and proportions; represent them in multiple ways.

o [MK~10] Describe techniques for identifying and solving problems involving ratios (rates), percentages, and proportions, then solve problems involving them.

Procedural Proficiency (Student)Student Proficiency Test #4<Ratios, Percentages, Proportions> Recognize R, P, & P Solve proportions Convert percents Solve word problems

Question(s) of the Week: When computing with parts of

numbers, which calculations are most easily done with decimals, and which are easiest with fractions?

Puzzle of the Week: p. 255 #22 in Sec. 6.2A; p. 257

#26 in Sec. 6.2BStandard(s) of the Week: GLE 1.1.6, Grade 6: “Apply

strategies and use computat’nl procedures to add and subtract non-negative decimals and

Monday [4.1] Discussion of self-evaluation & scoring

o Examine own description, ask “why is that true?”, “how do I know that?”, and “what should someone else see to recognize what I’m saying is true?”

o Use vocabulary to demonstrate professional knowledge

o At the same time, try to explain your formal descriptions in concrete terms an elem. ed. major OUTSIDE our class would understand.

Key notes: Estimationo What is estimation, and why do we do it?

Rapid determination of approximate size of an amount

Appropriate when accuracy doesn’t mattero Estimation is the true reason for knowing basic

number facts in the technological ageo Types of estimation

Range estimation: gives guaranteed low and high (high requires increasing even shorter numbers by 1 in the largest column present)

Front-end: one-column (w, w/o adjustment), two-column

o Why are there several ways to estimate? Reflect different needs for speed or accuracy.

Tuesday [4.1, 7.1, 6.1] Rounding vs. truncating

o Truncating means to erase all digits of a decimal beyond a specified place value.

o Rounding means to truncate a decimal, but adjust the last remaining digit based on the size of the first digit eliminated.

o Reason for using 5 as a cutoff is the midpoint of the digits from 0 to 9 is 4.5, so half the digits are below, half above. The significance is that in long sums, on average, half would round up and half down, so error due to rounding is minimized.

Key notes: Decimalso Naming process involves a mixture of “unit” words

(millions, hundredths) and blocks of digits “and” is strictly reserved for decimal point “th”-words are used to count out spaces

BEFORE trying to place the decimal digits confusion about absence of a “oneths” column

is due to asymmetry of NOTATION (where we put the decimal point), not asymmetry of the columns themselves

o In MANY other cultures, the roles of periods and

Page 26 of 35


fractions.” GLE 1.1.8, Grade 7: “Apply

estimation strategies involving addition and subtraction of integers and the four basic operations on non-negative decimals and fractions to predict results or determine reasonableness of answers.”

commas are reversed from what we’re used to. Key notes: Relating decimals and fractions

o The place-value name of the fraction (thousandths, etc.) describes the denominator of the fraction you could use instead.

o Converting between fractions and decimals is not equally difficult: Frac -> Dec: Perform long division

Common source of confusion: difference in order between writing a fraction for long division and for entering into a calculator – other cultures modify the long-division algorithm to avoid this

When will we be guaranteed a non-terminating decimal? [Factors of the denominator can only be factors of 10.]

Dec -> Frac: If decimal terminates, write decimal portion as whole number over the place-value name. If decimal repeats, must use the “shift & subtract” technique. Complexity of converting repeating

decimals reflects the difficulty of being forced to “capture” something infinite.

Wednesday [6.2, 6.3, 7.2] HW Presentations Key notes: Algorithms for fraction & decimal arith.

o Why do we need a common denominator for +, –? Addition and subtraction are rooted in counting,

so we must have the same size pieces to count each item as “one.”

o What is the equivalent idea with decimals? “Finding an LCD” means translating each

decimal into the same units by adding zeros to the end (e.g. make 2.51 into 2.5100 so it’s in “ten thousandths” like the other #s in a sum).

This allows us to extend our usual algorithm with whole numbers of “line up the right sides of the numbers” before adding.

If we now erase the extra zeros, what stands out is the decimal points are still aligned, hence the step in the standard algorithm for adding/ subtracting decimals.

Notice that this method preserves the algorithm for whole numbers AND addresses confusion about alignment.

The necessity of matching “same size pieces” can also be shown with a chip abacus or other manipulatives.

o The need to match “same size pieces” when adding or subtracting is why the relevant decimal algorithms REQUIRE lining up the decimal FIRST.

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o Why DON’T we need a common denominator for fraction , ÷? A fraction shows the number of pieces possessed

(numerator) and the size of each piece in terms of total number of pieces in a whole (denominator). When we multiply two fractions, we’re repeating both of these amounts somewhat independently. (Demonstrate with a rectangular array approach using a measurement model along the edges.)

o What is the equivalent idea with decimals? Since the decimal essentially reflects the

denominator, multiplying two decimals WITHOUT the decimal point is just multiplying numerators of fractions. Placing the decimal point at the end simply reflects calculating the denominator.

This is why we can ignore the decimal point without changing the digits in our answer!!

Thursday [6.2, 6.3, 8.1, 8.2, 9.1, 9.2] HW Presentations Key notes: Mixed-number arithmetic

o How can we add mixed numbers? What property are we using?

o What is problematic about subtracting mixed numbers? [“borrowing”] How can it be avoided easily? [Improper fractions!]

o What is the wrong, but seemingly reasonable way to compute (2 1/3) x (3 2/5)? Why is it reasonable? Why is it wrong? (Display calculation using

array approach w/ measurement model on edges; highlight FOIL lurking underneath)

Again, improper fractions avoid error. Mixed numbers also set up confusion with

algebra’s elimination of multiplication symbol.o What algorithms are there for division of mixed

numbers? Other surprising fraction algorithms:

o Dividing straight acrosso Common-denominator division

Key notes: Algorithms for integers, rational & real #so Failures of traditional addition and subtraction

algorithmso Use of patterns for handling signs separately in

multiplication and division

Friday [7.3, 7.4] – HW Due HW Q&A SPT #4 – Ratios, Percentages, and Proportions Intro to concepts of ratios, rates, and percentages

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Page 29 of 35



93/3

Summ

Ratios, Percents, Proportions(7.3, 7.4)Intro to Problem Solving(1.1, 1.2)

Math Knowledge Outcomes:o [MK9] Define, compare, and

contrast ratios (rates), percentages, and proportions; represent them in multiple ways.

o [MK10] Describe techniques for identifying and solving problems involving ratios (rates), percentages, and proportions, then solve problems involving them.

o [MK~2] Describe multiple problem solving strategies and use them to solve a variety of problems.

Procedural Proficiency (Teacher)Teacher Proficiency Test #4<??>

Robust Understanding: [MK6] Creating lesson on

alternative algorithms (Anal)

Question+Puzzle of the Week: (Resolving confusion over

describing parts using improper fractions or ratios)


“Understand the concepts of ratio and percent.”

GLE 1.1.4, Grade 6 evidence of learning: “Write or show and explain ratios in part/part and part/whole relationships using words, objects, pictures, models, and/or symbols.”

Monday [7.3, 7.4] Key notes: Ratios, Rates, and Percentages

o What are ratios for? [Comparisons] What are different ways of comparing two

quantities? Inequalities (more/less than) Difference (subtraction – amount more) Factor (multiplier – # of times more)

<Missing factor/multiplier idea versus missing addend/difference> What situations are best served by each of these

approaches? (Precision; q’tity vs. relative size)o Ratio notation: three ways to represent – 6 to 5, 6:5,

6/5; highlight significance of order when it comes to interpretation – class ratio of F:M vs. M:F & effect

o What are rates and percentages? How are they the same as, or different from ratios? Ratios – Comparison of two quantities

measured in the same units; Compares part to part or part to whole; Clue word of “to”; Written as frac or :

Rates – Comparison of two quantities measured in different units (like distance, time, money, volume, etc.); Compares part to part; Clue word of “per”; Write as a frac or :

Percentages – Standardized comparison of two quantities in the same units; Compares part to whole; Clue word of “of”; Written frac/100 or %

o Connection between ratios and concept of fairness – why isn’t it enough for things to simply be required to be equal? [Ethnicity & teachers example]

Tuesday [7.3] Assign HW problems to present Wed & Thu/Fri Thu will include some time to work in project groups Fri will include both TPT 4 & retake of SPT 3 or 4. Key notes: Concept of proportions

o Proportions are about equality of ratios. This leads to a number of connections and applications. Equality of ratios is the same idea as finding

equivalent fractions This is how we scale data up or down

(building objects from models, changing recipes, or recreating demographics)

Scaling also reflects role of proportions in describing fairness

Same cognition as linguistic analogies (“Grass : Sky :: Green : ______”) Why difficult?

o Solving proportions symbolically – first with whole

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numbers to reinforce equivalent fractions idea and then highlight “multiply diagonally” shortcut Note the likely confusion over fraction

multiplication, division, and solving proportions, especially in light of “cross multiply” phrase.

o Constructing proportions for a described situation Suppose your car travels approximately 320

miles on 14 gallons of gas. How much gas would be required to make a 581-mile trip?

“Want, Know, Relationship” strategy Role of finding pairwise relationships between

the known & unknown information Using pairs to create proportion – one pair

dictates a single fraction, and a second pair dictates placement of #s in second fraction

o Judging equivalence of proportions ("Have I set it up right?") – identifying when proportions give the same or different answers (equality of diagonals; pairs)

o “Cross multiply” vs. “Multiply across” vs. setting up the problem horizontally & vertically

Wednesday [7.3, 7.4] HW Presentations Practice setting up ratios (rates) & proportions

o [Fairness] In the 2001/2 academic year, Highline had 2,006 Asian/Pacific Islander students and 1,550 students of African American descent. If a single class had the same mix as the overall campus, how many Asian/PI students would be expected in a class with 5 AfrAm students?

o [Scaling] On a map, the distance between two towns is 2¾ inches. The legend shows a half-inch represents 30 miles. How far apart are they?

o [Sampling] Researchers at the Department of Fish and Wildlife decide they need to measure the fish population in a particular lake. They first go to the lake and catch 41 fish, which they tag and release back into the lake. One month later, the researchers return and catch 58 fish, of which 16 are tagged. How many fish might the researchers estimate are in the lake?

Key notes: Percentageso Review (oral) – What is a percentage? How is it

the same as, or different from a ratio? Advantages of ratios? [Can reveal original

raw numbers; highlights multiplier] Advantages of percentages? [Standardized,

so easier to compare]o Basic percents vs. percentage change vs.

percentage-point change Basic – I make $3000 a month and my rent is

$600. What percent of my income goes to rent?

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% chng – My rent rose from $580 to $620. This is an increase of what percent?

The president’s approval rating went from 51% to 32%, a drop of 19%.

o Word-problem set-up stratgies: 15 is 83% of what? “Equation translation” – “is” becomes =, “of”

becomes “‘is’ over ‘of’” – translate to proportion Shortcoming of these methods – they’re tricks

that obscure the reasoning; how do they translate to solving “real world” word problems

“Part, Whole, %” strategy

Thursday [7.4] HW Presentations Lecture [25 min] Project group work time

Friday [7.4, 1.1] – HW Due HW Presentations? [20 min] TPT #4 – ?? [10 min] SPT 3 or 4 retake

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103/12

Summ

Ratios, Percents, ProportionsProblem Solving(7.3, 7.4; 1.1, 1.2)

Math Knowledge Outcomes:o [MK5] use understanding of the

role of each operation and the different types of numbers to correctly set up and solve problems presented in words

o [MK6] explain how ratios, proportions, and percentages describe relationships between quantities

o [MK7] describe differences between genuine problem solving situations and routine exercises

o [MK8] describe problem solving strategies (like Polya’s 4-step process) and apply them to a wide variety of problems

Procedural Proficiency (Student)Student Proficiency Test Retake

Robust Understanding: [MK4] Creating lesson on

algorithms for × (Anal)

Question+Puzzle of the Week: (Resolving confusion over

describing parts using improper fractions or ratios)


“Understand the concepts of ratio and percent.”

GLE 1.1.4, Grade 6 evidence of learning: “Write or show and explain ratios in part/part and part/whole relationships using words, objects, pictures, models, and/or symbols.”

Monday [7.4] Lecture/Discussion: <Bring chips!>o Review (oral) – What is a percentage? How is it the

same as, or different from a ratio?o Advantages of ratios? [Can reveal original raw

numbers; highlights multiplier]o Advantages of percentages? [Standardized, so easier

to compare]o Basic percents vs. percentage change vs. percentage-

point change Basic – I make $3000 a month and my rent is

$600. What percent of my income goes to rent? % chng – My rent rose from $580 to $620. This

is an increase of what percent? The president’s approval rating went from 51%

to 32%, a drop of 19%.o Word-problem set-up stratgies: 15 is 83% of what?

“Equation translation” – “is” becomes =, “of” becomes

“‘is’ over ‘of’” – translate to proportionShortcoming of these methods – they’re tricks that obscure the reasoning; how do they translate to solving “real world” word problems “Part, Whole, %” strategy

o Terminating/nonterminating decimals & the relationship with fractions

Tuesday [7.4] Group project work timeo Check groups’ ideaso Ask about components of project, division of laboro Their questions for me & hypothetical student

questions

Wednesday [7.4, 1.1, 1.2] SPT #3 or 4 retake tomorrow Hand out final self-assessment assignment [15 min] Groups discuss HW/post solution attemptsPreferred:

[30 min] Groups present their work, I grade, then comment

Lecture/Discussion:o Practice with word problems

Thursday [] HW #10 turned in; reminder about project, self-eval [10 min] SPT #3 or 4 Retake Explain “Standard of the Week” in your own words Question/Puzzle of the Week

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111/28

Summ

Problem Solving(1.1, 1.2)

Math Knowledge Outcomes:

Procedural Proficiency (Teacher)Teacher Proficiency Test #2<WASL problems involving patterns – solving, analyzing>

Robust Understanding: [MK8] Written descriptions

of problem solving strategies (Analysis of ideas being used)

Question(s) of the Week: Why are prime numbers so

important?

Puzzle of the Week: Focus of entire assignment

Standard(s) of the Week: GLE 2.1.1, Grade 4 evidence of

learning: “Generate questions that would need to be answered in order to solve the problem.”

GLE 2.2.3, most grades: “Apply a variety of strategies to construct solutions.”

Monday [3.1] [10 min] Groups discuss HW/post solution attemptsPreferred: 1.1 A #4, 6/9, 10, 13/19


[15 min] “Problem Solving”o Is being a “good problem solver” an asset? Why or

why not? Does the answer depend on the problems being

“math” or “real life?” How are these different? [Definition of “classroom math” vs. data analy.]

o What are the attributes of a “good problem solver?” Persistent, patient, creative, resourceful, willing

to take risks/try, observant of patterns, adaptable, self-reflective, organized(?)

o Where are we taught this – both informally and formally? Why in math class? [Logos training – context-

independent strategies for data-gathering and reasoning]

Why use seemingly irrelevant number and shape puzzles rather than real-world stuff like reducing littering/the amount of garbage we produce; crime/justice (Future Problem Solving – 5th/6th); overpopulation? [Can experiment, manipulate, test, see results, control variables, limit emotional distraction – pathos]

o What are ways people approach a problem to which there is no answer already known?

Tuesday [1.1, 5.1] [30 min] Problem-solving strategies applied to HW,

WASL, and Sudoku (highlight Tetris tetrominos on p.10)o “Systematic ‘Guess & Check’” – Empirical/deductive

(scientific) approach [generates patterns to observe]o Model the situation – draw a picture, use

symbols/variables in place of quantitieso Use cause-and-effect (deductive) reasoning

Wednesday [1.2, 5.1, 5.2] TPT #2 tomorrow – WASL problems + analysis [15 min] Groups discuss HW/post solution attemptsPreferred:


Discussion

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o Inductive vs. deductive reasoning – pros and cons Inductive: Child needs to touch burner to know

it’s painful Deductive: Child sees burner make hot food,

knows hot food sometimes burns, concludes burner is unpleasantly hot

o Power of deductive reasoning: demonstrating certainty of that which cannot be shown through experiment p. 221’s proof that the number of primes is

infiniteo Demonstrating deductive nature of argument proving

divisibility rules

Thursday [5.2] HW #4 turned in; #5 assigned [self-eval] Explain “Standard of the Week” in your own words Question/Puzzle of the Week [20 min] TPT #2 – WASL problems + analysis

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Documents

Math 97 Pedagogy & Assessment Plans€¦ · Web viewWeekly Topic (& relevant sections of the text) Math Knowledge Outcomes