DRAFT September
13, 2014 jlewis
A core part of the course is a set of problem based activities
which lead to understanding concepts of algebraic thinking
including
1. Patterns
2. Variables, expressions, and equations
3. Functions
4. Multiple Representations
(NCTM Algebraic Thinking Yearbook 1999)
1.
Candle Problem (Driscoll, pp. 9, 95-97 and related materials). Cf.
491candle.pdf. Mathematics includes variables, rates, linear
functions, solving equations. Multiple representations - tables,
physical representation, graphing lines, symbolic representation.
Related problems: NCTM Year Book - Quinn and Larson) When Does a
Dog become older then its Owner? From a student in Fall 2003 When
does the Tortoise Catch the Hair?
2.
Tooth Pick Squares (Driscoll, p. 9, 160, and related materials).
Cf. 491tooth.pdf. Mathematics includes
points and lines, and developing formulas. Patterns, chunking the
problem, developing a function/formula.
Related problems: 491node.pdf - toothpick
squares and triangles from the point of view of graph theory.
Article on constructing tubes (cf. Fall 2003 Blackboard
Announcements, September 30, 2003).
Title: DEVELOPING ALGEBRAIC REASONING THROUGH GENERALIZATION.
Author(s): Lannin, John K. Source: Mathematics Teaching in the
Middle School;
Mar2003, Vol. 8 Issue 7, p343, 7p, 2 diagrams, 1c
Document Type: Article Subject(s): ALGEBRA
3.
Golden Apples ((Driscoll, p 22-24, 26 and related materials)
Mathematics includes predictions, repetitive processes, linear
functions, determining input from output, possible inputs for
whole number output. Patterns, developing rules, describing steps.
Related problems: Baldwin IMP
4.
Racetrack. Cf. 491roma.pdf Two cars traveling at constant speed on a track are side by side every
56 minutes. If, with the same speeds, one of the cars were
traveling in the opposite direction, the two cars would meet
every 8 minutes. How long does it take the faster car to complete
one lap on the track?
Mathematics includes variables, velocity, distance. Trials, Introducing
variables( including variables not stated explicitly), solving
equations.
5.
Digits and Multiplication. (Cf. 2003 Blackboard October 27, 2003 Announcement and 491place.pdf)
Discussion of Multiplication and your
carefully written solution of "reversed digits" problem.
Discussion of: Liping Ma, Multidigit Number Multiplication:
Dealing with Students' Mistakes, Chapter 2 of Knowing and
Teaching Elementary Mathematics, QA135.5.M22, 1999, pp. 28-54.
(Cf. October 21, 2003).
Mathematics includes representation of integers and chunking into
1's. 10's 100's, ….
Related problem: Multiplication of Binomials (Cf. October 21,
2003).
The next three problems are related. Note by Baldwin:
A proposal is to do first the orangy problem and then the Crystal task - making them write about both and trying to clarify how one
thinks about proportions. The water-wine problem fits into this
since teachers must come to grips with whether more means `more
dense' or just `more'.
I think `puzzle' problems have to be handled with care. Just
because it is a stimulating and fun problem doesn't mean it will
help teachers teach better or deepen a students understanding of
mathematics. But water/wine for example has algebraic solutions
and forces one to think about which data is really germane.
6.
Orangey Problem cf. orangey.pdf (Connected Mathematics -Comparing and Scaling?) Mathematics
includes ratios, proportions, percentages, scaling, nd comparing
representations. There are "two solutions".
7.
Crystal Task. (From SGMM - large pdf file and crystalmgm1.pdf)
Is there additional material from Small Group Mathematical Modeling (SGMM)
Project? Mathematics includes proportion, diameter, area, density, scaling, data
interpretation.
Water and Wine (Cf. Baldwin and 491place.pdf and H20.pdf or
420h20wine.pdf).
Mathematics includes careful computation, units, proportion, and a
descriptive solution.
Ratio and Proportions (Baldwin LaTeX
ratio1.pdf) Mathematics includes
proportion, compound proportion, rates, units, work, and solving equations.
10.
Replacement Problems (Baldwin
491replace.pdf) Mathematics includes variables, symbols, two
equations.
Related problem: Finding Prices (Driscolll p. 129)
Other Problems
11.
Weinzweig on Factors of 1000 !
12.
Postage Stamp and Lattices (Also Driscoll pp. 24-26, 93,
125-126, 148-149)
13.
Medicare Drug Benefit - jlewis work in progress on
percentages, piecewise linear relationships and expected value.
14.
Cooperative Learning (Baldwin, Dees, Foulser, Tartakoff
coopprob.pdf ) - the cooperative problems paper. It includes the
density jigsaw which I (JB) think is good although the teachers
may object that density is too abstract and so another jigsaw
maybe should be developed where one approximates a rate.
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