Imagine slicing the object into super-thin pieces that are vertical to the axis of twist, imag- ining each slice ......
QUANTITATIVE REASONING AND MATHEMATICAL MODELING1,2
-
Central Tenets of a Theory of Quantitative Reasoning A Quantity is in a Mind, it is not in the World saw
33
teach. -
3
th
34
-
n
m
nm
-
important ways.
-
35
-
-
of twistiness
n
n m
m
T T n,1) = nT
T
m) = mT
T
) = mnT
able th
36
-
-
2 3
-
3 4
-
5
-
ingless to them.
2
3
3
ters here.
after the cess. 6
Quantitative Operations are not the same as Numerical Operations (but they are related)
-
th
-
-
38
taller than Sister A.
tall.
39
have to be 82 cm tall.
50.
82 means.
40
-
Sister A.
84.
-
41
8
-
9
Dispositions That Allow Students to Create Algebra from Quantitative Reasoning -
42
-
A Disposition to Represent Calculations of their reasoning.
A Disposition to Propagate Information -
-
10
10
-
43
1)
Arithmetic Operation
Structure 11
2)
- If a = b * c, then c = a ÷ b
b = a ÷ c.
- If a = b ÷ c, then c = b ÷ a
b = a * c.
- If a = b + c, then c = a - b
b = a - c.
- If a = b - c, then c = b - a
b = a + c.
cal operations. 3)
-
11
44
A Disposition to Think with Abstract Units to -
O
O Together these mean that there are
O
O
O for Wants to Hit
O parametric
12
12
45
A Disposition to Reason with Magnitudes Q
Q,
the length of this
M
M M , then my height in
u
u is 3 times as . -
Quantitative Reasoning, Covariation, and Generalization13 operations that compose covariational reasoning are the very operations that enable one to see -
13
perspective.
46
Covariation
always
represents anticipates =
t), where t
over an interval. So, let D Let D
t
t ranges.
t
, which means that
where t
D
U t,t
e
,14
t
D anticipates that it will be
xe =x
te
, where
te
represents the interval
t,t e
t
15
16
14 am 15
16 within the bits as well.
t not see variables
xe , ye
xe , ye
x te , y te
-
its velocity at that location. -
-
-
-
48
-
-
49
-
18
-
openness
-
th th 19
in terms of SOH-CAH-TOA20
18 19 20 SOH-CAH-TOA: Sine is Opposite over H is Opposite over A
Cosine is A
50
H
Tangent
settings. -
21
-
Generalization -
21
51
-
-
-
-
-
ties to see mathematical similarity across settings.
52
Quantitative Reasoning and Modeling
as . This means that
-
V
x(11 2x)(8.5 2x) -
53
-
-
-
References
, 143-162. , 48-59. cept -
-
-
54
, 362-385. tions.
versa). proving. tion , 33-62.
operational time.
, 6-13. College Mathematics
, 282–300. , 431-449. -
concepts of trigonometry. -
55
ematics
, 499-511. , , 498–505.
, 61–82. , 2-24. algebraic of concrete materials in elementary mathematics. , 165–208. in the learning of mathematics
Research in Collegiate Mathematics The Mathematics , 33-35.
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