A Research Perspective on the Discussion Draft of Principles and Standards for School Mathematics Research Presession 1999

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Student Responses
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• 1) T 2(L 2) 2W
• 2) 4 2L 2W
• 3) (L 2)(W 2) LW

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4/3, and 3/2. ??????????
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Number of Bows
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?????? 1/2, 2 1/2, and 1/4.
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More and Better Mathematics
Grades 912

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More and Better Mathematics
Grades 912

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Interplay of Algebra and Geometry
c
a b
b
a
b
a b
a
b
a2 b2
c2


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• ?? Mentally
• ?? By hand
• ???????? Using a computer algebra system

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Reasoning in Geometry

• Since the base of each of the four small
  triangles is a midline, each side of the midpoint
  triangle should be half as long as the parallel
  side of the large triangle.
• Each midline cuts the altitude in half, so the
  height of each small triangle is half that of the
  large triangle.
• Dividing each of these lengths by 2 divides the
  area by 4, so the area of the small triangle is
  one-fourth the area of the large one.

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Reasoning in Geometry

• ???????What happens with a quadrilateral?

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Reasoning in Geometry
?????What happens with a pentagon?
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?????????Reasoning about properties of Numbers
?????a,b,c,d,e,f,g,h d ?, ??????ab? To c ? To
1/f ? To e ? To h ? ?????
2 2
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In Grades 912
Developing Flexible Problem Solvers

• Builds and deepens UNDERSTANDING of mathematical
  concepts
• Builds and promotes more and better mathematics
• Builds and strengthens stronger basics
• Builds and enhances flexible use of
  representations
• Develops through regular experiences with
  interesting, challenging problems

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Flexible and Resourceful Problem Solvers
? 8X8?????????(????? ???????????) Count only
those rectangles (including squares) whose sides
lie on grid lines.
For example, there are nine rectangles on a 2 x 2
board as shown below.
1
2
5
7
8
9
3
4
6
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Flexible and Resourceful Problem Solvers
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Flexible and Resourceful Problem Solvers
5 x 8
3 x 8
1 x 8
8 x 8
7 x 8
6 x 8
4 x 8
2 x 8
8 x 7
7 x 7
6 x 7
5 x 7
4 x 7
3 x 7
2 x 7
1 x 7
2 x 6
2 x 6
8 x 6
7 x 6
6 x 6
5 x 6
4 x 6
3 x 6
1 x 6
2 x 7
8 x 5
7 x 5
6 x 5
5 x 5
4 x 5
3 x 5
2 x 5
1 x 5
8 x 4
7 x 4
6 x 4
5 x 4
4 x 4
3 x 4
2 x 4
1 x 4
8 x 3
7 x 3
6 x 3
5 x 3
4 x 3
3 x 3
2 x 3
1 x 3
8 x 2
7 x 2
6 x 2
5 x 2
4 x 2
3 x 2
2 x 2
1 x 2
8 x 1
7 x 1
6 x 1
5 x 1
4 x 1
3 x 1
2 x 1
1 x 1
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Flexible and Resourceful Problem Solvers
. . .
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??? ABC???? A (5, 1), B (4, 7), and C
(8, 5). ?? y x ??????? ???ABC ??????M
, ?A?B?C ??A?B?C. ?? ??M???.
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???????????????
(a) The two telephone poles
(b) The vanishing point, the horizon, and the
center of the rectangle
(c) The diagonals of a rectangle determine the
location of the third telephone pole
(d) The three telephone poles in perspective
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• ???7?????????. ???????,???????????????????,
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• The county, which has a limited budget, wants to
  pave some roads so people can get from every town
  to every other town on paved roads, either
  directly or indirectly.
• But they want to minimize the total number of
  kilometers paved.
• Find a network of paved roads that fulfills these
  requirements.

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Discrete Mathematics Integrated in Geometry
Standard
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Data Analysis
???????????????
1200
1000
800
Box-Office Revenues (10 000s)
600
400
200
0
0
500
1000
1500
2000
2500
Number of Screens
49
??????? Exponential Equations Example
3
0
2
0
2
0
2
5
1
5
1
5
2
0
1
5
1
0
1
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1
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5
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5

5

5

1
0

1
5

2
0

2
5

3
0

5

5

1
0

1
5

5

5

1
0

1
5
g(x) 3 2x 4
h(x) 2 3x 1
k(x) 2 1.1x
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