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Where’s the Math?

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Where s the Math? Dr. Janet H. Caldwell Rowan University caldwell_at_rowan.edu Models that Make Math Meaningful Where s the Math? Fraction Models Models for ... – PowerPoint PPT presentation

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Title: Where’s the Math?


1
Wheres the Math?
  • Dr. Janet H. Caldwell
  • Rowan University
  • caldwell_at_rowan.edu

2
Models that Make Math Meaningful
3
Wheres the Math?
  • Fraction Models
  • Models for Multiplication
  • Models for Division
  • Decimals, Ratio Percent

4
Sketch what you first see
  • One-half
  • Two-thirds
  • Three-fourths
  • Three-fifths
  • One-sixth
  • Two and a quarter
  • Two and two-thirds

5
Sketch what you first see
  • One-half
  • Two-thirds
  • Three-fourths

6
Sketch what you first see
  • Three-fifths
  • One-sixth

7
Sketch what you first see
  • Two and a quarter
  • Two and two-thirds

8
What fraction is blue?
9
Part of a Set 3 of 13 pieces
10
Part of an Area 6 of 36 triangles
11
Part of an Area 1 of 6 hexagons
12
Part of an area Blue is 1/3 of largest piece
13
Fraction ModelsPart of a Whole
  • Set
  • Area or region
  • Circles
  • Clocks
  • Rectangles
  • Pattern blocks
  • Strips
  • Length
  • Number line
  • Ruler

14
Other Meanings for Fractions
  • Part-whole
  • Values - eg, money
  • Division
  • Ratio
  • Rate

15
Web Resources
  • Fraction model applet
  • Equivalent fractions
  • Fraction game

16
Playing Fraction Tracks
17
Make a triangle that is
  • ¼ green and ¾ red
  • 1/3 red and 2/3 green

18
  • Understanding is the key to remembering what
    is learned and being able to use it flexibly.
  • - Hiebert, in Lester Charles,
  • Teaching Mathematics through
  • Problem Solving, 2004.

19
Computational Fluency
20
Using Base Ten Blocks to Multiply
  • 24
  • x 3

12 60 72
21
Make an Array
  • 24
  • x 3
  • 12
  • 60
  • 72

22
A Harder Problem
  • 24
  • x 13
  • 12
  • 60
  • 40
  • 200
  • 312

23
Decimals
0.12 0.60 0.72
  • 3 x 0.24

0.3 x 0.6
24
Draw a picture that shows
25
Array
26
Mixed Numbers, too!
  • 8 x 3 ¾

8 x 3 24
24 6 30
27
1 2/3 x 2 ¼ ?
28
Algebra
  • (x 1) (x 2)
  • x2 2x x 2
  • x2 3x 2

x 2
x
1
29
Sidetrip to Geometry - Area
  • Counting squares on a grid
  • Whats the area?

30
Break it up
  • Yellow (L) ½ x 4 2
  • Blue 2 x 3 6
  • Yellow (R) ½ x 2 1
  • Orange ½ x 2 1
  • Red ½ x 4 2
  • 2 6 1 1 2
  • 12 square units

31
Make a Rectangle
  • Area of rectangle
  • 3 x 6 18 squares
  • Areas of triangles
  • UL ½ x 4 2
  • UR ½ x 2 1
  • LL ½ x 4 2
  • LR ½ x 2 1
  • Total 6 squares

Area of pentagon 18 6 12 sq.
32
So?
  • Find the area of a triangle with base 10 and
    height 5.

Area (10 x 5) / 2 25 sq. units
33
Fraction Division
  • What is the whole if half is 1¾?

Measurement model Need two pieces of size 1¾,
so find 1¾ x 2 3 ½

1¾ 2 1¾ x 2 3 ½
34
How many 1/2s are there in 1¾?
  • How many cakes can you make with 1 ¾ cups of
    sugar if each cake requires ½ cup?

Partitive Model (Sharing) 1 ¾ ½ 3 ½
35
Whats the length?
  • The area of a field is 1 ¾ square miles.
  • Its width is ½ mile.

1/2
Missing Factor Model ½ x ___ 1 ¾
36
Decimals
37
Percents
  • A group of students has 60 to spend on dinner.
    They know that the total cost, after adding tax
    and tip, will be 25 more than the food prices
    shown on the menu. How much can they spend on
    the food so that the total cost will be 60?

38
Percent Bar
  • A group of students has 60 to spend on dinner.
    They know that the total cost, after adding tax
    and tip, will be 25 more than the food prices
    shown on the menu. How much can they spend on
    the food so that the total cost will be 60?

60
x
100
125
39
Another Approach
  • A group of students has 60 to spend on dinner.
    They know that the total cost, after adding tax
    and tip, will be 25 more than the food prices
    shown on the menu. How much can they spend on
    the food so that the total cost will be 60?

40
More on Percent
  • Josie needs 40 for a new sweater. She has 24.
    What percent does she have of what she needs?

41
Using a Table
  • Josie needs 40 for a new sweater. She has 24.
    What percent does she have of what she needs?

42
Still more percent
  • Jamal has 48 of his homework done. He has done
    12 problems. How many problems did the teacher
    assign?

43
Still more percent
  • Jamal has 48 of his homework done. He has done
    12 problems. How many problems did the teacher
    assign?

44
SO?
  • Pictures
  • Manipulatives
  • Oral language
  • Written symbols
  • Tables
  • Graphs
  • Relevant situations
  • Which model(s) are most meaningful for my
    students?
  • Which models promote more powerful thinking?
  • In what order should I use selected models?

45
Wheres the Math?
  • Models help students explore concepts and build
    understanding
  • Models provide a context for students to solve
    problems and explain reasoning
  • Models provide opportunities for students to
    generalize conceptual understanding
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