1. Dr. Janet H. Caldwell Rowan University caldwell@rowan.edu
Where’s the Math?
Dr. Janet H. Caldwell
Rowan University

2. Models that Make Math Meaningful

3. Where’s 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 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 Models Part 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
Wins
Losses

15. Web Resources
Fraction model applet
Equivalent fractions
Fraction game

16. Playing Fraction Tracks
PURPOSE : Show how a game can engage students in using their understanding of concepts
SPEAKING POINTS
This task motivates students to think about the relationship of different fractions to the unit whole, compare fractional parts of a whole, and find equivalent fractions.
The goal is to move all of one’s markers to the right side of the fraction track board. On each move, the player is allowed to move one or more of the blue markers along the track so that all of the markers moved are equivalent to the amount shown in the fraction box. Clicking on “Finish Move” either lets the other player take a turn or returns all of the markers to their original position if the move was not correct.
REFERENCES
Principles and Standards, pp

17. Make a triangle that is:
¼ green and ¾ red
1/3 red and 2/3 green

18. - Hiebert, in Lester & Charles,
“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
I thought seven
25’s - that’s 175.
Then I need seven 3’s or 21. So the answer is = 196
7 x 20 is 140 and 7 x 8 is is 196
7 x 28
I did 7 x 30 first. That’s Then take off seven 2’s or 14. So it’s 196.
PURPOSE: An example of students’ strategies developing different models for multiplication that support the development of computational fluency
SPEAKING POINTS
Students exhibit computational fluency when they have flexibility in the computational methods they choose, understand and can explain the methods, and efficiently produce accurate answers. These methods for students in grades 3-5 should be based on the structure of the base-ten number system, properties of multiplication, and division and number relationships.
Fluency with whole-number computation depends on fluency with basic number combinations single-digit addition and multiplication pairs and their counterparts for subtraction and division. Fluency develops from understanding the meaning of the four operations and focusing on the development of strategies based on understanding.
REFERENCES
Principles and Standards: pp

20. Using Base Ten Blocks to Multiply24
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
3 x 0.24
= 0.72
0.3 x 0.6

24. Draw a picture that shows

25. Array
2 of 3 rows
3 of 4 in each row

26. Mixed Numbers, too!
8 x 3 ¾
8 x 3 = 24
= 30

27. 1 2/3 x 2 ¼ = ?

28. Algebra
(x + 1) (x + 2)
= x2 + 2x + x + 2
= x2 + 3x + 2 x x+ 1

29. Sidetrip to Geometry - Area
Counting squares on a grid
What’s 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
= 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
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. What’s the length?
The area of a field is 1 ¾ square miles.
Its width is ½ mile.
A = 1 ¾
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?
$60
Cost of Food
Tax
and Tip
PURPOSE: A major feature of the middle grades program.
SPEAKING POINTS:
Students need to have flexible views of rational numbers. This example illustrates how students’ consideration of numerical and geometric idea might be integrated. Here a geometric representation could be used a solve a problem about percents.
In this figure, a rectangular bar represents the total of $60. This total must cover the price of the food plus 25 percent more for tax and tip. To show this relationship, the bar is segmented into five equal parts, of which four represent the price of the food and one the tax and tip. Because there are five equal parts and the total of $60, each part must be $12. Therefore, the total price allowed for food is $48.
The representation in the figure could also help students see and understand that when one quantity is 125 percent of a second quantity, then the second is 80 percent of the first.
This type of visual representation for numerical quantities is quite adaptable and can be used to solve many problems involving fractions, percents, ratios and proportions.
REFERENCE: Chapter 6, Number and Operations, pp
Problem is from pp

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? x
$60
PURPOSE: A major feature of the middle grades program.
SPEAKING POINTS:
Students need to have flexible views of rational numbers. This example illustrates how students’ consideration of numerical and geometric idea might be integrated. Here a geometric representation could be used a solve a problem about percents.
In this figure, a rectangular bar represents the total of $60. This total must cover the price of the food plus 25 percent more for tax and tip. To show this relationship, the bar is segmented into five equal parts, of which four represent the price of the food and one the tax and tip. Because there are five equal parts and the total of $60, each part must be $12. Therefore, the total price allowed for food is $48.
The representation in the figure could also help students see and understand that when one quantity is 125 percent of a second quantity, then the second is 80 percent of the first.
This type of visual representation for numerical quantities is quite adaptable and can be used to solve many problems involving fractions, percents, ratios and proportions.
REFERENCE: Chapter 6, Number and Operations, pp
Problem is from pp
100%
125%

39. Another Approach Dinner $40 $4 $8 $48 Tax & tip $10 $1 $2 $12 Total
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?
Dinner
$40 $4 $8
$48
Tax & tip
$10 $1 $2
$12
Total
$50 $5
$60
PURPOSE: A major feature of the middle grades program.
SPEAKING POINTS:
Students need to have flexible views of rational numbers. This example illustrates how students’ consideration of numerical and geometric idea might be integrated. Here a geometric representation could be used a solve a problem about percents.
In this figure, a rectangular bar represents the total of $60. This total must cover the price of the food plus 25 percent more for tax and tip. To show this relationship, the bar is segmented into five equal parts, of which four represent the price of the food and one the tax and tip. Because there are five equal parts and the total of $60, each part must be $12. Therefore, the total price allowed for food is $48.
The representation in the figure could also help students see and understand that when one quantity is 125 percent of a second quantity, then the second is 80 percent of the first.
This type of visual representation for numerical quantities is quite adaptable and can be used to solve many problems involving fractions, percents, ratios and proportions.
REFERENCE: Chapter 6, Number and Operations, pp
Problem is from pp

40. More on Percent
Josie needs $40 for a new sweater. She has $24. What percent does she have of what she needs?
$40
100%
$24
? %

41. Using a Table
Josie needs $40 for a new sweater. She has $24. What percent does she have of what she needs?
Needs
$40
Has $4
$20
$24
Percent
10%
50%
60%

42. Still more percent
Jamal has 48% of his homework done. He has done 12 problems. How many problems did the teacher assign? 12 ?
48%
100%

43. Still more percent
Jamal has 48% of his homework done. He has done 12 problems. How many problems did the teacher assign?
% done
48%
Total
100 50 25
# done 48 24 12

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. Where’s 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