Where’s the Math? Dr. Janet H. Caldwell Rowan University [email protected].

Where’s the Math?Where’s the Math?

Dr. Janet H. CaldwellDr. Janet H. Caldwell

Rowan UniversityRowan University

[email protected]@rowan.edu

Models that Make Models that Make Math MeaningfulMath Meaningful


Fraction ModelsFraction Models Models for MultiplicationModels for Multiplication Models for DivisionModels for Division Decimals, Ratio & PercentDecimals, Ratio & Percent

Sketch what you first seeSketch what you first see

One-halfOne-half Two-thirdsTwo-thirds Three-fourthsThree-fourths Three-fifthsThree-fifths One-sixthOne-sixth Two and a quarterTwo and a quarter Two and two-thirdsTwo and two-thirds


One-halfOne-half Two-thirdsTwo-thirds Three-fourthsThree-fourths


Three-fifthsThree-fifths One-sixthOne-sixth


Two and a quarterTwo and a quarter Two and two-thirdsTwo and two-thirds

What fraction is blue?What fraction is blue?

Part of a SetPart of a Set 3 of 13 pieces 3 of 13 pieces

Part of an Area Part of an Area 6 of 36 triangles 6 of 36 triangles

Part of an AreaPart of an Area 1 of 6 hexagons 1 of 6 hexagons

Part of an area Part of an area Blue is 1/3 of largest Blue is 1/3 of largest piecepiece

Fraction ModelsFraction ModelsPart of a WholePart of a Whole

SetSet Area or regionArea or region

CirclesCircles ClocksClocks RectanglesRectangles Pattern blocksPattern blocks StripsStrips

LengthLength Number lineNumber line RulerRuler

Other Meanings for Other Meanings for FractionsFractions

Part-wholePart-whole Values - eg, moneyValues - eg, money

DivisionDivision RatioRatio RateRate

WinsLosses

Web ResourcesWeb Resources

Fraction model appleFraction model applett

Equivalent fractionsEquivalent fractions Fraction gameFraction game

Playing Fraction TracksPlaying Fraction Tracks

Make a triangle that is:Make a triangle that is:

¼ green and ¾ red¼ green and ¾ red

1/3 red and 2/3 green1/3 red and 2/3 green

““Understanding is the key to Understanding is the key to remembering what is learned remembering what is learned and being able to use it and being able to use it flexibly.”flexibly.”

- Hiebert, in Lester & Charles, - Hiebert, in Lester & Charles,

Teaching Mathematics through Teaching Mathematics through

Problem SolvingProblem Solving, 2004., 2004.

Computational FluencyComputational FluencyI thought seven 25’s - that’s 175.

Then I need seven 3’s or 21. So the answer

is 175 + 21 = 196

7 x 20 is 140

and 7 x 8 is 56

56 + 140 is 196

I did 7 x 30 first. That’s 210. Then take off seven 2’s or 14. So it’s 196.

7 x 28

Using Base Ten Blocks to Using Base Ten Blocks to MultiplyMultiply

2424

x 3x 3

1212

6060

7272

Make an ArrayMake an Array

2424

x 3x 3

1212

6060

7272

A Harder ProblemA Harder Problem

2424x 13x 13 1212 6060 4040 200200 312312

DecimalsDecimals

3 x 0.243 x 0.24

0.3 x 0.60.3 x 0.6

0.12 + 0.60 = 0.720.12 + 0.60 = 0.72

Draw a picture that Draw a picture that shows shows

4

3

3

2

ArrayArray

2 of 3 rows

3 of 4 in each row

2

1

12

6

43

32

ofrows

ofrows

areatotal

areashaded

Mixed Numbers, too!Mixed Numbers, too!

8 x 3 ¾8 x 3 ¾

8 x 3 = 248 x 3 = 24

64

24

4

38 x

24 + 6 = 3024 + 6 = 30

1 2/3 x 2 ¼ = ?1 2/3 x 2 ¼ = ?6

4

2

5 10

5 9 45 9 3x 3 3

3 4 12 12 4

AlgebraAlgebra

(x + 1) (x + 2) (x + 1) (x + 2)

= x= x22 + 2x + x + 2 + 2x + x + 2

= x= x22 + 3x + 2 + 3x + 2

x + 2

x+

1

Sidetrip to Geometry - Sidetrip to Geometry - AreaArea

Counting squares on Counting squares on a grida grid

What’s the area?What’s the area?

Break it upBreak it up

Yellow (L) = ½ x 4 = 2Yellow (L) = ½ x 4 = 2

Blue = 2 x 3 = 6Blue = 2 x 3 = 6

Yellow (R) = ½ x 2 = 1Yellow (R) = ½ x 2 = 1

Orange = ½ x 2 = 1Orange = ½ x 2 = 1

Red = ½ x 4 = 2Red = ½ x 4 = 2

2 + 6 + 1 + 1 + 2 2 + 6 + 1 + 1 + 2

= 12 square units= 12 square units

Make a RectangleMake a Rectangle

Area of rectangleArea of rectangle

= 3 x 6 = 18 squares= 3 x 6 = 18 squares

Areas of trianglesAreas of triangles

UL: ½ x 4 = 2UL: ½ x 4 = 2

UR: ½ x 2 = 1UR: ½ x 2 = 1

LL: ½ x 4 = 2LL: ½ x 4 = 2

LR: ½ x 2 = 1LR: ½ x 2 = 1

Total = 6 squaresTotal = 6 squares

Area of pentagon Area of pentagon

= 18 – 6 = 12 sq.= 18 – 6 = 12 sq.

So?So?

Find the area of a Find the area of a triangle with base 10 triangle with base 10 and height 5.and height 5.

5

10

Area = (10 x 5) / 2Area = (10 x 5) / 2

= 25 sq. units= 25 sq. units

Fraction DivisionFraction Division

What is the whole if What is the whole if half is 1¾?half is 1¾?

1¾ Measurement modelMeasurement model

Need two pieces ofNeed two pieces of

size 1¾, so findsize 1¾, so find

1¾ x 2 = 3 ½1¾ x 2 = 3 ½1¾ 1¾ ÷÷ 2 = 2 = 1¾ x 21¾ x 2

= 3 ½= 3 ½

How many 1/2s are there How many 1/2s are there in 1¾? in 1¾?

How many cakes How many cakes can you make with can you make with 1 ¾ cups of sugar 1 ¾ cups of sugar if each cake if each cake requires ½ cup?requires ½ cup?

Partitive Model (Sharing)Partitive Model (Sharing)

1 ¾ 1 ¾ ÷ ½ = 3 ½ ÷ ½ = 3 ½

What’s the length?What’s the length?

The area of a field is The area of a field is 1 ¾ square miles.1 ¾ square miles.

Its width is ½ mile.Its width is ½ mile.A = 1 ¾A = 1 ¾ 1/21/2

Missing Factor Missing Factor ModelModel

½ x ___ = 1 ¾½ x ___ = 1 ¾

DecimalsDecimals

Percents Percents

A group of students has $60 to spend on dinner. They A group of students has $60 to spend on dinner. They know that the total cost, after adding tax and tip, will be 25% know that the total cost, after adding tax and tip, will be 25% more than the food prices shown on the menu. How much more than the food prices shown on the menu. How much

can they spend on the food so that the total cost will be can they spend on the food so that the total cost will be $60?$60?



$60

Cost of Food Taxand Tip

Percent BarPercent Bar





$60x

100% 125%

Another ApproachAnother Approach

A group of students has $60 to spend on dinner. They A group of students has $60 to spend on dinner. They know that the total cost, after adding tax and tip, will be know that the total cost, after adding tax and tip, will be

25% more than the food prices shown on the menu. How 25% more than the food prices shown on the menu. How much can they spend on the food so that the total cost will much can they spend on the food so that the total cost will

be $60?be $60?

A group of students has $60 to spend on dinner. They A group of students has $60 to spend on dinner. They know that the total cost, after adding tax and tip, will be know that the total cost, after adding tax and tip, will be

25% more than the food prices shown on the menu. How 25% more than the food prices shown on the menu. How much can they spend on the food so that the total cost will much can they spend on the food so that the total cost will

be $60?be $60?

DinnerDinner $40$40 $4$4 $8$8 $48$48

Tax & tipTax & tip $10$10 $1$1 $2$2 $12$12

TotalTotal $50$50 $5$5 $10$10 $60$60

More on PercentMore on Percent

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

$40

100%

$24

? %

Using a TableUsing a Table

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

NeedsNeeds $40$40 $40$40 $40$40

HasHas $4$4 $20$20 $24$24

PercentPercent 10%10% 50%50% 60%60%

Still more percentStill more percent

Jamal has 48% of his homework done. He Jamal has 48% of his homework done. He has done 12 problems. How many has done 12 problems. How many problems did the teacher assign?problems did the teacher assign?

12 ?

48% 100%

Still more percentStill more percent

Jamal has 48% of his homework done. He Jamal has 48% of his homework done. He has done 12 problems. How many problems has done 12 problems. How many problems did the teacher assign?did the teacher assign?

% done% done 48%48% 48%48% 48%48%

TotalTotal 100100 5050 2525

# done# done 4848 2424 1212

SO?SO?

PicturesPictures ManipulativesManipulatives Oral languageOral language Written symbolsWritten symbols TablesTables GraphsGraphs Relevant situationsRelevant situations

Which model(s) are Which model(s) are most meaningful for most meaningful for my students?my students?

Which models Which models promote more promote more powerful thinking?powerful thinking?

In what order should In what order should I use selected I use selected models?models?


Models help students explore concepts Models help students explore concepts and build understandingand build understanding

Models provide a context for students to Models provide a context for students to solve problems and explain reasoning solve problems and explain reasoning

Models provide opportunities for students Models provide opportunities for students to generalize conceptual understanding to generalize conceptual understanding

Where’s the Math? Dr. Janet H. Caldwell Rowan University [email protected].

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