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

Models that Make Math Meaningful

Where’s the Math? Fraction Models Models for Multiplication Models for Division Decimals, Ratio & Percent

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

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

Sketch what you first see Three-fifths One-sixth

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

What fraction is blue?

Part of a Set 3 of 13 pieces

Part of an Area 6 of 36 triangles

Part of an Area 1 of 6 hexagons

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

Fraction Models Part of a Whole Set Area or region Circles Clocks Rectangles Pattern blocks Strips Length Number line Ruler

Other Meanings for Fractions Part-whole Values - eg, money Division Ratio Rate Wins Losses

Web Resources Fraction model applet Equivalent fractions Fraction game

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. 194-199

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

- 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.

Computational Fluency I 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 7 x 28 I did 7 x 30 first. That’s 210. 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. 148-156

Using Base Ten Blocks to Multiply 24 x 3 12 60 72

Make an Array 24 x 3 12 60 72

A Harder Problem 24 x 13 12 60 40 200 312

Decimals 3 x 0.24 0.12 + 0.60 = 0.72 0.3 x 0.6

Draw a picture that shows

Array 2 of 3 rows 3 of 4 in each row

Mixed Numbers, too! 8 x 3 ¾ 8 x 3 = 24 24 + 6 = 30

1 2/3 x 2 ¼ = ?

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

Sidetrip to Geometry - Area Counting squares on a grid What’s the area?

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

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.

So? Find the area of a triangle with base 10 and height 5. Area = (10 x 5) / 2 = 25 sq. units

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

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

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

Decimals

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. 214-225 Problem is from pp. 281-282.

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. 214-225 Problem is from pp. 281-282. 100% 125%

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. 214-225 Problem is from pp. 281-282.

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 ? %

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%

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%

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

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?

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