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

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Models that Make Math Meaningful

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

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

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

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Sketch what you first see Three-fifths One-sixth

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

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What fraction is blue?

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Part of a Set 3 of 13 pieces

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Part of an Area 6 of 36 triangles

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Part of an Area 1 of 6 hexagons

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

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

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

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

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Playing Fraction Tracks

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

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

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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 I did 7 x 30 first. That’s 210. Then take off seven 2’s or 14. So it’s 196. 7 x 28

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Using Base Ten Blocks to Multiply 24 x 3 126072

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Make an Array 24 x 3 126072

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

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

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Draw a picture that shows

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Array 2 of 3 rows 3 of 4 in each row

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

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1 2/3 x 2 ¼ = ?

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

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

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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 = 12 square units

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Make a Rectangle Area of rectangle = 3 x 6 = 18 squares Areas of triangles UL: ½ x 4 = 2 UL: ½ x 4 = 2 UR: ½ x 2 = 1 UR: ½ x 2 = 1 LL: ½ x 4 = 2 LL: ½ x 4 = 2 LR: ½ x 2 = 1 LR: ½ x 2 = 1 Total = 6 squares Total = 6 squares Area of pentagon = 18 – 6 = 12 sq. = 18 – 6 = 12 sq.

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

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Fraction Division What is the whole if half is 1¾? 1¾ Measurement model Need two pieces of size 1¾, so find size 1¾, so find 1¾ x 2 = 3 ½ 1¾ x 2 = 3 ½ 1¾ ÷ 2 =1¾ x 2 1¾ ÷ 2 = 1¾ x 2 = 3 ½ = 3 ½

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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 1 ¾ ÷ ½ = 3 ½

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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

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Decimals

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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

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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? $60x 100%125%

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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? Dinner$40$4$8$48 Tax & tip $10$1$2$12 Total$50$5$10$60

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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 ? %

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Using a Table Josie needs $40 for a new sweater. She has $24. What percent does she have of what she needs? Needs$40$40$40 Has$4$20$24 Percent10%50%60%

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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%

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Still more percent Jamal has 48% of his homework done. He has done 12 problems. How many problems did the teacher assign? % done 48%48%48% Total1005025 # done 482412

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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?

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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

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