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Where’s the Math?

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

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Where’s the Math?

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

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

  14. Wins Losses 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. 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. 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. 2 of 3 rows 3 of 4 in each row 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 • 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 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 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. A = 1 ¾ What’s 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. $60 Cost of Food Tax and Tip 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? x $60 100% 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. $40 $24 ? % 100% 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. ? 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?

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

  44. 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? SO?

  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

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