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Middle School Math Concepts: “Is this reasonable?”

Middle School Math Concepts: “Is this reasonable?”. Esther Kim estherkim39@gmail.com. 2011 API By Subgroup. Percent of students scoring proficient or advanced. Over 50% or under 50%?. Thumbs up if you think over 50% Thumbs down if you think under 50%. Over 50% or under 50%?. 2 – 6 + 3 – 1

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Middle School Math Concepts: “Is this reasonable?”

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  1. Middle School Math Concepts:“Is this reasonable?” Esther Kim estherkim39@gmail.com

  2. 2011 API By Subgroup

  3. Percent of students scoring proficient or advanced

  4. Over 50% or under 50%? Thumbs up if you think over 50% Thumbs down if you think under 50%

  5. Over 50% or under 50%? 2 – 6 + 3 – 1 -3 b) -2 c) 3 d) 6

  6. Over 50% or under 50%? Over 50% 2 – 6 + 3 – 1 -3 b) -2 c) 3 d) 6

  7. 81.5% of the students responded correctly. 2 – 6 + 3 – 1 -3 b) -2 c) 3 d) 6

  8. Over 50% or under 50%? 45% of 61.4 is a number between a) 3 and 30 b) 30 and 60 c) 60 and 240 d) 240 and 2800

  9. Over 50% or under 50%? Under 50% 45% of 61.4 is a number between a) 3 and 30 b) 30 and 60 c) 60 and 240 d) 240 and 2800

  10. Approximately 20% of the students chose C or D. 45% of 61.4 is a number between a) 3 and 30 b) 30 and 60 c) 60 and 240 d) 240 and 2800 (0.45)(61.4)=27.63

  11. Over 50% or under 50%? Which of the following is the largest? 3/4 b) 6/7 c) 12/13 d) 17/25

  12. Over 50% or under 50%? Under 50% Which of the following is the largest? 3/4 b) 6/7 c) 12/13 d) 17/25

  13. 31% of the students chose A. Under 50% Which of the following is the largest? 3/4 b) 6/7 c) 12/13 d) 17/25

  14. Over 50% or under 50%? (5/6)(30) 25 b) 36 c) 150 d) 156

  15. 70.8% of the students responded correctly. Over 50% (5/6)(30) 25 b) 36 c) 150 d) 156

  16. 35.8% of the students responded correctly, and 32.5% of the students chose A. Order the following from least to greatest. 3/8, 1/3, 0.36, 0.39 a) 1/3, 0.36, 0.39, 3/8 b) 0.36, 0.39, 3/8, 1/3 c) 1/3, 0.36, 3/8, 0.39 d) 0.36, 3/8, 0.39, 1/3

  17. What has worked? • Multiple Representations: One size does not fit all • Pictures, Charts, Tables • Verbal • Algebraic • Numeric • Connecting concepts within lessons

  18. Math 7

  19. Critical Concepts in Middle School Mathematics *Note: If you have an iPhone or Andriod phone with data access, please install the free Socrative Student Clicker app. My room # is 51016Once you’re in the “room” feel free to text in your thoughts and questions. Mark Ellis CSU Fullerton mellis@fullerton.edu

  20. Shifting Focus • Traditional U.S. • How can I teach my kids to get the answer to this problem? • Evaluate answers to determine proficiency. • Common Core • How can I use this problem to teach the mathematics of this unit? • Examine responses to uncover student thinking and inform next steps pedagogically as all students move toward big idea(s) of a unit.

  21. Reasoning about Division • Divide 365/4 (by hand) using three methods.

  22. Reasoning about Division

  23. Sense Making: Fraction Division • What question might this expression answer? • Find the quotient in a way that makes sense. • Share your reasoning with your neighbor(s) and/or send to Socrative #51016 • What justifies your method? • What prior knowledge is needed?

  24. Learning Trajectories K -2 3 - 6 7 - 12 Equal Partitioning Rates, proportional and linear relationships Unitizing in base 10 and in measurement Rational number Fractions Number line in Quantity and measurement Properties of Operations Rational Expressions From Phil Daro, CMC-S

  25. Two “Big Ideas” Equivalence Proportionality

  26. Misconception about Equality If you learned to interpret the equal sign as an operation, • 3 + 8 = • 23 x 7 = how would you make sense of these? • 4 x 97 = 4 (90 + 7) • 2x – 7 = x + 11

  27. Reasoning about One-half

  28. Euclid’s Algebra Challenge Model this geometrically, then with algebraic symbols. If a straight line segment be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. (Euclid, Elements, II.4, 300 B.C.)

  29. How Not to Learn Proportional Reasoning What is not developed when students “learn” this first? Why does this algorithm work?

  30. Reasoning Proportionally John’ s mixture was 3 spoonfuls of sugar and 12 spoonfuls of lemon juice. Mary’ s mixture was 4 spoonfuls of sugar and 13 spoonfuls of lemon juice. Whose lemonade is sweeter, John’ s or Mary’ s? Or would they taste equally sweet? Eva and Alex want to paint the door of their garage. They mix 2 cans of white paint and 3 cans of black paint to get a particular shade of gray. They then add one more can of each color. Will the new shade of gray be lighter, darker, or the same?

  31. Role of Technology http://www.skill-guru.com/gmat/wp-content/uploads/2011/02/thinking-cap.gif Support and advance mathematical sense-making, reasoning, problem solving, and communication (NCTM Position Statement, 2011) http://www.nctm.org/about/content.aspx?id=31734

  32. Rhythm Wheel: Exploring Multiplicative Reasoning After 4 loops of this wheel: How many individual sounds will be played? How do you know this? How many times will the “open” hand drum sound be played? How do you know? How many times more will the “open” hand drum sound be played than the “slap” drum sound? How do you know?

  33. Rhythm Wheel (part 2) How many loops would the 1st and 2nd wheels each need to make so they stop playing at the same time? Why? If the 2nd wheel does 18 loops, how many loops will the 1st wheel need to make so it stops at the same time? If the neck cowbell sound gets played 30 times, how many times will the open hand drum sound be played? Prove it! (Assume the wheels stop at the same time.) What if the neck is played x times?

  34. Proportionality and Linear Functions • If two quantities vary proportionally, that relationship can be represented as a linear function. • If two quantities vary proportionally, • the ratio of corresponding terms is constant, • the constant ratio can be expressed in lowest terms (a composite unit) or as a unit amount, and • the constant ratio is the slope of the related linear function. • When you graph the terms of equal ratios as ordered pairs (first term, second term) and connect the points, the graph is a straight line.

  35. Two Categories of Tools Communication & Collaboration Content Exploration

  36. Resources • Charles, R. (2005). Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics. http://www.authenticeducation.org/bigideas/sample_units/math_samples/BigIdeas_NCSM_Spr05v7.pdf • Learning Progressions/Trajectories in Mathematics • http://ime.math.arizona.edu/progressions/ • http://www.turnonccmath.com/ • http://www.cpre.org/images/stories/cpre_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf • Math Reasoning Inventory, https://mathreasoninginventory.com/ • Seigler, R. (2012). Knowledge of Fractions and Long Division Predicts Long-Term Math Success http://youtu.be/7YSj0mmjwBM and http://www.psychologicalscience.org/index.php/news/releases/knowledge-of-fractions-and-long-division-predicts-long-term-math-success.html • To Half or Not lesson, http://www.pbs.org/teachers/mathline/lessonplans/pdf/esmp/half.pdf

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