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2010 Mathematics Institute Patterns, Functions, and Algebra Grades 3-5

2010 Mathematics Institute Patterns, Functions, and Algebra Grades 3-5. Agenda. Introductions Big Topics Patterns Properties Lunch Equalities and Inequalities. Big Topics. Patterns Using Number Lines Properties Building Vocabulary Equations and Inequalities Keeping it Balanced.

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2010 Mathematics Institute Patterns, Functions, and Algebra Grades 3-5

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  1. 2010 Mathematics Institute Patterns, Functions, and Algebra Grades 3-5

  2. Agenda • Introductions • Big Topics • Patterns • Properties • Lunch • Equalities and Inequalities

  3. Big Topics • Patterns • Using Number Lines • Properties • Building Vocabulary • Equations and Inequalities • Keeping it Balanced

  4. Unpacking Patterns

  5. Multiplication shown on a Number Line2009 SOL 3.6 9 0 1 2 4 5 7 8 10 11 13 14 16 17 3 6 12 15 Write the multiplication number sentence that matches the hops “Factor Frog” made.

  6. Multiplication on a Number Line 2009 SOL 3.6 http://illuminations.nctm.org/LessonDetail.aspx?ID=L316

  7. Least Common Multiple2009 SOL 4.5a LCM 6 12 0 1 2 3 4 5 7 8 9 10 11 13 14 15 16 17

  8. Primes and Composites 2009 SOL 5.3 Welcome to the Bubble Gum Factory

  9. Primes and Composites 2009 SOL 5.3 Bubble Gum Factory Investigation At the Bubble Gum Factory, lengths of gum are stretched to larger lengths by putting them through stretching machines. There are 99 stretching machines, numbered 2 through 100.

  10. Primes and Composites 2009 SOL 5.3 We do not need a machine 1 because it does nothing to a piece of gum. • = Machine 2 stretches pieces of gum to twice its original length. =

  11. Primes and Composites 2009 SOL 5.3 Machine 3 triples the length and so forth. = So, machine 23, for example, will stretch a piece of gum to 23 times its original length. = Well…you get the point. 

  12. Primes and Composites 2009 SOL 5.3 Now It Is Your Job! An order has just come in for a piece of bubble gum 24 inches in length. The factory has pieces of gum that are only 1 inch in length, and machine number 24 is broken. *Is there any way to create a piece of bubble gum 24 inches in length by using other machines?

  13. Primes and Composites 2009 SOL 5.3 Figure out which machines are actually necessary. Do we need all of them?

  14. Primes and Composites 2009 SOL 5.3 We know that 2 is a necessary machine, but every even number has 2 as a factor…

  15. Primes and Composites2009 SOL 5.3 We also know that 3 is a necessary machine, but every third number has 3 as a factor.

  16. Primes and Composites2009 SOL 5.3 …and we also know that 5 is a necessary machine, but every fifth number has 5 as a factor.

  17. Primes and Composites2009 SOL 5.3 We know that 7 is a necessary machine and every seventh number has 7 as a factor.

  18. Primes and Composites2009 SOL 5.3 After exploring divisibility rules for 2, 3, 5, 7, 11, and 17, the prime numbers under 100 are revealed.

  19. Think/Pair/Share

  20. Properties Vocabulary

  21. Properties Vocabulary

  22. Commutative Property2009 SOL 3.20 Addition to the Standard: • Students have always had to • understand the property Now, they also have to name it

  23. Commutative Property2009 SOL 3.20 If students know: 4 + 5 Then they know : + 4 5

  24. Commutative Property2009 SOL 3.20 “It is not intuitively obvious that 3 x 8 is the same as 8 x 3 or that, in general, the order of the numbers makes no difference (the commutative or order property). A picture of 3 sets of 8 objects cannot immediately be seen as 8 piles of 3 objects. Eight hops of 3 land at 24, but it is not clear that 3 hops of 8 will land at the same point. The array, by contrast, is quite powerful in illustrating the order property. Students should draw or build arrays and use them to demonstrate why each array represents two different multiplications with the same product.” Van de Walle (2001)

  25. Commutative Property2009 SOL 3.20 - Given experiences with arrays, If students know 3 x 7 7 x 3 Then they can see that it is equal to -

  26. Commutative Property2009 SOL 3.20 6 x 2 = 2 x 6

  27. Automaticity If I asked you to multiply 56 x 36 using mental math, would you be able to do that with automaticity?

  28. Associative Property2009 SOL 4.16b Given a problem - (41 + 25) + 75 How can you make it an easier problem? 41 + (25 + 75) - Looking for friendly numbers

  29. Associative Property2009 SOL 4.16b Solving a volume problem - 5 (27 x 5) x 2 2 27 Becomes - 27 x (5 x 2)

  30. Distributive Property2009 SOL 5.19 Partial Products 3 x 24 = 3 x 20 + 3 x 4 http://www.glencoe.com/sites/common_assets/mathematics/ebook_assets/vmf/VMF-Interface.html

  31. Distributive Property2009 SOL 5.19 Slice It

  32. Think/Pair/Share

  33. Equations and Inequalities

  34. What does the equal sign mean?

  35. Equalities2009 SOL 4.16a http://illuminations.nctm.org/LessonDetail.aspx?ID=L183

  36. Equalities2009 SOL 3.20 4 4 3 3

  37. Inequalities2009 SOL 3.20 4 4 3 2

  38. Equalities2009 SOL 3.20 http://illuminations.nctm.org/ActivityDetail.aspx?id=26

  39. Equalities2009 SOL 4.16a What will the students say? 8 = 1 + 7 True or False? 2 + 3 = 2 x 3 3 + 5 = 5 + 3 7 x 4 = 4 + 4 + 4 + 4 9 = 9

  40. Equalities2009 SOL 4.16a True or False? Examples/Non-Examples

  41. Modeling One-step Linear Equations2009 SOL 5.18c Using your cups and candy corn, construct a model for J = 6

  42. Modeling One-step Linear Equations2009 SOL 5.18c

  43. Modeling One-step Linear Equations2009 SOL 5.18c Using your cups and candy corn, construct a model for J + 4 = 7

  44. Modeling One-step Linear Equations2009 SOL 5.18c

  45. Modeling One-step Linear Equations2009 SOL 5.18c

  46. Modeling One-step Linear Equations2009 SOL 5.18c B + 2 = 9

  47. Modeling One-step Linear Equations2009 SOL 5.18c

  48. Modeling One-step Linear Equations2009 SOL 5.18c B + 4 = 11

  49. Modeling One-step Linear Equations2009 SOL 5.18c http://illuminations.nctm.org/ActivityDetail.aspx?id=33

  50. Modeling One-step Linear Equations2009 SOL 5.18c http://illuminations.nctm.org/ActivityDetail.aspx?id=10

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