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2010 Mathematics Institute Patterns, Functions, and Algebra Grades 3-5 Fall 2010 Agenda • • • • • • Fall 2010 Introductions Big Topics Patterns Properties Lunch Equalities and Inequalities Big Topics I. Patterns • II. Using Number Lines Properties • Building Vocabulary III. Equations and Inequalities • Fall 2010 Keeping it Balanced Unpacking Patterns Fall 2010 Multiplication shown on a Number Line 2009 SOL 3.6 0 1 2 4 5 7 8 10 11 13 14 16 17 Write the multiplication number sentence that matches the hops “Factor Frog” made. 5 Fall 2010 Multiplication on a Number Line 2009 SOL 3.6 http://illuminations.nctm.org/LessonDetail.aspx?ID=L316 6 Fall 2010 LCM Least Common Multiple 2009 SOL 4.5a 0 1 2 3 4 5 7 8 9 10 11 13 14 15 16 17 7 Fall 2010 Primes and Composites 2009 SOL 5.3 Welcome to the Bubble Gum Factory Fall 2010 8 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. 9 Fall 2010 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. = 10 Fall 2010 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. 11 Fall 2010 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 12 machines? Fall 2010 Primes and Composites 2009 SOL 5.3 Figure out which machines are actually necessary. Do we need all of them? 13 Fall 2010 Primes and Composites 2009 SOL 5.3 We know that 2 is a necessary machine, but every even number has 2 as a factor… 14 Fall 2010 Primes and Composites 2009 SOL 5.3 We also know that 3 is a necessary machine, but every third number has 3 as a factor. 15 Fall 2010 Primes and Composites 2009 SOL 5.3 …and we also know that 5 is a necessary machine, but every fifth number has 5 as a factor. 16 Fall 2010 Primes and Composites 2009 SOL 5.3 We know that 7 is a necessary machine and every seventh number has 7 as a factor. 17 Fall 2010 Primes and Composites 2009 SOL 5.3 After exploring divisibility rules for 2, 3, 5, 7, 11, and 17, the prime numbers under 100 are revealed. 18 Fall 2010 Think/Pair/Share 19 Fall 2010 Properties Vocabulary Fall 2010 Properties Vocabulary Fall 2010 Commutative Property 2009 SOL 3.20 Addition to the Standard: -Students have always had to understand the property Now, they also have to name it 22 Fall 2010 Commutative Property 2009 SOL 3.20 If students know: 4 + 5 Then they know : 5 + 4 23 Fall 2010 Commutative Property 2009 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) 24 Fall 2010 Commutative Property 2009 SOL 3.20 - Given experiences with arrays, If students know 3 x 7 7 x Then they can see that it is equal to 25 Fall 2010 3 Commutative Property 2009 SOL 3.20 6x2 = 2x6 26 Fall 2010 Automaticity If I asked you to multiply 56 x 36 using mental math, would you be able to do that with automaticity? 27 Fall 2010 Associative Property 2009 SOL 4.16b Given a problem (41 + 25) + 75 How can you make it an easier problem? 41 + (25 + 75) - Looking for friendly numbers 28 Fall 2010 Associative Property 2009 SOL 4.16b Solving a volume problem - 5 27 Becomes - (27 x 5) x 2 2 27 x (5 x 2) 29 Fall 2010 Distributive Property 2009 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 30 Fall 2010 Distributive Property 2009 SOL 5.19 Slice It 31 Fall 2010 Think/Pair/Share Fall 2010 Equations and Inequalities Fall 2010 What does the equal sign mean? 34 Fall 2010 Equalities 2009 SOL 4.16a http://illuminations.nctm.org/LessonDetail.aspx?ID=L183 35 Fall 2010 Equalities 2009 SOL 3.20 36 Fall 2010 Inequalities 2009 SOL 3.20 37 Fall 2010 Equalities 2009 SOL 3.20 http://illuminations.nctm.org/ActivityDetail.aspx?id=26 38 Fall 2010 Equalities 2009 SOL 4.16a 8=1+7 2+3=2x3 3+5=5+3 7x4=4+4+4+4 9=9 39 Fall 2010 Equalities 2009 SOL 4.16a True or False? Examples/Non-Examples 40 Fall 2010 Modeling One-step Linear Equations 2009 SOL 5.18c Using your cups and candy corn, construct a model for J=6 41 Fall 2010 Modeling One-step Linear Equations 2009 SOL 5.18c 42 Fall 2010 Modeling One-step Linear Equations 2009 SOL 5.18c Using your cups and candy corn, construct a model for J+4=7 43 Fall 2010 Modeling One-step Linear Equations 2009 SOL 5.18c 44 Fall 2010 Modeling One-step Linear Equations 2009 SOL 5.18c 45 Fall 2010 Modeling One-step Linear Equations 2009 SOL 5.18c B+2=9 46 Fall 2010 Modeling One-step Linear Equations 2009 SOL 5.18c 47 Fall 2010 Modeling One-step Linear Equations 2009 SOL 5.18c B + 4 = 11 48 Fall 2010 Modeling One-step Linear Equations 2009 SOL 5.18c http://illuminations.nctm.org/ActivityDetail.aspx?id=33 49 Fall 2010 Modeling One-step Linear Equations 2009 SOL 5.18c http://illuminations.nctm.o rg/ActivityDetail.aspx?id= 10 50 Fall 2010 Think/Pair/Share Fall 2010 Questions? Visit Parking Lot 52 Fall 2010