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Math Hot Topic

Math Hot Topic. Multiplication Grades 3-5 September 20, 2012. Schedule Roster Internet Username & Password: k5math Folders & Handouts. Getting Started. Understand concepts of factors and multiples in the CCSSM Develop ways to build multiplication fluency

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Math Hot Topic

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  1. Math Hot Topic Multiplication Grades 3-5 September 20, 2012

  2. Schedule Roster Internet Username & Password: k5math Folders & Handouts Getting Started

  3. Understand concepts of factors and multiples in the CCSSM Develop ways to build multiplication fluency Understand concepts used in multiplying whole numbers and how these connect to alternate algorithms Goals of this Workshop

  4. www.corestandards.org

  5. CCSS Critical Areas of Study Instructional time should focus on: • Grade 3: Developing understanding of multiplication and division and strategies for multiplication within 100 • Grade 4: Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends • Grade 5: Developing understanding of the multiplication of fractions and of division of fractions in limited cases

  6. Developing “Understanding” • "Understand" is used in these standards to mean that students can explain the concept with mathematical reasoning, including concrete illustrations, mathematical representations, and example applications. Presentation by James Williams, NCCTM Leadership Seminar, 10-26-11

  7. Developing “Understanding” • Students who understand a concept can • use it to make sense of and explain quantitative situations • incorporate it into their own arguments and use it to evaluate the arguments of others • bring it to bear on the solutions to problems • make connections between it and related concepts

  8. Developing “Understanding” • Applying Common Core’s definition of “Understanding” forces us to move beyond algorithms, mnemonics and well rehearsed procedures Presentation by James Williams, NCCTM Leadership Seminar, 10-26-11

  9. Multiplication 3 x 2 = 6 • Describe this multiplication equation. • What does it mean? • Create a word problem to match the equation. • Draw two different models to represent the equation.

  10. Activity 1: Factors & Multiples Making Rectangles

  11. Making Rectangles • Using 12 square-inch tiles, make as many different rectangles as you can • How many different rectangles can you make? • Draw the rectangles on grid paper and cut them out

  12. Making Rectangles • Work together to make all possible rectangles for your assigned numbers • Draw the rectangles on grid paper and cut them out • Write the corresponding multiplication equation on the rectangle and attach the rectangles to the class chart

  13. Making Rectangles • Group 1: 1, 10, 18, 23 • Group 2: 2, 7, 9, 11, 12 • Group 3: 3, 13, 17, 24 • Group 4: 4, 8, 16, 20 • Group 5: 5, 15, 22, 25 • Group 6: 6, 14, 19, 21 *When you finish making your rectangles, check the rectangles of the next group.

  14. Questions to Ponder • Speculate the possible rectangles you will create • How can you be sure that you found all of possible the rectangles?

  15. Rectangles

  16. Chart Observations • List three observations about the chart

  17. Task Reflection • How did you work on this task with your group? • How did you ensure that you found all of the rectangles for your assigned numbers? • Describe your observations from the chart

  18. What Do You Think? • Which numbers have rectangles with 2 rows? Which rectangles have a side with two squares on them? Write the numbers from smallest to largest. • Which numbers have rectangles with 3 rows? • Which numbers have rectangles with 4 rows?

  19. Multiples • Defined operationally • Products of a given number • Numbers you say when you skip count by that number

  20. Multiples • Multiples of 4 4, 8, 12, 16, 20, 24, 28… • Multiples of 8 8, 16, 24, 32, 40, 48, 56… • How are the multiples of 4 and 8 related? Why? • Would 68 be a multiple of 2? How do you know? • Explain the importance of multiples.

  21. Are They Square? • Which numbers have rectangles that are squares? • What would be the next number after 25 that would create a square? Explain how you know

  22. Square Numbers • A number multiplied by itself (1x1=1, 2x2=4, 3x3=9, 4x4=16…) • Array forms a square

  23. What Are Your Ideas? • Which numbers have only one rectangle? List them from smallest to largest • What is the smallest number that has two different rectangles? Three different rectangles? Four different rectangles?

  24. Prime and Composite Numbers • Prime • A number with exactly two factors (itself and one) • Composite • A number that has more than two distinct factors

  25. Number Sort • Sort the following numbers into three categories • Numbers may fit into multiple categories 24 8 49 17 33 2 12 1 25 4 9 36 27 16 3 11 21 40 29

  26. Think about . . . • Is 1 a prime or composite number? • Neither! • A prime number has exactly two factors (itself & 1). The number 1 has only one factor (1 x 1). • Composite: Cannot be written as a product of 2 distinct factors • What about the number 0? • Neither! • 0 has an infinite number of divisors/factors • Cannot be written as a product of 2 distinct factors (neither of which is itself)

  27. Factorize http://illuminations.nctm.org/

  28. Factorize Game: Questions to Explore Consider these questions as you play the game: • Why do you think the length and width of the rectangles represent the factors of your numbers? • Which number has the most factorizations? Which has the fewest? Why do you think this is? • What kinds of numbers have only one factorization? What do the rectangles for these factorizations have in common? • If you double a number, what happens to the number of factorizations Do you notice a pattern?

  29. Foundations with MultiplicationSecond Grade Operations & Algebraic Thinking (2.OA.3 & 2.OA.4) Work with equal groups of objects to gain foundations for multiplication. 3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. 4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

  30. Foundations with MultiplicationThird Grade Operations and Algebraic Thinking (3.OA) Represent and solve problems involving multiplication and division. • Interpret products of whole numbers • Use multiplication within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities • Determine the unknown in a multiplication equation

  31. Foundations with MultiplicationThird Grade Operations and Algebraic Thinking (3.OA) Understand properties of multiplication and the relationship between multiplication and division. • Apply properties of operations (commutative, associative, distributive) Multiply and Divide within 100 • Fluently multiply within 100 and know all products of two one-digit numbers from memory

  32. Foundations with MultiplicationThird Grade Number and Operations in Base Ten (3.NBT) Use place value understanding and properties of operations to perform multi-digit arithmetic. • Multiply one-digit numbers by multiples of 10 in the range 10-90 using strategies based on place value and properties

  33. Foundations with MultiplicationThird Grade Measurement and Data (3.MD) Geometric measurement: understand concepts of area and relate area to multiplication and to addition. • Relate area to the operation of multiplication • Multiply side lengths to find areas of rectangles • Represent products as rectangular areas • Use area models to represent the distributive property

  34. Standards for Mathematical Content 4. Operations and Algebraic Thinking (4.OA.4) Gain familiarity with factors and multiples. 4. Find all factor pairs for a whole number in the range of 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range of 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range of 1-100 is prime or composite.

  35. Investigations Correlation Third Grade: • Unit 5 Equal Groups • Investigation 3 Arrays • Session 3.2 (square & prime numbers) Fourth Grade: • Unit 1 Factors, Multiples, and Arrays • Investigation 1 Representing Multiplication with Arrays • Session 1.3 (square, prime, & composite numbers) • Investigation 3 Finding Factors Fifth Grade: • Unit 1 Number Puzzles and Multiple Towers • Investigation 1 Finding Factors and Prime Factors • Session 1.2 (square, prime, & composite numbers)

  36. Activity 2: Games Games to Reinforce Multiplication Fluency

  37. Activity 2: The Factor Game Directions: • Each player uses a different colored chip • Player A selects a number and places a chip on it • Player B places chips on all of the proper factors of the selected number. Proper factors - all the factors of the number, except the number itself. The proper factors for 12 are 1, 2, 3, 4, and 6. • Player B selects a new number and Player A places a chip on the factors that are not already covered

  38. Activity 2: The Factor Game Directions: • Alternate between players until there are no factors left for the remaining numbers • If a player selects a number that has no factors left, the player loses a turn and does not get the points for the number selected • To determine the winner, add the numbers that are covered. The player with the greater total is the winner

  39. Activity 2: The Factor Game

  40. Activity 2: The Factor Game Score: Player A 18 Player B 21

  41. Activity 2: The Product Game Directions: • The object of the game is to get 4 in a row • To begin, Player 1 puts a paper clip on a number in the factor list of numbers 1-9 along the bottom of the game board • Player 2 then puts the other paper clip on any number in the factor list. The product of the two marked numbers is determined, and that product is covered with a blue marker • Player 1 moves either one of the paper clips on the factor list to another number, and the new product is covered with a red marker

  42. Activity 2: The Product Game

  43. Activity 2: The Product Game Directions: • Players take turns moving paper clips and marking each product with a red or blue marker, depending on which player made the product • If a product is already covered, the player does not get a square for that turn • Play continues until one player wins (getting four squares in a row vertically, horizontally or diagonally), or until all squares have been colored

  44. Activity 2: The Product Game

  45. Activity 2: The Product Game

  46. Activity 2: Pathways Directions: • Player 1 places a paperclip on two numbers and marks the product with her color marker • Player 2 moves one paperclip to another number and marks the product with a marker • Winner: First player to complete a continuous pathway in the same color across the board (a pathway may include boxes that share a common side or corner

  47. Activity 2: Pathways

  48. Activity 2: Target 300 Directions: • Object: Get closest to 300 after six rolls • Player 1 rolls the die and decides to multiply the number by 10, 20, 30, 40, or 50 • Record the multiplication sentence • Player 2 follows the same steps • Add each amount to keep a running total • At the end of six turns, compare scores to see whose total is closest to 300

  49. Activity 2: Target 300

  50. Games • The Factor Game • The Product Game • Pathways • Target 300 • Zang!

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