September 29th Class - Independent Study

Multiplication and Division

Repeated Addition is used to focus on the meaning of the operation of multiplication.

5 + 5 + 5 + 5 = 4 x 5  

4 groups of 5

Models for Multiplication

Properties of Multiplication

• Associative Property: grouping doesn't matter
• 2 x (3 x 4) = (2 x 3) x 4
• 35 x 4 = (7 x 5) x 4 = 7 x (5 x 4) .....this property can also break down larger #s
• Commutative Property: order doesn't matter
• 5 x 4 = 4 x 5
• 3 x 2 x 5 = 5 x 3 x 2
• Zero Property: zero times any number is zero
• 5x 0 = 0
• 0 x 4 = 0
• Identity Property: anything times one is itself
• 7 x 1 = 7
• Distributive Property: One of two factors can be split into two or more parts and each part multiplied separately and then added.
• 5 x 32 = 5(30 +2) 
• 5 x 38 = 5(40 - 2)

Mental Math 

• Rounding and Estimating: When multiplying and estimating round numbers to the leading digit place and then multiply.
• Benefits - rounding help make the multiplying process much easier to do mentally by making the numbers more compatable
• Disadvantages - when rounding, you are either going to have an estimate that is high or low, and students need to remember that it is an estimate not an exact answer.  Some student struggle with this concept and why to even estimate.  
• Example:
• 479 x 54 = 500 x 50 ~ 25,000
• Expanding: when multiplying a single digit by a number with a two or more digits, expand the multiplication by each place value in the larger number.
• 542 x 3 = (3 x 500) + (3 x 40) + (3 x 2)
• 1500 + 120 + 6
• =1626
• Break it Up: You can break up a number and multiply by a power of ten or hundred
• 256 x 4 + (250 x 4) + (6 x 4)
• Skip counting:  Sometimes it is just easier for some kids to learn the order and count by multiples.
• 2, 4, 6, 8, 10, 12
• 5, 10, 15, 20, 25, 30
• 4, 8, 12, 16, 

Thinking Strategies for Multiplication

• Stage 1 
• Introductory stage
• develops the meaning and interpretations of multiplication
• introduced using repeated addition and rectangular arrays
• Stage 2
• lasts through grades 2 - 3
• 2 goals: increasing calculational proficiency and deepening conceptual understanding
• students see plenty of word problems, mental math and worksheet problems
• begin committing 1-digit multiplication facts to memory
• Stage 3
• goal is simple: all students should have memorized the multiplications up to 9 x 9 by the end of third grade