How Many Times Larger? A Progression of Multiplicative Comparisons

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How Many Times Larger?
A Progression of Multiplicative Comparisons
Grades 3-5
NCCTM LEADERSHIP CONFERENCE
GREENSBORO, NC
MARCH 22,2013
MARTA GARCIA
WINDY TAYLOR
Bridging Major Work Standards
Think about this problem:
*How did you approach the problem?
*What representations did you use to solve the
problem?
*What mathematical ideas are embedded in this
problem?
$120 is shared among 3 persons A, B, and
C. If A receives $20 less than B, and B
receives 3 times as much money as C, how
much money does C receive?
Grade 3
3.NBT.3 Multiply one-digit whole numbers by
multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 ×
60) using strategies based on place value and
properties of operations.
Use base ten blocks to justify why 3 x 50
is ten times larger than 3 x 5.
Associative Property: Why ten times larger?
6 x 5 = 30
3 groups of 10
6 x 50
6 x 50 = 6 x (5 x 10 ) =
(6 x 5) x 10=
30 x 10=
( 3 x 10 ) x 10 = 3 x (10 x 10)
3 x 100
3 groups of 100
Modeling:
With base
ten blocks?
On a
number
line?
Analyzing Student Work
Let’s look at some work samples!
 Paige and Ben each babysat last weekend.
 Paige babysat three times as many hours as
Ben.
 Ben babysat for four hours.
 How many hours did Paige babysit?
What does it mean to be ten times bigger?
Student responses to how many times greater is
4 x 100 than 4 x 10?
Discuss each response: What understandings are students
bringing to this question? What misconceptions are present?
Student Responses
 4 x 100 is 360 times bigger than 4 x 10
 4 x 100 is 90 times bigger than 4 x 10
 4 x 100 is 100 times bigger than 4 x 10
Using Arrays to Compare
Try This !
 Build an array which is 3 times larger than a 2 x 4
array.
 Build an array which is 2 times smaller than a 4 x 10
array.
Fourth Grade
 4.OA.1 Interpret a multiplication equation as a comparison, e.g.,
interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7
and 7 times as many as 5. Represent verbal statements of
multiplicative comparisons as multiplication equations.
 4.OA.2 Multiply or divide to solve word problems involving
multiplicative comparison, e.g., by using drawings and equations
with a symbol for the unknown number to represent the problem,
distinguishing multiplicative comparison from additive
comparison.1
 4.NBT.1 Recognize that in a multi-digit whole number, a digit in
one place represents ten times what it represents in the place to its
right.
Grade 5
 5.NBT.1 Recognize that in a multi-digit number, a
digit in one place represents 10 times as much as it
represents in the place to its right and 1/10 of what it
represents in the place to its left.
 5.NBT.7 Add, subtract, multiply, and divide
decimals to hundredths, using concrete models or
drawings and strategies based on place value,
properties of operations, and/or the relationship
between addition and subtraction; relate the strategy
to a written method and explain the reasoning used
Decimals: What is happening to the size of the
numbers?
 Place the following decimals on a number line.
 Then discuss with a partner how your number lines
are alike or different.
1.14
0.089
0.3
0.04 0.25
What is happening to the products?
8 x 0.01 =
8 x 0.1 =
8x1=
8 x 10 =
8 x 100 =
3 x 2.5 =
 Estimate the product. Between what two whole
numbers will the product lie.
 Use base ten blocks and then the grid paper to model
the product of 3 x 2.5.
 How does the model justify that the product is 3
times larger than 2.5?
3.5 x 2.5=
 Estimate the product. Between what two whole
numbers will the product lie?
 Use base ten blocks and then grid paper to find the
product.
 How can you justify that the product is 3 ½ times
larger than 2.5?
Decimal Multiplication
 How are these ideas building across the grades so
that by fifth grade students can make sense of
multiplying and dividing with decimals?
 Looking back at the problem you solved at the
beginning of the session: How do these standards
support that type of reasoning?
Questions? Reflections?
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