Transcript Multiplication – equal groups, arrays, multiplicative

Multiplication & Division
Problem Situations
Something New?
June 23, 2011
Common Core Leadership in Mathematics (CCLM)
Summer Institute
DeAnn Huinker & Melissa Hedges
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Hell’s
Library
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Learning Intention
We are learning to:
Understand the differences among problem
types for multiplication and division.
Success Criteria
We will be successful when…
We can pose equal group and compare word
problems for multiplication and division.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Time to Fold Some Paper
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Pose a word problem for:
4x7=☐
Number of Groups
The total number of
objects in 4 groups of
7 objects each.
Group Size
Examine your word problem:
 What does the 4 mean?
 What does the 7 mean?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Pose a word problem for:
28 ÷ 4 = ☐
Examine your word problem:
What is unknown in your problem:
 Number of groups?
 Group size?
Tell a different story for 28 ÷ 4 = ☐.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Revisit Grade 3 expectations
3OA1
 Interpret products of whole numbers, e.g., 5 x 7 as the
total number of objects in 5 groups of 7 objects each.
3OA2
 Interpret whole-number quotients of whole numbers,
e.g., interpret 56 ÷ 8
 as the number of objects in each share when 56
objects are partitioned equally into 8 shares, or
 as a number of shares when 56 objects are
partitioned into equal shares of 8 objects each.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
28 ÷ 4 = ☐
Total
Number of shares
or
Number of objects
in each share
Division means to partition
to find the number of shares
or find the amount in each
share.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Something new?
Read, highlight, take note:
Grade 4 standards: 4OA1 & 4OA2
K-5 OA Progressions: p.29 1st paragraph
Shoulder partners:
Summarize the meaning of these standards
and implications for curriculum.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
4OA1
Interpret a multiplication equation as a comparison,
e.g., interpret 35 = 5 x 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.
4OA2
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.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
What’s the new thinking being
developed in 40A1 & 2?
Students in earlier grades learned to
compare quantities additively….
“This one is 32 feet higher than that one”
40 feet high, the
other 8 feet high
Students in Grade 4 learn to compare
these quantities multiplicatively…..
“This one is 5 times as high as that one.”
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Time to Fold More Paper
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Experiences with Compare Word Problems
Individually pose compare word problems for:
6x9=☐
54 ÷ 6 = ☐ with Group Size Unknown
54 ÷ 6 = ☐ with Number of Groups Unknown
Use Table 2, p. 89, of CCSSM and/or
use Table 3, p.23, of OA Progressions, if needed.
When you have a draft of each,
turn and check-in with a partner.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
7 x 8 = ☐ Total Unknown
56 ÷ 8 = ☐ Group Size Unknown
56 ÷ 8 = ☐ Number of Groups Unknown
Connie has 8 times as much money as
Melissa. Connie has $56. How much
money does Melissa have?
Unknown: Group Size
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
7 x 8 = ☐ Total Unknown
56 ÷ 8 = ☐ Group Size Unknown
56 ÷ 8 = ☐ Number of Groups Unknown
Hank is older than Kara. Hank is 56
years old. Kara is 8 years old. How
much older is Hank?
Is the answer 48 or 7?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Building Blocks of Algebra
 Understand problem situations
 Represent the situation with
objects or diagrams
 Represent quantitative relationships
with equations
 Use properties of operations as the
basis for strategies
p.13 OA Progressions
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Revisiting
Learning Intention and Success Criteria
We were learning to:
Understand the differences among
problem types for multiplication and division.
We will be successful when…
We can pose equal group and compare word
problems for multiplication and division.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Revisiting Meaning of Multiplication
Read, highlight, take note:
K-5 OA Progressions: p.33 last paragraph
How does the meaning of multiplication
expand across Grade 3 to 4 to 5?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011
Expanding View of Multiplication
 Grade 3: Equal groups
--discrete objects, arrays….
 Grade 4: Comparison situations
--continuous quantities
 Grade 5: Stretches or Shrinks (scale factor)
--Context for reasoning multiplicatively
with continuous quantities
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011