Transcript Teaching Multiplication (and Division) Conceptually

Teaching
Multiplication
(and Division)
Conceptually
Jenny C. Ray, Mathematics Specialist
Kentucky Department of Education
• 1. I can describe what it means and what it looks like to teach
multiplication (and division) conceptually.
• 2. I can describe how standards progress across grade levels,
giving details for the grade span in which I teach.
• 3. I can match assessment/learning tasks with the appropriate
Common Core Standard, and determine it’s congruency to the
standard.
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Professional Learning Targets…
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• Counting numbers in a set (K)
• Counting by tens (K)
• Understanding the numbers 10, 20, 30, 40, …, 90
refer to one, two, three, four, …, nine tens (1)
• Counting by fives (2)
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Prior Understandings—
Grades K-2
3
Prior Understandings
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• 2.G.2. Partition a rectangle into rows and columns of samesize squares and count to find the total number of them.
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• In Grade 3, instructional time should focus on four critical areas: (1)
developing understanding of multiplication and division and
strategies for multiplication and division within 100; …
• Students develop an understanding of the meanings of
multiplication and division of whole numbers through activities
and problems involving equal-sized groups, arrays, and area
models;
• multiplication is finding an unknown product, and division is finding
an unknown factor in these situations.
• For equal-sized group situations, division can require finding the
unknown number of groups or the unknown group size.
• Students use properties of operations to calculate products of whole
numbers, using increasingly sophisticated strategies based on these
properties to solve multiplication and division problems involving
single-digit factors.
• By comparing a variety of solution strategies, students learn the
relationship between multiplication and division.
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Grade 3 Introduction
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Commutative Property
• The array, however, can be quite powerful in illustrating the
commutative property.
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• It is not intuitively obvious that 3 x 8 = 8 x 3. 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 24.
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Distributive Property
& Area Models
3 x 7 =__
3
15
+
+
2
6
3x7=
3 x (5 + 2) =
(3 x 5) + (3 x 2)= 15 + 6 = 21
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Grade 3
•3.MD.7. Relate area to the operations of multiplication and addition.
• Multiply side lengths to find areas of rectangles with whole-number
side lengths in the context of solving real world and mathematical
problems, and represent whole-number products as rectangular
areas in mathematical reasoning.
• Use tiling to show in a concrete case that the area of a rectangle with
whole-number side lengths a and b + c is the sum of a × b and a ×
c. Use area models to represent the distributive property in
mathematical reasoning.
• Recognize area as additive. Find areas of rectilinear figures by
decomposing them into non-overlapping rectangles and adding the
areas of the non-overlapping parts, applying this technique to solve
real world problems.
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• Find the area of a rectangle with whole-number side lengths by tiling
it, and show that the area is the same as would be found by
multiplying the side lengths.
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Grade 3
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Represent and solve problems involving multiplication and division (Glossary-Table 2)
Not until 4th Grade
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Connections to other 3rd Grade Standards
9 x 80:
80 is ten 8’s. So, if I know that 8x9 is 72, then I have ten 72’s.
That equals 720.
Or..80 is 8 tens. So, if 10 x 9 = 90, then I know I have 8 of those
(90s). 90 + 90 + 90 + 90 + 90 + 90 + 90 + 90 = (800-80) = 720
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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.
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Sample Activity: Finding Factors
1. With a partner, choose one of the following
numbers:
12, 18, 24, 30, 36, or 48
2. Use equal sets, arrays, or number lines to find
as many multiplication expressions as possible
to represent your number.
3. For each multiplication expression, write its
equivalent addition expression showing your
groupings.
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(Elementary and Middle School Mathematics: Teaching Developmentally by
Van de Walle, Karp, Bay-Williams)
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Sample Activity: Finding Factors from
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Elementary and Middle School Mathematics: Teaching
Developmentally by Van de Walle, Karp, Bay-Williams
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Grade 4
A focus on teaching multiplication (and division)
conceptually
• In Grade 4, instructional time should focus on three critical areas: (1)
developing understanding and fluency with multi-digit multiplication,
and developing understanding of dividing to find quotients involving
multi-digit dividends
• They apply their understanding of models for multiplication (equal-sized
groups, arrays, area models), place value, and properties of operations,
in particular the distributive property, as they develop, discuss, and use
efficient, accurate, and generalizable methods to compute products of
multi-digit whole numbers.
• Depending on the numbers and the context, they select and accurately
apply appropriate methods to estimate or mentally calculate products.
• They develop fluency with efficient procedures for multiplying whole
numbers; understand and explain why the procedures work based on
place value and properties of operations; and use them to solve
problems.
• Students apply their understanding of models for division, place value,
properties of operations, and the relationship of division to
multiplication as they develop, discuss, and use efficient, accurate, and
generalizable procedures to find quotients involving multi-digit
dividends.
• They select and accurately apply appropriate methods to estimate and
mentally calculate quotients, and interpret remainders based upon the
context.
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Grade 4 Introduction
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Selected Standards…
• Multiply a whole number of up to four digits by a onedigit whole number, and multiply two two-digit numbers,
using strategies based on place value and the properties
of operations.
• Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.
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• 4.NBT.5.
• (Area models for this standard are directly linked to the
understanding of partitioning a rectangle into equal parts and
3.MD.7c)
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Area Models
30 + 8
150
40
600
160
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20 + 5
25 x 38= 950
750 + 200 = 950
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Partitioning Strategies for
Multiplication
27 x 4
4 x 20 = 80
4 x 7 = 28
•
•
•
•
267 x 7
7 x 200 = 1400
7 x 60 = 420
7 x 8 = 56
108
27 x 4
27 + 27 + 27 + 27
54
108
----------------------------------1820
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1876
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Invented Strategies– 35 x 12
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Selected Standards…
• Find whole-number quotients and remainders with up to
four-digit dividends and one-digit divisors, using
strategies based on place value, the properties of
operations, and/or the relationship between
multiplication and division.
• Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.
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• 4.NBT.6.
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Selected Standards…
• Solve multistep word problems posed with
whole numbers and having whole-number
answers using the four operations, including
problems in which remainders must be
interpreted.
• Represent these problems using equations with a
letter standing for the unknown quantity.
• Assess the reasonableness of answers using
mental computation and estimation strategies
including rounding.
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• 4.OA.3.
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Grade 5
A focus on teaching multiplication (and division)
conceptually
• (2) extending division to 2-digit divisors, integrating decimal fractions
into the place value system and developing understanding of operations
with decimals to hundredths, and developing fluency with whole
number and decimal operations;
• Students develop understanding of why division procedures work based
on the meaning of base-ten numerals and properties of operations.
• They finalize fluency with multi-digit addition, subtraction,
multiplication, and division.
• They apply their understandings of models for decimals, decimal
notation, and properties of operations to add and subtract decimals to
hundredths. They develop fluency in these computations, and make
reasonable estimates of their results. Students use the relationship
between decimals and fractions, as well as the relationship between
finite decimals and whole numbers (i.e., a finite decimal multiplied by
an appropriate power of 10 is a whole number), to understand and
explain why the procedures for multiplying and dividing finite
decimals make sense. They compute products and quotients of
decimals to hundredths efficiently and accurately.
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Grade 5 Introduction
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Grade 5 Selected Standards
• Explain patterns in the number of zeros of the product when
multiplying a number by powers of 10, and explain patterns in
the placement of the decimal point when a decimal is
multiplied or divided by a power of 10. Use whole-number
exponents to denote powers of 10.
• Fluently multiply multi-digit whole numbers using the
standard algorithm.
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5.NBT.2 and 5.NBT.5
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5.NF.5
Interpret multiplication as scaling (resizing), by:
•a. Comparing the size of a product to the size of one factor on
the basis of the size of the other factor, without performing the
indicated multiplication.
•b. Explaining why multiplying a given number by a fraction
greater than 1 results in a product greater than the given
number (recognizing multiplication by whole numbers greater
than 1 as a familiar case); explaining why multiplying a given
number by a fraction less than 1 results in a product smaller
than the given number; and relating the principle of fraction
equivalence a/b =(n×a)/(n×b) to the effect of multiplying a/b
by 1.
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Gr. 5 Selected Standards
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Scaling
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• Recognize that 3 x (25,421 + 376)
is 3 times larger than
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• 5.NBT.6.
• Find whole-number quotients of whole numbers
with up to four-digit dividends and two-digit
divisors, using strategies based on place value,
the properties of operations, and/or the
relationship between multiplication and division.
Illustrate and explain the calculation by using
equations, rectangular arrays, and/or area
models
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Gr. 5 Selected Standards
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• Rules with no reasons?
• No…ask students to reason.
• Ex: How many grams of fat are there in 7
servings of celery? Celery has 0 grams of fat.
• Ex: Note that on a number line, 5 hops of 0 land
at 0. Also, 0 hops of 20 also stays at 0.
• Arrays with factors of 1 are also worth
investigation to determine the identity property
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Zero and Identity Properties
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• Slice it Up…
• Each pair, please use the grid paper to make a
rectangle that has a total area greater than 10
square units.
• Make a slice through the rectangle and write an
equation that matches using the lengths and
widths of the smaller rectangles created.
• Continue this process until you have found all
the ways to “slice it up.”
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Distributive Property Activity
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14 x 25: An Area Model
5
80
20
200
50
10
*Sketch is not drawn to scale.
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+
+
4
20
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Algebra 1: Multiplying Binomials
4x
20
x2
5x
*Sketch is not drawn to scale.
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5
4
+
x +
x
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Vertical Team Activity
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• As a vertical team, determine the grade level appropriate for
each task given.
• After making a determination, look in the standards and
determine which standard(s) the task is illustrating.
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• I can describe what it means and what it looks like to teach
multiplication (and division) conceptually.
• I can describe how standards progress across grade levels,
giving details for the grade span in which I teach.
• I can match assessment/learning tasks with the appropriate
Common Core Standard, and determine it’s congruency to the
standard.
www.JennyRay.net
Assessing Today’s Targets…
Chart Paper Responses
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How may we continue to support
you?
• NOTE: Information was obtained from the following sources:
• Common Core State Standards for Mathematics
• Elementary and Middle School Mathematics (Van de Walle,
Karp, Bay-Williams)
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• [email protected]
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