Sit at tables with other 4th or 5th grade
teachers, not necessarily from your own
school.
Did you bring samples of student work on
the assessment?
When You Come In…
Multiplication and division follow a
learning progression across the grades.
3-4-5 Continuum
Starting Point
Middle School
2
1
1
2
2 1
5  8  (5  8)  (5  )  (  8)  (  )
3
4
4
3
3 4
High School
( x  3 )( x  5 )  x  3 x  5 x  15
2
One End Goal
• Purpose – Develop professional knowledge and effective
teaching strategies
• Outcomes – Responsive teaching, leading to greater
readiness for the next grade
• Processes – Collaborative discussion, videos, problem
types, learning progressions, diagnostic assessments,
resources, virtual manipulatives
P.O.P. for this session
• Textbooks like Everyday Math and EnVision are
important instructional resources, but:
• Professional knowledge is essential to know where each
child is on the learning progression and how to help them
move to the next step
• Professional knowledge includes both deep content
knowledge and a broad repertoire of teaching strategies
Professional Knowledge
• Content knowledge: CCSS, Learning Progressions,
Common Error Patterns
• Instructional strategies: IES Practice Guide, ConcreteRepresentational-Abstract (CRA)
Professional Knowledge
1) What must a student know in order to know
multiplication (and division)?
2) How is multiplication of decimals like multiplication of
whole numbers?
3) How is multiplication of fractions like multiplication of
whole numbers?
4) How do we help all students become successful with
multiplication?
Key Questions
Critical Area 1: Developing understanding and fluency
with multi-digit multiplication, and developing
understanding of dividing to find quotients involving multidigit dividends;
• Use the four operations with whole numbers to solve
problems.
• Generalize place value understanding for multi-digit
whole numbers.
• Use place value understanding and properties of
operations to perform multi-digit arithmetic. (SBAC)
th
Common Core for 4 grade
Critical Area 2: Developing an understanding of fraction
equivalence, addition and subtraction of fractions with like
denominators, and multiplication of fractions by whole numbers;
• Extend understanding of fraction equivalence and ordering.
• Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers.
• Understand decimal notation for fractions, and compare
decimal fractions. (SBAC)
Critical Area 3: Understanding that geometric figures can be
analyzed and classified based on their properties, such as having
parallel sides, perpendicular sides, particular angle measures, and
symmetry.
th
Common Core for 4 grade
Critical Area 1: 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;
• Understand the place value system.
• Perform operations with multi-digit whole numbers and
with decimals to hundredths. (SBAC)
th
Common Core for 5 Grade
Critical Area 2: Developing fluency with addition and
subtraction of fractions, and developing understanding of the
multiplication of fractions and of division of fractions in limited
cases (unit fractions divided by whole numbers and whole
numbers divided by unit fractions);
• Use equivalent fractions as a strategy to add and subtract
fractions.
• Apply and extend previous understandings of multiplication
and division to multiply and divide fractions. (SBAC)
Critical Area 3: developing understanding of volume.
• Geometric measurement: understand concepts of volume and
relate volume to multiplication and to addition. (SBAC)
th
Common Core for 5 Grade
• Using the Common Core Standards for 3rd-5th grade,
make a map of the learning progression for multiplication
and division.
• Decide among yourselves what the “end goal” is.
Common Core Standards
• Instruction during the intervention should be explicit and
systematic. This includes providing models of proficient
problem solving, verbalization of thought processes, guided
practice, corrective feedback, and frequent cumulative review.
• Interventions should include instruction on solving word
problems that is based on common underlying structures.
• Intervention materials should include opportunities for
students to work with visual representations of mathematical
ideas and interventionists should be proficient in the use of
visual representations of mathematical ideas.
Key Instructional Strategies
• Interventions at all grade levels should devote about 10
minutes in each session to building fluent retrieval of
basic arithmetic facts.
• Include motivational strategies in tier 2 and tier 3
interventions.
Key Instructional Strategies
• Teachers provide clear models for solving a problem type
using an array of examples.
• Students receive extensive practice in use of newly
learned strategies and skills.
• Students are provided with opportunities to think aloud
(i.e., talk through the decisions they make and the steps
they take).
• Students are provided with extensive feedback.
National Mathematics Advisory Panel, 2008
Explicit Instruction
• Ensure that instructional materials are systematic and
explicit. In particular, they should include numerous clear
models of easy and difficult problems, with
accompanying teacher think-alouds.
• Provide students with opportunities to solve problems in a
group and communicate problem-solving strategies.
• Ensure that instructional materials include cumulative
review in each session.
Explicit Instruction
The NMAP notes that this does not mean that all
mathematics instruction should be explicit.
So what’s the difference?
1) Do you see a pattern? (3.OA.9)
2) Can you make a connection?
3) Can you solve problems independently?
Explicit Instruction
• You need to paint the walls of the cafetorium in your
school. You have to figure out how much paint is needed.
The walls of the room are 24 feet high. You measure each
wall along the bottom, and get these measurements: Wall
1: 53 feet long. Wall 2: 34 feet long, Wall 3: 53 feet long,
Wall 4: 34 feet long. There are also two doors in the gym,
each is 10 feet high by 8 feet wide, and you don’t have to
paint them. How many square feet of paint would you
need for one coat on all the walls?
When?
• Teach students about the structure of various problem
types, how to categorize problems based on structure, and
how to determine appropriate solutions for each problem
type.
• Teach students to recognize the common underlying
structure between familiar and unfamiliar problems and
to transfer known solution methods from familiar to
unfamiliar problems.
Underlying structure
Visual Representations
• Moving from concrete to representational to abstract
• … from objects to pictures to symbols
The C-R-A Approach
Knowledge of the content is interwoven with knowledge of
effective teaching strategies to produce deep learning.
Braided Knowledge
• CCSS
• Learning Progressions
• Common Error Patterns
Content
1. Concepts of multiplication and division need to be
understood through problem situations.
2. Fact family relationships are developed by solving many
problems. Students often develop strategies for quickly
producing a fact.
3. Fluency comes gradually.
4. Number sentences should always be written when solving
problems – even if they can be solved by alternative methods
(e.g. repeated adding) – to connect the symbols to the
operations. Some problems involve “missing factors” such as
5 people will ride in each car. How many cars do we need for
35 people? 5 x ___ = 35.
Basic concepts
Multiplication is
1. Equal groups of… (3.OA.1)
2. So many times larger (or smaller)… (4.OA.1)
Basic problem types
Multiplication is: Equal groups of…
• Megan has 5 bags of cookies. There are 3 cookies in each bag.
How many cookies does Megan have all together?
• Rate problems: I eat 3 bananas a day. How many have I eaten
after 5 days?
• Price problems: One piece of candy costs 5 cents. How much
does 8 pieces cost?
• Array problems: There are 4 rows of desks in a classroom with
8 desks in each row. How many desks are there altogether?
• Combination problems: At the ice cream store they sell 7
flavors of ice cream and 3 kinds of cones. How many different
combinations of one flavor and one kind of cone are there?
Basic problem types
Multiplication is: So many times larger (or smaller)…
• Multiplicative comparison problems: My mom is 6 times older
than I am. I am 7 years old. How old is my mom?
Basic problem types
When students are given these kinds of problems and asked to
work them in any way they can (and given manipulatives), they
generally use these kinds of strategies:
• Skip counting with smaller numbers, then counting on
• Strategies for deriving facts, such as knowing doubles, “double
- doubles” for multiplying by 4, “times ten then 1/2” for
multiplying by 5, etc.
• Some combinations are easier to remember, such as the square
numbers. A strategy based on the square numbers is the “off 1”
strategy: the product of the two numbers “around” another (6
and 4 are “around” 5) is one less than the square of the middle
number.
• Knowing the fact families for products of numbers results in
knowing the division facts.
Foundational teaching
Are there more
steps to this?
Or are some out
of order?
Or does the
order matter?
Or are some
inconsequential?
Learning Progression
for multi-digit numbers
Area Models
Area Models
• National Library of Virtual Manipulatives
• http://nlvm.usu.edu/ 3-5 grade Number and Operations
Rectangle Multiplication
Area Models
Objects
Pictures
Symbols
C-R-A for multiplication
• How would you connect the learning progressions
(content) with the instructional strategies (practices)?
Look back through the powerpoint slides at the content,
look at the instructional strategies on the walls.
• What changes should we see in our instruction?
How do you connect this?
• Looking at the results of the little assessment, can you
spot areas that need attention? Whole class or small
group?
Formative feedback
Intervention resources
National Library of Virtual Manipulatives
http://nlvm.usu.edu/
3-5 Number and Operations – Base Blocks Decimals
3-5 Number and Operations – Percent Grids
How would you modify your C-R-A?
Multiplying decimals
Formative feedback
National Library of Virtual Manipulatives
http://nlvm.usu.edu/
3-5 grade Number and Operations Fractions – Rectangle
Multiplication
How would you modify your C-R-A?
Multiplying fractions
Division is
1. partitive (fair shares) (3.OA.2)
Megan has 15 cookies. She put the cookies into 5 bags
with the same number of cookies in each bag. How
many cookies are in each bag?
2. measurement (goes into)
Megan has 15 cookies. She puts 3 cookies in each bag.
How many bags can she fill?
Basic division concepts
• The remainder is simply left over and not taken into
account (ignored)
• The remainder means an extra is needed
• The remainder is the answer to the problem
• The answer includes a fractional part
• Nat’l Library of Virtual Manip’s: Rectangle division
Remainders…
C-R-A for division
C-R-A for division
How can you show
this procedure
symbolically?
C-R-A for division
• What did you learn from the assessments?
Formative feedback
• Choose something from today’s session (or a couple
things) to try before the next session. Come back with
examples of students’ work from what you tried.
• Next session Dec. 5, here.
Something to try…