Transcript Blue Swirls - Diane Culbertson

Day One Training
Objectives
• Develop strategies aimed at increasing
understanding of basic facts
• Facilitate the interpretations of multiplication
situations
• Use models to represent multiplication
• Develop effective estimation strategies for
multiplication
• Identify misconceptions associated
with multiplication
Using only the numbers 1,2,3,4,and 5 once
for each problem, write the number sentence
which would give you the greatest result.
+
=
-
=
x
=
÷
=
What understanding would our
students need to successfully
complete these problems?
• How many of our students understand
we can use the inverse operation to
solve each problem?
• Is this the same for
addition/subtraction?
Relating it to the Standards
Third Grade: Cluster Heading
• Represent and solve problems involving multiplication and division
• Understand properties of multiplication and the relationship between
multiplication and division.
• Multiply and divide within 100.
Fourth Grade: Cluster Heading
• Use the four operations with whole numbers to solve problems.
• Generate and analyze patterns
• Gain familiarity with factors and multiples.
Fifth Grade: Cluster Heading
• Write and interpret numerical expressions.
• Perform operations with multi-digit whole numbers.
• Apply and extend previous understandings of multiplication and division.
Throwing Out The Old Ideas
• In order for students to understand
math, they must understand the rules of
math.
• If students do not understand addition
and subtraction, they will not be able to
understand multiplication.
• If a student can not multiply, they can
not divide.
Developing Operation Number
Sense
In order for students to
effectively use multiplication
and division when dealing with
real world settings-they must
be able to connect the
meanings of multiplication and
division to each other.
Van de Walle, John. Elementary and middle school mathematics: teaching developmentally
Teaching Student-Centered Mathematics Series Pearson /Allyn and Bacon, 2007
Misconception or Error?
http://www.meridianschools.org/Staff/Di
strictCurriculum/MoreResources/Math/A
ll%20Grades/Misconceptions_Error%20
2[1].pdf
Creating the
Misconceptions.
The mathematical ideas and procedures
(or rules) a student learns may be correct
or they may be full of misconceptions.
However, we need to understand the
process of learning these procedures is
basically the same.
3-3-2 Method
CRA supports understanding underlying mathematical
concepts before learning “rules,” that is, moving from a
concrete model of chips or blocks for multiplication to an
abstract representation such as 4 x 3 = 12.
– Concrete. The “doing” stage using concrete
objects to model problems
– Representational. The “seeing” stage using
representations of the objects to model problems
– Abstract. The “symbolic” stage using abstract
symbols to model problems
Is it always about the RIGHT
answer?
The techniques students use to solve
problems will provide you with insight into
their understanding.
YOUR TURN
There are the same number of counters in each
row and in each column. How many counters are
covered by the paper?
Assessment Tools http://nzmaths.co.nz/
Student A
The student counts each counter in
the top row of the picture, then each
counter another four times. His
answer: 29.
Student B
The student counts each counter in
the top row and says (6). He then
counts the number in the column (4)
then multiplies orally to get 24.
Student C
The student nods his head at each of
the counters in the top row, then
counts the number in the column. He
almost instantly responds, 30.
Student A;
The student counts each counter in the top row of the
picture, then each counter another four times. His
answer: 29.
Student B:
The student counts each counter in the top row and
says (6). He then counts the number in the column (4)
then multiplies orally to get 24.
Student C:
The student nods his head at each of the counters in
the top row, then counts the number in the column. He
almost instantly responds, 30.
Student A;
The student counts each counter in the top row of
the picture, then each counter another four times.
His answer: 29.
Knowledge: Appears to understand
repeated addition, some evidence of arrays,
but dealing with ones not groups. Student may
not have 1:1 correspondence, may not have
the skill of number/counting sequence.
Strategy: He attempts to count all of the
counters, so he still relies on concrete models.
He also miscounts to get 29.
What kind of strategies could you use to help
develop his understanding?
1. Questions to identify errors and
misconceptions?
 Display five rows of two. How many rows are there? How many
dots are there in each? How can you describe the array?
 Turn the array 90 degrees. How has the array changed? What is
the same and what is different?
2.
What kind of activity might you use to
address this error?
Student B:
The student counts each counter in the top row and
says (6). He then counts the number in the column
(4) then multiplies orally to get 24.
Knowledge: Appears to understand the array concept
for multiplication. He also knows his number facts.
Strategy: Counts and uses number facts orally. Has
difficulty with the 1:1 correspondence of the dots
covered.
Do you think student B has a good
understanding of the array? How might you go
about making sure?
Describe this array to me. Write a number
sentence to match the array.
Make up a number sentence that matches
another array and draw the array.
Student C:
The student nods his head at each of the counters
in the top row, then counts the number in the
column. He almost instantly responds, 30.
Knowledge: Demonstrates understanding of number
facts and arrays. Appears to have 1:1
correspondence.
Strategy: Counts along the top edge and then down
the side. Developing mental computation strategies.
Students A, B and C are all the same age and in
the third grade. Each of the students have had
the same teachers for math instruction since they
started school. So why is their math knowledge
so different?
Is it always about the RIGHT
answer?
Rule 1:
The teacher is always right.
Rule 2:
There is only one right answer.
Rule 3:
See rule number one.
A class of 25 students is going
on a field trip to a science
museum. They plan to visit the
exhibits in small groups of 2, 3,
or 4 students along with a
chaperone. What is the fewest
number of chaperones they will
need?
Common Problem Solving
Strategies
 Guess and Check
 Model it
 Draw it (this includes drawing pictures or
diagrams)
 Make a List or Table
 Think (this includes using skills you know
already)
Engaging?
While playing “Game
Hunter”, I found out it
takes 7 stars to
become invisible. The
directions say you
have to become
invisible 12 times
before gaining
immunity. How many
stars will I have to
capture for this level?
Maria purchased
five books with 12
stamps in each.
How many stamps
did she buy?
Life is one big Word Problem!
Vonda Stamm
Contextual or Real World Problems allow…
 The student to develop meanings for the
operations based on the problem.
 The teacher to anticipate the modeling a
student may use in solving the problem.
 Students to “connect” to the subject
matter.
Steps for designing effective
learning problems.
1. The problem must begin on the
student’s level.
2. Must be centered on what you want
students to learn.
3. The problem must require justifications
and explanations for answers and
methods.
Van de Walle, John. Elementary and middle school mathematics: teaching developmentally
Teaching Student-Centered Mathematics Series Pearson /Allyn and Bacon, 2007
This is Your PROBLEM!
1. Choose a sealed envelope. Open to
reveal a number.
2. You and your partner will make up a
word problem. Your answer will be the
number you found in the envelope.
3. The problem will contain one solution.
4. Problems should involve multiplication
and/or division.
Idea adapted from “The Problem” Figuring it Out, level 4.
http:NZMATH.com
Time for a BREAK!
Capability
Misunderstandings
• Some people are just born with a math
gene
• Males are better in math than females
• Special education students can not do
math, nor do they need to.
CAN ALL STUDENTS DO MATH? YES!.
Let’s Do a Foldable!
Multiplication,
Factors, Product,
Array, Division,
Quotient,
Dividend, Divisor,
Remainder, Equal,
Perform operations with multi-digit whole
numbers and with decimals to
hundredths.
5. Fluently multiply multi-digit whole numbers using
the standard algorithm.
Addressing the BIG IDEAS
1. Develop the relationship between the base
ten number system and multiplicative
structure.
2. Model and construct multiplication strategies.
3. Demonstrate strategies of composing and
decomposing numbers.
4. Check for continual concept understanding.
National Council of Teachers of Mathematics. (2000). Principles and
standards for school mathematics. Reston, VA: The Council, p.9
I learned my facts by memorizing
them…why can’t they?
Children who memorize facts without
understanding them often are not sure what
they know or how to use it.
Drill is good for students who understand
how to use math but just need additional
practice.
If drill worked for all children, we wouldn’t
have students struggling.
Multiplication Facts
Are they needed?
YES
Do students need to know all of
their facts to be successful?
NO
Strategies for Multiplication
Facts
“Multiplication facts can and
should be mastered by
relating new facts to existing
knowledge”
J. Van De Walle
Van de Walle, J.A., & Lovin, L.H. (2006)
Teaching Student Centered Mathematics
Volume II (3-5), Boston: Pearson.
Critical Multiplication and Division
Facts
•Zero’s
•One’s
•Two’s
•Five’s
•Ten’s
Doubles
• Facts that have 2 as a factor are
equivalent to the addition doubles.
Students that know their addition
doubles, know these facts.
• Our job is to help them see this
relationship
2x7=7+7
7x2=7+7
Doubles
x
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
10 12 14 16 18
3
0
3
6
9
12 15 18 21 24 27
4
0
4
8
12 16 20 24 28 32 36
5
0
5
10 15 20 25 30 35 40 45
6
0
6
12 18 24 30 36 42 48 54
7
0
7
14 21 28 35 42 49 56 63
8
0
8
16 24 32 40 48 56 64 72
9
0
9
18 27 36 45 54 63 72 81
Fives Facts
• Skip Counting by fives
• Connect counting by fives
to rows of 5 dots
• Use the clock minute
hand
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5
10
15
20
25
30
So far ….36 facts
x
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
10 12 14 16 18
3
0
3
6
9
12 15 18 21 24 27
4
0
4
8
12 16 20 24 28 32 36
5
0
5
10 15 20 25 30 35 40 45
6
0
6
12 18 24 30 36 42 48 54
7
0
7
14 21 28 35 42 49 56 63
8
0
8
16 24 32 40 48 56 64 72
9
0
9
18 27 36 45 54 63 72 81
Zeros and Ones
• 36 facts have at least one factor of
0 or 1.
• Students sometimes confuse them
with addition facts that a 0 or 1.
• Use Learning Problems!!!
Van de Walle, J.A., & Lovin, L.H. (2006)
Teaching Student Centered Mathematics
Volume II (3-5), Boston: Pearson.
After twos, fives, zeros, and ones
x
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
10 12 14 16 18
3
0
3
6
9
12 15 18 21 24 27
4
0
4
8
12 16 20 24 28 32 36
5
0
5
10 15 20 25 30 35 40 45
6
0
6
12 18 24 30 36 42 48 54
7
0
7
14 21 28 35 42 49 56 63
8
0
8
16 24 32 40 48 56 64 72
9
0
9
18 27 36 45 54 63 72 81
Naughty Nines
It is all about the patterns!
After twos, fives, zeros, ones and nines
x
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
10 12 14 16 18
3
0
3
6
9
12 15 18 21 24 27
4
0
4
8
12 16 20 24 28 32 36
5
0
5
10 15 20 25 30 35 40 45
6
0
6
12 18 24 30 36 42 48 54
7
0
7
14 21 28 35 42 49 56 63
8
0
8
16 24 32 40 48 56 64 72
9
0
9
18 27 36 45 54 63 72 81
Only 15
facts left!
Student Work Examples
It’s that time….
Multiplication Sort
1. Lay out the 8 cards that have the multiplication
problems so that each problem can be seen.
2. Shuffle the remaining cards and turn them face
down in a pile.
3. Take turns turning over a card and matching it with
the multiplication problem that it represents.
 Additional games for use could include matching,
rummy, spoons, etc.
Vonda Stamm, Making Math Magic LLC
Three spiders on a web each
have 8 legs. How many legs
are there altogether?
3 + 8 = 11
So Where are the Problems?
• Making connections to real life often
challenges students since we use
“times” to describe multiplication.
• Seeing multiplication as “groups of
groups”.
• Not allowing students to model what
they see or think when addressing a
problem.
Let’s start with some
multiplication vocabulary.
• Numbers that we multiply together are
called factors.
3 x 7 = 21
factors
What are Factors?
Numeracy Professional Development Projects. Teaching Multiplication and Division. New
Zealand: Ministry of Education 2008, p. 14.
Let’s start with some
multiplication vocabulary.
• Numbers that we multiply together are
called factors.
• The answer in a multiplication is called
the product.
3 x 7 = 21
factors
product
Let’s look at the two main
types of problems…
• Equal groups: involves combining sets of equal size.
• Multiplicative Comparison: involves a comparison between
two quantities in which one is described as a multiple of the
other. Students must understand expressions such as “3
times as many” in order to be successful in this type of
problem.
Assessment Questions
Equal Groups and Repeated
Addition
How many hats are there?
5 + 5 + 5 = 15
or
3 x 5 = 15
How many fish?
4 + 4 + 4 = 12
or
3 x 4 = 12
Question Activity
7 x 8 = 56
The Properties
6 + 7 + 4=
The commutative property of
multiplication simply means it does not
matter which number is first when you
write the problem.
3x5=5x3
The numbers can be switched around and
the answer is the same.
Associative Property
Changing the grouping of the factors does not change
the product.
Understand properties of
multiplication and the relationship
between multiplication and division.
3.OA.5: Apply properties of operations as strategies to multiply and
divide. (Note: Students need not use formal terms for these properties.)
Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known.
(Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5
= 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative
property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one
can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56.
(Distributive property.)
1. Cut out each
of the
descriptions and
place in your
foldable under
the tab of the
correct property
name.
2. Write a
numerical
example the
illustrates the
property.
Your turn
1.6 + 7 + 4 =
2.19 + 33 + 81 =
3.(6 x 4) x 5 =
Let’s Take a Break
What happens when students
can’t see how the problems 8 x
5 and 2 x 8 are related!
• Use your grid paper to draw and color a
5 by 8 rectangle; write the
multiplication problem 5 x 8 on the
grids; then cut it out.
• Now draw and color (using a different
color) a 2 by 8 rectangle; write 2 x 8 on
the grids and cut out this rectangle.
• Your next job is to tape the two
rectangles together to form a new
rectangle.
5x8
What are the
dimensions of the
new rectangle?
2x8
What multiplication
fact did we model?
Vonda Stamm, Making Math Magic, LLC
Composing Numbers
If I build 5 x 6 and 2 x 6, what new rectangle
can I make?
If I build a 7 x 2 and a 7 x 1, what new
rectangle can I make?
If I build a 5 x 4, a 2 x 4, and a 1 x 4, what
new rectangle can I make?
Spinning for a Winner
“Students with number sense
naturally DECOMPOSE, it is a
natural way to find a solution to any
type of problem” NCTM
1.What does it mean to decompose
numbers?
2.Why is it important for students to
be able to decompose?
http://standards.nctm.org/document/chapter3/numb.htm
A student in your class is struggling with
multiplication. You write the following on
the board for review:
7x8=
What can you do to help this student
become successful?
Decomposing Using the
Factor 7 Strategy
Multiplying 7 = Multiplying by 5 +
Multiplying by 2.
7x8=?
(5 x 8) and (2 x 8)
40 + 16 = 56
Your Turn
1.
2.
3.
4.
7x6=
3x8=
12 x 13 =
19 x 6 =
Distributive Property
The sum of two numbers times a third number is
equal to the sum of each addend times the third
number.
3(2 + 7 – 5)=
3(9-5)
= 3(2) + 3(7) + (3)(-5)
3(4)
=
12
= 12
6
+ 21
- 15
Distributive Property
When using the distributive property, you are
distributing something as you separate or break it
into parts.
The distributive property makes numbers easier to
work with.
http://math.springbranchisd.com/
2 Groups of 3 Yellow and 2
Red
2 (3y + 4r)
2 (3y) + 2 (4r)
6y + 8r