Transcript Addition and Subtraction fluency - elementary-math

Happy Spring!
1) Addition and subtraction (last month)
2) Multiplication and division (today)
3) Measuring length, working
with time and money (Apr. 17)
Agenda
• Your experiences in the last few weeks
• Establishing fluency with addition and subtraction
• The different ways that multiplication is represented and
learned
– A sequential approach
– Number talks and other fluency approaches
– Internet resources for practice and drill
• Connection of multiplication to area of rectangles
• Extensions to division
2nd grade critical areas
(1) extend understanding of base-ten notation;
(2) build fluency with addition and subtraction;
(3) use standard units of measure; and
(4) describe and analyze shapes.
3rd grade critical areas
(1) develop understanding of multiplication and division and
strategies for multiplication and division within 100;
(2) develop understanding of fractions, especially unit fractions
(fractions with numerator 1);
(3) develop understanding of the structure of rectangular
arrays and of area;
(4) describe and analyze two-dimensional shapes.
How do we get to fluency?
Addition and Subtraction fluency
2.OA.2 Fluently add and subtract within 20 using
mental strategies. By end of Grade 2, know from
memory all sums of two one-digit numbers.
Teaching strategy:
1. Lots of real-world problems to make sure students
know the concepts
2. Focus on combinations that aren’t known.
3. Practice in various ways to develop useful and
efficient strategies
4. Drill in ways that reward speed
See wiki and
fluency packet
1. Real-world problems
Joining, separating, comparing, part-whole
• Lucy has 8 fish. She wants to buy 5 more fish. How many fish
would Lucy have then?
• TJ had 13 chocolate chip cookies. At lunch she ate 5 of those
cookies. How many cookies did TJ have left?
• Mark has 3 mice. Joy has 7 mice. Joy has how many more
mice than Mark?
• 10 children were playing soccer. 6 were boys and the rest
were girls. How many girls were playing soccer?
2. Focus on specific combinations
3. Develop useful and efficient
strategies
sums to 5
If you know the sums to 5, like 3+2, you can find other
sums like 3+4, because 3+4 = 3+2 plus 2 more
sums to 10
If you know that 6+4 = 10, then you can figure out 6+5,
because it’s 1 more than 6+4
doubles
plus one
6+7 = (6+6) + 1 more
doubles
plus two
7+9 = (7+7) + 2 more
nines
Add ten then subtract 1
3. Practice
Sum Search
Math Squares
• Always follow the small group work with a whole class
discussion where students explain their methods.
Number Talks 8+6
Fluency – Practice and Drill
• “Practice” refers to lessons that are problembased and that encourage students to develop
flexible and useful strategies that are
personally meaningful.
• “Drill” is repetitive non-problem-based activity
to help children become facile with strategies
they know already in order to internalize
(remember) the fact combinations.
From Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally
4. Drill
http://www.fun4thebrain.com/addition.html
Print triangle flash cards
Addition and Subtraction fluency
2.NBT.5 Fluently add and subtract within 100 using
strategies based on place value, properties of
operations, and/or the relationship between addition
and subtraction.
Teaching strategy:
1. C
2. R
3. A
4. Practice with corrective feedback (I do it, we do it,
you do it – Explain your thinking)
“…using strategies based
on place value, properties
of operations, and/or the
relationship between
addition and subtraction.”
What’s a problem for
which this solution
strategy would be an easy
approach?
• Johanna has 12 fish in her aquarium. Over the
summer, she gets some more fish from the pet
store. Now she has 32 fish. How many more
fish did she get over the summer?
• 12 + ___ = 32
Multiplication and Division
• A silent activity: Write all the things you know about
multiplication.
• Discussion: Go around the table and discuss what
you wrote, one item at a time.
2nd Grade
2.OA.4 Use addition to find the total number of objects
arranged in rectangular arrays with up to 5 rows and up
to 5 columns; write an equation to express the total as
a sum of equal addends.
2.G.2 Partition a rectangle into rows and columns of
same-size squares and count to find the total number
of them.
3rd Grade
3.OA.1 Interpret products of whole numbers, e.g.,
interpret 5 × 7 as the total number of objects in 5
groups of 7 objects each. For example, describe a
context in which a total number of objects can be
expressed as 5 × 7.
3.OA.3 Use multiplication and division within 100 to
solve word problems in situations involving equal
groups, arrays, and measurement quantities, e.g., by
using drawings and equations with a symbol for the
unknown number to represent the problem.
Multiplication – Equal Groups
• Skip counting
Acquisition-Fluency-Generalization
Acquisition
1) modeling word problems
2) skip counting in arrays
3) using visual representations
4) developing the connection to area
Whole-class activities
Developing Number Concepts
Book 3: Place Value,
Multiplication and Division
by Kathy Richardson
Dale Seymour Publications, 1999
$34.95 through Math Perspectives –
Teacher Development Center
Act. 2-23 Lots of Rectangles
The child selects a card, fills the rectangular shape with cubes or
tiles, and writes the multiplication equation that describes the
array that is formed.
Looking for equal groups in the real world
I see six panes in the window by my desk. Are there any other
windows in our room that have six panes? I see two more. Yes,
we have three windows, each with six panes. Six panes and six
panes and six panes.
6’s
8’s
panes in watercolors
each
in each set
window
Six chairs at
each of our
tables
4’s
tires on a
car
legs on a
horse
2’s
eyes
eyebrows
Acting out multiplication stories
1: Using real objects
• 4 chairs around 3 tables, 2 pencils for 4
children, 4 stacks of 3 books, 5 boxes of
crayons with 8 in each box
Acting out multiplication stories
2: Using counters with word problems
There are 4 houses on Letitia’s street. The family in
each house has 2 cars. How many cars is that in all?
• Use your counters to show only the cars. How many cars are
in front of the first house?
• How many cars are in the front of the second house? Show
this with counters.
• Have you shown the cars for all the houses? Show me the cars
in front of the next house.
• You made four groups of two cars. How many cars is that in
all?
Other story problems
• Tim had three dogs. He gave each dog two bones. How many
bones did he give all his dogs?
• Five girls went to the library. They each checked out three
books. How many books did they check out altogether?
• There are five children in Dale’s family. Each child gets to
carve one pumpkin for Halloween. How many jack-o’lanterns will they have?
• Robin’s mother went shopping for school clothes for her
three children. She bought three shirts for each child. How
many shirts did she buy?
Other types of multiplication problems
Rate problems
A baby elephant gains 4 pounds each day…
Price problems
How much would 5 pieces of bubble gum cost
if…?
Combination problems
The Friendly Old Ice Cream Shop has 3 types of
ice cream cones. They also have 4 flavors of ice
cream.
Other types of multiplication problems
Array and Area problems (symmetric problems)
For the second grade play, the chairs have been
put into 4 rows with 6 chairs in each row…
A candy maker has a pan of fudge that
measures 8 inches on one side and 9 inches on
another side. If the fudge is cut into square
pieces 1 inch on each side…?
Building models of multiplication problems
• Towers or stacks
• Rows
• Groups or piles
Build 2 rows of 3. How many altogether?
Build 3 piles of 5. How many?
Build 5 piles of 3. How many?
Build 4 stacks of 2. How many?
Build 2 stacks of 4. How many?
Modeling the recording of multiplication
experiences
Misty stacks books into two piles.
She puts four in each pile.
How many stacks is Misty making? Two.
2 stacks of
How many books in each pile? Four.
2 stacks of 4
How many books altogether? Eight.
2 stacks of 4 = 8
Introducing the multiplication sign
2 stacks of 4 = 8
2x4=8
Read the sign as “stacks of” (not “times”) (or groups of,
rows of, piles of)
Interpreting symbols
Write 4x3. Let’s make rows this time.
How many stacks will you make?
How many in each stack?
Acting out stories to go with
multiplication problems
What story can you tell for this problem?
Learning to write the multiplication sign
• Copying equations
• Writing equations and checking
Acquisition-Fluency-Generalization
Fluency
1) independent and small group activities
2) games
3) developing strategies that build to fluency
3.OA.7 Fluently multiply and divide within 100, using strategies
such as the relationship between multiplication and division
(e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or
properties of operations. By the end of Grade 3, know from
memory all products of one-digit numbers.
3.OA.5 Apply properties of operations as strategies to multiply
and divide. 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.)
(Students need not use formal terms for these properties.)
Doubling and Halving
1 x 16
2x8
4x4
8x2
16 x 1
Number Talks
Independent activities
Developing Number Concepts
Book 3: Place Value,
Multiplication and Division
by Kathy Richardson
Dale Seymour Publications, 1999
$34.95 through Math Perspectives –
Teacher Development Center
How many cups
How many groups?
How many rows?
How many towers?
Interpreting multiplication problems
Copying and solving problems
Making up their own
Problems for partners
Roll and Multiply (Patterns)
Discovering Patterns: Cupfuls
Fluency – two steps
1.Practice – to learn strategies
• The Product Game; Number Talks
• Word problems clustered on 5’s (ORIGO)
2. Drill – to habituate the combinations
• Some element of competition, where quick
recall is important
• Multiplication call-out; Computer games
Number Talks Multiplication String: 7x7
Practice to develop strategies
The numbers in the problems should be built up gradually and deliberately.
While many teachers already focus on one set of “facts” at a time, this
focused approach to story problems helps students see patterns and begin to
develop strategies. Here is a set of “times 5” problems that would be used
together over the course of several days:
• There are bags of 5 apples for sale. If you buy 3 bags, how many apples
will you have?
• It takes 5 minutes to fill a wheelbarrow with soil. How long will it take to
fill 6 wheelbarrows?
• There are 5 rows of 8 chairs. How many people can be seated?
• Nine cats each had 5 kittens. How many kittens is that altogether?
• When Ben places 4 shoes end to end, they measure 1 yard. How many of
these shoes would be lined up to measure 5 yards?
• Jacob wants to plant 7 rows of 5 seeds. How many seeds is that?
Fluency – Practice and Drill
• “Practice” refers to lessons that are problembased and that encourage students to develop
flexible and useful strategies that are
personally meaningful.
• “Drill” is repetitive non-problem-based activity
to help children become facile with strategies
they know already in order to internalize
(remember) the fact combinations.
From Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally
Games to develop recall strategies
http://nlvm.usu.edu
Multiplication Table V.M.
multiplication.com
coolmath-games.com
Primary Krypto @ Illuminations
Recall strategies for multiplication
1) Skip counting (number line, arrays)
2) Known facts plus or minus (6x3)
Number Talks 7x7
3) Double-doubles
Box of Facts
Drill
Most 3rd graders won’t be ready for extensive drill to
habituate the multiplication and division combinations.
They should continue to play practice games. By the
end of third grade, most of the combinations should be
available to all students through strategies or memory.
Some drill might be necessary with some combinations
for some students. Multiplication call-out
and timed games are appropriate.
http://www.helpingwithmath.com/printables/others/fac0201fact_triangle01.htm
Early work with multiplication should mainly be devoted to
conceptual understanding that multiplication is a shorthand
notation for denoting multiple addition. Multiplication
situations should be presented to children, and they should
then use materials (beans, counters, etc.) to demonstrate
the problem given and to generate an answer (a product).
Do not be concerned with habituating the facts too early. It
is essential that children understand that the multiplication
problems expresses a relationship between the numbers
involved and that they own the meaning of the symbolism –
that the first factor in the problem denotes the number of
sets and the second factor denotes the number or objects
contained in each set. The product is then the total number
of objects when the sets are joined.
As children progress in conceptualizing the meaning of
multiplication and can express what the multiplication
situation means, it is time to begin habituating the facts.
We do not like the word “memorizing” because it
communicates that the fact is a paired association, of 2 with
4 in the above example, but the student could be devoid of
an understanding of what the multiplication symbol means.
Instead, after much work with material objects, the
association of 2 with 4 is habituated so the child can
develop speed and accuracy in recalling the fact.
Habituation does not contribute to understanding; it
presumes that understanding precedes the habituation,
and the fact is habituated in order for the child to gain
speed in working with multiplication situations.
Acquisition-Fluency-Generalization
Generalization
1) division (unknown factor problems)
2) solving two-step problems
3) developing the connection to area
4) distributive property
Two-step problems
3.OA.8 Solve two-step word problems using the four
operations. 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.
(This standard is limited to problems posed with whole numbers
and having whole-number answers; students should know how
to perform operations in the conventional order when there are
no parentheses to specify a particular order (Order of
Operations)).
A shape puzzle
• Fill this shape with two different color tiles or
counters, making two rectangles.
• Represent the total number of tiles or
counters with a number sentence that uses
multiplication.
Act. 2-24, with classroom vignette on pp. 142-143
Area standards
3.MD.5 Recognize area as an attribute of plane figures and
understand concepts of area measurement.
a. A square with side length 1 unit, called “a unit square,” is
said to have “one square unit” of area, and can be used to
measure area.
b. A plane figure which can be covered without gaps or
overlaps by n unit squares is said to have an area of n square
units.
3.MD.6 Measure areas by counting unit squares (square cm,
square m, square in, square ft, and improvised units).
Area standards
3.MD.7 Relate area to the operations of multiplication and
addition.
a. 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.
b. Multiply side lengths to find areas of rectangles with wholenumber side lengths in the context of solving real world and
mathematical problems, and represent whole-number
products as rectangular areas in mathematical reasoning.
3
4
Area standards
3.MD.7 Relate area to the operations of
multiplication and addition. (cont’d)
c. 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.
d. 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.
Division
3.OA.2 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. For example, describe a context in which a
number of shares or a number of groups can be
expressed as 56 ÷ 8.
Division by Partitioning
• There are 6 pieces of candy. If two children
share the candy, how many pieces does each
child get?
Also known as “fair shares” division.
Division by Measurement
• There are 6 pieces of candy. Each person is to
get 2 pieces. How many people can get candy?
Also known as “repeated subtraction.”
Write Two Division Stories
• Measurement division – how many in a group
is known
• Partitive division – how many groups is known
20 marbles
Multiplication and division are taught separately in most
traditional programs, with multiplication preceding division. It is
important, however, to combine multiplication and division soon
after multiplication has been introduced in order to help
students see how they are related.
In most curricula, these topics are a main focus of third grade
with continued development in the fourth and fifth grades.
3.OA.4 Determine the unknown whole number in a
multiplication or division equation relating three whole
numbers. For example, determine the unknown number that
makes the equation true in each of the equations 8 × ? = 48,
5 = __÷ 3, 6 × 6 = ?
3.OA.6 Understand division as an unknown-factor problem. For
example, divide 32 ÷ 8 by finding the number that makes 32
when multiplied by 8.
Division Activities
Developing Number Concepts
Book 3: Place Value,
Multiplication and Division
by Kathy Richardson
Dale Seymour Publications, 1999
$34.95 through Math Perspectives –
Teacher Development Center
Division Activities
• Given what you’ve seen about multiplication
activities, what can you imagine for division
activities?
Remainders
• What’s the story that this picture represents?
The remainder is simply left over and not taken into account (ignored)
It takes 3 eggs to make a cake. How many cakes can you make with 17
eggs?
The remainder means an extra is needed
20 people are going to a movie. 6 people can ride in each car. How many
cars are needed to get all 20 people to the movie?
The remainder is the answer to the problem
Ms. Baker has 17 cupcakes. She wants to share them equally among her 3
children so that no one gets more than anyone else. If she gives each child
as many cupcakes as possible, how many cupcakes will be left over for Ms.
Baker to eat?
The answer includes a fractional part
9 cookies are being shared equally among 4 people. How much does each
person get?
So much more than “fact families”
Subtraction and division are the inverses of
addition and multiplication – they are related in
fact families. “Missing addend” and “missing
factor” problems highlight this.
We have 5 bags of candy with the same
amount in each bag. 35 pieces of candy
altogether. How many in each bag?
This can be thought of as 5 x ? = 35.