### CGI for addition and subtraction - elementary-math

Goals for today
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Share “what works” with each other
Get new resources and do some planning
Go inside other’s classrooms
Think about how to provide intervention support
as needed (“Daily 5”)
3 sessions
2) Multiplication and division
(Mar. 20)
3) Measuring length, geometric
shapes, working with time and
money (Apr. 17)
1) Addition and subtraction fluency builds on a strong
foundation of solving structured problems and using
mental math.
2) Multiplication fluency is rooted in 2nd grade “skip
counting” and 3rd grade mental math.
3) Great resources are available for early numeracy
activities.
4) Diagnostic assessments can help students who are
behind their classmates.
Key Teaching Strategies
1) Teach the underlying structure of word problems to promote
problem solving ability
2) Use visual representations often
3) Verbalize your thought processes when explaining
procedures
4) Have students verbalize their thinking to each other by
explaining what they’re doing and why
5) Provide lots of practice with feedback
6) Use frequent cumulative review
7) Emphasize reasoning, looking for patterns, and problem
solving
Framework
• Cognitively Guided Instruction
• Add/Subtract Situations – Strategies – Fluency
• Tens and Ones – Place Value – Multi-digit Operations
Resources
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Van de Walle and Lovin
Kathy Richardson
Grayson Wheatley
Elementary Math Resources wiki
Georgia and New York lesson plans
Classroom Visits
• Teaching Channel
• Number Talks
Sharing and Planning
• Always share
• Keep track of “What I Want to Try”
• Make and take – Daily 5
CCSS Collaborative Cards
Carla is packing treat bags for Halloween. Each treat bag has to
have exactly the same items in it or the neighborhood kids will
complain! She has 99 Hershey’s Kisses and 108 Jelly Beans.
a) What is the greatest number of treat bags she can make if
she wants to use all the treats she has? Explain.
b) How many of each kind of treat is in one bag?
In your class, how many can…
• Add and subtract single digit numbers fluently?
• Add multi-digit numbers using strategies?
• Solve word problems involving single-digit or multi-digit
numbers?
• Fluently add two- or three-digit numbers without
regrouping?
• Fluently add two- or three-digit numbers with
regrouping?
Purpose of these sessions
• Purpose: Develop professional knowledge –
deep content understanding and ability to use
effective teaching strategies
• Outcome: Responsive teaching, leading to
(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.
(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.
and subtract within 10, e.g., by using objects or drawings
to represent the problem.
K: Fluently add and subtract within 5.
1: Use addition and subtraction within 20 to solve word
problems involving situations of adding to, taking from,
putting together, taking apart, and comparing, with
unknowns in all positions, e.g., by using objects, drawings,
and equations.
1: Add and subtract within 20, demonstrating fluency for
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.
Children’s Mathematics
Cognitively Guided Instruction
Carpenter, Fennema, Franke, Levi and Empson
What operation is this?
• Steven had 4 toy cars. He wanted 9. How many more toy
cars would Steven need to have 9 altogether?
• Show how a kindergarten or 1st grade student might solve
this.
Modeling the Action
does Eliz have left?
• Liz has 3 marbles. How many more marbles does she need
to buy to have 8 marbles?
• Liz has 3 fish. Tom has 8 fish. How many more fish does
Tom have than Liz?
Rachel’s Problems
• Try each of the problems. Think about how students might
model the action in the problem.
• Discuss your solutions with a partner.
• As you watch the video, think about which problems seem
harder for Rachel.
learning of mathematics
• Very young children know how to solve math problems.
• Children develop mathematical understanding and
acquire fluency with whole number computation by
solving a variety of problems in any way that they
choose.
problem solving strategies by watching their classmates
solve problems.
Problem Types
• solve word problems involving situations of adding to,
taking from, putting together, taking apart, and
comparing, with unknowns in all positions
CCSS
CGI
Join
Taking from, taking apart
Separate
Comparing
Compare
Part-Part-Whole
Where is the unknown?
1. Lucy has 8 fish.
more fish. How many
fish would Lucy have
then?
3. Janelle has 7 trolls
in her collection. How
many more does she
11 trolls?
chocolate chip
ate 5 of those
left?
6. 11 children were
playing in the
sandbox. Some
children went home.
There were 3 children
still playing in the
sandbox. How many
children went home?
money. He spent \$9
on a video game.
Now he has \$7 left.
How much money did
Max have to start
with?
Problem Types - Action
Join
Separate
Result
Unknown
Change
Unknown
Start
Unknown
5+2=
8–3=
5+=7
8–=5
+2=7
–3=5
No-action problems
Part-part-whole problems
• 6 boys and 4 girls were playing soccer. How many
children were playing soccer?
• 10 children were playing soccer. 6 were boys and the rest
were girls. How many girls were playing soccer?
10
6
No-action problems
Comparison problems
• Willy has 12 crayons. Lucy has 7 crayons. How many
more crayons does Willy have than Lucy?
Are some more difficult?
• There are 14 hats in the closet. 6 are red and the rest are
green. How many green hats are in the closet?
• 14 birds were in a tree. 6 flew away. How many birds
were left?
Try this twice a week
• Present a problem to the whole class, let them work on it
individually, then have several students present their
approaches.
• For older children, use some two-digit problems that
don’t require regrouping and some that do.
• Keep track of students’ solutions.
Solution Strategies
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Direct modeling of the action in the problem
Counting strategies
Derived facts
Fluency
The structure of a problem determines how difficult it is for
children to solve and determines their initial solution
strategies.
Video of children using counting and derived facts.
Effects of CGI
• Teachers prepared in CGI spend more time having their
students solve problems, listen more to their students,
and are more likely to expect students to find multiple
solution strategies to problems than teachers who are
not prepared in CGI (Carpenter, Fennema, Peterson,
Chiang, & Loef, 1989).
• CGI also results in improved performance by primarygrade students on both standardized and problemsolving tests (Carpenter et al., 1989; Fennema,
Carpenter, & Peterson, 1989; Peterson et al., 1989).
Fluency with “math facts”
• The use of manipulatives, counting and derived-fact
strategies eventually grows into knowledge of most math
facts.
• Explicit instruction on strategies can be helpful for
building math facts that haven’t come naturally through
problem solving.
Explicit strategy instruction
For addition and subtraction, children can derive unknown
facts using a combination of
• anchor sums to 10 (6 + 4, 7 + 3, 8 + 2) – (7 + 6 can be
derived by knowing that 7 + 3 is 10, and 3 more is 13)
• doubles (7 + 6 can be derived by knowing that 7 + 7 is 14,
so the answer is one less)
Number Talks 6+8
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 6+7 = (6+6) + 1 more
plus one
doubles 7+9 = (7+7) + 2 more
plus two
nines
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 Developme
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
Kathy Richardson, Developing Number Concepts Bk. 2, p. 136-137
Drill
Once strategies are learned, students can often be induced
into remembering combinations by playing competitive
games that require quick recall of number combinations,
such as “addition war” or “subtraction war” or games
where pieces are moved around game boards by rolling
dice and calling out sums or differences.
Vary the drill
Print triangle flash cards
Summarize and Plan
• What are the critical elements of 2nd grade math instruction
(and 3rd grade practice and intervention) that support
children’s development of single-digit adding and subtracting?
• Word problems – various types that grow in complexity
• Children’s use of strategies to solve them – developmental
• Number Talks to build mental math and further develop
computation strategies
• Games for practice that reward quick recall
Does your curriculum emphasize these things?
What are your favorite games for developing fluency?
one digit number, and adding a two-digit number and a multiple
of 10, using concrete models or drawings and strategies based
on place value, properties of operations, and/or the relationship
between addition and subtraction; relate the strategy to a
written method and explain the reasoning used. Understand
and ones; and sometimes it is necessary to compose a ten.
2: Fluently add and subtract within 100 using strategies based
on place value, properties of operations, and/or the relationship
2: Add and subtract within 1000, using concrete models or
drawings and strategies based on place value, properties of
operations, and/or the relationship between addition and
subtraction; relate the strategy to a written method. Understand
subtracts hundreds and hundreds, tens and tens, ones and ones;
and sometimes it is necessary to compose or decompose tens or
hundreds.
3: Fluently add and subtract within 1000 using strategies and
algorithms based on place value, properties of operations,
and/or the relationship between addition and subtraction. (A
range of algorithms may be used.)
A classroom view
• Counting Collections to 100
Place value
K.NBT.1 Compose and decompose numbers from 11 to 19 into
ten ones and some further ones, e.g., by using objects or
drawings, and record each composition or decomposition by a
drawing or equation (such as 18 = 10 + 8); understand that these
numbers are composed of ten ones and one, two, three, four,
five, six, seven, eight, or nine ones.
1.NBT.2 Understand that the two digits of a two-digit number
represent amounts of tens and ones. Understand the following
as special cases:
a. 10 can be thought of as a bundle of ten ones — called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one,
two, three, four, five, six, seven, eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one,
two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
2.NBT.1 Understand that the three digits of a three-digit
number represent amounts of hundreds, tens, and ones; e.g.,
706 equals 7 hundreds, 0 tens, and 6 ones. Understand the
following as special cases:
a. 100 can be thought of as a bundle of ten tens — called a
“hundred.”
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900
refer to one, two, three, four, five, six, seven, eight, or nine
hundreds (and 0 tens and 0 ones).
Another classroom
• Making Math Fun with Place Value Games
Multi-digit Progression
1. Mental strategies
2. Place value representations (e.g. base 10 blocks, pictures)
3. Algorithms
Multi-digit Problems
1. Separating, result unknown
Peter had 28 cookies. He ate 13 of them. How many did he
have left? Write this as a number sentence: 28 – 13 = ____
There were 51 geese in the farmer’s field. 28 of the geese
flew away. How many geese were left in the field?
2. Comparing two amounts (height, weight, quantity)
There are 18 girls on a soccer team and 5 boys. How many
more girls are there than boys on the soccer team?
Multi-digit Problems
3. Part-whole where a part is unknown
There are 23 players on a soccer team. 18 are girls and the
rest are boys. How many boys are on the soccer team?
4. Distance between two points on a number line
(difference in age, distance between mileposts)
How far is it on the number line between 27 and 42?
Multi-digit Problems
• There were 51 geese in the farmer’s field. 28 of the
geese flew away. How many geese were left in the field?
• There were 28 girls and 35 boys on the playground at
recess. How many children were there on the playground
at recess?
• Misha has 34 dollars. How many dollars does she have to
earn to have 47 dollars?
• Strategies? Counting single units. Direct modeling with
tens and ones. Invented algorithms: Incrementing by
tens and then ones, Combining tens and ones,
Compensating.
Mental Math
Base Ten Concepts
Using objects grouped by ten:
• There are 10 popsicle sticks in each of these 5 bundles,
and 3 loose popsicle sticks. How many popsicle sticks are
there all together?
• Students’ strategies?
• The extension: The teacher puts out one more bundle of
ten popsicle sticks and asks students “Now how many
popsicle sticks are there all together?” What strategies
would students use to answer this?
Development of Algorithms
• The C-R-A approach is used
to develop meaning for
algorithms.
• Without meaning, students
can’t generalize the
algorithm to more complex
problems.
Practice? Drill?
• What would be good kinds of practice for double digit
work?
• Good kinds of drill?
Typical Learning Problems
• You can differentiate instruction appropriately for your
students by adjusting the numbers in the problems.
• Try this twice a week before our next session.
• Come back with stories to tell!
• You can also create learning centers around these
problems, number strategy games and fluency practice.
When children work at centers during “Math Daily 5” you
can bring a small group together who needs additional
support from you.
• http://elementary-math-resources.wiki.inghamisd.org/2-3
Summarize and Plan
• What are the critical elements of 2nd grade math instruction
(and 3rd grade practice and intervention) that support
children’s development of multi-digit adding and subtracting?
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Word problems – same types, larger numbers
Children’s use of strategies to solve them – developmental
Place value understanding
C-R-A
Does your curriculum emphasize these things?
Do your students have both conceptual understanding and
procedural skill?