Content Session 11
July 15, 2009
Georgia Performance Standards
Instruction and assessment should include the use
of manipulatives and appropriate technology.
Topics should be represented in multiple ways
including concrete/pictorial, verbal/written,
numeric/data-based, graphical, and symbolic.
Concepts should be introduced and used in the
context of real world phenomena.
(emphasis added)
Georgia Performance Standards
Process Standards 5
Students will represent mathematics in
multiple ways.
a. Create and use representations to organize, record,
and communicate mathematical ideas.
b. Select, apply, and translate among mathematical
representations to solve problems.
c. Use representations to model and interpret
physical, social, and mathematical phenomena.
Representations
• Tools for teachers to
–
–
–
–
present problems
explain ideas
demonstrate procedures
etc.
• Tools for students to
– organize, record and communicate ideas
– solve problems
– model various phenomena
Representations include
• concrete/pictorial
• verbal/written
• numeric/data-based
• graphical
• symbolic
Representation “fluency” means:
• students are familiar with a variety of
representations - i.e., students are able
to create and interpret a variety of
representations
• students are able to freely move across
a variety of representations
• students are able to select an
appropriate representation
Visual Representations
• Pictures
• Diagrams
• Graphs
For the school sports festival, two
fourth grade teams are making 40
posters altogether. Team A will
make 8 more posters than Team B.
How many posters will each team
make?
How might you represent this
problem visually?
M2N2
• M2N2. Students will build fluency with
multi-digit addition and subtraction.
b. Understand and use the inverse relation
between addition and subtraction to solve
problems and check solutions.
M1N3
Students will add and subtract numbers less than
100, as well as understand and use the inverse
relationship between addition and subtraction.
h. Solve and create word problems involving addition
and subtraction to 100 without regrouping. Use words,
pictures and concrete models to interpret story
problems and reflect the combining of sets as addition
and taking away or comparing elements of sets as
subtraction.
Can you represent this problem?
• Carrie had some candies. Her friend, Kim,
gave her 8 more candies. Now, Carrie has
14 candies. How many candies did Carrie
have at first?
Can you represent this problem?
• Carrie had some candies. Her friend, Kim,
gave her 8 more candies. Now, Carrie has
14 candies. How many candies did Carrie
have at first?
?
Carrie had
at first
Can you represent this problem?
• Carrie had some candies. Her friend, Kim,
gave her 8 more candies. Now, Carrie has
14 candies. How many candies did Carrie
have at first?
?
Carrie had
at first
8
Kim gave
Can you represent this problem?
• Carrie had some candies. Her friend, Kim,
gave her 8 more candies. Now, Carrie has
14 candies. How many candies did Carrie
have at first?
Carrie now has 14
?
Carrie had
at first
8
Kim gave
Can you represent this problem?
• Carrie had some candies. Her friend, Kim,
gave her 8 more candies. Now, Carrie has
14 candies. How many candies did Carrie
have at first?
Carrie now has 14
?
Carrie had
at first
14 = ? + 8;
8
Kim gave
? = 14 - 8
Can you represent this problem?
• Juan has some marbles. Tony has 8 more
marbles than Juan does. If Tony has 14
marbles, how many marbles does Juan have?
Juan
Tony
?
8
14
multiplication & division
double number lines
• There are 4 apples on each plate. If there
are 6 plates, how many apples are there
altogether?
multiplication & division
double number lines
• There are 4 apples on each plate. If there
are 6 plates, how many apples are there
altogether?
multiplication & division
double number lines
• There are 4 apples on each plate. If there
are 6 plates, how many apples are there
altogether?
• There are 24 children in a classroom and 6
large round tables. How many children
should be seated at each table if there must
be the same number of children at each
table?
• There are 24 children in a classroom and 6
large round tables. How many children
should be seated at each table if there must
be the same number of children at each
table?
• There are 24 students in a class. A
hexagonal table can seat 6 students. How
many hexagonal tables do we need to seat
all students?
• There are 24 students in a class. A
hexagonal table can seat 6 students. How
many hexagonal tables do we need to seat
all students?
Can you represent these problems?
Problem 1 Paul is building a book case. If each shelf
can hold 15 books and there are 5 shelves in the book
case, how many books can be placed on in the book
case?
Problem 2 Willie has a board that is 32 feet long. If he
cuts the board into 4 equal length pieces, how long will
each piece be?
Problem 3 Carlos bought 8 packages of gum. If each
package has 6 pieces of gum, how many pieces of gum
did Carlos buy altogether?
Problem 4 Lynn bought 28 chocolate bars. They were
in packages of 4. How many packages did Lynn buy?
How can you represent this
problem in a diagram?
Book 5B
How about this problem?
A little more challenging problem
A percent problem
Concluding Thoughts
• We can help students develop
representation fluency.
• Developing representation fluency takes
time.
• We must make developing representations
a specific focus of our instruction.