Let’s Get Started…
Directions: Use the whole numbers 1 through 9
only one time each to find the largest (or
smallest) possible values for x.
www.openmiddle.com
1
Learning Targets
• I can model algebraic expressions and
equations using concrete materials.
• I can solve equations using concrete materials.
• I can solve equations and problems using
representations.
2
Concrete-Representational-Abstract
• The C-R-A model is instructional approach in
mathematics.
• It can enhance student performance
• Promote student learning and retention of
conceptual knowledge
• Supports understanding of underlying concepts,
before learning rules and procedures
Concrete Models
Studies have shown that “students who use
concrete materials develop more precise and
more comprehensive mental representations,
often show more motivation and on-task
behavior, understand mathematical ideas, and
better apply these ideas to life situations.”
Hauser , Jane. Concrete-representational-abstract instructional approach. Retrieved April 9, 2009, from the
Access Center: Improving Outcomes for all Students K-8.
Website: http://www.k8accesscenter.org/training_resources/CRA_Instructional_Approach.asp
4
Solving Equations the Concrete Way
Count off 1, 2, 3
• 1’s begin with Cindy
• 2’s begin with Magdalena
• 3’s begin with Stacy
Egg Timer
Using Bar Models to Solve Equations
(Representational)
3x + 7 = 2x + 20
• Bar Model
• Decompose the Equation
• Algebraic Method
6
Using Bar Models to Solve Equations with
Variables on Both Sides (Systems)
What is the value of showing students different
ways to solve equations?
• Bar Model
• Decompose the Equation
• Algebraic Method
7
Representations for Solving Word Problems
• “The goal for model drawing is to build a
pictorial bridge to abstract thinking.”
Forsten, Char . Step-by-Step Model Drawing Solving Problems the Singapore Way. Peterborough: Crystal Springs Books, 2010. Print.
• “Model drawing is wonderful for kids who are
struggling with abstract math, because it literally
gives them a picture of what’s happening in the
problem. Model drawing gives word problems a
visual context.”
Walker, Lorraine. Model Drawing for Challenging Word Problems Finding Solutions the Singapore Way. Peterborough: Crystal Springs
Books, 2010. Print.
8
Model Drawing for Word Problems
If 3/8 of a sum of money is $384, how much is
1/4 of the money?
Model Drawing for Word Problems
2/5 of all the boys from Central School and 3/4
of all the girls in the school attended the
baseball game. If the number of boys at the
game was the same as the number of girls, what
fraction of the school's students attended the
game?
Model Drawing for Word Problems
Cheryl had 1/12 as many books as Brenda. Later
Cheryl bought 18 more books. If Brenda has 72
books, what will be the ratio of Cheryl's books to
Brenda's after Cheryl's purchase?
Model Drawing for Word Problems
The ratio of Mia's balloons to Kiran's was 3:5.
After Mia was handed 21 more balloons she had
twice as many balloons as Kiran. How many
balloons did Mia have initially?
Model Drawing for Word Problems
I think of a number, double it, and add 12. My
answer is 54. What number am I thinking of?
Model Drawing for Word Problems
A pencil costs $2 less than a notebook. A pen
costs 3 times as much as a pencil.
The pen costs $9. What is the total price of a
pen, pencil and notebook?
Model Drawing for Word Problems
Jake takes a board that is 50 inches long and
cuts it into two pieces, one of which is 16 inches
longer than the other. How long is each piece?
Model Drawing for Word Problems
There were 24 boys and 20 girls in a chess club
last year. This year the number of boys increased
by 25% but the number of girls decreased by
10%. Was there an increase or decrease in
overall membership? Find the overall percent
change in membership of the club.
Model Drawing for Word Problems
Sandy was buying new notebooks and pens for
her home office. At one store she bought 2
notebooks and 2 pens for $13.14. At another
store she found exact same products and
pricing; this time she paid $25.08 for 5
notebooks and 2 pens. How much did each
notebook cost?
Model Drawing for Word Problems
There are 73 children at the playground. Onethird of the boys and 2/7 of the girls are
swimming in the pool. If there are 22 kids in the
pool in all, how many boys are there at the
playground?
Model Drawing for Word Problems
The sum of two whole numbers is 28. When one
of the numbers is tripled and added to the other
number, the sum is then 50. What are the two
numbers?
Model Drawing for Word Problems
• Complete the handout.
• For your assigned problem, model with a
drawing.
• Work with your group.
• Discuss your strategies and thoughts as you
work.
20
Model Drawing for Word Problems
• How might the bar model help students?
• How might the bar model challenge students?
• The bar model may be a new concept for
many students, how might you introduce this
strategy to students?
• Think about how you would transition from
the bar model to a numeric or an algebraic
model (representation to abstract).
21
Model Drawing for Word Problems
• Online Tool for Creating Bar Models
• http://www.mathplayground.com/thinkingblo
cks.html
22
Day 1 Learning Targets
• I can model algebraic expressions and
equations using concrete materials.
• I can solve equations using concrete materials.
• I can solve equations and problems using
representations.
23