### constructing and deconstructing for algebra readiness

```CONSTRUCTING AND
DECONSTRUCTING FOR
Andrea Tamarazio & Steve Graser , Erie 1 BOCES
April 9, 2013
Algebra Warm Up
Complete the Frayer Model
independently then create a final
table version.
Post the table Frayer Model along
the wall.
Algebra Vocabulary Brainstorm
Does This Help?
Does This Help?
Will This Help?
How Can This Help?
Will This Help?
Nomials
2
3xy
5x - 1
3 x + 5y2 - 3
Expression vs. Equation
2x – 6
2x – 6 = 5
Write Algebraic Expressions for The
Word Phrases
 A number s plus 2
 A number decreased by 1
 31 less than a number
 A number b increased by 7
 The sum of a number and 6
 Scott’s brother is two years
younger than Scott
Write Algebraic Expressions for These
Word Phrases
 A number s plus 2
s+2
 A number decreased by 1
n-1
 31 less than a number
n - 31
 A number b increased by 7
b+7
 The sum of a number and 6
n+6
 Scott’s brother is two years younger than
Scott
s-2
Show Me
 eighteen
 Compare 24 and 15
 One less than thirteen
 Two numbers that make 20
 Three numbers that make thirty-seven
 The sum of nine and twelve
Bar Diagrams / Tape Diagrams
 Part Whole Model
 Comparison
Why Learn Algebra
Learning Algebra
 All students should learn algebra beginning in pre-kindergarten
“By viewing algebra as a strand in the curriculum from pre-kindergarten on,
teachers can help students build a solid foundation of understanding and
experiences as preparation for more sophisticated work in algebra in the
NCTM, 2000
 Primary algebra should not be a watered-down version of the standard
high school course that most of us took; instead the goal in Grades K - 2 is
to develop students’ understanding for the order and situations, predict
beyond the information at hand, and make generalizations.
from Lessons for Algebraic Thinking Grades K- 2 by von Rotz and Burns
Algebra Experience?
 Students in Grades K - 2 have knowledge about and are developing
comfort with number patterns
 Patterns provide a useful foundation on which teachers can develop
students’ algebraic thinking
 Students need experience creating, recognizing, and extending
patterns
 Relating to items in which they are familiar with
 This foundational skill leads to the conceptual understanding of
functions in future years.
from Lessons for Algebraic Thinking Grades K- 2 by von Rotz and Burns
Connecting to CCLS
standards connect to foundational skills within algebra.
On Chart Paper, make a grade level list.
NYSED Recommended Models for
 Area Model / Arrays
 Number Bonds
 Bar Diagrams / Tape Diagrams
Number Towers
Show me how many groups of four make 24.
Write and solve an equation aligned to the prompt
above.
Area Model / Array Example
6
24
Array Model Video
X
Area Model
X
48
Area Model Video
y
Number Bonds
Number Bond Video
Number Bonds
Number Bonds
Bar Diagrams / Tape Diagrams
 Part Whole Model
 Comparison
Bar Diagram Video
Bar Diagram / Tape Diagram # 1
Sara has 5 stamps. Mark brings her 4 more
stamps. How many stamps does Sara have now?
Bar Diagram / Tape Diagram #2
Sara has 16 stamps. Mark brings her 4 more
stamps. How many stamps does Sara have now?
Bar Diagram / Tape Diagram #3
Sara brought 4 apples to school. After Mark
brings her some more apples, she has 9 apples
altogether. How many apples did Mark bring her?
Bar Diagram / Tape Diagram #4
Matteo has 5 toy cars. Josiah has 2 more than
Matteo. How many toy cars do Matteo and Josiah
have altogether?
Bar Modeling – Key Points
 When building proficiency in tape diagraming skills start
with simple accessible situations and add complexities
one at a time.
 Develop habits of mind in students to reflect on the size
of bars relative to one another.
 Part-whole models are more helpful when modeling
situations where you are given information relative to a
whole.
 Compare to models are best when comparing quantities.
Writing Word Problems with Tape
Diagrams
 Tape diagrams are well suited for problems that
provide information relative to the whole or
comparative information of two or more quantities.
 Visual fraction models includes: tape diagrams,
number line diagrams, and area models.
 When designing a word problem that is well
supported by a tape diagram, sketch the diagram for
the problem before or as your write the problem
itself.
Key Points – Tape Diagrams
 Use of tape diagrams, as described in the progressions documents
provides visualization of relationships between quantities thereby
promoting conceptual understanding, provides coherence through
mathematical practice.
 Proficiency in the tape diagram method can be developed in students and
teachers new to the process through a natural development of problems
and representations.
 Content knowledge directed by the standards and the progressions is
required to provide coherent and balanced instruction.
Bar Diagram / Tape Diagram #5
Jasmine had 328 gumballs. Then, she gave 132
gumballs to her friend. How many gumballs does
Jasmine have now?
Bar Diagram / Tape Diagram #6
Jose has 4 paper clips. Harry has twice as many
paper clips as Jose. How many paper clips does
Harry have?
Bar Diagram / Tape Diagram #7
Jose has 4 paper clips. Harry has twice as many
paper clips as Jose. How many paper clips do they
have altogether?
Bar Diagram / Tape Diagram #8
William’s weight is 40 kg. He is 4 times as heavy as
his youngest brother Sean. What is Sean’s weight?
Bar Diagram / Tape Diagram #9
Jamal has 8 more marbles than Thomas. They
have 20 marbles altogether. How many marbles
does Thomas have?
Bar Diagram / Tape Diagram #10
The total weight of a football and 10 tennis balls is
1 kg. If the weight of each tennis ball is 60 g, find
the weight of the football.
Bar Diagram / Tape Diagram #11
Two pears and a pineapple cost \$2. Two pears and
three pineapples cost \$4.50. Find the cost of a
pineapple.
Bar Diagram / Tape Diagram #12
David spent 2/5 of his money on a storybook. The
storybook cost \$20 how much did he have at first?
Bar Diagram / Tape Diagram #13
Alex bought some chairs. One third of them were
red and one fourth of them were blue. The
remaining chairs were yellow. What fraction of
the chairs were yellow?
Bar Diagram / Tape Diagram #14
Jim had 360 stamps. He sold 1/3 of them on
Monday and ¼ of the remainder on Tuesday. How
many stamps did he sell on Tuesday?
Bar Diagram / Tape Diagram #15
Max spent 3/5 of his money in a shop and ¼ of the
remainder in another shop. What fraction of his
money was left? If he had \$90 left, how much did
he have at first?
Bar Diagram / Tape Diagram #16
Henry bought 280 blue and red paper cups. He
used 1/3 of the blue ones and ½ of the red ones at
a party. If he had an equal number of blue cups
and red cups left, how many cups did he use
altogether?
Bar Diagram / Tape Diagram #17
A club had 600 members. 60% of them were
males. When 200 new members joined the club,
the percentage of male members was reduced to
50%. How many of the new members were
males?
Bar Diagram / Tape Diagram #18
The ratio of the length of Tom’s rope to the length
of Jan’s rope was 3:1. The ratio of the length of
Maxwell’s rope to the length of Jan’s rope was 4:1.
If Tom, Maxwell and Jan have 80 feet of rope
altogether, how many feet of rope does Tom
have?
Exit Ticket
Create a model to represent the CCLS aligned
standards that you previously identified at your