Transcript Day 2 - Longwood University

Longwood University Professional Development Seminar
Algebra, Number Sense, and
Mathematical Connections
in Grades 3-5

Blanton, Maria, et. al. Developing Essential

Jacob, Bill, and Catherine Twomey Fosnot.

Understanding of Algebraic Thinking: Grades
3-5. Reston, Va.: NCTM, 2011
The California Frog-Jumping Contest:
Algebra. Portsmouth, NH: Heinemann, 2007
Russell, Susan Jo, Deborah Schifter, & Virginia
Bastable. Connecting Arithmetic to Algebra.
Portsmouth, NH: Heinemann, 2011



Cuevas, Gilbert and Karol Yeatts. Navigating
through Algebra in Grades 3-5. Reston, VA:
NCTM, 2001
Wickett, Maryann, et. al. Lessons for Algebraic
Thinking: Grades 3-5. Sausalito, CA: Math
Solutions, 2002
Von Rotz, Leyani and Marilyn Burns. Lessons
for Algebraic Thinking: Grades K-2. Sausalito,
CA: Math Solutions, 2002

Bamberger, Honi J. and Christine Oberdorf.

2010
Collins, Anne and Linda Dacey. Xs and Whys

Activities to Undo Math Misconceptions,
Grades 3-5. Portsmouth, NH: Heineman,
of Algebra: Key Ideas and Common
Misconceptions. Portland, ME: Stenhouse,
2011
Mirra, Amy. Focus in Grades 3-5: Teaching
with Curriculum Focal Points. Reston, VA:
NCTM, 2008

Subtraction problems that help students think
about what happens when they add two odd
numbers.
There is no other decision that teachers make
that has a greater impact on students’
opportunity to learn and their perceptions about
what mathematics is than the selection or
creation of the tasks with which the teacher
engages students in studying mathematics.
---Glenda Lappan & Diane Briars, 1995
1.
2.
3.
Conceptual Phase
◦ Explores topic with concrete models;
◦ Invents own strategies and solutions.
Connecting Phase
◦ Builds relationships between language,
concrete models, and written symbols and
procedures.
Symbolic Phase
◦ Understands the connections between a
procedure and underlying rationale.
---Baroody,
A.J. with Coslick, R.T. (1998), Fostering Children’s
Mathematical Power: An Investigative Approach to K-8 Mathematics
Instruction, p. 3-8.
A mathematical statement that uses an
equal sign to show that two quantities are
equivalent is called an equation.
---Blanton, et al., 2011, p. 25.

What number might your students place in
the box when asked to solve the following
task?
9+3=+4

The equals sign is a symbol that represents a
relationship of equivalence.
--Blanton et al., 2011, p. 25.

“The repetitive use of arithmetic tasks where
children compute an expression then write
their answers immediately after the = symbol
can build a misconception in their thinking
about what equality means. Many children fail
to see the algebraic role of = as signaling a
relationship between quantities, such as 9 + 3
is equivalent to, or the same as  + 4” (p. 23).
--Blanton, 2008.

Many students marked these equations as
incorrect. Can you guess why?
7 = 5 + 2
7 = 7
4 + 6 = 3 + 7

When asking students to find a sum, instead
of having them express the sum as one
number, ask them to express it as the sum of
two other numbers.
25 + 37 =
+



Make a matching game for students to form
equations with equivalent expressions
Make a concentration game with equivalent
expressions
Is This True? (Bamberger & Oberdorf, 2010,
p. 51)
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
Find as many ways as you can to partition the set
of diamonds and record each pattern using an
equation.
For example, 4 + 4 + 4 + 4 + 9 = 25
What is the value of the left side of your
equation?
What is the value of the right side of your
equation?
How do you know that you have written a correct
equation?
 Cuevas & Yeatts, 2001, pp. 48-50


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
Examine your patterns and the patterns of your partner
and identify two equations that demonstrate a
particular property.
On your paper write each of your original equations.
Then write a new equation which combines the two
equations and demonstrates the property. Which
property does your new equation demonstrate? How
do you know?
Under your new equation include the drawings you
partitioned that match with each side of your new
equation.
Explain how you know your new equation is true.

“Two quantities can relate to each other in
one of three ways:
◦ (1) they can be equal,
◦ (2) one quantity can be larger than the other , or
◦ (3) one quantity can be smaller than the other” (p.
39).
--Blanton et al., 2011


< and >, = and ≠
Greater than, less than, equal to and not
equal to



You and your partner each grab two handfuls
of cubes.
When you grab your cubes put each handful
on a separate plate.
Record the number of cubes in each handful.
Von Rotz & Burns, 2002
pp. 138-156

Each partner then writes an expression for the
number of cubes they grabbed.
◦ (For example: 15 + 22, if you grabbed 15 cubes and then
22 cubes)

Write an appropriate equation or inequality that
represents the relationship between the number
of cubes you grabbed and the number your
partner grabbed.
(e.g., 15+22 >13 + 21)


How do you know your equation or inequality
is true? Can you explain without calculating
the total number of cubes each of you
grabbed?
Can you write another equation or inequality
that also represents the relationship between
the number of cubes you grabbed and the
number your partner grabbed. How do you
know this equation or inequality is true?
 Equations
can be used to
represent problem situations.

--Blanton et al., 2011, p. 30.
“Using
equations to reason about, represent, and
communicate relationships between quantities is a
cornerstone of algebra” (p. 25).
“Writing
equations that represent the situation in
arithmetic problems builds a foundation for writing
equations in algebra” (p. 31).
---Blanton, et al., 2011




Write a story context for 27 + 39.
How does your story illustrate the meaning of
addition?
Solve your story in 2 different ways: use
manipulatives, drawings, mental math, open
number lines, and equations.
Share and discuss your work with a partner.



How can the story explain why
27 + 39 = 26 + 40?
Justify how this equation represents a
relationship of equivalence.
Share and discuss your work with a partner.



Write a story to model 54 – 18.
How can the story explain why
54 – 18 = 56 – 20?
Solve your story in 2 different ways:
use manipulatives, drawings, mental
math, open number lines, and
equations.
Number line representations for 54 – 18.
36
18
36
54
18
54
 Equations
can be reasoned
about in their entirety rather
than as a series of
computations to execute.

--Blanton et al., 2011, p. 26.
Task: Make a conjecture
Are these number sentences true?
2+5=3+4
19 + 6 = 20 + 5
How do you know they are equal?
27 + 34 and 30 + 31
2+5=3+4
19 + 6 = 20 + 5
If you add an amount to one number and
subtract it from the other, the total doesn’t
change.
If you add an amount to one addend and
subtract it from the other, the sum remains the
same.
If a + b = c, then (a + n) + (b – n) = c
(a + n) + (b – n) = (a + b)
Now try the same generalization with
subtraction
9-3 = 8-4
Why doesn’t this work?
Create a few more examples.
Use one of the representations (drawings, cubes, or
number lines) to talk about the conjecture in
general; that is, use the representation, but do not
use the numbers in the specific instances.
Here are a few ways to express the conjecture:
1. If you take away more, you end up with less.
2. If you increase the second number in a
subtraction expression, you decrease the difference
by the same amount.
3. If (a – b) = c, then a – (b + n) = c – n
1.
2.
3.
Use a specific problem and informal
reasoning using the context of the problem.
Make a general statement or conjecture.
Use formal algebraic notation – variables
and equations.
---Blanton, et al., 2011, p. 18
“There
is much, much more to the development of
the ability to solve equations than moving up step by
step”
--- Fosnot & Jacob, 2010, p. 94.
Nonstandard strategies for solving equations are
“
particularly relevant to algebra in grades 3-5 because
they allow students to reason intuitively about an
equation in its entirety”
---Blanton, et al., 2011, p. 28
Video link
Providing
Create
regular routines to set up habits for math explanations.
variations within routines to highlight various aspects of a claim
or to call attention to an unstated assumption.
Giving
students multiple opportunities to clarify for themselves the
ideas they are working to express.
Encouraging
representations such as cubes, diagrams, drawings, and
story contexts to provide tools for expressing ideas. (continued…)
Insisting
Giving
students explain what they mean by ‘it’ or ‘this.’
many students the opportunity to state a claim in their own
words and how they do this: individually or in pairs, orally or in writing.
Refining
language and offering vocabulary as needed.
---Russell, Schifter, & Bastable, (2011), p. 49.
Variables are versatile tools that are
used to describe mathematical ideas in
succinct ways.
---Blanton, et al., 2011, p. 32.


What is a variable?
How would you describe the role played by
the variable t in each of the following:
◦ t + 4 = 3t – 6
◦ y = tx + 2
◦ 3 + (t + 5) = (3 + t) + 5
---Blanton, et al., 2011, p. 34
A variable can represent:
1. a
number in a generalized pattern.
2. a fixed but unknown number.
3. a quantity that varies, especially in relation to
another quantity.
4. a parameter.
5. an arbitrary or abstract placeholder in an
algebraic process.
--Blanton et al., 2011, pp. 32-34.
Make a conjecture that describes why
all these examples are true.
2×6=4×3
5 × 16 = 10 × 8
32 × 50 = 16 × 100
If you double one factor in a
multiplication expression and halve
the other, the product remains the
same.
(a × 2) × (b ÷ 2) = a × b
“Mathematical proofs are important because
they provide insights into the mathematical
relationships that underlie generalizations. By
engaging in proof, students learn not just
that claims are true, but why they are true…
the types of proofs that elementary-aged
students can construct are representationbased [for example: using a number line,
objects, or a story context]”
---Russell, Schifter, & Bastable, (2011), p. 56.
 The meaning of the operation(s) involved in the
conjecture is represented in diagrams,
manipulatives, or story contexts.
 The representation can accommodate a class of
instances (for example, all whole numbers).
 The conclusion of the conjecture follows from
the structure of the representation; that is, the
representation shows why the statement must
be true.
How would you prove
2x6=4x3?
1.What does each argument show that the
student understands about proving the
conjecture?
2. What more would the student need to
do to move toward proving this
conjecture?
I figured out that 2 times 6 equals 4
times 3, and also 8 times 10 equals 4
times 20. So it works.
Argument #2:
I did a story context. I
have 2 stacks of
books, and each one
has 6 books. That’s 12
books. Then I have 4
stacks of books, and
each one only has 3
books. That’s 12, too.
So they’re the same.
Argument #3:
I have 2 stacks of
books, and each one
has 6 books. But the
stacks were too heavy
to carry, so I put each
stack in half. Now there
are 4 stacks and each
has 3 books. So when I
doubled the number of
stacks, there was only
half of the books in a
stack than there was
before.
Argument #4:
See this is a 2 by 6,
and this is a 4 by
3, and they both
have 12.
Argument #5:
I cut the 2 by 6 in half,
and I put one piece
underneath. It’s half
across the top, but now
it’s twice as tall. It’s all
the same stuff I started
with, like if this was a
carpet and I cut it and
moved it around.
1. Draw an unknown amount as one ‘jump’ on a
number line. Label it j.
2. If this is one jump, what does 3j look like?
3. How about one jump and seven steps?
4. What do three jumps and one step backward
look like?

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Day One: Frog Jumping Lesson; Look at Appendices A
& B; Complete Appendix C
‘Jump and step’ on classroom number line
Draw ‘jumps and steps’ on an open number line for
Appendix C activity
4j + 8 = 52
What if j + 7 = 23? How many steps are equal to one
jump? Use an open number line to represent this
equation. Solve for j.
 (Jacob & Fosnot, 2007, pp. 15-20)

“Reasoning with properties of equality and of
number and operations [p.16 properties] to
solve equations with a single variable can
provide a foundation for understanding how
to solve more complex equations”


--Blanton et al., p. 29
4 properties of equality:
◦ Addition property of equality
◦ Subtraction property of equality
◦ Multiplication property of equality
◦ Division property of equality
+
Reading Expressions
Place some of the
Algebra Tiles on the
Basic Mat
1
1
1
1
X
Combine like terms
and read the algebraic
expression.
Answer
X+4
_
National Library of Virtual Manipulatives
http://nlvm.usu.edu/
+
X
1 1
1
1
+
=
1
1
1
1
1
1
1
1
1
1
1
1
_
What equation is modeled on the Equation Mat?
Answer:
_
X + 4 = 12
Model with Algebra Tiles
2X
X+5
5–X
4X – 2X + 3
2(X+3)
Write the expression for
this Cup and Chip model.
=X
=1
Make connections between the symbols and algebra
tiles to model the following:
Model the 3 different balance scales using algebra tiles.
Solve the equations using the tiles and write out
the steps of your actions. NO PENCILS!
(Collins & Dacey, 2011, p. A17)
Teams of three teachers model at least 3 different
equations using algebra tiles, Chips and Cups
model, and balance model. Solve the equations
using the manipulatives, and write out the steps of
your actions.
(Collins & Dacey, 2011, p. A20)

http://www.borenson.com

Make a conjecture that describes how the
perimeter of a square varies with the length of
the side of the square.
1cm


In words: The perimeter of the square is 4 times
the length of the side of the square.
With symbols: p = 4s where p is the perimeter
and s is the length of the side of the square.

“A parameter can be thought of as a
quantity whose value determines the
characteristics or behaviors of other
quantities” (p. 33).
--Blanton et al., 2011
◦ Every week Diondra’s Dad gives her money for
helping with chores around the house. Diondra is
saving her money to buy a bicycle.
 Write an equation that represents the amount of
money Diondra saves (s) if her Dad gives her
d (dollars) in w (weeks).
 How would this equation be different if Diondra’s Dad
gives her $5 for helping with chores each week? How
about $15? How about $20? (Of course she has to do
more chores for more money.)
 If the bike Diondra wants costs $300, what is the
fewest number of weeks Diondra must do chores in
order to buy her new bike? Explain your thinking.
3 + (t + 5) = (3 + t) + 5
t is thought of as an abstract symbol that can be
manipulated. It does not represent a particular number
under a particular circumstance.

3 x 5 + 3 x 6 = 3(5 + 6)
Cherry
Orange
Base 10




12
12
17
21
X
x
x
x
3
14
23
23
and
Algebra Tiles



17 * 23 = (10 + 7) x (20 +3)
= 10 (20 +3) + 7(20+3)
= (10x20) + (10x3) + (7x20)+ (7x3)
=
200 + 30 +
140 +
21
=
391
Model with your base ten blocks to see the four
partial products; build a 17 x 23 rectangle.
Now try 12 x 22: write the equation that shows
the distributive property, build the rectangular
model, draw a sketch, and state the product.

For each of the following expressions use the
distributive property to find an equivalent
expression. Then model with algebra tiles or
Hands-on-Equations, make a sketch, and write an
equation that shows the two expressions are equal.
a) 3(x - 2)
b) 4(x + 3)
1. Draw one jump and 2 steps. What else could it
look like?
2. So how about two jumps and four steps?
3. What about 2(j + 2) on the same double
number line, with the previous problem.
4. What about 3(j + 2)?
5. What about 3j + 6 on the same double
number line?
--- Fosnot and Jacob (2010). Young mathematicians at work:
Constructing algebra. Portsmouth, NH: Heinemann. (p.166)

Use a double open HUMAN number line to
show why the following is an equation.
(3 x 2) + (3 x 5) = 3 (2 + 5)

Use a double open number line to show why
the following is an equation.
(12 x 4) - (12 x 3) = 12 (4 - 3)
“Regardless of what interpretation is
given to a variable, it is important to
develop an appreciation for the
complexity associated with a thorough
understanding of variables” (p. 36).
--Blanton et al., 2011

“The same variable used more than
once in the same equation must
represent identical values in all
instances, but different variables may
represent the same value.” (p.37).
--Blanton et al., 2011
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

Be sure to use the term equation when
working with things that are =
Be sure to use the term inequality when
working with things that are >, <, or ≠
Equations and Inequalities are both types of
mathematical sentences.
8+4=5+7
5=4+1
6X0=6
 Who
would like to explain their thinking
about the first sentence? How about the
second equation?
 How
could we rewrite the third sentence to
make it a true sentence?
---Wickett, et al., 2002, p. 29
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Take the next few minutes to think of at least
one example of a mathematical sentence that
is true and one that is false.
Make two columns on the board, one for true
sentences and one for false sentences. Take
turns sharing your sentences and discussing
why they are true or false.
What about this one… 6 x 3 ÷ 2 = 4 + 5?
---Wickett, et al., 2002, p. 32
5 +  = 13
 What
could we put in the box to make this
statement a true statement?
is called an open sentence because it is
not true or false.
 This
 Is
there anything else we could put in the box
to make it a true statement?
---Wickett, et al., 2002, p. 33
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

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Would 7 x 6 =  be an open sentence?
Can anyone think of a value for the  that
would make the statement true?
Can anyone think of a value for the  that
would make the statement false?
How about 4 +  = 12?
---Wickett, et al., 2002, p. 34



It is an open sentence if whether or not it is
true or false depends on the what is in the
box.
Now think of three open sentences that you
can share with the class.
Share the sentences and decide if they are
open sentences or not. For the open
sentences decide what has to be in the box
for the statement to be true.
What about  - 4 = 3?
I think the following sentence is the same as
the one above but different in another way.
What am I thinking?
What if I wrote  - 4 =3?
What if I wrote x – 4 = 3?
Is this still an open sentence?
Can you use any letter for the variable?
---Wickett, et al., 2002, p. 35
How is this open sentence different from the
others we have been discussing?
 +  = 10
 Is it still an open sentence if it has more than
one box?

What could we put in the boxes to make this a
true statement?

When you use a variable in more than one place
in a sentence, it has to take on the same value.

---Wickett, et al., 2002, p. 36

How could I make this open sentence false
and still use the rule that the box has to
stand for the same value?
 X =16

What could I place in the box to make the
sentence true?
=8
---Wickett, et al., 2002, p. 37



You can have more than one variable. What
values will make this open sentence true?
 +  = 10
The variables are different so they do not
have to be the same value.
It is possible that the different variables have
the same value.
---Wickett, et al., 2002, p. 37

What value for the variable will make this
open sentence true?
(  x 5) + 3 – 20 = 8
---Wickett, et al., 2002, p. 41



Share explanations of how you decided on
the value of each shape.
Can each of the circles have a different value?
How do you know?
Can the circle and the triangle have the same
value? How do you know?

-- (2009) Focus in Grades 3-5 Teaching with Curriculum Focal
Points. Reston, VA: NCTM.

Read:

Do:
◦ Blanton et al. (2011): pp. 25-38
◦ Jacob & Fosnot (2007) Day Two, pp. 21-26;
and Day 6 and Day 7, pp. 45-53
◦ Wickett et al. (2002): Read pages 38 - 42 and focus
on the student work. The students were asked to
write 5 open sentences and tell how to make the
statement true. Steve had incorrect solutions to #3
and #5. What are the correct solutions? Justin has an
error in his first equation. What is it? Choose one of
Tessa’s equations and explain how you know her
equation is true for the given value.