1.
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Can my students use variables
to express relationships between numbers?
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Example Grade K-3:
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How
many different ways can you put ten counters into these two
containers?
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V-2.1, 2.4
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Here
is a number sentence that can represent all the different ways
that you get ten.
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What
number can you put in the box and what number can you put in the
triangle to make the sentence true?
Can you find two other numbers?
Any more?
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As
students give pairs of numbers, the numbers can be listed in a
table and then plotted onto a large coordinate graph.
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Do
you see a pattern? Can
you use the pattern to mark another point that will work?
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Example Grade 3-4:
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Consider
the following statements. Which
are true?
Which are false? Which
are open? Why?
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V-2.1, 2.3
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3
+ 11 = 986
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r
+ 14.29 = Î
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(3
x r)
+ 2 = 14
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r
+ 2 = 7
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2
x 2 ½ = 5
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For
the open sentences, find values that will make them true and
values that will make them false.
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Example Grade 5-6:
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Consider p
+ Î
= 10.
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V-2.1, 2.4
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If
you put 3 1/2 in the p,
what would go in the Î?
Where would the point be on the graph?
What other pairs and points are on the graph?
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What
would the graph look like if we changed the sentence to
p
- Î
= 10?
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Example Grade 7-8:
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Consider
and describe the solution sets for each of the following:
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V-2.1
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3
r
+ r
=
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5
- x = x-5
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x
+ 1 = 3x + 3
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7
– x > 9
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A
+ A = 2 ·
A
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B
+ 4 =
7 + 2B
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2.
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Can my students
maintain the equality relationship between the quantities by
making the same change to both quantities?
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Example Grade K-3:
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a.
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Each
of you has ten raisins. When
you have eaten half of your raisins, will each of you have the
same number?
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V-2.1
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b.
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Linda
has four pebbles in one hand and five pebbles in the other hand.
Paul has two pebbles in one hand and seven in the other
hand. Jane has four
pebbles in one hand and three in the other hand.
Steve has nine pebbles in one hand and none in the other
hand. Which of the
children have the same number of pebbles?
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Example Grade 3-4:
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Tara
and Armando have the same number of peanuts.
Tara has two unopened packages of peanuts which both
contain the same number of peanuts and three loose peanuts.
If Tara eats
one of her peanuts and Armando eats one of his, does Tara still
have the same number of peanuts as Armando?
If Tara gets two more peanuts, what does Armando have to
get so he still has the same number as Tara?
If Armando’s peanuts are doubled, how many more packages
and loose ones does Tara need to get in order to have the same
number as Armando? How
can we use symbols to write down what is happening when we don’t
know how many peanuts Tara has?
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V-2.1
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If
Armando has 35 peanuts, what can we do to both sets of peanuts
(adding more? Taking some away?
Taking half of both?) that will help us find out how many
peanuts there are in one of Tara’s packages?
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Example Grade 5-6:
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Which
of the following is illustrated by going from
Figure
1 to Figure 2
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V-2.2
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a.
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Addition
and subtraction property of equality?
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b.
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Multiplication
and division property of equality?
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Figure 1
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Figure 2
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Example Grade 7-8:
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x
+ 4 = 7 can be transformed into x + 5 = 8 by
adding one to both sides.
Is it possible to transform x + 4 = 7 into each of the
following and if so, how?
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V-2.1, 2.5
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x
- 5 = 2
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5x
+ 20 = 35
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x
- 10 = -7
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½x
+ 2 = 3 ½
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x
= 3
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.25
(x + 4) = 7.25
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2x
+ 4 = 14
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How
many others can you find?
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3.
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Can my students
use the properties of operations on numbers to operate on
variables?
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Example Grade K-3:
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a.
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Get
two counters and put four counters with them.
How many do you have?
Will the number be the same or different if you start with
four and then add two?
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V-2.1, 2.4
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b.
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Get
six counters and take away two.
What number is left? Is
that the same number as you would have if you started with two and
tried to take away six?
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c.
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Show
me three groups of four? Do
you have the same number or a different number when you have four
groups of three?
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d.
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Divide
nine counters into three piles.
How many in each pile?
Can you divide three counters into nine piles?
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Example Grade 3-4:
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Consider
these sentences: 20 ·
0 = 0, 6 ·
0 = 0, 5 ·
0 = 0.
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V-1.3, 2.4
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OK
for 3-4, 5-6
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Which
of the following sentences go with the sentences above?
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6
+ 0 = 6
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7
+ 1 = 8
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16
+ 2 = 2 + 16
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8
+ 1 = 9
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4 ·
0 = 0
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0
+ 7 = 7
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22
+ 0 = 22
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9 ·
0 = 0
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Example Grade 5-6:
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Consider
these sentences: 20 ·
0 = 0, 6 ·
0 = 0, 5 ·
0 = 0.
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V-1.3, 2.4
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OK
for 3-4, 5-6
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Which
of the following sentences go with the sentences above?
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6
+ 0 = 6
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7
+ 1 = 8
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16
+ 2 = 2 + 16
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8
+ 1 = 9
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4 ·
0 = 0
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0
+ 7 = 7
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22
+ 0 = 22
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9 ·
0 = 0
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Why
do they go together? Can
you make up more? Could
you replace all the sentences that go together with one sentence
with a variable in it?
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Example Grade 7-8:
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Decide
if each of the following is true for all numbers.
Then find examples which illustrate whether they are true
or false.
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V-1.3, 2.4
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a
+ 0 = a
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a
+ b = b + a
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a(b
+ c) = ab + c
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a
+ 1 = 1
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a ·
0 = a
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