1.
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Can
my students identify specific attributes needed for sorting and
classifying things?
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Example Grade K-3:
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Look
at all the keys in this pile.
Which keys can go together? Can you sort them by size?
By color? How
else can you sort them? What’s
another idea? Another
idea?
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VI-2.2
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Example Grade 3-4:
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Choose
a set of numbers and sort them into these categories:
(a) primes; (b) multiples of 3 and 8; (c) an odd number or
a multiple of 3. Can
a number belong to more than one of these categories?
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VI-2.2
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Example Grade 5-6:
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Is
everything on earth either animal, vegetable, or mineral?
How
many species of butterflies are there?
What is it that distinguishes one species from another?
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VI-2.2
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Example Grade 7-8:
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How
would you classify different kinds of numbers such as even, odd,
primes, composites, rationals, negative, factors, multiples?
Can a number belong to more than one of these
classifications?
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VI-2.2
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2.
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Can
my students make precise mathematical statements using words like
all, some, none, every, or, and, many, if … then, if and only
if, necessary and sufficient?
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Example Grade K-3
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a.
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Can
you tell me something that is true about all of the buttons in
this pile? Can you tell me something that is true about some of these
buttons? Can you find
any red buttons in the pile of buttons?
Any small buttons? Are
there any buttons in the pile that are both red and small?
Give me the buttons that are red or small. Put the buttons in the box if they are not red and not
small.
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VI-2.2
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b.
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It
is time to line up to go outside for recess.
You may line up if you are wearing a belt.
Line up if you have buttons on your clothing or if you have
pockets.
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c.
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You
may go to the library if you have your library book at school and
have finished your work.
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Example Grade 3-4:
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Fill
in the blanks with all,
some or none to make
the following statements true for our class.
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VI-2.2
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OK
for 3-4, 5-6
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_________of
the students are boys.
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_________of
the students are either boys or
girls.
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_________have
two eyes.
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_________have
one or two arms.
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_________have
naturally blue hair.
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_________do
not have blond hair.
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_________triangles
have three sides.
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_________of
the factors of 12 are 2, 3, and 4.
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Example Grade 5-6:
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a.
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We
have a rule for our class: If
it is raining at recess, we stay inside; if it is not raining at
recess, we go outside. We
have another rule: If
we want to finish our work, we do not go outside for recess.
On Tuesday, we stayed inside for recess.
Do you know whether it was raining on Tuesday?
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VI-2.2
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b.
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Use
an example to explain the difference between these two situations:
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If
it rains, then I'm going to take my umbrella.
I'm
going to take my umbrella if and only if it rains.
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Example Grade 7-8:
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a.
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A
polygon with six sides is called a hexagon.
Are all hexagons six-sided?
Are all six-sided polygons hexagons?
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II-1.3, VI-2.2
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II-1.3, VI-2.2
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b.
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A
rectangle with equal sides is a square.
Are all rectangles squares?
Are all squares rectangles? Are all equal-sided figures
squares?
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IV-3.3, VI-2.2
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c.
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A
natural number is prime if and only if it has exactly two factors
(which are natural numbers) namely one and itself.
What does that mean? From
this definition do you think that 1 is a prime?
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3.
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Can
my students use a series of logical arguments to reach a valid
conclusion?
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Example Grade K-3:
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Look
at these pictures: a
bat, a bird, a mouse, a giraffe, a butterfly.
I am thinking of one of these creatures.
It can fly. Do
you know for sure which one I am thinking of?
Do you know what it might be?
Do you know for sure what it is not?
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VI-2.5
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Here
are three marbles: two
red ones and one white one. I
hid all three marbles and now I am showing you the white one.
I have another one of the three marbles in my other hand.
Do you know what color it is?
How do you know?
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One
of the bags contain triangles.
Another bag contains circles.
A third bag contains rectangles.
If you reach in one bag and find out what it contains, what
do you know about the other bags?
What don’t you know?
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Example Grade 3-4:
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Consider
these sentences:
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IV-3.3, VI-2.2
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1.
Five and six are even numbers.
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2.
Five is not an even number.
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Is
sentence one true or false? Is
sentence two true or false? Can
a sentence be both true and false?
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Example Grade 5-6:
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You
have your choice of four pets.
You are told that collies eat more than poodles, boxers eat
more than collies, and boxers eat less than dalmatians.
Which pets would eat the least amount of food?
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VI-2.5
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Example Grade 7-8:
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Consider
the following statements:
1.
2, 13, 16 are even numbers.
2.
27, 36, 45 are multiples of nine.
3.
22 = 4, 23 = 16, 24 = 32.
4.
/144
= 12, /49
£
7, /5
<
2.
Why
are some of the above statements false?
Is
sentence one true or false? Is
sentence two true or false?
Can
a sentence be both true and false?
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IV-3.5, VI-2.5
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*Logic
is not a separate category in the
“Michigan Curriculum Framework for Mathematics”, but the
content appears in other strands.
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