January 12, 2011
Monica Hartman
A rug is made of very small knots that
each fills one square millimeter.
How many knots will the rug makers have to
tie to make one square meter?
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4.NF.7: Compare two decimals to hundredths
by reasoning about their size.
4.MD.2: Use the four operations to solve
word problems involving distances, intervals
of time, liquid volumes, masses of objects,
and money, including problems that require
expressing measurements given in a larger
unit in terms of a smaller unit.
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5.NBT.1: Recognize that in a multi-digit
number, a digit in one place represents 10
times as much as it represents in the place to
its right and 1/10 of what it represents in the
place to its left. (N.ME.05.08)
5.MD.1: Convert like measurement units
within a given measurement system (e.g.,
convert 5cm to .05m), and use multi-step
real world problems
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Use your fingers, hands, and arms to show:
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mm
cm
dm
m
square
square
square
square
meter
decimeter
centimeter
millimeter
Prerequisite Knowledge
or Use Calculators
10 x 10 = 100
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10 x 100 = 1,000
10 x 1,000 = 10,000
 10 x 10,000 = 100,000
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100 x 100 =
10,000
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100 x 1,000 =
100,000
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1,000 x 1,000 =
1,000,000
Representing and Solving the
Problem
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Cut out a square decimeter on your grid paper.
How should you place the square decimeters to
make a square meter?
Solve the problem in your group. Make sure
everyone in your group understands the problem
and can explain it.
If you are finished ahead of time, solve the problem
in another way.
A rug is made of very small knots that
each fill one square millimeter.
In 1 square centimeter there are
100 knots (10 mm x 10 mm)
How many knots will the rug
makers have to tie to make
one square centimeter?
1 square decimeter
In 1 square decimeter there are 100 x 100
knots or 10,000.
A rug is made of very small knots that
each fill one square millimeter.
1 square decimeter,
with 10,000 knots
How many knots will the rug
makers have to tie to make
one square meter?
1 square meter
In 1 square meter there are 10,000
x 100 knots or 1,000,000.
Another Representation
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How many centimeters are long each side of a
square meter?
100
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How many millimeters are along each side of the
square meter ?
100 x 10 = 1,000
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How many square millimeters are in a square
meter?
1,000 x 1,000 = 1,000,000
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How easy/difficult was the rug problem for
you?
What did you do when you got “stuck”?
Was that strategy successful or not? Why?
Was there something we did in class that
helped you?
What do you think was the “big idea” in this
problem?
Were you able to learn from or contribute to
the learning of others?
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1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning
of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
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Read Lappan article silently in your group.
Get ready to share one idea from the article with
the others in your group.
With your group, brainstorm ways that you can
help your students develop a good mathematical
disposition.
Choose a strategy that you will try for the new year.
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List all the ways that they have seen fractions
developed in your own learning.
List all the uses for fractions in the real world.
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Students have difficulty transferring from
shaded circles to fraction of a length.
Children should explore fractions across all
three models.
1)
2)
3)
4)
5)
The relationship to the whole unit is
immediate.
Size comparisons are easily made in the
ordering of fractions
Common denominators are apparent in
addition and subtraction
Equivalent fractions can be easily generated.
All four of the fundamental operations can
be represented by folding strips.
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Fractions that are compared must be on
strips of equal lengths.
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Loop the strip, placing the ends together, and
crease.
Label each part ½.
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Loop the strip one and on-half times,
bringing the ends opposite one another, and
crease.
Label each part 1/3
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Fold two equal parts.
Fold each of them in half for a crisp fold.
Label each section, fold back and forth.
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Loop the ends all the way
around and to opposite sides
of the loop, where it will be
creased.
The ends need to be at the
creases, so before creasing,
flatten the ring and adjust if
necessary.
Fold back and forth (zig-zag
fold) to assure that all five
sections are of equal length.
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Fold the 1/3s in half or fold the 1/2s into
thirds.
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Fold, crease, and label two more strips to
complete the set.
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4.NF.1: Explain why a fraction a/b is
equivalent to a fraction (n*a)/(n*b) by using
visual fraction models, with attention to how
the number and size of the parts differ even
though the two fractions themselves are the
same size. Use this principle to recognize
and generate equivalent fractions.
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4.NF.2: Compare two fractions with different
numerators and different denominators, e.g.,
by creating common denominators or
numerators, or by comparing to a benchmark
fraction such as ½. Recognize that
comparisons are valid only when the two
fractions refer to the same whole. Record the
results of comparisons with symbols >, =, <,
and justify the conclusions, e.g., by using a
visual fraction model.
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4.NF.3: Understand a fraction a/b with a>1 as a
sum of fractions 1/b.
a) Understand addition and subtraction of fractions as
joining and separating parts referring to the same
whole.
b) Decompose a fraction into a sum of fractions with
the same denominator in more than one way,
recording each decomposition by an equation.
Justify decompositions, e.g., by using a visual
fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8
or 3/8 = 1/8 + 2/8;
2 1/8 = 1 + 1 +1/8 or 2 1/8 = 8/8 + 8/8 + 1/8
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5.NF.1: Add and subtract fractions with unlike
denominators (including mixed numbers) by
replacing given fractions with equivalent
fractions in such a way to produce an
equivalent sum or difference of fractions with
like denominators. For example, 2/3 + 5/4
= 8/12 + 15/12 = 23/12. (in general, a/b +
c/d = (ad + bc)/bd.
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5.NF.2: Solve word problems involving
addition and subtraction of fractions referring
to the same whole, including cases of unlike
denominators, e.g., by using visual fraction
models or equations to represent the
problem. Use benchmark fractions and
number sense of fractions to estimate
mentally and assess the reasonableness of
answers. For example, recognize an incorrect
result 2/5 + 1/2 = 3/7 by observing that 3/7
is less than 1/2.
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Ask students to move the cards around to
learn how they can be arranged to show
decimal numbers.
Have students select the cards that can be
used to show the decimal number 0.268.
Ask students to place the cards on top of
each other to display the numbering standard
form.
Model the same decimal number on the
decimal mat.
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Which decimal number is greater?
0.6 or 0.28
Have students find the Decimal Seret Code
cards for 0.3 and .03 and .003. Using the
cards and the < and> signs inequality signs
ask them to arrange all possible comparisons
of the three decimal numbers. Repeat the
activity using the cards for .8, .08 and .008.
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Find the Decimal Secret Code cards for 0.3
and 0.03 and 0.003.
Use the > and < signs to write all possible
comparisons.
0.3 > 0.03
0.3 > 0.003
0.03 > 0.003
0.03 < 0.3
0.003 < 0.3
0.003 < 0.03
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Find the Decimal Secret Code cards for 0.8,
0.08 and 0.008.
Use the > and < signs to write all possible
comparisons.
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0.8 > 0.08
0.8 > 0.008
0.08 > 0.008
0.08 < 0.8
0.008 < 0.8
0.008 < 0.08
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4.OA.3 Solve multistep word problems posed
with whole numbers and having whole
number answers using the four operations,
including problems in which remainders must
be interpreted. Represent these problems
using equations with a letter standing for the
unknown quantity. Assess the
reasonableness of answers using mental
computation and estimation strategies,
including rounding.
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Amy has 5 times as many stickers as Ben.
Together they have 54 stickers. How many
more stickers does Amy have than Ben?
Go to Thinking Blocks to see how they
represent this problem.
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Continue to explore Thinking Blocks.
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Can these ideas be useful without technology?
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Design a rich open-ended problem that can
be solved in multiple ways.
Give this problem to your students.
Use the problem solving structures we use in
class.
Allow time for student reflection.
Bring samples of students’ work and
reflection next time.