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Big Idea:
Numeric reasoning involves fluency and facility with numbers. |
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4.1.5 A.
Number Sense |
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Descriptive Statement: Number sense is an intuitive feel
for numbers and a common sense approach to using them. It is a
comfort with what numbers represent that comes from investigating
their characteristics and using them in diverse situations. It
involves an understanding of how different types of numbers, such as
fractions and decimals, are related to each other, and how each can
best be used to describe a particular situation. It subsumes the
more traditional category of school mathematics curriculum called
numeration and thus includes the important concepts of place value,
number base, magnitude, and approximation and estimation. |
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Essential Questions |
Enduring Understandings |
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- How do mathematical ideas interconnect and
build on one another to produce a coherent whole? (4.5C1; 4.5C6)
- How can we compare and contrast numbers? (4.5A4)
- How can counting, measuring, or labeling help to make sense of the
world around us? |
- One representation may sometimes be more helpful than another;
and, used together, multiple representations give a fuller
understanding of a problem.
- A quantity can be represented numerically in various ways. Problem
solving depends upon choosing wise ways.
- Numeric fluency includes both the understanding of and the ability
to appropriately use numbers. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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1.
Use real-life experiences, physical materials, and technology to construct
meanings for numbers (unless otherwise noted, all indicators for grade 5 pertain
to these sets of numbers as well): All fractions as part of a whole, as subset
of a set, as a location on a number line, and as divisions of whole numbers; All
decimals. |
It is important to note that the sets of numbers
specified in this CPI also apply to the other grade 5 mathematics
CPIs, including for example 4.1.5A3 and 4.1.5A6 below.
Sample Short Constructed Response (SCR)
Item: Four friends have three brownies left over from a
party. They would like to split them equally. How much should each
of them receive? (Answer: 75% or .75 or 3/4 of a brownie) |
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2. Recognize the decimal nature of
United States currency and compute with money. |
Assessment Focus:
The emphasis in statewide assessment is on the computation.
Sample Multiple Choice (MC)
Item: Debbie has a $5 bill. She wants to purchase a notebook for
75’ and a pen for 50’. How much money will Debbie have left after
purchasing the notebook and the pen?
a. $1.25 b. $2.75 * c. $3.75 d. $4.25
Sample Short Constructed
Response (SCR) Item: Juliette has a $5 bill. She wants to
purchase a notebook for 75’ and a pen for 50’. How much money will
Juliette have left after purchasing the notebook and the pen?
(Answer: $3.75) |
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3.
Demonstrate a sense of the relative magnitudes of numbers. |
Instructional/Assessment Focus:
Refers not only to whole numbers, but also to fractions and
decimals, as specified in 4.1.5A1.
Sample MC Item: If these fractions were graphed on the number
line, which of them would be closest to zero?
a. 3/5
b. 1/4
c.
3/20
* d.
1/10 |
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4.
Use whole numbers, fractions, and decimals to represent equivalent
forms of the same number |
Sample MC Item: Which of the following is
equivalent to 3/4?
a. .25
b. 4/3
c. .85
* d. 9/12
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5.
Develop and apply number theory concepts in problem solving
situations: Primes, factors, multiples. |
Assessment Focus:
The emphasis in statewide assessment is on application.
Sample MC Item: How many numbers between
20 and 50 have no remainder when divided by 6?
a. 3
b. 4
* c. 5
d. 6 |
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6.
Compare and order numbers. |
Instructional/Assessment Focus:
Refers not only to whole numbers, but also to fractions and decimals, as
specified in 4.1.5A1.
Sample SCR Item: State
a number that is between 1/3 and 0.36.
Acceptable answers would
include various representations of Real Numbers between 1/3 and .36
(e.g., 0.34, 0.334, 0.35, 7/20, etc.)
Sample Extended Constructed
Response (ECR) Item: On the number line in your answer folder,
plot points for the following numbers.
4/5, 0.6
Label each point.
Name two different rational numbers that are greater than 0.6 and
less than 4/5. (Write one of your numbers in fractional form and
write the other number in decimal form.)
Explain how you know that each of your numbers is greater than 0.6
and less than 4/5.
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4.1.5 B. Numerical Operations |
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Descriptive Statement: Numerical Operations are an
essential part of the mathematics curriculum, especially in the
elementary grades. Students must be able to select and apply various
computational methods, including mental math, pencil-and-paper
techniques, and the use of calculators. Students must understand how
to add, subtract, multiply, and divide whole numbers, fractions,
decimals, and other kinds of numbers. With the availability of
calculators that perform these operations quickly and accurately,
the instructional emphasis now is on understanding the meanings and
uses of these operations, and on estimation and mental skills,
rather than solely on the development of paper-and-pencil
proficiency. |
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Essential Questions |
Enduring Understandings |
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- What makes a computational strategy both
effective and efficient? (4.5D1)
- How do operations affect numbers?
- How do mathematical representations reflect the needs of society
across cultures? (An essential question with broad applicability
across multiple standards) (4.5C5) |
- Computational fluency includes
understanding not only the meaning, but also the appropriate use of
numerical operations.
- The magnitude of numbers affects the outcome of operations on
them.
- In many cases, there are multiple algorithms for finding a
mathematical solution, and those algorithms are frequently
associated with different cultures. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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1.
Recognize the appropriate use of
each arithmetic operation in problem situations. |
Instructional/Assessment Focus:
The intent is that students not only recognize the appropriate
use of arithmetic operations in the work of others, but that they
also be able to appropriately use those operations themselves. |
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2.
Construct, use, and explain
procedures for performing addition and subtraction with fractions and decimals
with:
·
Pencil-and-paper
·
Mental math
·
Calculator |
"Construct" here means "develop" an algorithm or
process. Sample SCR Item: Paulas
tractor holds 3 liters of gasoline. Toms tractor holds 2.4 liters.
How much more does one tractor hold than the other?
(Answer: 0.6 liters)
Sample ECR Item: Joe had a pizza party.
He ordered 8 pizzas, each cut into 8 slices. When his friends went
home, he had 1/4 of a pepperoni pizza, 5/8 of a mushroom pizza, 1/2
of a cheese pizza, and 1/8 of a veggie pizza left over. How much
pizza was left over in all?
Show one way to get the answer to this problem. Explain your
method.
Show another way to get the answer to this problem. Explain your
method.
Sample SCR Item: A fifth-grade class
will perform an act for the spring talent show. Two-thirds of the
class of 24 students want to perform a skit. The rest of the
students in the class want to sing a song. The teacher decided that
3/4 of the students must agree on an act before the decision will be
final. How many more students would have to choose a skit before 3/4
of the students agree on it?
(Answer: 2 students) |
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3.
Use an efficient
and accurate pencil-and-paper procedure for division of a 3-digit number by a
2-digit number. |
Sample SCR Item: A gallon contains 128
ounces. Paul wants to divide three gallons of apple cider equally
among the two dozen friends at his party. How much apple cider will
each friend receive? (Answer: 16 oz.) |
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4.
Select pencil-and-paper,
mental math, or a calculator as the appropriate computational method in a given
situation depending on the context and numbers. |
Assessment of this CPI is generally within the
context of one or more of the other content CPIs. |
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5.
Check the
reasonableness of results of computations. |
Instructional/Assessment Focus:
Includes
Identifying unreasonable answers obtained using a calculator;
Using inverse operations to check solutions;
Reasoning (4.5D2) and communication (4.5B2);
Solving problems (4.5A2)** involving this recognition; and
Application to all fractions and decimals, as specified in
4.1.5A1.Sample ECR Item: The
fifth grade at Park Middle School is taking a field trip using buses
that hold 36 passengers each.. There are three classes of 25
students each, and 5 adults (teachers or parents) will accompany
each of the three classes.
The Principal wants to order 2 and 1/2 buses;
the Superintendent wants to order 2 buses; and the fifth-grade
teachers want to order 3 buses. Which suggestion is most reasonable
and why? Explain your reasoning. |
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6.
Understand and use
the various relationships among operations and properties of operations. |
"Use" here means "apply." The "properties of
operations" referred to include those specifically listed in
4.3.2D1, 4.3.3D1, or 4.3.4D1 (commutative properties, identity
elements, associative properties, and multiplication or division by
zero). Assessment Focus:
The emphasis in statewide assessment is on the "use" or
"apply," rather than on the "understand." |
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4.1.5 C. Estimation |
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Descriptive Statement: Estimation is a process that is
used constantly by mathematically capable adults, and one that can
be easily mastered by children. It involves an educated guess about
a quantity or an intelligent prediction of the outcome of a
computation. The growing use of calculators makes it more important
than ever that students know when a computed answer is reasonable;
the best way to make that determination is through the use of strong
estimation skills. Equally important is an awareness of the many
situations in which an approximate answer is as good as, or even
preferable to, an exact one. Students can learn to make these
judgments and use mathematics more powerfully as a result. |
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Essential Questions |
Enduring Understandings |
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- How can we decide when to use an exact
answer and when to use an estimate? |
- Context is critical when using
estimation. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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1.
Use a variety of estimation
strategies for both number and computation. |
"Number" here refers to "quantities."
Assessment of this CPI is generally within the
context of one or more of the other content CPIs. |
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2.
Recognize when an
estimate is appropriate, and understand the usefulness of an estimate as
distinct from an exact answer. |
"Understand" implies "explain," consistent with
4.5B1 and 4.5B2. This is an area of
focus in grade 4 and may be assessed at a higher level of
understanding in grade 5. |
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3.
Determine the reasonableness of an answer by estimating the result of
operations. |
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4.
Determine whether a given estimate is an overestimate or an underestimate. |
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