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Big Idea:
Numeric reasoning involves fluency and facility with numbers. |
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4.1 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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By the end of Grade 4: |
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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 4 pertain to these
sets of numbers as well).
·
Whole numbers
through millions
·
Commonly used
fractions (denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 16) as part of a whole,
as a subset of a set, and as a location on a number line
·
Decimals through
hundredths
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Instructional/Assessment Focus:
• It is important to note that the sets of numbers specified in this
CPI also apply to the other grade 4 mathematics CPIs (e.g., 4.1.4A3
and 4.1.4A6 below).
Sample Assessment Item:
• Extended Constructed Response (ECR): A class of 24 students will
perform an act for the spring talent show. In the class, 2/3 of the
students want to perform a skit. The rest of the students 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 of the students want to perform a skit?
• How many more students would have to choose a skit before 3/4 of
the students agree on it?
• Show all of your work and explain your answer.
(Note: Students may draw a picture in response to this question;
they are not expected to use formal algorithms for working with
fractions at this grade level.)
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2.
Demonstrate an understanding of place value concepts. |
Sample Assessment Item:
Multiple Choice (MC): Using the digits 1 - 7 only once, what is the
largest even number you can make with a 5 in the thousands place?
a. 7,654,321
b.
7,654,312
* c.
7,645,312
d.
7,435,216 |
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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 4.1.4A1.
Sample Assessment Item:
MC: If the following fractions were graphed on a number line, which
fraction would be closest to zero?
a. 2/3
b. 1/4
c. 3/8
d.
1/10 |
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4.
Understand the
various uses of numbers.
·
Counting,
measuring, labeling (e.g., numbers on baseball uniforms), locating (e.g., Room
235 is on the second floor)
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5.
Use concrete and
pictorial models to relate whole numbers, commonly used fractions, and decimals
to each other, and to represent equivalent forms of the same number. |
Sample Assessment Item:
• SCR: How many wholes are there in 16/8? ________
(This item would appear on a non-calculator portion of the statewide
assessment. Answer: two or 2) |
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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 4.1.4A1.
Sample Assessment Item:
MC: Which of the following shows the decimals in order from least to
greatest?
a. 0.5, 0.45, 0.54
* b. 0.45, 0.5, 0.54
c. 0.54, 0.5, 0.45
d. 0.45, 0.54, 0.5
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7.
Explore settings
that give rise to negative numbers.
·
Temperatures
below 0o, debts
·
Extension of the
number line
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Instructional/Assessment Focus:
• Students should have the opportunity to explore settings that give
rise to negative numbers (e.g., temperatures below 0o, debts, games
that involve negative numbers). This would include the use of a
thermometer in science experiments.
• This content should be introduced at this grade level, but mastery
of the content is not assessed in statewide assessment at this grade
level. |
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4.1 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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By the end of Grade 4: |
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1.
Develop the meanings of the four basic arithmetic operations by modeling and
discussing a large variety of problems.
·
Addition and subtraction: joining, separating, comparing
·
Multiplication: repeated addition, area/array
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Division: repeated subtraction, sharing
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2.
Develop proficiency with basic multiplication and division number
facts using a variety of fact strategies (such as “skip counting” and “repeated
subtraction”) and then commit them to memory. |
Sample Assessment Item:
• MC: Mrs. Kinney bought batteries in packs of 4 for the students’
science experiments. Which of these could be the total number of
batteries that she bought?
a. 22
b. 26
* c.
28
d. 30 |
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3.
Construct, use, and explain procedures for performing whole number calculations
and with:
·
Pencil-and-paper
·
Mental math
·
Calculator
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Sample Assessment Items:
• MC: Find the exact value of 568 ÷ 4.
a. 564
* b.
142
c. 140
d. 112
(This item would appear on a non-calculator portion of the statewide
assessment.)
• SCR: Find the exact answer: 568 ÷ 4 = ______
(This item would appear on a non-calculator portion of the statewide
assessment. Answer: 142)
• SCR: Find the exact answer: 4 x 25 x 9 = ______
(This item would appear on a non-calculator portion of the statewide
assessment. Answer: 900)
• SCR: Find the exact answer: 800 - 301 = ______
(This item would appear on a non-calculator portion of the statewide
assessment. Answer: 499)
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4.
Use efficient and accurate pencil-and-paper procedures for
computation with whole numbers.
·
Addition of 3-digit numbers
·
Subtraction of 3-digit numbers
·
Multiplication of 2-digit numbers
·
Division of 3-digit numbers by 1-digit numbers
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Sample Assessment Items:
MC: 20 x 70 =
a. 14
b. 140
* c.
1,400
d.
14,000
(This item would appear on a non-calculator portion of the statewide
assessment.)
MC: 810 - 18 =
a. 828
b. 808
c. 802
* d.
792
(This item would appear on a non-calculator portion of the statewide
assessment.)
MC: 56 X 74
a. 120
b. 130
c.
3,144
* d.
4,144
(This item would appear on a non-calculator portion of the statewide
assessment.)
• SCR: If 942 trading cards are divided equally among 3 students,
how many trading cards would each receive? ______
(This item would appear on a non-calculator portion of the statewide
assessment. Answer: 314 trading cards)
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5.
Construct and use procedures for performing decimal addition and
subtraction. |
Instructional/Assessment Focus:
• This content should be introduced at this grade level, with
decimals through hundredths (as specified in 4.1.4A1), but statewide
assessment of the content is limited at this grade level. Much of
the assessment of this CPI will be within the context of CPI
4.1.4B6. |
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6.
Count and perform simple computations with money.
·
Standard dollars and cents notation
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Sample Assessment Item:
• ECR: Sarah goes to the store to buy some food for an afternoon
snack. She buys a bottle of orange juice for $1.67, a bag of
pretzels for $0.89, and 2 apples for $0.45 each. She must also pay
$0.16 tax.
• How much does Sarah have to pay in all? Show your work.
• What bills and coins would Sarah give to the salesperson to pay
for the food using exact change? |
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7.
Select pencil-and-paper, mental math, or a calculator as the appropriate
computational method in a given situation depending on the context and numbers. |
Sample Assessment Item:
• Sample MC Item: Find the exact answer: 4 x 25 x 9 =
a. 90
b. 100
c. 360
* d.
900
(This item would appear on a non-calculator portion of the statewide
assessment.) |
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8.
Check the reasonableness of results of computations. |
Instructional/Assessment Focus:
• Includes:
o identifying unreasonable answers obtained using a calculator;
o the use of inverse operations to check solutions;
o reasoning (4.5D2) and communication (4.5B2)**;
o solving problems (4.5A2)** involving this recognition; and
• Is most applicable to whole numbers at this grade level, rather
than to fractions or decimals. |
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9.
Use concrete models to explore addition and subtraction with fractions. |
Instructional/Assessment Focus:
• The intent at grade 4 is that students be provided with
opportunities to develop a better conceptual understanding of
fractions and non-algorithmic addition and subtraction using visual
models (either physical or electronic). Formal algorithmic
procedures for adding and subtracting fractions are an area of focus
in grade 5. |
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10.
Understand and use the inverse relationships between addition and subtraction
and between multiplication and division. |
"Use"
here means "apply."
Assessment Focus:
The emphasis in statewide assessment is on the "use" or "apply,"
rather than on the "understand."
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4.1 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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By the end of Grade 4: |
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1.
Judge without counting whether a
set of objects has less than, more than, or the same number of objects as a
reference set. |
Instructional/Assessment Focus:
• This is an area of focus in grade 3 and may be assessed at a
higher level of understanding in grade 4. |
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2.
Construct and use
a variety of estimation strategies (e.g., rounding and mental math) for
estimating both quantities and the results of computations. |
Instructional/Assessment Focus:
• An area of focus in grade 3 for whole-number addition and
subtraction, this CPI is an area of focus in grade 4 for
whole-number multiplication and division and also for addition and
subtraction of decimals.
Sample Assessment Items:
• MC: Estimate 39 X 11. The product is between which numbers?
a. 30 and 80
* b.
300 and 800
c.
3,000 and 8,000
d.
30,000 and 80,000
(This item would appear on a non-calculator portion of the statewide
assessment.)
• MC: Estimate 756 ÷ 8. The quotient is between which numbers?
a. 8 and 10
b. 11
and 13
*c. 80
and 100
d. 110
and 130
(This item would appear on a non-calculator portion of the statewide
assessment.) |
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3. Recognize when
an estimate is appropriate, and understand the usefulness of an estimate as
distinct from an exact answer. |
Instructional/Assessment Focus:
• Assessment of this CPI and demonstration of this understanding is
frequently within the context of one or more of the other content
CPIs.
Sample Assessment Items:
• MC: For which of the following would it generally be better to
calculate the exact answer than to estimate?
a. The number of words in a composition
* b. The number of runs scored by a team in a baseball game
c. The number of steps taken on your way to school
d. The number of miles you traveled on your vacation
• MC: In which of the following situations would it be better to
estimate than to calculate the exact answer?
a. To feed your family hamburgers, you need the number of family
members
* b. To purchase paint for a wall, you want the area of the wall in
square feet
c. To give a customer change, you want the cost of the items
purchased
d. To buy theater tickets, you want the number of people attending
the show
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4.
Use estimation to
determine whether the result of a computation (either by calculator or by hand)
is reasonable. |
Sample Assessment Items:
• ECR: Kelly predicted that each of the 24 fourth grade students in
her class would use 52 sheets of composition paper during the coming
month. Sam told Kelly that 24 x 52 = 2284. Use estimation to explain
if you think Sam is right or wrong and why.
• ECR: John had $4.70 to purchase a binder. However, he found a
cheaper binder in the store for $3.27. Amy told John that $4.70 –
$3.27 = $2.20. Use estimation to explain why you think Amy is right
or wrong.
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