Standard 4.1 Number and Numerical Operations

All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.

Big Idea: Numeric reasoning involves fluency and facility with numbers.

4.1 A. Number Sense

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.

Essential Questions

Enduring Understandings

- 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.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

By the end of Grade 4:

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

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.)

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

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

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)

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)

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

7.         Explore settings that give rise to negative numbers.

·        Temperatures below 0^o, debts

·        Extension of the number line

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.

4.1 B. Numerical Operations

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.

Essential Questions

Enduring Understandings

-  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.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

By the end of Grade 4:

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

·        Division: repeated subtraction, sharing

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

3.         Construct, use, and explain procedures for performing whole number calculations and with:

·        Pencil-and-paper

·        Mental math

·        Calculator

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)

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

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)

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.

6.         Count and perform simple computations with money.

·        Standard dollars and cents notation

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?

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.)

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.

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.

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."

4.1 C. Estimation

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.

Essential Questions

Enduring Understandings

-  How can we decide when to use an exact answer and when to use an estimate?

-  Context is critical when using estimation.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

By the end of Grade 4:

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.

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.)

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

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.

Standard 4.2 Geometry and Measurement

All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.

Big Idea Geometry: Spatial sense and geometric relationships are a means to solve problems and make sense of a variety of phenomena.
Big Idea Measurement: Measurement is a tool to quantify a variety of phenomena.

4.2 A. Geometric Properties

Descriptive Statement: This includes identifying, describing and classifying standard geometric object, describing and comparing properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity.

Essential Questions

Enduring Understandings

- How can spatial relationships be described by careful use of geometric language?

- How do geometric relationships help in solving problems and/or make sense of phenomena?

- Geometric properties can be used to construct geometric figures. (4.5D1; 4.5D2; 4.5E3)

- Geometric relationships provide a means to make sense of a variety of phenomena.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

By the end of Grade 4:

1.       Identify and describe spatial relationships of two or more objects in space.

·        Direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?)

·        Relative shapes and sizes

·        Shadows (projections) of everyday objects

2.         Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them.

·        Vertex, edge, face, side, angle

·        3D figures – cube, rectangular prism, sphere, cone, cylinder, and pyramid

·        2D figures – square, rectangle, circle, triangle, quadrilateral, pentagon, hexagon, octagon

·        Inclusive relationships – squares are rectangles, cubes are rectangular prisms

3.        Identify and describe relationships among two-dimensional shapes.

·      Congruence

·      Lines of symmetry

4.         Understand and apply concepts involving lines, angles, and circles.

·        Point, line, line segment, endpoint

·        Parallel, perpendicular

·        Angles – acute, right, obtuse

·        Circles – diameter, radius, center

5.         Recognize, describe, extend, and create space-filling patterns.

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.

4.2 B. Transforming Shapes

Descriptive Statement:  This includes identifying, describing and classifying standard geometric object, describing and comparing properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity.

Essential Questions

Enduring Understandings

- What situations can be analyzed using transformations and symmetries? (4.5E1; 4.5E2; 4.5E3)

- Shape and area can be conserved during mathematical transformations..

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

By the end of Grade 4:

1.        Use simple shapes to cover an area (tessellations).

Suggested Instructional/Assessment Strategies:
• This content provides an opportunity to integrate mathematics with the visual arts. Students can:
• view prints by M.C. Escher and see how tessellations can become a famous art form;
•engage in problem solving as they discover the different ways they can tessellate polygons from pattern blocks or geoblocks;
•tessellate shapes using slides, rotations, and reflections; or
•explore various figures (including, but not limited to, those mentioned in CPIs 4.2.3A2 and 4.2.4A2) as they try to tessellate kites, ovals, parallelograms, rhombi, triangles, pentagons, hexagons, circles, or rectangles.

2.         Describe and use geometric transformations (slide, flip, turn).

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.

3.         Investigate the occurrence of geometry in nature and art.

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.

4.2 C. Coordinate Geometry

Descriptive Statement: Coordinate geometry provides an important connection between geometry and algebra. It facilitates the visualization of algebraic relationships, as well as an analytical understanding of geometry.

Essential Questions

Enduring Understandings

- How can geometric/algebraic relationships best be represented and verified? (4.5C2; 4.5D2; 4.5E1; 4.5E2; 4.5F5)

- Reasoning and/or proof can be used to verify or refute conjectures or theorems in geometry (4.5D1; 4.5D3; 4.5D4; 4.5D5; 4.5F5)

- Coordinate geometry can be used to represent and verify geometric/algebraic relationships.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

By the end of Grade 4:

1.        Locate and name points in the first quadrant on a coordinate grid.

2.         Use coordinates to give or follow directions from one point to another on a map or grid.

4.2 D. Units Of Measurement

Descriptive Statement: Measurement helps describe our world using numbers. An understanding of how we attach numbers to real-world phenomena, familiarity with common measurement units (e.g., inches, liters, and miles per hour), and a practical knowledge of measurement tools and techniques are critical for students' understanding of the world around them.

Essential Questions

Enduring Understandings

- How can measurements be used to solve problems? (4.5A6)

- Everyday objects have a variety of attributes, each of which can be measured in many ways.
 

-What we measure affects how we measure it. (4.5A4; 4.5A6)

- Measurements can be used to describe, compare, and make sense of phenomena.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

By the end of Grade 4:

1.         Understand that everyday objects have a variety of attributes, each of which can be measured in many ways.

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.

2.         Select and use appropriate standard units of measure and measurement tools to solve real-life problems

·        Length – fractions of an inch (1/8, 1/4, 1/2), mile, decimeter, kilometer

·        Area – square inch, square centimeter

·        Volume – cubic inch, cubic centimeter

·        Weight – ounce

·        Capacity – fluid ounce, cup, gallon, milliliter

Sample Assessment Item:
• MC: What is the most reasonable estimate of the length of a city’s swimming pool?
a. 1 meter

* b. 25 meters

c. 1 kilometer

d. 25 kilometers

3.         Develop and use personal referents to approximate standard units of measure (e.g., a common paper clip is about an inch long).

Instructional/Assessment Focus:
• This CPI is largely an instructional CPI. Assessment of this CPI is generally within the context of one or more of the other content CPIs.

Suggested Instructional/Assessment Strategy:
• Students identify parts of their body that are the same length as 10 centimeters and use them to measure the length of their pencil.

4.         Incorporate estimation in measurement activities (e.g., estimate before measuring).

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.

5.         Solve problems involving elapsed time.

4.2 E. Measuring Geometric Objects

Descriptive Statement: This area focuses on applying the knowledge and understandings of units of measurement in order to actually perform measurement. While students will eventually apply formulas, it is important they develop and apply strategies that derive from their understanding of the attributes. In addition to measuring objects directly, students apply indirect measurement skills, using, for example, similar triangles and trigonometry.

Essential Questions

Enduring Understandings

- How can measurements be used to solve problems? (4.5A6)

- Everyday objects have a variety of attributes, each of which can be measured in many ways.

- What we measure affects how we measure it. (4.5A4; 4.5A6)**

- Measurements can be used to describe, compare, and make sense of phenomena.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

By the end of Grade 4

1.         Determine the area of simple two-dimensional shapes on a square grid.