# Courses of Study : Mathematics

Operations and Algebraic Thinking
Represent and solve problems involving multiplication and division.
 Mathematics (2016) Grade(s): 3 All Resources: 2 Learning Activities: 2
1 ) Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. [3-OA1]

Example: Describe a context in which a total number of objects can be expressed as 5 x 7.

NAEP Framework
NAEP Statement::
4NPO3e: Interpret whole number operations and the relationships between them.

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.1- Find the sum of equal groups of objects using repeated addition (sums up to 30).

 Mathematics (2016) Grade(s): 3 All Resources: 1 Learning Activities: 1
2 ) Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. [3-OA2]

Example: Describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

NAEP Framework
NAEP Statement::
4NPO3e: Interpret whole number operations and the relationships between them.

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.2- Divide a group of items into smaller equal groups (limit given group to fifteen items or less; limit equal groups to two, three, four, five, ten).

 Mathematics (2016) Grade(s): 3 All Resources: 2 Learning Activities: 2
3 ) Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Appendix A, Table 2.) [3-OA3]

NAEP Framework
NAEP Statement::
4A3a: Use letters and symbols to represent an unknown quantity in a simple mathematical expression.

NAEP Statement::
4NPO3f: Solve application problems involving numbers and operations.

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

 Mathematics (2016) Grade(s): 3 All Resources: 1 Learning Activities: 1
4 ) Determine the unknown whole number in a multiplication or division equation relating three whole numbers. [3-OA4] NAEP Framework
NAEP Statement::
4A4a: Find the value of the unknown in a whole number sentence.

NAEP Statement::
4NPO5b: Identify factors of whole numbers.

NAEP Statement::
4NPO5e: Apply basic properties of operations.

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

Understand properties of multiplication and the relationship between multiplication and division.
 Mathematics (2016) Grade(s): 3 All Resources: 0
5 ) Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) [3-OA5]

Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication)

3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication)

Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property)

NAEP Framework
NAEP Statement::
8NPO3a: Perform computations with rational numbers.

 Mathematics (2016) Grade(s): 3 All Resources: 0
6 ) Understand division as an unknown-factor problem. [3-OA6]

Example: Find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

NAEP Framework
NAEP Statement::
4NPO3e: Interpret whole number operations and the relationships between them.

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

Multiply and divide within 100.
 Mathematics (2016) Grade(s): 3 All Resources: 1 Learning Activities: 1
7 ) Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. [3-OA7]

NAEP Framework
NAEP Statement::
4NPO3c: Divide whole numbers:
• Up to three digits by one digit with paper and pencil computation, or
• Up to five digits by two digits with use of calculator.

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

NAEP Statement::
8NPO3d: Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a rational number by:
• Zero, or
• A number less than zero, or
• A number between zero and one,
• One, or
• A number greater than one.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.7- Multiply and divide one-digit numbers using repeated addition or repeated subtraction where the products are within twenty and the factors are one, two, three, four, five, or ten using multiplication and division tools.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.
 Mathematics (2016) Grade(s): 3 All Resources: 2 Learning Activities: 1 Lesson Plans: 1
8 ) Solve two-step word problems using the four operations. 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. (This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).) [3-OA8]

NAEP Framework
NAEP Statement::
4NPO3f: Solve application problems involving numbers and operations.

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.8- Solve one-step real-world problems using addition or subtraction without regrouping.

 Mathematics (2016) Grade(s): 3 All Resources: 1 Learning Activities: 1
9 ) Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. [3-OA9]

Example: Observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

NAEP Framework
NAEP Statement::
4A1a: Recognize, describe, or extend numerical patterns.

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.9- Extend numeric and non-numeric patterns of two terms.

Number and Operations in Base Ten
Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)
 Mathematics (2016) Grade(s): 3 All Resources: 1 Lesson Plans: 1
10 ) Use place value understanding to round whole numbers to the nearest 10 or 100. [3-NBT1]

NAEP Framework
NAEP Statement::
8NPO3a: Perform computations with rational numbers.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.10- Use decade numbers (0, 10, 20, 30) as benchmarks to demonstrate understanding of place value for rounding numbers 0 to 34 using rounding tools (charts, number lines).

 Mathematics (2016) Grade(s): 3 All Resources: 3 Learning Activities: 1 Lesson Plans: 2
11 ) Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. [3-NBT2]

NAEP Framework
NAEP Statement::
• Whole numbers, or
• Fractions with like denominators, or
• Decimals through hundredths.

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.11- Add and subtract one- and two-digit numbers up to 30 (no regrouping).

 Mathematics (2016) Grade(s): 3 All Resources: 0
12 ) Multiply one-digit whole numbers by multiples of 10 in the range 10 - 90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations. [3-NBT3]

NAEP Framework
NAEP Statement::
4NPO3b: Multiply whole numbers:
• No larger than two digit by two digit with paper and pencil computation, or
• Larger numbers with use of calculator.

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

NAEP Statement::
8NPO3d: Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a rational number by:
• Zero, or
• A number less than zero, or
• A number between zero and one,
• One, or
• A number greater than one.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.12- Relate groups of ten to multiplying by ten up to 100, using objects, skip counting by tens.

Number and Operations — Fractions
(Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.)
Develop understanding of fractions as numbers.
 Mathematics (2016) Grade(s): 3 All Resources: 0
13 ) Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts and size 1/b. [3-NF1]

NAEP Framework
NAEP Statement::
8NPO1h: Order or compare rational numbers (fractions, decimals, percents, or integers) using various models and representations (e.g., number line).

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.13- Use models to represent unit fractions as parts of a whole (halves and fourths).

 Mathematics (2016) Grade(s): 3 All Resources: 0
14 ) Understand a fraction as a number on the number line; represent fractions on a number line diagram. [3-NF2]

a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. [3-NF2a]

b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. [3-NF2b]

NAEP Framework
NAEP Statement::
4NPO1e: Connect model, number word, or number using various models and representations for whole numbers, fractions, and decimals.

NAEP Statement::
4NPO3d: Describe the effect of operations on size (whole numbers).

NAEP Statement::
8NPO1b: Model or describe rational numbers or numerical relationships using number lines and diagrams.

NAEP Statement::
8NPO1h: Order or compare rational numbers (fractions, decimals, percents, or integers) using various models and representations (e.g., number line).

NAEP Statement::
8NPO3a: Perform computations with rational numbers.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.14- Use a number line to represent halves and fourths.

 Mathematics (2016) Grade(s): 3 All Resources: 1 Learning Activities: 1
15 ) Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. [3-NF3]

a. Understand two fractions as equivalent (equal) if they are the same size or the same point on a number line. [3-NF3a]

b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model. [3-NF3b]

c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. [3-NF3c]

Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. [3-NF3d]

NAEP Framework
NAEP Statement::
4NPO3d: Describe the effect of operations on size (whole numbers).

NAEP Statement::
4NPO6a: Explain or justify a mathematical concept or relationship (e.g., explain why 15 is an odd number or why 7-3 is not the same as 3-7).

NAEP Statement::
8NPO1d: Write or rename rational numbers.

NAEP Statement::
8NPO1h: Order or compare rational numbers (fractions, decimals, percents, or integers) using various models and representations (e.g., number line).

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.15- Compare fractions.
M.AAS.3.15a- Use models to identify two equivalent fractions (limit to fourths and halves).
M.AAS.3.15b- Recognize two equivalent fractions (limit to fourths and halves).
M.AAS.3.15c- Use models of fourths and halves to make a whole.

Measurement and Data
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
 Mathematics (2016) Grade(s): 3 All Resources: 0
16 ) Tell and write time to the nearest minute, and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. [3-MD1]

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.16- Tell time to the nearest half hour on a clock.

 Mathematics (2016) Grade(s): 3 All Resources: 2 Lesson Plans: 2
17 ) Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (Excludes compound units such as cm3 and finding the geometric volume of a container.) Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. (Excludes multiplicative comparison problems (problems involving notions of "times as much").) (See Appendix A, Table 2.) [3-MD2]

NAEP Framework
NAEP Statement::
4M1c: Estimate the size of an object with respect to a given measurement attribute (e.g., length, perimeter, or area using a grid).

NAEP Statement::
8M1c: Estimate the size of an object with respect to a given measurement attribute (e.g., area).

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.17- Identify the appropriate measurement tool to measure liquid; identify the appropriate standard unit of measurement (grams, kilograms, and liters).

Represent and interpret data.
 Mathematics (2016) Grade(s): 3 All Resources: 8 Learning Activities: 2 Lesson Plans: 4 Unit Plans: 2
18 ) Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. [3-MD3]

Example: Draw a bar graph in which each square in the bar graph might represent 5 pets.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.18- Use a pictograph or bar graph to answer questions about data (limit to three categories).

 Mathematics (2016) Grade(s): 3 All Resources: 2 Lesson Plans: 1 Unit Plans: 1
19 ) Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot where the horizontal scale is marked off in appropriate units — whole numbers, halves, or quarters. [3-MD4]

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.19- Measure lengths of objects using standard tools (rulers, yardsticks, meter sticks). Limit to whole numbers.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
 Mathematics (2016) Grade(s): 3 All Resources: 0
20 ) Recognize area as an attribute of plane figures, and understand concepts of area measurement. [3-MD5]

a. A square with side length 1 unit called "a unit square," is said to have "one square unit" of area and can be used to measure area. [3-MD5a]

b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. [3-MD5b]

NAEP Framework
NAEP Statement::
4M1g: Solve problems involving area of squares and rectangles.

NAEP Statement::
8M1e: Select or use appropriate measurement instrument to determine or create a given length, area, volume, angle, weight, or mass.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.20- Identify a model that demonstrates area and/or recognize one square unit of area as a "unit square" to use when measuring area.

 Mathematics (2016) Grade(s): 3 All Resources: 0
21 ) Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). [3-MD6]

NAEP Framework
NAEP Statement::
4M1g: Solve problems involving area of squares and rectangles.

NAEP Statement::
8M1e: Select or use appropriate measurement instrument to determine or create a given length, area, volume, angle, weight, or mass.

 Mathematics (2016) Grade(s): 3 All Resources: 0
22 ) Relate area to the operations of multiplication and addition. [3-MD7]

a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. [3-MD7a]

b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real-world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. [3-MD7b]

c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a x b and a x c. Use area models to represent the distributive property in mathematical reasoning. [3-MD7c]

d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into nonoverlapping rectangles and adding the areas of the nonoverlapping parts, applying this technique to solve real-world problems. [3-MD7d]

NAEP Framework
NAEP Statement::
4M1g: Solve problems involving area of squares and rectangles.

NAEP Statement::
4NPO3f: Solve application problems involving numbers and operations.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.22- Find the area of a rectangle with side lengths of no more than one, two, three, four, or five.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
 Mathematics (2016) Grade(s): 3 All Resources: 0
23 ) Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. [3-MD8]

NAEP Framework
NAEP Statement::
4M1f: Solve problems involving perimeter of plane figures.

NAEP Statement::
4NPO3f: Solve application problems involving numbers and operations.

NAEP Statement::
8M1f: Solve mathematical or real-world problems involving perimeter or area of plane figures such as triangles, rectangles, circles, or composite figures.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.23- Find the perimeter of a rectangle with lengths limited to one to ten units.

Geometry
Reason with shapes and their attributes.
 Mathematics (2016) Grade(s): 3 All Resources: 0
24 ) Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. [3-G1]

NAEP Framework
NAEP Statement::
4G1f: Describe attributes of two- and three-dimensional shapes.

NAEP Statement::
4G3a: Analyze or describe patterns of geometric figures by increasing number of sides, changing size or orientation (e.g., polygons with more and more sides).

NAEP Statement::
8G1b: Identify a geometric object given a written description of its properties.

NAEP Statement::
8G3f: Describe or analyze simple properties of, or relationships between, triangles, quadrilaterals, and other polygonal plane figures.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.3.24- Identify two-dimensional shapes by their attributes (triangle, rectangle, square, circle).

 Mathematics (2016) Grade(s): 3 All Resources: 1 Learning Activities: 1
25 ) Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. [3-G2]

Example: Partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.