# Courses of Study : Mathematics

Operations and Algebraic Thinking
Write and interpret numerical expressions.
 Mathematics (2016) Grade(s): 5 All Resources: 4 Learning Activities: 4
1 ) Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. [5-OA1]

 Mathematics (2016) Grade(s): 5 All Resources: 1 Learning Activities: 1
2 ) Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. [5-OA2]

Examples: Express the calculation "add 8 and 7, then multiply by 2" as 2 x (8 + 7). Recognize that 3 x (18,932 + 921) is three times as large as 18,932 + 921, without having to calculate the indicated sum or product.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.2- Select the expression that represents a given calculation (include parentheses).

Analyze patterns and relationships.
 Mathematics (2016) Grade(s): 5 All Resources: 0
3 ) Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. [5-OA3]

Example: Given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

NAEP Framework
NAEP Statement::
4A1b: Given a pattern or sequence, construct or explain a rule that can generate the terms of the pattern or sequence.

NAEP Statement::
4A1c: Given a description, extend or find a missing term in a pattern or sequence.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.3- Extend an addition or subtraction number pattern given one rule and the starting point of less than 10.

Number and Operations in Base Ten
Understand the place value system.
 Mathematics (2016) Grade(s): 5 All Resources: 0
4 ) 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. [5-NBT1]

NAEP Framework
NAEP Statement::
8NPO1a: Use place value to model and describe integers and decimals.

 Mathematics (2016) Grade(s): 5 All Resources: 0
5 ) Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. [5-NBT2]

NAEP Framework
NAEP Statement::
4A1b: Given a pattern or sequence, construct or explain a rule that can generate the terms of the pattern or sequence.

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::
8NPO1a: Use place value to model and describe integers and decimals.

 Mathematics (2016) Grade(s): 5 All Resources: 2 Lesson Plans: 1 Unit Plans: 1
6 ) Read, write, and compare decimals to thousandths. [5-NBT3]

a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000). [5-NBT3a]

b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. [5-NBT3b]

NAEP Framework
NAEP Statement::
8NPO1a: Use place value to model and describe integers and decimals.

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.5.6- Compare numbers, including decimals up to hundredths.

 Mathematics (2016) Grade(s): 5 All Resources: 3 Learning Activities: 1 Lesson Plans: 1 Unit Plans: 1
7 ) Use place value understanding to round decimals to any place. [5-NBT4]

NAEP Framework
NAEP Statement::
8NPO1a: Use place value to model and describe integers and decimals.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.7- Round three-digit whole numbers from 100 to 949 to the nearest 10 or 100 and round decimals to the nearest hundredths using dollars and cents.

Perform operations with multi-digit whole numbers and with decimals to hundredths.
 Mathematics (2016) Grade(s): 5 All Resources: 0
8 ) Fluently multiply multi-digit whole numbers using the standard algorithm. [5-NBT5]

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.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.8- Multiply a two-digit number by a one-digit number with regrouping.

 Mathematics (2016) Grade(s): 5 All Resources: 0
9 ) Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. [5-NBT6]

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.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.9- Divide a three-digit number by a one-digit number with no remainder.

 Mathematics (2016) Grade(s): 5 All Resources: 2 Learning Activities: 2
10 ) Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method, and explain the reasoning used. [5-NBT7]

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.5.10- Add and subtract two-digit numbers with regrouping (include numbers with decimals of tenths).

Number and Operations — Fractions
Use equivalent fractions as a strategy to add and subtract fractions.
 Mathematics (2016) Grade(s): 5 All Resources: 3 Learning Activities: 2 Lesson Plans: 1
11 ) Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. [5-NF1]

Example: 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

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

NAEP Statement::
8NPO6b: Provide a mathematical argument to explain operations with two or more fractions.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.11- Use a model to add and subtract fractional parts with like denominators.

 Mathematics (2016) Grade(s): 5 All Resources: 1 Lesson Plans: 1
12 ) 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. [5-NF2]

Example: Recognize an incorrect result 2/5 + 1/2 = 3/7 by observing that 3/7 < 1/2.

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

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

NAEP Statement::
8NPO6b: Provide a mathematical argument to explain operations with two or more fractions.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.12- Solve word problems involving addition and subtraction of fractions with like denominators.

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
 Mathematics (2016) Grade(s): 5 All Resources: 0
13 ) Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. [5-NF3]

Examples: Interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get' Between which two whole numbers does your answer lie'

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

NAEP Statement::
4NPO4a: Use simple ratios to describe problem situations.

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.

NAEP Statement::
8NPO6b: Provide a mathematical argument to explain operations with two or more fractions.

 Mathematics (2016) Grade(s): 5 All Resources: 3 Learning Activities: 3
14 ) Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. [5-NF4]

a. Interpret the product (a/b) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q ÷ b. [5-NF4a]

Example: Use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.)

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. [5-NF4b]

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

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.

NAEP Statement::
8NPO5e: Apply basic properties of operations.

NAEP Statement::
8NPO6b: Provide a mathematical argument to explain operations with two or more fractions.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.14- Find the product of unit fractions (with denominators of 2, 3, 4, 5, 10).

 Mathematics (2016) Grade(s): 5 All Resources: 0
15 ) Interpret multiplication as scaling (resizing), by: [5-NF5]

a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. [5-NF5a]

b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case), explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number, and relating the principle of fraction equivalence a/b = (n x a)/(n x b) to the effect of multiplying by 1. [5-NF5b]

NAEP Framework
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::
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.

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.

 Mathematics (2016) Grade(s): 5 All Resources: 1 Learning Activities: 1
16 ) Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. [5-NF6]

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

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

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

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.16- Use a model to solve one-step real-world problems involving multiplying a whole number by a unit fraction.

 Mathematics (2016) Grade(s): 5 All Resources: 2 Learning Activities: 2
17 ) Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general by reasoning about the relationship between multiplication and division. However, division of a fraction by a fraction is not a requirement at this grade.) [5-NF7]

a. Interpret division of a unit fraction by a nonzero whole number, and compute such quotients. [5-NF7a]

Example: Create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. [5-NF7b]

Example: Create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4.

c. Solve real-world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. [5-NF7c]

Examples: How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally' How many 1/3 -cup servings are in 2 cups of raisins'

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

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.

NAEP Statement::
8NPO5e: Apply basic properties of operations.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.17- Divide a whole number by a unit fraction.

Measurement and Data
Convert like measurement units within a given measurement system.
 Mathematics (2016) Grade(s): 5 All Resources: 2 Learning Activities: 2
18 ) Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep, real-world problems. [5-MD1]

NAEP Framework
NAEP Statement::
4M2b: Solve problems involving conversions within the same measurement system such as conversions involving inches and feet or hours and minutes.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.18a- Given a smaller unit of measurement, determine how many smaller units it would take to make the larger unit.

Represent and interpret data.
 Mathematics (2016) Grade(s): 5 All Resources: 0
19 ) Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. [5-MD2]

Example: Given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

NAEP Framework
NAEP Statement::
4DASP1b: For a given set of data, complete a graph (limits of time make it difficult to construct graphs completely).

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.19- Interpret data on a bar graph, pictograph, or line plot to display a data set of measurements in fractions of a unit (limited to halves).

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
 Mathematics (2016) Grade(s): 5 All Resources: 1 Lesson Plans: 1
20 ) Recognize volume as an attribute of solid figures, and understand concepts of volume measurement. [5-MD3]

a. A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume. [5-MD3a]

b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. [5-MD3b]

NAEP Framework
NAEP Statement::
8M1h: Solve problems involving volume or surface area of rectangular solids, cylinders, prisms, or composite shapes.

 Mathematics (2016) Grade(s): 5 All Resources: 1 Learning Activities: 1
21 ) Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. [5-MD4]

NAEP Framework
NAEP Statement::
8M1h: Solve problems involving volume or surface area of rectangular solids, cylinders, prisms, or composite shapes.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.21- Determine the volume of a three- dimensional figure by counting unit cubes (where at least the width or height or depth is 1).

 Mathematics (2016) Grade(s): 5 All Resources: 1 Lesson Plans: 1
22 ) Relate volume to the operations of multiplication and addition, and solve real-world and mathematical problems involving volume. [5-MD5]

a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. [5-MD5a]

b. Apply the formulas V = l x w x h and V = B x h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems. [5-MD5b]

c. Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the nonoverlapping parts, applying this technique to solve real-world problems. [5-MD5c]

NAEP Framework
NAEP Statement::
8A4e: Use and evaluate common formulas (e.g., relationship between a circle's circumference and diameter [C = pi d], distance and time under constant speed).

NAEP Statement::
8M1h: Solve problems involving volume or surface area of rectangular solids, cylinders, prisms, or composite shapes.

Geometry
Graph points on the coordinate plane to solve real-world and mathematical problems.
 Mathematics (2016) Grade(s): 5 All Resources: 1 Lesson Plans: 1
23 ) Use a pair of perpendicular number lines, called axes, to define a coordinate system with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). [5-G1]

NAEP Framework
NAEP Statement::
4A2c: Graph or interpret points with whole number or letter coordinates on grids or in the first quadrant of the coordinate plane.

NAEP Statement::
4G4a: Describe relative positions of points and lines using the geometric ideas of parallelism or perpendicularity.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.23- Identify quadrant 1 and the origin on a coordinate system grid.

 Mathematics (2016) Grade(s): 5 All Resources: 2 Learning Activities: 1 Lesson Plans: 1
24 ) Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. [5-G2]

NAEP Framework
NAEP Statement::
4A2c: Graph or interpret points with whole number or letter coordinates on grids or in the first quadrant of the coordinate plane.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.5.24- Identify the coordinate values of a point with whole number coordinates in quadrant 1 (x and y values limited to 5 or less).

Classify two-dimensional figures into categories based on their properties.
 Mathematics (2016) Grade(s): 5 All Resources: 1 Learning Activities: 1
25 ) Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. [5-G3]

Example: All rectangles have four right angles, and squares are rectangles, so all squares have four right angles.

NAEP Framework
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::
4G5a: Distinguish which objects in a collection satisfy a given geometric definition and explain choices.

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.5.25- Classify two-dimensional figures and identify the attributes (angles, number of sides, corners) they have in common.

 Mathematics (2016) Grade(s): 5 All Resources: 1 Learning Activities: 1
26 ) Classify two-dimensional figures in a hierarchy based on properties. [5-G4]

NAEP Framework
NAEP Statement::
4G5a: Distinguish which objects in a collection satisfy a given geometric definition and explain choices.

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