Numbers and Operations in Base Ten

Equations and Expressions

The Number System

Statistics and Probability

Geometry

Functions

8.EE.A: Work with radicals and integer exponents

8.EE.C: Analyze and solve linear equations and pairs of simultaneous linear equaitons

8.NS.1 : Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

8.EE.B: Understand the connections between proportional relationships, lines and linear equations

8.NS.2 : Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., pi^2). For example, by truncating the decimal expansion of sqrt(2), show that sqrt(2) is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Notes

• Single digit multiplication facts and their related quotients take a while to teach because there are no general strategies, rather; there are patterns (mult. table) and specific strategies for certiain numbers (e.g. 9)
• There are three types of Multiplication and division situations, equal groups, arrays, comparison (multiplicative relationships)
• Three levels of mult: 1)making and counting 2)skip counting 3) associative and distribtive to compose or decompose

Know that numbers that are not rational are called irrational.

Understand a rational number to be any number that is the value of a ratio of two integers.

3.NBT.A: Use place value understanding and properties of operations to perform multi-digit arithmetic.

8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^2 × 3^-5 = 3^-3 = 1/3^3 = 1/27.

C

P/F

3.OA.5: Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

8.EE.7: Solve linear equations in one variable.

Understand when two numerical expressions involving exponents are equivalent.

Bill's Blog: Now, I think I know where the teacher is coming from (you probably do too). The teacher has in mind that the student should be thinking of 10 groups of 2, and 3.OA.5 does suggest this would be written as 10×2:

Apply properties of exponents when multiplying or dividing terms with the same base. Know when properties are not applicable.

Apply commutative property of multiplication as strategies to multiply

Apply associative property of multiplication as strategies to multiply

Apply distributive property as strategies to multiply and divide

C

8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

8 x 40 = 40 x 8 (which wouldn't have been allowed in 3.NBT.3 alone)

C; P/F

3.NBT.1: Use place value understanding to round whole numbers to the nearest 10 or 100. (3.NBT footnote: A range of algorithms may be used.)

8.G.9: Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real world and mathematical problems.

8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

C; P/F

8.EE.2: Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that sqrt(2) is irrational.

Solve a linear equation with rational coefficients

Understand the representations of having one, infinite, or no solution.

Identify linear equations in one variable with one, no, or infinite solutions by inspection.

Identify linear equations in one variable with one, no, or infinite solutions by transforming given equations into simpler forms, ie using the distributive property and collecting like terms.

Know that the square root of 2 is irrational.

C; P/F

Know and apply properties of roots to include negative solutions when appropriate.

Recognize and evaluate the roots of perfect squares and cubes.

Represent the solutions of simple quadratic and cubic expressions by using square and cube roots.

C

Demonstrate that rational numbers expressed as decimals either terminate or repeat eventually.

C; P/F

3.OA.6 Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Use place value understanding to round whole numbers to the nearest 100

Use place value understanding to round whole numbers to the nearest 10

Use place value understanding to round whole numbers to the nearest 100

Use place value understanding to round whole numbers to the nearest 10

3.NBT.3: Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. (3.NBT footnote: A range of algorithms may be used.)

3.NBT.1: Use place value understanding to round whole numbers to the nearest 10 or 100. (3.NBT footnote: A range of algorithms may be used.)

Understand the number of objects being partitioned (dividend) as equivalent to the product of number of groups and number of objects in each group.

Represent a division problem as an equivalent multiplication problem with an unknown factor

Understand the dividend represents the product of the divisor and quotient.

Understand division as the inverse of multiplication and vice versa

3.NBT.2: 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.NBT footnote: A range of algorithms may be used.)

Understand that every number has a decimal expansion.

C; A

Bill's Blog: " 3.NBT.3 is about 4 x 80, and 3.OA.5 allows for students reasoning from this to 80 x 4. But the emphasis in Grade 3 would be on 4 x 80."

8.EE.8: Analyze and solve pairs of simultaneous linear equations.

8.EE.3: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 10^8 and the population of the world as 7 × 10^9, and determine that the world population is more than 20 times larger.

Compare quantities using multiplicative comparison, scientific notation, and properties of integer exponents.

Estimate quantities using scientific notation, converting between standard form and scientific notation.

8.EE.8a: (conceptual)Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Missing addend problems are fair game (PARCC question

Use place value understanding to round whole numbers to the nearest 100

Use place value understanding to round whole numbers to the nearest 10

Focus on methods that generalize readily to larger numbers so that these methods can be extended to 1,000,000 in 4th grade

Add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 1000 (“students use written methods without concrete models or drawings, though concrete models or drawings can be used with explanations to overcome errors and to continue to build understanding as needed.” Progressions)

8.EE.8b: (Conceptual, Procedural/Fluency) Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

Convert a decimal expansion which repeats eventually into a rational number.

8.EE.8c: (Application)Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Application

P/F; C ; A

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Interpret scientific notation that has been generated by technology, i.e. calculators.

Add, subtract, multiply, and divide numbers written in scientific notation.

Choose units of appropriate size for measures of very large or very small quantities, and express those amounts in scientific notation.

Convert between decimal and scientific notation as needed when solving problems.

describing and analyzing two-dimensional shapes.

2) developing understanding of fractions, especially unit fractions

3) developing understanding of the structure of rectangular arrays and of area

1) developing understanding of multiplication and division and strategies for multiplication and division within 100

Operations and Algebraic Thinking

Measurement and Data

Geometry

Numbers and Operations in Base Ten

Numbers and Operations - Fractions

3.OA.C: Multiply and divide within 100

3.OA.D: Solve problems involving the four operations, and identify explain patterns in arithmetic

3.OA.A: Represent and solve problems involving multiplication and division

3.OA.B: Understand properties of multiplication and the relationship between multiplication and division

3.MD.B: Represent and interpret data

3.MD.C: Geometric measurement: understand concepts of area and relate area to multiplication and to addition

3.MD.D: Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measurements.

3.MD.A: Solve Problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Notes

• Single digit multiplication facts and their related quotients take a while to teach because there are no general strategies, rather; there are patterns (mult. table) and specific strategies for certiain numbers (e.g. 9)
• There are three types of Multiplication and division situations, equal groups, arrays, comparison (multiplicative relationships)
• Three levels of mult: 1)making and counting 2)skip counting 3) associative and distribtive to compose or decompose

3.OA.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. (3.OA.8 footnote: 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.NBT.A: Use place value understanding and properties of operations to perform multi-digit arithmetic.

3.MD.5: Recognize area as an attribute of plane figures and understand concepts of area measurement.

3.OA.7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 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.OA.1: Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

C

3.OA.5: Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Use addition and subtraction within 1,000, and multiplication and division within 100 to solve two step real- world problems

Determine if an answer is reasonable using mental computation and estimation strategies including rounding.

Represent a two-step word problem using equations with a letter standing for the unknown quantity.

Explain if an answer is reasonable using mental computation and estimation strategies including rounding.

Distinguish between the number of groups and the size of groups

Interpret products of whole-number as the total number of objects in n groups of n objects each

P/F; A

Represent a situation with a multiplication expression; represent a multiplication expression with a situation.

Bill's Blog: Now, I think I know where the teacher is coming from (you probably do too). The teacher has in mind that the student should be thinking of 10 groups of 2, and 3.OA.5 does suggest this would be written as 10×2:

A

Apply commutative property of multiplication as strategies to multiply

Apply associative property of multiplication as strategies to multiply

Apply distributive property as strategies to multiply and divide

3.MD.1: 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.MD.3: 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. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

Fluently multiply within 100 (up to 10 x 10)

Use strategies such as the relationship between multiplication and division to multiply and divide within 100

Fluently divide within 100 (up to 10 x 10)

Evidence Statements:

• limited to 60 minutes
• no more than 20% require determining an interval from clock readings having difference hour values.
• for number line part: "only the answer is required" and "tasks do not involve reading start/stop times from a clock nor calculating elapsed time"

Evidence statements:

• no more than 10 items in 2-5 categories
• category shouldn't have a numerical representation
• students shouldn't have to create the entire graph, but rather complete or otherwise demonstrate knowledge of creation

8 x 40 = 40 x 8 (which wouldn't have been allowed in 3.NBT.3 alone)

Tell time to the nearest minute

Solve word problems involving addition of time intervals in minutes

Solve word problems involving subtraction of time intervals in minutes

Represent word problems involving addition and subtraction of time intervals in minutes on a number line diagram.

Measure time intervals in minutes (within 60 minutes), including finding elapsed time

Solve two-step "how many more" and "how many less" problems using information

Solve one-step "how many more" and "how many less" problems using information presented in scaled bar graphs.

Draw scaled bar graph to represent a data set with several categories.

Draw a scaled picture graph to represent a data set with several categories.

C

3.MD.6: Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

3.OA.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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

C; P/F

3.OA.9: Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Represent a situation with a division expression; represent a division expression with a situation.

C; P/F

Measure area by counting unit squares

Interpret whole-number quotients as the number of groups when a set number of objects are partitioned

Interpret whole-number quotients as the number of objects in each group when partitioned into equal groups; OR Interpret the number of objects in each group after being partitioned as the quotient of a division problem

Recognize that the number of tiles along a side corresponds to the length of the side

C

C; P/F

3.OA.6 Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Identify arithmetic patterns

Identify arithmetic patterns in the addition table or multiplication table

explain arithmetic patterns using properties of operations.

3.NBT.3: Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. (3.NBT footnote: A range of algorithms may be used.)

3.NBT.1: Use place value understanding to round whole numbers to the nearest 10 or 100. (3.NBT footnote: A range of algorithms may be used.)

Understand the number of objects being partitioned (dividend) as equivalent to the product of number of groups and number of objects in each group.

Represent a division problem as an equivalent multiplication problem with an unknown factor

Understand the dividend represents the product of the divisor and quotient.

Understand division as the inverse of multiplication and vice versa

3.NBT.2: 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.NBT footnote: A range of algorithms may be used.)

A

Bill's Blog: " 3.NBT.3 is about 4 x 80, and 3.OA.5 allows for students reasoning from this to 80 x 4. But the emphasis in Grade 3 would be on 4 x 80."

3.OA.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. (3.OA.3 footnote: See CCSS Glossary, Table 2.)

C

3.MD.7: Relate area to the operations of multiplication and addition.

Using drawings to represent multiplication and division problems

Using equations with a symbol for the unknown number to represent multiplication and division problems

Use multiplication within 100 (up to 10 x 10) to solve word problems in situations involving equal groups, arrays, and measurement quantities (including area)

Use division within 100 (up to 10 x 10) to solve word problems in situations involving equal groups, arrays, and measurement quantities (including area)

Manipulate rectangular arrays to concretely demonstrate the arithmetic properties

Connect extensive work with rectangular arrays and multiplication to eventually discover the area formula for a rectangle

Drawing area models by completing rows and columns in figures

Use properties to validate that area stays the same, despite different dimensions

Apply tiling and multiplication skills to determine all whole number possibilities for the side lengths of rectangles given their areas

Missing addend problems are fair game (PARCC question

C

P/F; A

3.MD.7c

3.MD.7d

3.MD.7a: 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.

Use place value understanding to round whole numbers to the nearest 100

Use place value understanding to round whole numbers to the nearest 10

3.MD.7b: 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.MD.2: Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (3.MD.2 footnote: Excludes compound units such as cm^3 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.(3.MD.2 footnote: Excludes multiplicative comparison problems(problems involving notions of "times as much"; see CCSS Glossary, Table 2).

3.MD.4: 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.

Focus on methods that generalize readily to larger numbers so that these methods can be extended to 1,000,000 in 4th grade

Add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluently add and subtract within 1000 (“students use written methods without concrete models or drawings, though concrete models or drawings can be used with explanations to overcome errors and to continue to build understanding as needed.” Progressions)

Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of real world and mathematical problems

Represent whole-number products as rectangular areas in mathematical reasoning (draw a rectangle from a given area)

Tile a rectangle to find the area

Draw an area models to represent given dimensions

Show that the area found by tiling is the same as would be found by multiplying the side lengths; make the connection between rectangular arrays and multiplication

Determine all whole number possibilities for the side lengths of rectangles given their areas (EngageNY)

Multiply side lengths to find areas of rectangles with whole-number side lengths

Represent word problems involving addition and subtraction of time intervals in minutes on a number line diagram.

P/F

3.OA.4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = __÷ 3, 6 × 6 = ?.

Determine the unknown whole number, in any place, in a multiplication equation relating three whole number

Determine the unknown whole number, in any place, in a division equation relating three whole number

3.MD.3: 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. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

Solve two-step "how many more" and "how many less" problems using information

Solve one-step "how many more" and "how many less" problems using information presented in scaled bar graphs.

Draw scaled bar graph to represent a data set with several categories.

Draw a scaled picture graph to represent a data set with several categories.