D2L Corporation
2014-06-26T02:54:06-04:00
2014-06-26T02:54:06-04:00
D2L Corporation
Alaska Mathematics Standards
The mathematics standards prepare Alaska students to be competitive on the national and world stage. These standards are a set of specific, rigorous expectations that build students' conceptual understanding, mathematical language, and application of processes and procedures coherently from one grade to the next so all students will be prepared for post-secondary experiences. The focus areas for each grade level and each conceptual category narrative establish a depth of knowledge as opposed to a breadth of knowledge across multiple standards in each grade level or content area.
2014-06-21
2012
Alaska Board of Education & Early Development
Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education.
MP.1
1.
Standard
Make sense of problems and persevere in solving them.
MP.2
2.
Standard
Reason abstractly and quantitatively.
MP.3
3.
Standard
Construct viable arguments and critique the reasoning of others.
MP.4
4.
Standard
Model with mathematics.
MP.5
5.
Standard
Use appropriate tools strategically.
MP.6
6.
Standard
Attend to precision.
MP.7
7.
Standard
Look for and make use of structure.
MP.8
8.
Standard
Look for and express regularity in repeated reasoning.
1.
Domain
Counting and Cardinality
Cluster
Know number names and the count sequence.
K.CC.1
Standard
Count to 100 by ones and by tens.
K.CC.2
Standard
Count forward beginning from a given number within the known sequence.
K.CC.3
Standard
Write numbers from 0 to 20. Represent a number of objects with a written numeral 0 - 20 (with 0 representing a count of no objects).
Cluster
Know ordinal names and counting flexibility.
1.CC.1
Standard
Skip count by 2s and 5s.
1.CC.2
Standard
Use ordinal numbers correctly when identifying object position (e.g., first, second, third, etc.).
1.CC.3
Standard
Order numbers from 1-100. Demonstrate ability in counting forward and backward.
Cluster
Count to tell the number of objects.
K.CC.4
Standard
Understand the relationship between numbers and quantities; connect counting to cardinality.
K.CC.4.a
a.
Standard
When counting objects, say the number names in standard order, pairing each object with one and only one number name and each number name with one and only one object.
K.CC.4.b
b.
Standard
Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
K.CC.4.c
c.
Standard
Understand that each successive number name refers to a quantity that is one larger.
K.CC.5
Standard
Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.
1.CC.4
Standard
Count a large quantity of objects by grouping into 10s and counting by 10s and 1s to find the quantity.
Cluster
Compare numbers.
K.CC.6
Standard
Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group (e.g., by using matching, counting, or estimating strategies).
K.CC.7
Standard
Compare and order two numbers between 1 and 10 presented as written numerals.
1.CC.5
Standard
Use the symbols for greater than, less than or equal to when comparing two numbers or groups of objects.
1.CC.6
Standard
Estimate how many and how much in a given set to 20 and then verify estimate by counting.
2.
Domain
Operations and Algebraic Thinking
Cluster
Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
K.OA.1
Standard
Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps) acting out situations, verbal explanations, expressions, or equations.
K.OA.2
Standard
Add or subtract whole numbers to 10 (e.g., by using objects or drawings to solve word problems).
K.OA.3
Standard
Decompose numbers less than or equal to 10 into pairs in more than one way (e.g., by using objects or drawings, and record each decomposition by a drawing or equation).
For example, 5 = 2 + 3 and 5 = 4 + 1.
K.OA.4
Standard
For any number from 1 - 4, find the number that makes 5 when added to the given number and, for any number from 1 - 9, find the number that makes 10 when added to the given number (e.g., by using objects, drawings or 10 frames) and record the answer with a drawing or equation.
K.OA.5
Standard
Fluently add and subtract numbers up to 5.
Cluster
Identify and continue patterns.
K.OA.6
Standard
Recognize, identify and continue simple patterns of color, shape, and size.
Cluster
Represent and solve problems involving addition and subtraction.
1.OA.1
Standard
Use addition and subtraction strategies to solve word problems (using numbers up to 20), involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions, using a number line (e.g., by using objects, drawings and equations). Record and explain using equation symbols and a symbol for the unknown number to represent the problem.
1.OA.2
Standard
Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 (e.g., by using objects, drawings and equations). Record and explain using equation symbols and a symbol for the unknown number to represent the problem.
2.OA.1
Standard
Use addition and subtraction strategies to estimate, then solve one- and two-step word problems (using numbers up to 100) involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions (e.g., by using objects, drawings and equations). Record and explain using equation symbols and a symbol for the unknown number to represent the problem.
Cluster
Understand and apply properties of operations and the relationship between addition and subtraction.
1.OA.3
Standard
Apply properties of operations as strategies to add and subtract. (Students need not know the name of the property.)
For example: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known (Commutative property of addition). To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12 (Associative property of addition). Demonstrate that when adding zero to any number, the quantity does not change (Identity property of addition).
1.OA.4
Standard
Understand subtraction as an unknown-addend problem.
For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.
Cluster
Add and subtract using numbers up to 20.
1.OA.5
Standard
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
1.OA.6
Standard
Add and subtract using numbers up to 20, demonstrating fluency for addition and subtraction up to 10. Use strategies such as<ul><li>counting on</li><li>making ten(8 + 6 = 8 + 2 + 4 = 10 + 4 = 14)</li><li>decomposing a number leading to a ten (13 - 4 = 13 - 3 - 1 = 10 - 1 = 9)</li><li>using the relationship between addition and subtraction, such as fact families, (8 + 4 = 12 and 12 - 8 = 4) </li><li>creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).</li></ul>
2.OA.2
Standard
Fluently add and subtract using numbers up to 20 using mental strategies. Know from memory all sums of two one-digit numbers.
Cluster
Work with addition and subtraction equations.
1.OA.7
Standard
Understand the meaning of the equal sign (e.g., read equal sign as "same as") and determine if equations involving addition and subtraction are true or false.
For example, which of the following equations are true and which are false?6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2).
1.OA.8
Standard
Determine the unknown whole number in an addition or subtraction equation.
For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 6 + 6 = ?, 5 = ? - 3.
Cluster
Identify and continue patterns.
1.OA.9
Standard
Identify, continue and label patterns (e.g., aabb, abab). Create patterns using number, shape, size, rhythm or color.
Cluster
Work with equal groups of objects to gain foundations for multiplication.
2.OA.3
Standard
Determine whether a group of objects (up to 20) is odd or even (e.g., by pairing objects and comparing, counting by 2s). Model an even number as two equal groups of objects and then write an equation as a sum of two equal addends.
2.OA.4
Standard
Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns. Write an equation to express the total as repeated addition (e.g., array of 4 by 5 would be 5 + 5 + 5 + 5 = 20).
Cluster
Identify and continue patterns.
2.OA.5
Standard
Identify, continue and label number patterns (e.g., aabb, abab). Describe a rule that determines and continues a sequence or pattern.
Cluster
Represent and solve problems involving multiplication and division.
3.OA.1
Standard
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, show objects in rectangular arrays or describe a context in which a total number of objects can be expressed as 5 × 7.
3.OA.2
Standard
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, deconstruct rectangular arrays or describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3.OA.3
Standard
Use multiplication and division numbers up to 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.4
Standard
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 x ? = 48, 5 = ? ÷ 3, 6 x 6 = ?
Cluster
Understand properties of multiplication and the relationship between multiplication and division.
3.OA.5
Standard
Make, test, support, draw conclusions and justify conjectures about properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.)<ul><li>Commutative property of multiplication: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known.</li><li>Associative 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.</li><li>Distributive property: 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.</li><li>Inverse property (relationship) of multiplication and division.</li></ul>
3.OA.6
Standard
Understand division as an unknown-factor problem.
For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
Cluster
Multiply and divide up to 100.
3.OA.7
Standard
Fluently multiply and divide numbers up to 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.
Cluster
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.8
Standard
Solve and create two-step word problems using any of the four operations. Represent these problems using equations with a symbol (box, circle, question mark) standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
3.OA.9
Standard
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.
Cluster
Use the four operations with whole numbers to solve problems.
4.OA.1
Standard
Interpret a multiplication equation as a comparison e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 groups of 7 and 7 groups of 5 (Commutative property). Represent verbal statements of multiplicative comparisons as multiplication equations.
4.OA.2
Standard
Multiply or divide to solve word problems involving multiplicative comparison (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem or missing numbers in an array). Distinguish multiplicative comparison from additive comparison.
4.OA.3
Standard
Solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. 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.
Cluster
Gain familiarity with factors and multiples.
4.OA.4
Standard
<ul><li>Find all factor pairs for a whole number in the range 1–100.</li><li> Explain the correlation/differences between multiples and factors.</li><li>Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number.</li><li>Determine whether a given whole number in the range 1–100 is prime or composite.</li></ul>
Cluster
Generate and analyze patterns.
4.OA.5
Standard
Generate a number, shape pattern, table, t-chart, or input/output function that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Be able to express the pattern in algebraic terms.
For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
4.OA.6
Standard
Extend patterns that use addition, subtraction, multiplication, division or symbols, up to 10 terms, represented by models (function machines), tables, sequences, or in problem situations. (L)
Cluster
Write and interpret numerical expressions.
5.OA.1
Standard
Use parentheses to construct numerical expressions, and evaluate numerical expressions with these symbols.
5.OA.2
Standard
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.
For example, express the calculation "add 8 and 7, then multiply by 2" as 2 x (8 + 7). Recognizing that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
Cluster
Analyze patterns and relationships.
5.OA.3
Standard
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.
For 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.
3.
Domain
Number and Operations in Base Ten
Cluster
Work with numbers 11-19 to gain foundations for place value.
K.NBT.1
Standard
Compose and decompose numbers from 11 to 19 into ten ones and some further ones (e.g., by using objects or drawings) and record each composition and decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight or nine ones.
Cluster
Extend the counting sequence.
1.NBT.1
Standard
Count to 120. In this range, read, write and order numerals and represent a number of objects with a written numeral.
Cluster
Understand place value.
1.NBT.2
Standard
Model and identify place value positions of two digit numbers. Include:
1.NBT.2.a
a.
Standard
10 can be thought of as a bundle of ten ones, called a "ten".
1.NBT.2.b
b.
Standard
The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight or nine ones.
1.NBT.2.c
c.
Standard
The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90, refer to one, two, three, four, five, six, seven, eight or nine tens (and 0 ones).
1.NBT.3
Standard
Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, <.
2.NBT.1
Standard
Model and identify place value positions of three digit numbers. Include:
2.NBT.1.a
a.
Standard
100 can be thought of as a bundle of ten tens --called a "hundred".
2.NBT.1.b
b.
Standard
The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
2.NBT.2
Standard
Count up to 1000, skip-count by 5s, 10s and 100s.
2.NBT.3
Standard
Read, write, order up to 1000 using base-ten numerals, number names and expanded form.
2.NBT.4
Standard
Compare two three-digit numbers based on the meanings of the hundreds, tens and ones digits, using >, =, < symbols to record the results.
Cluster
Use place value understanding and properties of operations to add and subtract.
1.NBT.4
Standard
Add using numbers up to 100 including adding a two-digit number and a one-digit number and adding a two-digit number and a multiple of 10. Use:<ul><li>concrete models or drawings and strategies based on place value</li><li>properties of operations</li><li> and/or relationship between addition and subtraction.</li></ul>Relate the strategy to a written method and explain the reasoning used. Demonstrate in adding two-digit numbers, tens and tens are added, ones and ones are added and sometimes it is necessary to compose a ten from ten ones.
1.NBT.5
Standard
Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
1.NBT.6
Standard
Subtract multiples of 10 up to 100. Use:<ul><li>concrete models or drawings</li><li>strategies based on place value</li><li>properties of operations</li><li>and/or the relationship between addition and subtraction.</li></ul> Relate the strategy to a written method and explain the reasoning used.
2.NBT.5
Standard
Fluently add and subtract using numbers up to 100. Use:<ul><li>strategies based on place value</li><li>properties of operations</li><li>and/or the relationship between addition and subtraction.</li></ul>
2.NBT.6
Standard
Add up to four two-digit numbers using strategies based on place value and properties of operations.
2.NBT.7
Standard
Add and subtract using numbers up to 1000. Use:<ul><li>concrete models or drawings and strategies based on place value</li><li>properties of operations </li><li> and/or relationship between addition and subtraction.</li></ul> Relate the strategy to a written method and explain the reasoning used. Demonstrate in adding or subtracting three-digit numbers, hundreds and hundreds are added or subtracted, tens and tens are added or subtracted, ones and ones are added or subtracted and sometimes it is necessary to compose a ten from ten ones or a hundred from ten tens.
2.NBT.8
Standard
Mentally add 10 or 100 to a given number 100-900 and mentally subtract 10 or 100 from a given number.
2.NBT.9
Standard
Explain or illustrate the processes of addition or subtraction and their relationship using place value and the properties of operations.
Cluster
Use place value understanding and properties of operations to perform multi-digit arithmetic.
3.NBT.1
Standard
Use place value understanding to round whole numbers to the nearest 10 or 100.
3.NBT.2
Standard
Use strategies and/or algorithms to fluently add and subtract with numbers up to 1000, demonstrating understanding of place value, properties of operations, and/or the relationship between addition and subtraction.
3.NBT.3
Standard
Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 x 80, 10 x 60) using strategies based on place value and properties of operations.
Cluster
Generalize place value understanding for multi-digit whole numbers.
4.NBT.1
Standard
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
4.NBT.2
Standard
Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on the value of the digits in each place, using >, =, and < symbols to record the results of comparisons.
4.NBT.3
Standard
Use place value understanding to round multi-digit whole numbers to any place using a variety of estimation methods; be able to describe, compare, and contrast solutions.
Cluster
Use place value understanding and properties of operations to perform multi-digit arithmetic.
4.NBT.4
Standard
Fluently add and subtract multi-digit whole numbers using any algorithm. Verify the reasonableness of the results.
4.NBT.5
Standard
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
4.NBT.6
Standard
Find whole-number quotients and remainders with up to four-digit dividends and one-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.
Cluster
Understand the place value system.
5.NBT.1
Standard
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.NBT.2
Standard
Explain and extend the patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain and extend the 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.NBT.3
Standard
Read, write, and compare decimals to thousandths.
5.NBT.3.a
a.
Standard
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 (1/10) + 9 (1/100) + 2 (1/1000)].
5.NBT.3.b
b.
Standard
Compare two decimals to thousandths place based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
5.NBT.4
Standard
Use place values understanding to round decimals to any place.
Cluster
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.5
Standard
Fluently multiply multi-digit whole numbers using a standard algorithm.
5.NBT.6
Standard
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, number lines, real life situations, and/or area models.
5.NBT.7
Standard
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 the operations. Relate the strategy to a written method and explain their reasoning in getting their answers.
4.
Domain
Measurement and Data
Cluster
Describe and compare measurable attributes.
K.MD.1
Standard
Describe measurable attributes of objects (e.g., length or weight). Match measuring tools to attribute (e.g., ruler to length). Describe several measurable attributes of a single object.
K.MD.2
Standard
Make comparisons between two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference.
For example, directly compare the heights of two children and describe one child as taller/shorter.
Cluster
Classify objects and count the number of objects in each category.
K.MD.3
Standard
Classify objects into given categories (attributes). Count the number of objects in each category (limit category counts to be less than or equal to 10).
Cluster
Work with time and money.
K.MD.4
Standard
Name in sequence the days of the week.
K.MD.5
Standard
Tell time to the hour using both analog and digital clocks.
K.MD.6
Standard
Identify coins by name.
Cluster
Measure lengths indirectly and by iterating length units.
1.MD.1
Standard
Measure and compare three objects using standard or non-standard units.
1.MD.2
Standard
Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.
Cluster
Work with time and money.
1.MD.3
Standard
Tell and write time in half hours using both analog and digital clocks.
1.MD.4
Standard
Read a calendar distinguishing yesterday, today and tomorrow. Read and write a date.
1.MD.5
Standard
Recognize and read money symbols including $ and ¢.
1.MD.6
Standard
Identify values of coins (e.g., nickel = 5 cents, quarter = 25 cents). Identify equivalent values of coins up to $1 (e.g., 5 pennies = 1 nickel, 5 nickels = 1 quarter).
Cluster
Represent and interpret data.
1.MD.7
Standard
Organize, represent and interpret data with up to three categories. Ask and answer comparison and quantity questions about the data.
Cluster
Measure and estimate lengths in standard units.
2.MD.1
Standard
Measure the length of an object by selecting and using standard tools such as rulers, yardsticks, meter sticks, and measuring tapes.
2.MD.2
Standard
Measure the length of an object twice using different length units for the two measurements. Describe how the two measurements relate to the size of the unit chosen.
2.MD.3
Standard
Estimate, measure and draw lengths using whole units of inches, feet, yards, centimeters and meters.
2.MD.4
Standard
Measure to compare lengths of two objects, expressing the difference in terms of a standard length unit.
Cluster
Relate addition and subtraction to length.
2.MD.5
Standard
Solve addition and subtraction word problems using numbers up to 100 involving length that are given in the same units (e.g., by using drawings of rulers). Write an equation with a symbol for the unknown to represent the problem.
2.MD.6
Standard
Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1,2, …, and represent whole-number sums and differences within 100 on a number line diagram.
Cluster
Work with time and money.
2.MD.7
Standard
Tell and write time to the nearest five minutes using a.m. and p.m. from analog and digital clocks.
2.MD.8
Standard
Solve word problems involving dollar bills and coins using the $ and ¢ symbols appropriately.
Cluster
Represent and interpret data.
2.MD.9
Standard
Collect, record, interpret, represent, and describe data in a table, graph or line plot.
2.MD.10
Standard
Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart and compare problems using information presented in a bar graph.
Cluster
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
3.MD.1
Standard
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 or hours (e.g., by representing the problem on a number line diagram or clock).
3.MD.2
Standard
Estimate and measure 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 and create 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."])
3.MD.3
Standard
Select an appropriate unit of English, metric, or non-standard measurement to estimate the length, time, weight, or temperature (L)
Cluster
Represent and interpret data.
3.MD.4
Standard
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.
3.MD.5
Standard
Measure and record lengths using rulers marked with halves and fourths of an inch. Make a line plot with the data, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.
3.MD.6
Standard
Explain the classification of data from real-world problems shown in graphical representations. Use the terms minimum and maximum. (L)
Cluster
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
3.MD.7
Standard
Recognize area as an attribute of plane figures and understand concepts of area measurement.
3.MD.7.a
a.
Standard
A square with side length 1 unit is said to have "one square unit" and can be used to measure area.
3.MD.7.b
b.
Standard
Demonstrate that a plane figure which can be covered without gaps or overlaps by n (e.g., 6) unit squares is said to have an area of n (e.g., 6) square units.
3.MD.8
Standard
Measure areas by tiling with unit squares (square centimeters, square meters, square inches, square feet, and improvised units).
3.MD.9
Standard
Relate area to the operations of multiplication and addition.
3.MD.9.a
a.
Standard
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.
For example, after tiling rectangles, develop a rule for finding the area of any rectangle.
3.MD.9.b
b.
Standard
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.9.c
c.
Standard
Use area models (rectangular arrays) to represent the distributive property in mathematical reasoning. 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 × b and a × c.
3.MD.9.d
d.
Standard
Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
For example, the area of a 7 by 8 rectangle can be determined by decomposing it into a 7 by 3 rectangle and a 7 by 5 rectangle.
Cluster
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
3.MD.10
Standard
Solve real world and mathematical problems involving perimeters of polygons, including:<ul><li>finding the perimeter given the side lengths,</li><li>finding an unknown side length,</li><li>exhibiting rectangles with the same perimeter and different areas,</li><li>exhibiting rectangles with the same area and different perimeters.</li></ul>
Cluster
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit, and involving time.
4.MD.1
Standard
Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.
For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4-ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36).
4.MD.2
Standard
Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
4.MD.3
Standard
Apply the area and perimeter formulas for rectangles in real world and mathematical problems.
For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
4.MD.4
Standard
Solve real-world problems involving elapsed time between U.S. time zones (including Alaska Standard time). (L)
Cluster
Represent and interpret data.
4.MD.5
Standard
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.
For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.
4.MD.6
Standard
Explain the classification of data from real-world problems shown in graphical representations including the use of terms range and mode with a given set of data. (L)
Cluster
Geometric measurement: understand concepts of angle and measure angles.
4.MD.7
Standard
Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand the following concepts of angle measurement:
4.MD.7.a
a.
Standard
An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles.
4.MD.7.b
b.
Standard
An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
4.MD.8
Standard
Measure and draw angles in whole-number degrees using a protractor. Estimate and sketch angles of specified measure.
4.MD.9
Standard
Recognize angle measure as additive. When an angle is divided into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems (e.g., by using an equation with a symbol for the unknown angle measure).
Cluster
Convert like measurement units within a given measurement system and solve problems involving time.
5.MD.1
Standard
Identify, estimate measure, and convert equivalent measures within systems English length (inches, feet, yards, miles) weight (ounces, pounds, tons) volume (fluid ounces, cups, pints, quarts, gallons) temperature (Fahrenheit) Metric length (millimeters, centimeters, meters, kilometers) volume (milliliters, liters), temperature (Celsius), (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems using appropriate tools.
5.MD.2
Standard
Solve real-world problems involving elapsed time between world time zones. (L)
Cluster
Represent and interpret data.
5.MD.3
Standard
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving information presented in line plots.
For 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.
5.MD.4
Standard
Explain the classification of data from real-world problems shown in graphical representations including the use of terms mean and median with a given set of data. (L)
Cluster
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
5.MD.5
Standard
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
5.MD.5.a
a.
Standard
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.MD.5.b
b.
Standard
A solid figure that can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
5.MD.5.c
c.
Standard
Estimate and measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and non-standard units.
5.MD.6
Standard
Estimate and measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and non-standard units.
5.MD.7
Standard
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
5.MD.7.a
a.
Standard
Estimate and 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. Demonstrate the associative property of multiplication by using the product of three whole numbers to find volumes (length x width x height).
5.MD.7.b
b.
Standard
Apply the formulas V = l × w × h andV = b × 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.MD.7.c
c.
Standard
Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
5.
Domain
Number and Operations—Fractions
Cluster
Develop understanding of fractions as numbers.
3.NF.1
Standard
Understand a fraction 1/b (e.g., 1/4) as the quantity formed by 1 part when a whole is partitioned into b (e.g., 4) equal parts; understand a fraction a/b (e.g., 2/4) as the quantity formed by a (e.g., 2) parts of size 1/b. (e.g., 1/4)
3.NF.2
Standard
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
3.NF.2.a
a.
Standard
Represent a fraction 1/b (e.g., 1/4) on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b (e.g., 4) equal parts. Recognize that each part has size 1/b (e.g., 1/4) and that the endpoint of the part based at 0 locates the number 1/b (e.g., 1/4) on the number line.
3.NF.2.b
b.
Standard
Represent a fraction a/b (e.g., 2/8) on a number line diagram or ruler by marking off a lengths 1/b (e.g., 1/8) from 0. Recognize that the resulting interval has size a/b (e.g., 2/8) and that its endpoint locates the number a/b (e.g., 2/8) on the number line.
3.NF.3
Standard
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.3.a
a.
Standard
Understand two fractions as equivalent if they are the same size (modeled) or the same point on a number line.
3.NF.3.b
b.
Standard
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.NF.3.c
c.
Standard
Express and model whole numbers as fractions, and recognize and construct fractions that are equivalent to whole numbers.
For example: 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.
3.NF.3.d
d.
Standard
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).
Cluster
Extend understanding of fraction equivalence and ordering.
4.NF.1
Standard
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
4.NF.2
Standard
Compare two fractions with different numerators and different denominators (e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2). Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions (e.g., by using a visual fraction model).
Cluster
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
4.NF.3
Standard
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
4.NF.3.a
a.
Standard
Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
4.NF.3.b
b.
Standard
Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions (e.g., by using a visual fraction model).
For example: 3/8 = 1/8 + 1/8 + 1/8 ;3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
4.NF.3.c
c.
Standard
Add and subtract mixed numbers with like denominators (e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction).
4.NF.3.d
d.
Standard
Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators (e.g., by using visual fraction models and equations to represent the problem).
4.NF.4
Standard
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
4.NF.4.a
a.
Standard
Understand a fraction a/b as a multiple of 1/b.
For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
4.NF.4.b
b.
Standard
Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.
For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
4.NF.4.c
c.
Standard
Solve word problems involving multiplication of a fraction by a whole number (e.g., by using visual fraction models and equations to represent the problem). Check for the reasonableness of the answer.
For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
Cluster
Understand decimal notation for fractions, and compare decimal fractions.
4.NF.5
Standard
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.
For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
4.NF.6
Standard
Use decimal notation for fractions with denominators 10 or 100.
For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
4.NF.7
Standard
Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions (e.g., by using a visual model).
Cluster
Use equivalent fractions as a strategy to add and subtract fractions.
5.NF.1
Standard
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.
For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
5.NF.2
Standard
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 check the reasonableness of answers.
For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
Cluster
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.3
Standard
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).
For example, 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 what two whole numbers does your answer lie?
5.NF.4
Standard
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.4.a
a.
Standard
Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.
For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
5.NF.4.b
b.
Standard
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.NF.5
Standard
Interpret multiplication as scaling (resizing), by:
5.NF.5.a
a.
Standard
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.NF.5.b
b.
Standard
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 × a)/(n × b) to the effect of multiplying a/b by 1. (Division of a fraction by a fraction is not a requirement at this grade.)
5.NF.6
Standard
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.NF.7
Standard
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
5.NF.7.a
a.
Standard
Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
For 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) × 4 = 1/3.
5.NF.7.b
b.
Standard
Interpret division of a whole number by a unit fraction, and compute such quotients.
For 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 × (1/5) = 4.
5.NF.7.c
c.
Standard
Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions (e.g., by using visual fraction models and equations to represent the problem).
For example, 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?
6.
Domain
Geometry
Cluster
Identify and describe shapes.
K.G.1
Standard
Describe objects in the environment using names of shapes and describe their relative positions (e.g., above, below, beside, in front of, behind, next to).
K.G.2
Standard
Name shapes regardless of their orientation or overall size.
K.G.3
Standard
Identify shapes as two-dimensional (flat) or three-dimensional (solid).
Cluster
Analyze, compare, create, and compose shapes.
K.G.4
Standard
Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices), and other attributes (e.g., having sides of equal lengths).
K.G.5
Standard
Build shapes (e.g., using sticks and clay) and draw shapes.
K.G.6
Standard
Put together two-dimensional shapes to form larger shapes (e.g., join two triangles with full sides touching to make a rectangle).
Cluster
Reason with shapes and their attributes.
1.G.1
Standard
Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes. Identify shapes that have non-defining attributes (e.g., color, orientation, overall size). Build and draw shapes given specified attributes.
1.G.2
Standard
Compose (put together) two-dimensional or three-dimensional shapes to create a larger, composite shape, and compose new shapes from the composite shape.
1.G.3
Standard
Partition circles and rectangles into two and four equal shares. Describe the shares using the words, halves, fourths, and quarters and phrases half of, fourth of and quarter of. Describe the whole as two of or four of the shares. Understand for these examples that decomposing (break apart) into more equal shares creates smaller shares.
2.G.1
Standard
Identify and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces compared visually, not by measuring. Identify triangles, quadrilaterals, pentagons, hexagons and cubes.
2.G.2
Standard
Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
2.G.3
Standard
Partition circles and rectangles into shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
3.G.1
Standard
Categorize shapes by different attribute classifications and recognize that shared attributes can define a larger category. Generalize to create examples or non-examples.
3.G.2
Standard
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.
For 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.
Cluster
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
4.G.1
Standard
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular, parallel, and intersecting line segments. Identify these in two-dimensional (plane) figures.
4.G.2
Standard
Classify two-dimensional (plane) figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
4.G.3
Standard
Recognize a line of symmetry for a two-dimensional (plane) figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
Cluster
Graph points on the coordinate plane to solve real-world and mathematical problems.
5.G.1
Standard
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.G.2
Standard
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.
Cluster
Classify two-dimensional (plane) figures into categories based on their properties.
5.G.3
Standard
Understand that attributes belonging to a category of two-dimensional (plane) figures also belong to all subcategories of that category.
For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
5.G.4
Standard
Classify two-dimensional (plane) figures in a hierarchy based on attributes and properties.
Cluster
Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.1
Standard
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing or decomposing into other polygons (e.g., rectangles and triangles). Apply these techniques in the context of solving real-world and mathematical problems.
6.G.2
Standard
Apply the standard formulas to find volumes of prisms. Use the attributes and properties (including shapes of bases) of prisms to identify, compare or describe three-dimensional figures including prisms and cylinders.
6.G.3
Standard
Draw polygons in the coordinate plane given coordinates for the vertices; determine the length of a side joining the coordinates of vertices with the same first or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
6.G.4
Standard
Represent three-dimensional figures (e.g., prisms) using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
6.G.5
Standard
Identify, compare or describe attributes and properties of circles (radius, and diameter). (L)
Cluster
Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.1
Standard
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.2
Standard
Draw (freehand, with ruler and protractor, and with technology) geometric shapes including polygons and circles with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
7.G.3
Standard
Describe the two-dimensional figures, i.e., cross-section, that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Cluster
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
7.G.4
Standard
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
7.G.5
Standard
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
7.G.6
Objective
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Cluster
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.1
Standard
Through experimentation, verify the properties of rotations, reflections, and translations (transformations) to figures on a coordinate plane).
8.G.1.a
a.
Standard
Lines are taken to lines, and line segments to line segments of the same length.
8.G.1.b
b.
Standard
Angles are taken to angles of the same measure.
8.G.1.c
c.
Standard
Parallel lines are taken to parallel lines.
8.G.2
Standard
Demonstrate understanding of congruence by applying a sequence of translations, reflections, and rotations on two-dimensional figures. Given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.3
Standard
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.4
Standard
Demonstrate understanding of similarity, by applying a sequence of translations, reflections, rotations, and dilations on two-dimensional figures. Describe a sequence that exhibits the similarity between them.
8.G.5
Standard
Justify using informal arguments to establish facts about<ul><li>the angle sum of triangles (sum of the interior angles of a triangle is 180º),</li><li>measures of exterior angles of triangles,</li><li>angles created when parallel lines are cut be a transversal (e.g., alternate interior angles), and</li><li>angle-angle criterion for similarity of triangles.</li></ul>
Cluster
Understand and apply the Pythagorean Theorem.
8.G.6
Standard
Explain the Pythagorean Theorem and its converse.
8.G.7
Standard
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.8
Standard
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Cluster
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
8.G.9
Standard
Identify and apply the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
7.
Domain
Ratios and Proportional Relationships
Cluster
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.1
Standard
Write and describe the relationship in real life context between two quantities using ratio language.
For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."
6.RP.2
Standard
Understand the concept of a unit rate (a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship) and apply it to solve real world problems (e.g., unit pricing, constant speed).
For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."
6.RP.3
Standard
Use ratio and rate reasoning to solve real-world and mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).
6.RP.3.a
a.
Standard
Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios, and understand equivalencies.
6.RP.3.b
b.
Standard
Solve unit rate problems including those involving unit pricing and constant speed.
For example, if it took 7 hours to mow 4 lawns, then at that rate how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
6.RP.3.c
c.
Standard
Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
6.RP.3.d
d.
Standard
Use ratio reasoning to convert measurement units between given measurement systems (e.g., convert kilometers to miles); manipulate and transform units appropriately when multiplying or dividing quantities.
Cluster
Analyze proportional relationships and use them to solve real-world and mathematical problems.
7.RP.1
Standard
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour or apply a given scale factor to find missing dimensions of similar figures.
7.RP.2
Standard
Recognize and represent proportional relationships between quantities. Make basic inferences or logical predictions from proportional relationships.
7.RP.2.a
a.
Standard
Decide whether two quantities are in a proportional relationship (e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin).
6.RP.2.b
b.
Standard
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships in real world situations.
7.RP.2.c
c.
Standard
Represent proportional relationships by equations and multiple representations such as tables, graphs, diagrams, sequences, and contextual situations.
For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
7.RP.2.d
d.
Standard
Understand the concept of unit rate and show it on a coordinate plane. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
7.RP.3
Standard
Use proportional relationships to solve multistep ratio and percent problems.
Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
8.
Domain
The Number System
Cluster
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
6.NS.1
Standard
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions (e.g., by using visual fraction models and equations to represent the problem).
For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Cluster
Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.2
Standard
Fluently multiply and divide multi-digit whole numbers using the standard algorithm. Express the remainder as a whole number, decimal, or simplified fraction; explain or justify your choice based on the context of the problem.
6.NS.3
Standard
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Express the remainder as a terminating decimal, or a repeating decimal, or rounded to a designated place value.
6.NS.4
Standard
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
For example, express 36 + 8 as 4 (9 + 2).
Cluster
Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.5
Standard
Understand that positive and negative numbers describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explain the meaning of 0 in each situation.
6.NS.6
Standard
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
6.NS.6.a
a.
Standard
Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; Recognize that the opposite of the opposite of a number is the number itself [e.g., –(–3) = 3] and that 0 is its own opposite.
6.NS.6.b
b.
Standard
Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
6.NS.6.c
c.
Standard
Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.7
Standard
Understand ordering and absolute value of rational numbers.
6.NS.7.a
a.
Standard
Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.
For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
6.NS.7.b
b.
Standard
Write, interpret, and explain statements of order for rational numbers in real-world contexts.
For example, write –3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC.
6.NS.7.c
c.
Standard
Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
6.NS.7.d
d.
Standard
Distinguish comparisons of absolute value from statements about order.
For example, recognize that an account balance less than -30 dollars represents a debt greater than 30 dollars.
6.NS.8
Standard
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
Cluster
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7.NS.1
Standard
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
7.NS.1.a
a.
Standard
Show that a number and its opposite have a sum of 0 (additive inverses). Describe situations in which opposite quantities combine to make 0.
For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
7.NS.1.b
b.
Standard
Understand addition of rational numbers (p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative). Interpret sums of rational numbers by describing real-world contexts.
7.NS.1.c
c.
Standard
Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
7.NS.1.d
d.
Standard
Apply properties of operations as strategies to add and subtract rational numbers.
7.NS.2
Standard
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers and use equivalent representations.
7.NS.2.a
a.
Standard
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
7.NS.2.b
b.
Standard
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.
7.NS.2.c
c.
Standard
Apply and name properties of operations used as strategies to multiply and divide rational numbers.
7.NS.2.d
d.
Standard
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
7.NS.2.e
e.
Standard
Convert between equivalent fractions, decimals, or percents.
7.NS.3
Standard
Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.
Cluster
Know that there are numbers that are not rational, and approximate them by rational numbers.
8.NS.1
Standard
Classify real numbers as either rational (the ratio of two integers, a terminating decimal number, or a repeating decimal number) or irrational.
8.NS.2
Standard
Order real numbers, using approximations of irrational numbers, locating them on a number line.
For example, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
8.NS.3
Standard
Identify or write the prime factorization of a number using exponents. (L)
9.
Domain
Expressions and Equations
Cluster
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.1
Standard
Write and evaluate numerical expressions involving whole-number exponents.
For example, multiply by powers of 10 and products of numbers using exponents. (7³ = 7•7•7)
6.EE.2
Standard
Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.2.a
a.
Standard
Write expressions that record operations with numbers and with letters standing for numbers.
For example, express the calculation "Subtract y from 5" as 5 – y.
6.EE.2.b
b.
Standard
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.
For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
6.EE.2.c
c.
Standard
Evaluate expressions and formulas. Include formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order with or without parentheses. (Order of Operations)
6.EE.3
Standard
Apply the properties of operations to generate equivalent expressions. Model (e.g., manipulatives, graph paper) and apply the distributive, commutative, identity, and inverse properties with integers and variables by writing equivalent expressions.
For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x.
6.EE.4
Standard
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).
For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
Cluster
Reason about and solve one-variable equations and inequalities.
6.EE.5
Standard
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
For example: does 5 make 3x > 7 true?
6.EE.6
Standard
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.7
Standard
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
6.EE.8
Standard
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Cluster
Represent and analyze quantitative relationships between dependent and independent variables.
6.EE.9
Standard
Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Cluster
Use properties of operations to generate equivalent expressions.
7.EE.1
Standard
Apply properties of operations as strategies to add, subtract, factor, expand and simplify linear expressions with rational coefficients.
7.EE.2
Standard
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.
For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."
Cluster
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.3
Standard
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form and assess the reasonableness of answers using mental computation and estimation strategies.
For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
7.EE.4
Standard
Use variables to represent quantities in a real-world or mathematical problem, and construct multi-step equations and inequalities to solve problems by reasoning about the quantities.
7.EE.4.a
a.
Standard
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers.
For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
7.EE.4.b
b.
Standard
Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.
For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
Cluster
Work with radicals and integer exponents.
8.EE.1
Standard
Apply the properties (product, quotient, power, zero, negative exponents, and rational exponents) of integer exponents to generate equivalent numerical expressions.
For example, 3² × 3<sup>–5</sup> = 3-³ = 1/3³ = 1/27.
8.EE.2
Standard
Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.EE.3
Standard
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<sup>8</sup> and the population of the world as 7 × 10<sup>9</sup>, and determine that the world population is more than 20 times larger.
8.EE.4
Standard
Perform operations with numbers expressed in scientific notation, including problems where both standard notation and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.
Cluster
Understand the connections between proportional relationships, lines, and linear equations.
8.EE.5
Standard
Graph linear equations such as y = mx + b, interpreting m as the slope or rate of change of the graph and b as the y-intercept or starting value. Compare two different proportional relationships represented in different ways.
For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8.EE.6
Standard
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Cluster
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.7
Standard
Solve linear equations in one variable.
8.EE.7.a
a.
Standard
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.EE.7.b
b.
Standard
Solve linear equations with rational coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms.
8.EE.8
Standard
Analyze and solve systems of linear equations.
8.EE.8.a
a.
Standard
Show that the solution to a system of two linear equations in two variables is the intersection of the graphs of those equations because points of intersection satisfy both equations simultaneously.
8.EE.8.b
b.
Standard
Solve systems of two linear equations in two variables and estimate solutions by graphing the equations. Simple cases may be done by inspection.
For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
8.EE.8.c
c.
Standard
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.
10.
Domain
Functions
Cluster
Define, evaluate, and compare functions.
8.F.1
Standard
Understand that a function is a rule that assigns to each input (the domain) exactly one output (the range). The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
For example, use the vertical line test to determine functions and non-functions.
8.F.2
Standard
Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
8.F.3
Standard
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Cluster
Use functions to model relationships between quantities.
8.F.4
Standard
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
8.F.5
Standard
Given a verbal description between two quantities, sketch a graph. Conversely, given a graph, describe a possible real-world example.
For example, graph the position of an accelerating car or tossing a ball in the air.
11.
Domain
Statistics and Probability
Cluster
Develop understanding of statistical variability.
6.SP.1
Standard
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.
For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages.
6.SP.2
Standard
Understand that a set of data has a distribution that can be described by its center (mean, median, or mode), spread (range), and overall shape and can be used to answer a statistical question.
6.SP.3
Standard
Recognize that a measure of center (mean, median, or mode) for a numerical data set summarizes all of its values with a single number, while a measure of variation (range) describes how its values vary with a single number.
Cluster
Summarize and describe distributions.
6.SP.4
Standard
Display numerical data in plots on a number line, including dot or line plots, histograms and box (box and whisker) plots.
6.SP.5
Standard
Summarize numerical data sets in relation to their context, such as by:
6.SP.5.a
a.
Standard
Reporting the number of observations (occurrences).
6.SP.5.b
b.
Standard
Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
6.SP.5.c
c.
Standard
Giving quantitative measures of center (median and/or mean) and variability (interquartile range), as well as describing any overall pattern and any outliers with reference to the context in which the data were gathered.
6.SP.5.d
d.
Standard
Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
6.SP.6
Standard
Analyze whether a game is mathematically fair or unfair by explaining the probability of all possible outcomes. (L)
6.SP.7
Standard
Solve or identify solutions to problems involving possible combinations (e.g., if ice cream sundaes come in 3 flavors with 2 possible toppings, how many different sundaes can be made using only one flavor of ice cream with one topping?) (L)
Cluster
Use random sampling to draw inferences about a population.
7.SP.1
Standard
Understand that statistics can be used to gain information about a population by examining a reasonably sized sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
7.SP.2
Standard
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.
For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Cluster
Draw informal comparative inferences about two populations.
7.SP.3
Standard
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.
For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
7.SP.4
Standard
Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
Cluster
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.5
Standard
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
7.SP.6
Standard
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7.SP.7
Standard
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
7.SP.7.a
a.
Standard
Design a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
7.SP.7.b
b.
Standard
Design a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
7.SP.8
Standard
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
7.SP.8.a
a.
Standard
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
7.SP.8.b
b.
Standard
Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
7.SP.8.c
c.
Standard
Design and use a simulation to generate frequencies for compound events.
For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
Cluster
Investigate patterns of association in bivariate data.
8.SP.1
Standard
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.2
Standard
Explain why straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
8.SP.3
Standard
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and y-intercept.
For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
8.SP.4
Standard
Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects and use relative frequencies to describe possible association between the two variables.
For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
1.
Conceptual Category
Number and Quantity
Domain
The Real Number System
Cluster
Extend the properties of exponents to rational exponents.
N-RN.1
Standard
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
For example, we define 5<sup>1/3</sup> to be the cube root of 5 because we want (5<sup>1/3</sup>)³ = 5(<sup>1/3</sup>)³ to hold, so (5<sup>1/3</sup>)³ must equal 5.
N-RN.2
Standard
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
For example: Write equivalent representations that utilize both positive and negative exponents.
Cluster
Use properties of rational and irrational numbers.
N-RN.3
Standard
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Domain
Quantities
Cluster
Reason quantitatively and use units to solve problems.
N-Q.1
Standard
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
N-Q.2
Standard
Define appropriate quantities for the purpose of descriptive modeling.
N-Q.3
Standard
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Domain
The Complex Number System
Cluster
Perform arithmetic operations with complex numbers.
N-CN.1
Standard
Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.
N-CN.2
Standard
Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
N-CN.3
Standard
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Cluster
Represent complex numbers and their operations on the complex plane.
N-CN.4
Standard
(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
N-CN.5
Standard
(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
For example, (1 – √3i)³ = 8 because (1 – √3i) has modulus 2 and argument 120°.
N-CN.6
Standard
(+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
Cluster
Use complex numbers in polynomial identities and equations.
N-CN.7
Standard
Solve quadratic equations with real coefficients that have complex solutions.
N-CN.8
Standard
(+) Extend polynomial identities to the complex numbers.
For example, rewrite x² + 4 as (x + 2i)(x – 2i).
N-CN.9
Standard
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Domain
Vector and Matrix Quantities
Cluster
Represent and model with vector quantities.
N-VM.1
Standard
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
N-VM.2
Standard
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
N-VM.3
Standard
(+) Solve problems involving velocity and other quantities that can be represented by vectors.
Cluster
Perform operations on vectors.
N-VM.4
Standard
(+) Add and subtract vectors.
N-VM.4.a
a.
Standard
Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
N-VM.4.b
b.
Standard
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
N-VM.4.c
c.
Standard
Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
N-VM.5
Standard
(+) Multiply a vector by a scalar.
N-VM.5.a
a.
Standard
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v<sub>x</sub>, v<sub>y</sub>) = (cv<sub>x</sub>, cv<sub>y</sub>).
N-VM.5.b
b.
Standard
Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
Cluster
Perform operations on matrices and use matrices in applications.
N-VM.6
Standard
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
N-VM.7
Standard
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
N-VM.8
Standard
(+) Add, subtract, and multiply matrices of appropriate dimensions.
N-VM.9
Standard
(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
N-VM.10
Standard
(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
N-VM.11
Standard
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
N-VM.12
Standard
(+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.
2.
Conceptual Category
Algebra
Domain
Seeing Structure in Expressions
Cluster
Interpret the structure of expressions.
A-SSE.1
Standard
Interpret expressions that represent a quantity in terms of its context.
A-SSE.1.a
a.
Standard
Interpret parts of an expression, such as terms, factors, and coefficients.
A-SSE.1.b
b.
Standard
Interpret complicated expressions by viewing one or more of their parts as a single entity.
For example, interpret P(1+r)<sup>n</sup> as the product of P and a factor not depending on P.
A-SSE.2
Standard
Use the structure of an expression to identify ways to rewrite it.
For example, see x<sup>4</sup> – y<sup>4</sup> as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²).
Cluster
Write expressions in equivalent forms to solve problems.
A-SSE.3
Standard
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A-SSE.3.a
a.
Standard
Factor a quadratic expression to reveal the zeros of the function it defines.
For example, x² + 4x +3 = (x +3)(x +1).
A-SSE.3.b
b.
Standard
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
For example, x² + 4x + 3 = (x + 2)² - 1.
A-SSE.3.c
c.
Standard
Use the properties of exponents to transform expressions for exponential functions.
For example the expression 1.15t can be rewritten as (1.15<sup>1/12</sup>)<sup>12t</sup> ≈ 1.012<sup>12t</sup> to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
A-SSE.4
Standard
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
For example, calculate mortgage payments.*
Domain
Arithmetic with Polynomials and Rational Expressions
Cluster
Perform arithmetic operations on polynomials.
A-APR.1
Standard
Add, subtract, and multiply polynomials. Understand that polynomials form a system similar to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.
Cluster
Understand the relationship between zeros and factors of polynomials.
A-APR.2
Standard
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
A-APR.3
Standard
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Cluster
Use polynomial identities to solve problems.
A-APR.4
Standard
Prove polynomial identities and use them to describe numerical relationships.
For example, the polynomial identity (x² + y²)² = (x² – y²)² + (2xy)² can be used to generate Pythagorean triples.
A-APR.5
Standard
(+) Know and apply the Binomial Theorem for the expansion of (x + y)<sup>n</sup> in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.
Cluster
Rewrite rational expressions.
A-APR.6
Standard
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
A-APR.7
Standard
(+) Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system similar to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.
Domain
Creating Equations and Inequalities
Cluster
Create equations and inequalities that describe numbers or relationships.
A-CED.1
Standard
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
A-CED.2
Standard
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A-CED.3
Standard
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
For example, represent inequalities describing cost constraints in various situations.
A-CED.4
Standard
Rearrange formulas (literal equations) to highlight a quantity of interest, using the same reasoning as in solving equations.
For example, rearrange Ohm's law V = IR to highlight resistance R.
Domain
Reasoning with Equations and Inequalities
Cluster
Understand solving equations as a process of reasoning and explain the reasoning.
A-REI.1
Standard
Apply properties of mathematics to justify steps in solving equations in one variable.
A-REI.2
Standard
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Cluster
Solve equations and inequalities in one variable.
A-REI.3
Standard
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
A-REI.4
Standard
Solve quadratic equations in one variable.
A-REI.4.a
a.
Standard
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.
A-REI.4.b
b.
Standard
Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
Cluster
Solve systems of equations.
A-REI.5
Standard
Show that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
A-REI.6
Standard
Solve systems of linear equations exactly and approximately, e.g., with graphs or algebraically, focusing on pairs of linear equations in two variables.
A-REI.7
Standard
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3.
A-REI.8
Standard
(+) Represent a system of linear equations as a single matrix equation in a vector variable.
A-REI.9
Standard
(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
Cluster
Represent and solve equations and inequalities graphically.
A-REI.10
Standard
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
A-REI.11
Standard
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
A-REI.12
Standard
Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
3.
Conceptual Category
Functions
Domain
Interpreting Functions
Cluster
Understand the concept of a function and use function notation.
F-IF.1
Standard
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
F-IF.2
Standard
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.3
Standard
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥1.
Cluster
Interpret functions that arise in applications in terms of the context.
F-IF.4
Standard
For a function that models a relationship between two quantities,<ul><li>interpret key features of graphs and tables in terms of the quantities, and</li><li>sketch graphs showing key features given a verbal description of the relationship.</li></ul>Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
F-IF.5
Standard
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then negative numbers would be an inappropriate domain for the function.*
F-IF.6
Standard
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Cluster
Analyze functions using different representations.
F-IF.7
Standard
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.7.a
a.
Standard
Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-IF.7.b
b.
Standard
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
F-IF.7.c
c.
Standard
Graph polynomial functions, identifying zeros (using technology) or algebraic methods when suitable factorizations are available, and showing end behavior.
F-IF.7.d
d.
Standard
(+) Graph rational functions, identifying zeros and discontinuities (asymptotes/holes) using technology, and algebraic methods when suitable factorizations are available, and showing end behavior.
F-IF.7.e
e.
Standard
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F-IF.8
Standard
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F-IF.8,a
a.
Standard
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
F-IF.8.b
b.
Standard
Use the properties of exponents to interpret expressions for exponential functions.
For example, identify percent rate of change in functions such as y = (1.02)<sup>t</sup>, y = (0.97)<sup>t</sup>, y = (1.01)<sup>12t</sup>, y = (1.2)<sup>t/10</sup>, and classify them as representing exponential growth or decay.
F-IF.9
Standard
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Domain
Building Functions
Cluster
Build a function that models a relationship between two quantities.
F-BF.1
Standard
Write a function that describes a relationship between two quantities.
F-BF.1.a
a.
Standard
Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.1.b
b.
Standard
Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
F-BF.1.c
c.
Standard
(+) Compose functions.
For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
F-BF.2
Standard
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Cluster
Build new functions from existing functions.
F-BF.3
Standard
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
F-BF.4
Standard
Find inverse functions.
F-BF.4.a
a.
Standard
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
For example, f(x) = 2x³ for x > 0 or f(x) = (x + 1)/(x – 1) for x ≠ 1.
F-BF.4.b
b.
Standard
(+) Verify by composition that one function is the inverse of another.
F-BF.4.c
c.
Standard
(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
F-BF.4.d
d.
Standard
(+) Produce an invertible function from a non-invertible function by restricting the domain.
F-BF.5
Standard
(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Domain
Linear, Quadratic, and Exponential Models
Cluster
Construct and compare linear, quadratic, and exponential models and solve problems.
F-LE.1
Standard
Distinguish between situations that can be modeled with linear functions and with exponential functions.
F-LE.1.a
a.
Standard
Show that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F-LE.1.b
b.
Standard
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.1.c
c.
Standard
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
F-LE.2
Standard
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or input-output table of values.
F-LE.3
Standard
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
F-LE.4
Standard
For exponential models, express as a logarithm the solution to ab<sup>ct</sup> = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Cluster
Interpret expressions for functions in terms of the situation they model.
F-LE.5
Standard
Interpret the parameters in a linear or exponential function in terms of a context.
Domain
Trigonometric Functions
Cluster
Extend the domain of trigonometric functions using the unit circle.
F-TF.1
Standard
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
F-TF.2
Standard
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
F-TF.3
Standard
(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for π-x, π +x, and 2π –x in terms of their values for x, where x is any real number.
F-TF.4
Standard
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Cluster
Model periodic phenomena with trigonometric functions.
F-TF.5
Standard
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
F-TF.6
Standard
(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
F-TF.7
Standard
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
Cluster
Prove and apply trigonometric identities.
F-TF.8
Standard
Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to calculate trigonometric ratios.
F-TF.9
Standard
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
4.
Conceptual Category
Modeling
Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by an asterisk (*).
5.
Conceptual Category
Geometry
Domain
Congruence
Cluster
Experiment with transformations in the plane.
G-CO.1
Standard
Demonstrates understanding of key geometrical definitions, including angle, circle, perpendicular line, parallel line, line segment, and transformations in Euclidian geometry. Understand undefined notions of point, line, distance along a line, and distance around a circular arc.
G-CO.2
Standard
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
G-CO.3
Standard
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G-CO.4
Standard
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G-CO.5
Standard
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Cluster
Understand congruence in terms of rigid motions.
G-CO.6
Standard
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
G-CO.7
Standard
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G-CO.8
Standard
Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS, and HL) follow from the definition of congruence in terms of rigid motions.
Cluster
Prove geometric theorems.
G-CO.9
Standard
Using methods of proof including direct, indirect, and counter examples to prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
G-CO.10
Standard
Using methods of proof including direct, indirect, and counter examples to prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G-CO.11
Standard
Using methods of proof including direct, indirect, and counter examples to prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Cluster
Make geometric constructions.
G-CO.12
Standard
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
G-CO.13
Standard
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Domain
Similarity, Right Triangles, and Trigonometry
Cluster
Understand similarity in terms of similarity transformations.
G-SRT.1
Standard
Verify experimentally the properties of dilations given by a center and a scale factor:
G-SRT.1.a
a.
Standard
A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
G-SRT.1.b
b.
Standard
The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G-SRT.2
Standard
Given two figures, use the definition of similarity in terms of transformations to explain whether or not they are similar.
G-SRT.3
Objective
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Cluster
Prove theorems involving similarity.
G-SRT.4
Standard
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely.
G-SRT.5
Standard
Apply congruence and similarity properties and prove relationships involving triangles and other geometric figures.
Cluster
Define trigonometric ratios and solve problems involving right triangles.
G-SRT.6
Standard
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G-SRT.7
Standard
Explain and use the relationship between the sine and cosine of complementary angles.
G-SRT.8
Standard
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Cluster
Apply trigonometry to general triangles.
G-SRT.9
Standard
(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
G-SRT.10
Standard
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
G-SRT.11
Standard
(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Domain
Circles
Cluster
Understand and apply theorems about circles.
G-C.1
Standard
Prove that all circles are similar.
G-C.2
Standard
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
G-C.3
Standard
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
G-C.4
Standard
(+) Construct a tangent line from a point outside a given circle to the circle.
Cluster
Find arc lengths and areas of sectors of circles.
G-C.5
Standard
Use and apply the concepts of arc length and areas of sectors of circles. Determine or derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Domain
Expressing Geometric Properties with Equations
Cluster
Translate between the geometric description and the equation for a conic section.
G-GPE.1
Standard
Determine or derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
G-GPE.2
Standard
Determine or derive the equation of a parabola given a focus and directrix.
G-GPE.3
Standard
(+) Derive the equations of ellipses and hyperbolas given foci and directrices.
Cluster
Use coordinates to prove simple geometric theorems algebraically.
G-GPE.4
Standard
Perform simple coordinate proofs.
For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
G-GPE.5
Standard
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
G-GPE.6
Standard
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G-GPE.7
Standard
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
Domain
Geometric Measurement and Dimension
Cluster
Explain volume formulas and use them to solve problems.
G-GMD.1
Standard
Explain how to find the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
G-GMD.2
Standard
(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
G-GMD.3
Standard
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
For example: Solve problems requiring determination of a dimension not given.*
Cluster
Visualize relationships between two-dimensional and three-dimensional objects.
G-GMD.4
Standard
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Domain
Modeling with Geometry
Cluster
Apply geometric concepts in modeling situations.
G-MG.1
Standard
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
G-MG.2
Standard
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
G-MG.3
Standard
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
6.
Conceptual Category
Statistics and Probability
Domain
Interpreting Categorical and Quantitative Data
Cluster
Summarize, represent, and interpret data on a single count or measurement variable.
S-ID.1
Standard
Represent data with plots on the real number line (dot plots, histograms, and box plots).
S-ID.2
Standard
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
S-ID.3
Standard
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
For example: Justify why median price of homes or income is used instead of the mean.
S-ID.4
Standard
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
Cluster
Summarize, represent, and interpret data on two categorical and quantitative variables.
S-ID.5
Standard
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
S-ID.6
Standard
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
5-ID.6.a
a.
Standard
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
5-ID.6.b
b.
Standard
Informally assess the fit of a function by plotting and analyzing residuals.
For example: Describe solutions to problems that require interpolation and extrapolation.
5-ID.6.c
c.
Standard
Fit a linear function for a scatter plot that suggests a linear association.
Cluster
Interpret linear models.
S-ID.7
Standard
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
S-ID.8
Standard
Compute (using technology) and interpret the correlation coefficient of a linear fit.
S-ID.9
Standard
Distinguish between correlation and causation.
Domain
Making Inferences and Justifying Conclusions
Cluster
Understand and evaluate random processes underlying statistical experiments.
S-IC.1
Standard
Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
S-IC.2
Standard
Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
Cluster
Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
S-IC.3
Standard
Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
S-IC.4
Standard
Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
S-IC.5
Standard
Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
S-IC.6
Standard
Evaluate reports based on data.
Domain
Conditional Probability and the Rules of Probability
Cluster
Understand independence and conditional probability and use them to interpret data.
S-CP.1
Standard
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
S-CP.2
Standard
Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
S-CP.3
Standard
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
S-CP.4
Standard
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in 10th grade. Do the same for other subjects and compare the results.
S-CP.5
Standard
Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Cluster
Use the rules of probability to compute probabilities of compound events in a uniform probability model.
S-CP.6
Standard
Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
S-CP.7
Standard
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
S-CP.8
Standard
(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
S-CP.9
Standard
(+) Use permutations and combinations to compute probabilities of compound events and solve problems.
Domain
Using Probability to Make Decisions
Cluster
Calculate expected values and use them to solve problems.
S-MD.1
Standard
(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
S-MD.2
Standard
(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
S-MD.3
Standard
(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.
For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
S-MD.4
Standard
(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.
For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
Cluster
Use probability to evaluate outcomes of decisions.
S-MD.5
Standard
(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
5-MD.5.a
a.
Standard
Find the expected payoff for a game of chance.
For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
5-MD.5.b
b.
Standard
Evaluate and compare strategies on the basis of expected values.
For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
S-MD.6
Standard
(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
S-MD.7
Standard
(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).