Standards Database (Arizona K-12 Math)

Click this text to show the ‘Standards Database’ summary.

Below is a collection of all 369 standards (Kindergarten-Algebra2). They have been taken from each of the ADE’s Standard Placemats and compiled here.

The standards can be easily filtered by using any of the three filtering categories below (Grade/Domain/Topic). While both “Grade” and “Domain” are official categories of standards, “Topic” is one that I created as it’s my preferred method of thinking about standards, and I think the most intuitive to the average person.

Some standards are straightforward like this Kindergarten one: K.CC.A.1: Count to 100 by ones and by tens. (There is practically speaking only one goal for a student to meet). Other standards, however, are more complicated, like this one: 3.OA.D.8: [1] Solve two-step word problems using the four operations. [2] Represent these problems using equations with a letter standing for the unknown quantity. [3] Utilize understanding of the Order of Operations when there are no parentheses.

That’s all one single standard. For more complex standards like this, numbers in brackets [] have been added to separate distinct goals. These different goals will be highlighted in orange depending on which “Topic” checkbox is checked.

If a standard has nothing highlighted, the entire standard is relevant to the “Topic” selected. (Standards that have nothing relevant to the filter will be hidden entirely).

[*Some level of judgment has been used in deciding what is a separate goal vs. not, so there is some room for different interpretations. (For example, the Kindergarten standard mentioned earlier, K.CC.A.1: Count to 100 by ones and by tens, could be argued to be two separate goals – counting by ones and counting by tens, but for me these feel closely related enough [and small enough in scope] to be considered one goal).].

K.CC.A.1: Count to 100 by ones and by tens.
K.CC.A.2: Count forward from a given number other than one, within the known sequence (e.g., “Starting at the number 5, count up to 11.”).
K.CC.A.3: [1] Write numbers from 0 to 20. [2] Represent a number of objects with a written numeral 0 to 20 (with 0 representing a count of no objects).
K.CC.B.4: Understand the relationship between numbers and quantities; connect counting to cardinality.
a. [1] When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object (one to one correspondence).
b. [2] Understand that the last number name said tells the number of objects counted. [3] The number of objects is the same regardless of their arrangement or the order in which they were counted (cardinality).
c. [4] Understand that each successive number name refers to a quantity that is one larger (hierarchical inclusion).
K.CC.B.5: [1] Count to answer questions about “How many?” when 20 or fewer objects are arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; [2] given a number from 1 to 20, count out that many objects.
K.CC.C.6: Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group. (Include groups with up to ten objects.)
K.CC.C.7: Compare two numbers between 0 and 10 presented as written numerals
K.OA.A.1: Represent addition and subtraction concretely. See Table 1.
K.OA.A.2: [1] Solve addition and subtraction word problems and [2] add and subtract within 10. See Table 1.
K.OA.A.3: Decompose numbers less than or equal to 10 into pairs in more than one way (e.g., using fingers, objects, symbols, tally marks, drawings, expressions).
K.OA.A.4: For any number from 1 to 9, find the number that makes 10 when added to the given number (e.g., using fingers, objects, symbols, tally marks, drawings, or equation).
K.OA.A.5: Fluently add and subtract within 5.
K.NBT.A.1: Compose and decompose numbers from 11 to 19 into ten ones and additional ones by using objects, drawings and/or equations. Understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones (e.g., 18 = 10 + 8).
K.NBT.B.2: Demonstrate understanding of addition and subtraction within 10 using place value. See Table 1.
K.MD.A.1: Describe measurable attributes of a single object (e.g., length and weight).
K.MD.A.2: Directly compare two objects with a measurable attribute in common to see which object has “more of” or “less of” the attribute, and describe the difference (e.g., directly compare the length of 10 cubes to a pencil and describe one as longer or shorter).
K.MD.B.3: Classify objects into given categories; count the number in each category and sort the categories by count. (Note: limit category counts to be less than or equal to 10.)
K.G.A.1: [1] Describe objects in the environment using names of shapes, and [2] describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
K.G.A.2: Correctly name shapes regardless of their orientation or overall size (e.g., circle, triangle, square, rectangle, rhombus, trapezoid, hexagon, cube, cone, cylinder, sphere).
K.G.A.3: Identify shapes as two-dimensional (lying in a plane, flat) or three-dimensional (solid).
K.G.B.4: Analyze and compare two-dimensional 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/corners), and other attributes (e.g., having sides of equal length).
K.G.B.5: Model shapes in the world by building shapes from components (e.g., use sticks and clay balls) and drawing shapes.
K.G.B.6: Use simple shapes to form composite shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”
1.OA.A.1: [1] Use addition and subtraction within 20 [2] to solve word problems with unknowns in all positions (e.g., by using objects, drawings, and/or equations with a symbol for the unknown number to represent the problem). See Table 1.
1.OA.A.2: [1] Solve word problems that call for [2] addition of three whole numbers whose sum is less than or equal to 20 (e.g., by using objects, drawings, and/or equations with a symbol for the unknown number to represent the problem). See Table 1.
1.OA.B.3: [1] Apply properties of operations (commutative and associative properties of addition) as strategies to [2] add and subtract through 20. (Students need not use formal terms for these properties.)
1.OA.B.4: Understand subtraction as an unknown-addend problem within 20 (e.g., subtract 10 – 8 by finding the number that makes 10 when added to 8).
1.OA.C.5: Relate counting to addition and subtraction (e.g., by using counting on 2 to add 2).
1.OA.C.6: Fluently add and subtract within 10.
1.OA.D.7: Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false (e.g., Which of the following equations are true and which are false? 6 + 1 = 6 – 1, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2).
1.OA.D.8: Determine the unknown whole number in an addition or subtraction equation relating three whole numbers (e.g., determine the unknown number that makes the equation true in each of the equations 8 + = 11, 5 = – 3, 6 + 6 = ).
1.NBT.A.1: [1] Count to 120 by 1’s, 2’s, and 10’s starting at any number less than 100. [2] In this range, read and write numerals and represent a number of objects with a written numeral.
1.NBT.B.2: Understand that the two digits of a two-digit number represent groups of tens and ones. Understand the following as special cases:
a. [1] 10 can be thought of as a group of ten ones — called a “ten”.
b. [2] The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones
c. [3] 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.B.3: Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
1.NBT.C.4: Demonstrate understanding of addition within 100, connecting objects or drawings to strategies based on place value (including multiples of 10), properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written form. See Table 1.
1.NBT.C.5: Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count.
1.NBT.C.6: Subtract multiples of 10 in the range of 10 to 90 (positive or zero differences), using objects or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written form.
1.MD.A.1: [1] Order three objects by length; [2] compare the lengths of two objects indirectly by using a third object.
1.MD.A.2: 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. (Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.)
1.MD.B.3a: Tell and write time in hours and half-hours using analog and digital clocks.
1.MD.B.3b: Identify coins by name and value (pennies, nickels, dimes and quarters).
1.MD.C.4: [1] Organize, represent, and interpret data with up to three categories; [2] ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
1.G.A.1: [1] Distinguish between defining attributes (triangles are closed and 3 sided) versus non-defining attributes (color, orientation, overall size) for two-dimensional shapes; [2] build and draw shapes that possess defining attributes.
1.G.A.2: Compose two-dimensional shapes or three-dimensional shapes to create a composite shape.
1.G.A.3: [1] Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters. Describe the whole as two of, or four of the shares. [2] Understand that decomposing into more equal shares creates smaller shares.
2.OA.A.1: [1] Use addition and subtraction within 100 to [2] solve one and two-step word problems. [3] Represent a word problem as an equation with a symbol for the unknown. See Table 1.
2.OA.B.2: [1] Fluently add and subtract within 20. [2] By the end of Grade 2, know from memory all sums of two one-digit numbers.
2.OA.C.3: Determine whether a group of objects (up to 20) has an odd or even number of members (e.g., by pairing objects or counting them by 2’s).
2.OA.C.4: [1] Use addition to find the total number of objects arranged in rectangular arrays (with up to 5 rows and 5 columns). [2] Write an equation to express the total as a sum of equal addends.
2.NBT.A.1: Understand that the three digits of a three-digit number represent groups of hundreds, tens, and ones (e.g., 706 equals 7 hundreds, 0 tens, and 6 ones and also equals 70 tens and 6 ones).Understand the following as special cases:
a. [1] 100 can be thought of as a group of ten tens—called a “hundred.”
b. [2] 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.A.2: [1] Count within 1000; [2] skip count by 5’s, 10’s and 100’s.
2.NBT.A.3: [1] Read and write numbers up to 1000 using base-ten numerals, number names, and [2] expanded form.
2.NBT.A.4: Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
2.NBT.B.5: Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
2.NBT.B.6: Add up to three two-digit numbers using strategies based on place value and properties of operations.
2.NBT.B.7: Demonstrate understanding of addition and subtraction within 1000, connecting objects or drawings to strategies based on place value (including multiples of 10), properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written form. See Table 1.
2.NTB.B.8: Mentally add 10 or 100 to a given number in the range of 100 and 900, and mentally subtract 10 or 100 from a given number in the range of 100 and 900.
2.NBT.B.9: Explain why addition and subtraction strategies work, using place value and the properties of operations. (Explanations may be supported by drawings or objects.)
2.MD.A.1: Measure the length of an object by selecting and using appropriate tools (e.g., ruler, meter stick, yardstick, measuring tape).
2.MD.A.2: Measure the length of an object twice, using different standard length units for the two measurements; describe how the two measurements relate to the size of the unit chosen. Understand that depending on the size of the unit, the number of units for the same length varies.
2.MD.A.3: Estimate lengths using units of inches, feet, centimeters, and meters.
2.MD.A.4: Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.
2.MD.B.5: [1] Use addition and subtraction within 100 to [2] solve word problems involving lengths that are given in the same unit. See Table 1.
2.MD.B.6: [1] Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and [2] represent whole-number sums and differences within 100 on a number line diagram.
2.MD.C.7: Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
2.MD.C.8: Solve word problems involving collections of money, including dollar bills, quarters, dimes, nickels, and pennies. Record the total using $ and ¢ appropriately. See Table 1.
2.MD.D.9: Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.
2.MD.D.10: 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 the graph. See Table 1.
2.G.A.1: [1] Identify and describe specified attributes of two-dimensional and three-dimensional shapes, according to the number and shape of faces, number of angles, and the number of sides and/or vertices. [2] Draw two-dimensional shapes based on the specified attributes (e.g., triangles, quadrilaterals, pentagons, and hexagons).
2.G.A.2: Partition a rectangle into rows and columns of same-size rectangles and count to find the total number of rectangles.
2.G.A.3: [1] Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, fourths, half of, third of, fourth of, and describe the whole as two halves, three thirds, or four fourths. [2] Recognize that equal shares of identical wholes need not have the same shape.
3.OA.A.1: Interpret products of whole numbers as the total number of objects in equal groups (e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each).
3.OA.A.2: Interpret whole number quotients of whole numbers (e.g., interpret 56 ÷ 8 as the number of objects in each group when 56 objects are partitioned equally into 8 groups, or as a number of groups when 56 objects are partitioned into equal groups of 8 objects each). See Table 2.
3.OA.A.3: [1] Use multiplication and division within 100 to [2] solve word problems in situations involving equal groups, arrays, and measurement quantities. See Table 2.
3.OA.A.4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 x = 48, 5 = ÷ 3, 6 x 6 =. See Table 2.
3.OA.B.5: Apply properties of operations as strategies to multiply and divide. Properties include commutative and associative properties of multiplication and the distributive property. (Students do not need to use the formal terms for these properties.)
3.OA.B.6: Understand division as an unknown-factor problem (e.g., find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8).
3.OA.C.7: [1] Fluently multiply and divide within 100. [2] By the end of Grade 3, know from memory all multiplication products through 10 x 10 and division quotients when both the quotient and divisor are less than or equal to 10.
3.OA.D.8: [1] Solve two-step word problems using the four operations. [2] Represent these problems using equations with a letter standing for the unknown quantity. [3] Utilize understanding of the Order of Operations when there are no parentheses.
3.OA.D.9: Identify patterns in the addition table and the multiplication table and explain them using properties of operations (e.g. observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends).
3.OA.D.10: When solving problems, assess the reasonableness of answers using mental computation and estimation strategies including rounding.
3.NBT.A.1: Use place value understanding to round whole numbers to the nearest 10 or 100.
3.NBT.A.2: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
3.NBT.A.3: Multiply one-digit whole numbers by multiples of 10 in the range 10 to 90 using strategies based on place value and the properties of operations (e.g., 9 x 80, 5 x 60).
3.NF.A.1: [1] Understand a fraction (1/b) as the quantity formed by one part when a whole is partitioned into b equal parts; [2] understand a fraction a/b as the quantity formed by a parts of size 1/b.
3.NF.A.2: Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. [1] Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Understand that each part has size 1/b and that the end point of the part based at 0 locates the number 1/b on the number line.
b. [2] Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Understand that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line including values greater than 1.
c. [3] Understand a fraction 1/b as a special type of fraction that can be referred to as a unit fraction (e.g. 1/2, 1/4).
3.NF.A.3: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. [1] Understand two fractions as equivalent if they have the same relative size compared to 1 whole. b. [2] Recognize and generate simple equivalent fractions. [3] Explain why the fractions are equivalent. c. [4] Express whole numbers as fractions, and [5] recognize fractions that are equivalent to whole numbers. d. [6] Compare two fractions with the same numerator or the same denominator by reasoning about their size. Understand that comparisons are valid only when the two fractions refer to the same whole. Record results of comparisons with the symbols >, =, or <, and justify conclusions.
3.MD.A.1a: [1] Tell and write time to the nearest minute and [2] measure time intervals in minutes. [3] Solve word problems involving addition and subtraction of time intervals in minutes (e.g., representing the problem on a number line diagram).
3.MD.A.1b: Solve word problems involving money through $20.00, using symbols $, “.”, ₵.
3.MD.A.2: [1] Measure and estimate liquid volumes and masses of objects using metric units. (Excludes compound units such as cm3 and finding the geometric volume of a container.) [2] Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units. Excludes multiplicative comparison problems (problems involving notions of “times as much”). See Table 2.
3.MD.B.3: Create 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. See Table 1.
3.MD.B.4: Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch to the nearest quarter-inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.
3.MD.C.5: Understand area as an attribute of plane figures and understand concepts of area measurement.
a. [1] A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.
b. [2] A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
3.MD.C.6: Measure areas by counting unit squares (e.g., square cm, square m, square in, square ft, and improvised units).
3.MD.C.7: Relate area to the operations of multiplication and addition. a. [1] 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.
b. Multiply side lengths to find areas of rectangles with whole number side lengths in the context of solving [2] real-world and [3] mathematical problems, and [4] represent whole-number products as rectangular areas in mathematical reasoning.
c. [5] Use tiling to show that the area of a rectangle with whole number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
d. [6] Understand that rectilinear figures can be decomposed into non-overlapping rectangles and that the sum of the areas of these rectangles is identical to the area of the original rectilinear figure. [7] Apply this technique to solve problems in real-world contexts.
3.MD.C.8: Solve [1] real-world and [2] mathematical problems involving perimeters of plane figures and areas of rectangles, including finding the perimeter given the side lengths, finding an unknown side length. [3] Represent rectangles with the same perimeter and different areas or with the same area and different perimeters.
3.G.A.1: Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples quadrilaterals that do not belong to any of these subcategories.
3.G.A.2: Partition shapes into b parts with equal areas. Express the area of each part as a unit fraction 1/b of the whole. (Grade 3 expectations are limited to fractions with denominators b = 2,3,4,6,8.)
4.OA.A.1: [1] Represent verbal statements of multiplicative comparisons as multiplication equations. [2] Interpret a multiplication equation as a comparison (e.g., 35 is the number of objects in 5 groups, each containing 7 objects, and is also the number of objects in 7 groups, each containing 5 objects).
4.OA.A.2: [1] Multiply or divide within 1000 to [2] solve word problems involving multiplicative comparison (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison). See Table 2.
4.OA.A.3: [1] Solve multi-step word problems using the four operations, including problems in which remainders must be interpreted. [2] Understand how the remainder is a fraction of the divisor. [3]Represent these problems using equations with a letter standing for the unknown quantity.
4.OA.B.4: [1] Find all factor pairs for a whole number in the range 1 to 100 and [2] understand that a whole number is a multiple of each of its factors.
4.OA.C.5: Generate a number pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself and explain the pattern informally (e.g., 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).
4.OA.C.6: When solving problems, assess the reasonableness of answers using mental computation and estimation strategies including rounding.
4.NBT.A.1: Apply concepts of place value, multiplication, and division to understand that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
4.NBT.A.2: [1] Read and write multi-digit whole numbers using base-ten numerals, number names, and [2] expanded form. [3] Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
4.NBT.A.3: Use place value understanding to round multi-digit whole numbers to any place.
4.NBT.B.4: Fluently add and subtract multi-digit whole numbers using a standard algorithm.
4.NBT.B.5: [2] multiply two two-digit numbers, using strategies based on place value and the properties of operations. 4.NBT.B.6: Demonstrate understanding of division by finding whole number quotients and remainders with up to four-digit dividends and one-digit divisors.
4.NF.A.1: Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x 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 understand and generate equivalent fractions.
4.NF.A.2: Compare two fractions with different numerators and different denominators (e.g., by creating common denominators or numerators and by comparing to a benchmark fraction).
a. Understand that comparisons are valid only when the two fractions refer to the same size whole.
b. Record the results of comparisons with symbols >, =, or <, and justify the conclusions.
4.NF.B.3: Understand a fraction a/b with a > 1 as a sum of unit fractions (1/b).
a. [1] Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
b. [2] Decompose a fraction into a sum of fractions with the same denominator in more than one way (e.g., 3/8 = 1/8 + 1/8+1/8; 3/8 = 2/8 + 1/8; 2 1/8 = 1 + 1 + 1/8 + or 2 1/8 = 8/8 + 8/8 + 1/8).
c. [3] Add and subtract mixed numbers with like denominators (e.g., by using properties of operations and the relationship between addition and subtraction and/or by replacing each mixed number with an equivalent fraction).
d. [4] Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.
4.NF.B.4: Build fractions from unit fractions.
a. [1] Understand a fraction a/b as a multiple of a unit fraction 1/b. In general, a/b=a x 1/b.
b. Understand a multiple of a/b as a multiple of a unit fraction 1/b, and use this understanding to multiply a whole number by a fraction. In general, n x (a/b)=(nxa)/b.
c. [2] Solve word problems involving multiplication of a whole number by a fraction. 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?
4.NF.C.5: 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 (tenths) and 100 (hundredths). For example, express 3/10 as 30/100, and and 3/10 + 4/100 = 34/100. (Note: Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators, in general, is not a requirement at this grade.)
4.NF.C.6: [1] Use decimal notation for fractions with denominators 10 (tenths) or 100 (hundredths), and [2] locate these decimals on a number line.
4.NF.C.7: Compare two decimals to hundredths by reasoning about their size. Understand that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <.
4.MD.A.1: [1] Know relative sizes of measurement units within one system of units which could include km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. [2] Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit and in a smaller unit in terms of a larger unit. 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. [3] Generate a conversion table for feet and inches listing the number pairs (1,12), (2,24), (3,36).
4.MD.A.2: Use the four operations to solve word problems and problems in real-world context involving distances, intervals of time (hr, min, sec), liquid volumes, masses of objects, and money, including decimals and problems involving fractions with like denominators, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using a variety of representations, including number lines that feature a measurement scale.
4.MD.A.3: [1] Apply the area and perimeter formulas for rectangles in mathematical problems and [2] problems in real-world contexts including problems with unknown side lengths. See Table 2.
4.MD.B.4: [1] Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). [2] Solve problems involving addition and subtraction of fractions by using information presented in line plots.
4.MD.C.5: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
a. 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.
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
4.MD.C.6: [1] Measure angles in whole-number degrees using a protractor. [2] Sketch angles of specified measure.
4.MD.C.7: [1] Understand angle measures as additive. (When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts.) [2] Solve addition and subtraction problems to find unknown angles on a diagram within mathematical problems [3] as well as problems in real-world contexts.
4.G.A.1: [1] Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. [2] Identify these in two-dimensional figures.
4.G.A.2: Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size (e.g., understand right triangles as a category, and identify right triangles).
4.G.A.3: Recognize a line of symmetry for a two-dimensional 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.
5.OA.A.1: Use parentheses and brackets in numerical expressions, and evaluate expressions with these symbols (Order of Operations).
5.OA.A.2: [1] Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them (e.g., express the calculation “add 8 and 7, then multiply by 2” as 2 x (8 + 7). [2] Recognize that 3 x (18,932 + 921) is three times as large as 18,932 + 921, without having to calculate the indicated sum or product).
5.OA.B.3: Generate two numerical patterns using two given rules (e.g., generate terms in the resulting sequences). Identify and explain the 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 (e.g., 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).
5.OA.B.4: Understand primes have only two factors and decompose numbers into prime factors.
5.NBT.A.1: Apply concepts of place value, multiplication, and division to understand 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.A.2: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.
5.NBT.A.3: Read, write, and compare decimals to thousandths.
a. [1] Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.
b. [2] Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the the results of comparisons.
5.NBT.A.4: Use place value understanding to round decimals to any place.
5.NBT.B.5: Fluently multiply multi-digit whole numbers using a standard algorithm.
5.NBT.B.6: Apply and extend understanding of division to find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors.
5.NBT.B.7: Add, subtract, multiply, and divide decimals to hundredths, connecting objects or drawings to strategies based on place value, properties of operations, and/or the relationship between operations. Relate the strategy to a written form.
5.NF.A.1: 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 (e.g., 2/3 + 5/4 = 8/12 + 15/12 = 23/12).
5.NF.A.2: Solve [1] word problems involving [2] addition and subtraction of fractions referring to the same whole, including cases of unlike denominators by using a variety of representations, equations, and visual models to represent the problem. [3] Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers (e.g. recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2).
5.NF.B.3: [1] Interpret a fraction as the number that results from dividing the whole number numerator by the whole number denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers. 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.B.4: Apply and extend previous understandings of multiplication to multiply a fraction by a whole number and a fraction by a fraction.
a. [1] Interpret the product (a/b) x q as a parts of a partition of q into b equal parts. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation.
b. [2] Interpret the product of a fraction multiplied by a fraction (a/b) x (c/d). Use a visual fraction model and create a story context for this equation. For example, use a visual fraction model to show (2/3) x (4/5) = 8/15, and create a story context for this equation. In general, (a/b) x (c/d) = ac/bd.
c. [3] 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.B.5: Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number; 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=(nxa)/(nxb) to the effect of multiplying a/b by 1.
5.NF.B.6: [1] Solve problems in real-world contexts involving [2] multiplication of fractions, including mixed numbers, by using a variety of representations including equations and models.
5.NF.B.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
a. [1] Interpret division of a unit fraction by a non-zero whole number, and [2] compute such quotients. Use the relationship between multiplication and division to justify conclusions.
b. [3] Interpret division of a whole number by a unit fraction, and [4] 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 justify conclusions (e.g., 4 ÷ (1/5) = 20 because 20 x (1/5) = 4).
c. [5] Solve problems in real-world context involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, using a variety of representations.
5.MD.A.1: [1] Convert among different-sized standard measurement units within a given measurement system, and [2] use these conversions in solving multi-step, real-world problems.
5.MD.B.2: [1] Make a line plot to display a data set of measurements in fractions of a unit (1/8, 1/2, 3/4). [2] Use operations on fractions for this grade to 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.C.3: Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
5.MD.C.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
5.MD.C.5: Relate volume to the operations of multiplication and addition and solve mathematical problems and problems in real-world contexts involving volume.
a. [1] Find the volume of a right rectangular prism with whole-number side lengths by packing it with 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. [2] Represent threefold whole-number products as volumes (e.g., to represent the associative property of multiplication).
b. [3] Understand and use the formulas V = l x w x h and V = B x h, where in this case B is the area of the base (B = l x w), for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths to solve mathematical problems and [4] problems in real-world contexts.
c. [5] Understand volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms, applying this technique to solve mathematical problems and [6] problems in real-world contexts.
5.G.A.1: [1] Understand and describe a coordinate system as perpendicular number lines, called axes, that intersect at the origin (0 , 0). [2] Identify a given point in the first quadrant of the coordinate plane using an ordered pair of numbers, called coordinates. Understand that the first number (x) indicates the distance traveled on the horizontal axis, and the second number (y) indicates the distance traveled on the vertical axis.
5.G.A.2: Represent [1] real-world and [2] mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
5.G.B.3: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
5.G.B.4: Classify two-dimensional figures in a hierarchy based on properties.
6.RP.A.1: [1] Understand the concept of a ratio as comparing two quantities multiplicatively or joining/composing the two quantities in a way that preserves a multiplicative relationship. [2] Use ratio language to describe a ratio relationship between two quantities. For example, “There were 2/3 as many men as women at the concert.”
6.RP.A.2: Understand the concept of a unit rate a/b associated with a ratio a : b with b ≠ 0, and use rate language (e.g., for every, for each, for each 1, per) in the context of a ratio relationship. (Complex fraction notation is not an expectation for unit rates in this grade level.)
6.RP.A.3: Use ratio and rate reasoning to solve mathematical problems and problems in real-world context (e.g., by reasoning about data collected from measurements, tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).
a. [1] 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.
b. [2] Solve unit rate problems including those involving unit pricing and constant speed.
c. [3] Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity).
d. [4] Solve percent problems with the unknown in all positions of the equation.
e. [5] Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
6.NS.A.1: Interpret and compute quotients of fractions to solve [1] mathematical problems and [2] problems in real-world context involving division of fractions by fractions 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.
6.NS.B.2: Fluently divide multi-digit numbers using a standard algorithm.
6.NS.B.3: Fluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation.
6.NS.B.4: Use previous understanding of factors to find the greatest common factor and the least common multiple.
a. [1] Find the greatest common factor of two whole numbers less than or equal to 100.
b. [2] Find the least common multiple of two whole numbers less than or equal to 12.
c. [3] Use the distributive property to express a sum of two whole numbers 1 to 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).
6.NS.C.5: [1] Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. [2] Use positive and negative numbers to represent quantities in real-world context, explaining the meaning of 0 in each situation.
6.NS.C.6: Understand a rational number can be represented 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.
a. [1] Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; [2] recognize that the opposite of the opposite of a number is the number itself and that 0 is its own opposite.
b. [3] Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; [4] recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
c. [5] Find and position integers and other rational numbers on a horizontal or vertical number line diagram; [6] find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.C.7: Understand ordering and absolute value of rational numbers.
a. [1] Interpret statements of inequality as statements about the relative position of two numbers on a number line.
b. [2] Write, interpret, and explain statements of order for rational numbers in real-world context.
c. [3] Understand the absolute value of a rational number as its distance from 0 on the number line; [4] interpret absolute value as magnitude for a positive or negative quantity in real-world context.
d. [5] Distinguish comparisons of absolute value from statements about order in mathematical problems and problems in real-world context.
6.NS.C.8: Solve [1] mathematical problems and [2] problems in real-world context 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.
6.EE.A.1: Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.2: Write, read, and evaluate algebraic expressions.
a. [1] Write expressions that record operations with numbers and variables.
b. [2] Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, and coefficient); [3] view one or more parts of an expression as a single entity.
c. [4] Evaluate expressions given specific values of their variables. Include expressions that arise from formulas used to solve mathematical problems and [5] problems in real-world context. [6] Perform arithmetic operations, including those involving [7] whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
6.EE.A.3: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x.
6.EE.A.4: Identify when two expressions are equivalent. For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
6.EE.B.5: [1] Understand solving an equation or inequality as a process of reasoning to find the value(s) of the variables that make that equation or inequality true. [2] Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.B.6: Use variables to represent numbers and write expressions when solving [1] mathematical problems and problems in real-world context; [3] understand that a variable can represent an unknown number or any number in a specified set.
6.EE.B.7: Solve [1] mathematical problems and [2] problems in real-world context by writing and solving equations of the form x + p = q, x – p = q, px = q, and x/p = q for cases in which p, q and x are all non-negative rational numbers.
6.EE.B.8: [1] Write an inequality of the form x > c, x < c, x ≥ c, or x ≤ c to represent a constraint or condition to solve mathematical problems and [2] problems in real-world context. [3] Recognize that inequalities have infinitely many solutions; represent solutions of such inequalities on number lines.
6.EE.C.9: [1] Use variables to represent two quantities that change in relationship to one another to solve mathematical problems and [2] problems in real-world context. [3] Write an equation to express one quantity (the dependent variable) in terms of the other quantity (the independent variable). [4] Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
6.G.A.1: [1] Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques to solve mathematical problems and [2] problems in real-world context.
6.G.A.2: [1] Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. [2] Understand and use the formula V = B · h, where in this case, B is the area of the base (B = l x w) to find volumes of right rectangular prisms with fractional edge lengths in mathematical problems and [3] problems in real-world context.
6.G.A.3: [1] Draw polygons in the coordinate plane given coordinates for the vertices; [2] use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques to solve mathematical problems and [3] problems in a real-world context.
6.G.A.4: [1] Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques to solve mathematical problems and [2] problems in real-world context.
6.SP.A.1: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for variability 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.A.2: Understand that a set of data collected to answer a statistical question has a distribution whose general characteristics can be described by its center, spread, and overall shape.
6.SP.A.3: [1] Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, [2] while a measure of variation uses a single number to describe the spread of the data set.
6.SP.B.4: Display and interpret numerical data by creating plots on a number line including [1] histograms, [2] dot plots, and [3] box plots.
6.SP.B.5: Summarize numerical data sets in relation to their context by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation including how it was measured and its units of measurement.
c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
d. 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.
7.RP.A.1: Compute unit rates associated with ratios involving both simple and complex fractions, including ratios of quantities measured in like or different units.
7.RP.A.2: Recognize and represent proportional relationships between quantities.
a. [1] 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).
b. [2] Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
c. [3] Represent proportional relationships by equations. 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.
d. [4] 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.A.3: Use proportional relationships to solve multi-step ratio and percent problems (e.g., simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error).
7.NS.A.1: Add and subtract integers and other rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
a. [1] Describe situations in which opposite quantities combine to make 0.
b. [2] Understand 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. Show that a number and its opposite have a sum of 0 (are additive inverses). [3] Interpret sums of rational numbers by describing real-world context.
c. [4] Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). [5] 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 context.
d. [6] Apply properties of operations as strategies to add and subtract rational numbers.
7.NS.A.2: Multiply and divide integers and other rational numbers.
a. [1] 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 [2] rules for multiplying signed numbers. [3] Interpret products of rational numbers by describing real-world context.
b. [4] 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. [5] If p and q are integers, then –(p/q) = (–p)/q = p/(–q). [6] Interpret quotients of rational numbers by describing real-world context.
c. [7] Apply properties of operations as strategies to multiply and divide rational numbers.
d. [8] Convert a rational number to decimal form using long division; know that the decimal form of a rational number terminates in 0’s or eventually repeats.
7.NS.A.3: Solve [1] mathematical problems and [2] problems in real-world context involving the four operations with rational numbers. [3] Computations with rational numbers extend the rules for manipulating fractions to complex fractions where a/b ÷ c/d when a,b,c,and d are all integers and b,c, and d ≠ 0.
7.EE.A.1: Apply properties of operations as strategies to [1] add, subtract, [2] factor, and [3] expand linear expressions with rational coefficients.
7.EE.A.2: Rewrite an expression in different forms, and understand the relationship between the different forms and their meanings in a problem context. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
7.EE.B.3: Solve multi-step [1] mathematical problems and [2] problems in real-world context posed with positive and negative rational numbers in any form. Convert between forms as appropriate and assess the reasonableness of answers. 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 per hour.
7.EE.B.4: Use variables to represent quantities in mathematical problems and problems in real-world context, and construct simple equations and inequalities to solve problems.
a. [1] 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. [2] Solve equations of these forms fluently. [3] Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
b. Solve [4] word problems leading to [5] inequalities of the form px+q > r or px+q < r, where p, q, and r are rational numbers. [6] Graph the solution set of the inequality and [7] interpret it in the context of the problem.
7.G.A.1: Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.A.2: Draw geometric shapes with given conditions using a variety of methods. 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.A.3: Describe the two-dimensional figures that result from slicing three-dimensional figures.
7.G.B.4: [1] Understand and use the formulas for the area and circumference of a circle to solve problems; [2] give an informal derivation of the relationship between the circumference and area of a circle.
7.G.B.5: Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle in a figure.
7.G.B.6: Solve [1] mathematical problems and [2] problems in a real-world context involving area of two-dimensional objects composed of triangles, quadrilaterals, and other polygons. Solve [3] mathematical problems and [4] problems in real-world context involving volume and surface area of three-dimensional objects composed of cubes and right prisms.
7.SP.A.1: [1] Understand that statistics can be used to gain information about a population by examining a sample of the population; [2] generalizations about a population from a sample are valid only if the sample is representative of that population. [3] Understand that random sampling tends to produce representative samples and support valid inferences.
7.SP.A.2: [1] Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. [2] 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. [3] Gauge how far off the estimate or prediction might be.
7.SP.B.3: 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.B.4: 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.
7.SP.C.5: 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.C.6: 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.C.7: 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.
a. [1] Develop 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.
b. [2] Develop 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?
8.NS.A.1: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion. Know that numbers whose decimal expansions do not terminate in zeros or in a repeating sequence of fixed digits are called irrational.
8.NS.A.2: [1] Use rational approximations of irrational numbers to compare the size of irrational numbers. [2] Locate them approximately on a number line diagram, and estimate their values.
8.NS.A.3: Understand that given any two distinct rational numbers, a < b, there exist a rational number c and an irrational number d such that a < c < b and a < d < b. Given any two distinct irrational numbers, a < b, there exist a rational number c and an irrational number d such that a < c < b and a < d < b.
8.EE.A.1: Understand and apply the properties of integer exponents to generate equivalent numerical expressions.
8.EE.A.2: [1] Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3= p, where p is a positive rational number. [2] Know that √2 is irrational.
a. [3] Evaluate square roots of perfect squares less than or equal to 225.
b. [4] Evaluate cube roots of perfect cubes less than or equal to 1000.
8.EE.A.3: [1] Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, [2] and express how many times larger or smaller one is than the other.
8.EE.A.4: [1] Perform operations with numbers expressed in scientific notation including problems where both decimal and scientific notation are used. [2] Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.
8.EE.B.5: [1] Graph proportional relationships interpreting the unit rate as the slope of the graph. [2] 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.B.6: [1] 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. [2] 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 (0, b).
8.EE.C.7: [1] Fluently solve linear equations and inequalities in one variable.
a. [2] Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solution. 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).
b. [3] Solve linear equations and inequalities with rational number coefficients, including solutions that require expanding expressions using the distributive property and collecting like terms.
8.EE.C.8: Analyze and solve pairs of simultaneous linear equations.
a. [1] Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
b. [2] Solve systems of two linear equations in two variables algebraically, and [3] estimate solutions by graphing the equations including cases of no solution and infinite number of solutions. [4] Solve simple cases by inspection.
c. Solve mathematical problems and [5] problems in real-world context leading to two linear equations in two variables.
8.F.A.1: [1] Understand that a function is a rule that assigns to each input exactly one output. [2] The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.).
8.F.A.2: 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.A.3: 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 in not linear because its graph contains the points (1,1), (2,4), and (3,9) which are not on a straight line.
8.F.B.4: [1] Given a description of a situation, generate a function to model a linear relationship between two quantities. [2] Determine the rate of change and initial value of the function from a description of a relationship [3] or from two (x, y) values, including reading these from a table or a graph. [4] Track how the values of the two quantities change together. [5] Interpret the rate of change and initial value of a linear function in terms of the situation it models, its graph, or its table of values.
8.F.B.5: [1] Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). [2] Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
8.G.A.1: Verify experimentally the properties of rotations, reflections, and translations. Properties include: lines are taken to lines, line segments are taken to line segments of the same length, angles are taken to angles of the same measure, parallel lines are taken to parallel lines.
8.G.A.2: Understand that a two-dimensional figure is congruent to another if one can be obtained from the other by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that demonstrates congruence.
8.G.A.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.A.4: Understand that a two-dimensional figure is similar to another if, and only if, one can be obtained from the other by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that demonstrates similarity.
8.G.A.5: Use informal arguments to establish facts [1] about the angle sum and exterior angle of triangles, [2] about the angles created when parallel lines are cut by a transversal, and [3] the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
8.G.B.6: Understand the Pythagorean Theorem and its converse.
8.G.B.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in [1] real-world context and [2] mathematical problems in two and three dimensions.
8.G.B.8: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
8.G.C.9: Understand and use formulas for volumes of cones, cylinders and spheres and use them to solve [1] real-world context and mathematical problems.
8.SP.A.1: Construct and interpret scatter plots for bivariate measurement data to investigate and describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.A.2: Know that 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.A.3: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
8.SP.A.4: [1] Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. [2] Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. [3] Use relative frequencies calculated for rows or columns to describe possible association between the two variables.
8.SP.B.5: [1] Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. [2] Understand that the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. b. [3] Represent sample spaces for compound events using organized lists, tables, tree diagrams and other methods. [4] Identify the outcomes in the sample space which compose the event. c. [5] Design and use a simulation to generate frequencies for compound events.
A1.N-RN.B.3 : [1] Explain why the sum or product of two rational numbers is rational; [2] that the sum of a rational number and an irrational number is irrational; and [3] that the product of a nonzero rational number and an irrational number is irrational.
A1.N-Q.A.1 : [1] Use units as a way to understand problems and to guide the solution of multi-step problems; [2] choose an interpret units consistently in formulas; [3] choose and interpret the scale and the origin in graphs and data displays, include utilizing real world context.
A1.N-Q.A.2: Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.
A1.N-Q.A.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.
A1.A-SSE.A.1 : Interpret expressions that represent a quantity in terms of its context.
a. [1] Interpret parts of an expression, such as terms, factors, and coefficients.
b. [2] Interpret expressions by viewing one or more of their parts as a single entity.
A1.A-SSE.A.2: Use structure to identify ways to rewrite numerical and polynomial expressions. Focus on polynomial multiplication and factoring patterns.
A1.A-SSE.B.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
a. [1] Factor a quadratic expression to reveal the zeros of the function it defines.
b. [2] Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
A1.A-APR.A.1 : Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
A1.A-APR.B.3 : [1] Identify zeros of polynomials when suitable factorizations are available and [2] use the zeros to construct a rough graph of the function defined by the polynomial. Focus on quadratic and cubic polynomials in which linear and quadratic factors are available.
A1.A-CED.A.1: Create equations and inequalities in one variable and use them to solve problems. Include problem-solving opportunities utilizing real-world context. Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).
A1.A-CED.A.2: [1] Create equations in two or more variables to represent relationships between quantities; [2] graph equations on coordinate axes with labels and scales.
A1.A-CED.A.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
A1.A-CED.A.4: Rearrange formulas 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.
A1.A-REI.A.1: Explain each step in solving linear and quadratic equations as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A1.A-REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
A1.A-REI.B.4: Solve quadratic equations in one variable.
a. [1] Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – k)2 = q that has the same solutions. [2] Derive the quadratic formula from this form.
b. Solve quadratic equations by [3] inspection (e.g., x2 = 49), [4] taking square roots, [5] completing the square, [6] the quadratic formula and [7] factoring, as appropriate to the initial form of the equation. Focus on solutions for quadratic equations that have real roots. Include cases that recognize when a quadratic equation has no real solutions.
A1.A-REI.C.5: Prove 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.
A1.A-REI.C.6: Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables. Include problem solving opportunities utilizing real-world context.
A1.A-REI.D.10: 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.
A1.A-REI.D.11: [1] 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); [2] find the solutions approximately (e.g., using technology to graph the functions, make tables of values, or find successive approximations). Focus on cases where f(x) and/or g(x) are linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).
A1.A-REI.D.12: [1] 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 [2] graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
A1.F-IF.A.1 : 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).
A1.F-IF.A.2: [1] Evaluate a function for inputs in the domain and [2] interpret statements that use function notation in terms of a context.
A1.F-IF.A.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
A1.F-IF.B.4: For a function that models a relationship between two quantities, [1] interpret key features of graphs and tables in terms of the quantities, and [2] sketch graphs showing key features given a verbal description of the relationship. Include problem-solving opportunities utilizing real-world context. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums. Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).
A1.F-IF.B.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
A1.F-IF.B.6: [1] Calculate and interpret the average rate of change of a continuous function (presented symbolically or as a table) on a closed interval. [2] Estimate the rate of change from a graph. Include problem-solving opportunities utilizing real-world context. Focus on linear, quadratic, exponential and piecewisedefined functions (limited to absolute value and step).
A1.F-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).
A1.F-IF.C.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square of a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
A1.F-IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).
A1.F-BF.A.1 : [1] Write a function that describes a relationship between two quantities. [2] Determine an explicit expression, a recursive process, or steps for calculation from real-world context. Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).
A1.F-BF.B.3: [1] Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), and f(x+k) for specific values of k (both positive and negative); [2] find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph. Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).
A1.F-LE.A.1 : Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. [1] Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
b. [2] Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. [3] Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
A1.F-LE.A.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given [1] a graph, [2] a description of a relationship, or [3] input/output pairs.
A1.F-LE.A.3: Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.
A1.F-LE.B.5: Interpret the parameters in a linear or exponential function with integer exponents utilizing real world context.
A1.S-ID.A.1: Represent real-value data with plots for the purpose of comparing two or more data sets.
A1.S-ID.A.2: 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.
A1.S-ID.A.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of outliers if present.
A1.S-ID.B.5: [1] Summarize categorical data for two categories in two-way frequency tables. [2] Interpret relative frequencies in the context of the data, including joint, marginal, and conditional relative frequencies. [3] Recognize possible associations and trends in the data.
A1.S-ID.B.6: Represent data on two quantitative variables on a scatter plot and describe how the quantities are related.
a. [1] Fit a function to the data; [2] use functions fitted to data to solve problems in the context of the data. Focus on linear models.
b. [3] Informally assess the fit of a function by plotting and analyzing residuals
A1.S-ID.C.7: Interpret the slope as a rate of change and the constant term of a linear model in the context of the data.
A1.S-ID.C.8: Compute and interpret the correlation coefficient of a linear relationship.
A1.S-ID.C.9: Distinguish between correlation and causation.
A1.S-CP.A.1: Describe events as subsets of a sample space using characteristics of the outcomes, or as unions, intersections, or complements of other events.
A1.S-CP.A.2: Use the Multiplication Rule for independent events to 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.
G.N-Q.A.1 : [1] Use units as a way to understand problems and to guide the solution of multi-step problems; [2] choose and interpret units consistently in formulas; [3] choose and interpret the scale and the origin in graphs and data displays, include utilizing real-world context.
G.N-Q.A.2: Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing realworld context.
G.N-Q.A.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.
G.G-CO.A.1 : Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G.G-CO.A.2: [1] Represent and describe transformations in the plane as functions that take points in the plane as inputs and give other points as outputs. [2] Compare transformations that preserve distance and angle to those that do not.
G.G-CO.A.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G.G-CO.A.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.G-CO.A.5: [1] Given a geometric figure and a rotation, reflection, or translation draw the transformed figure. [2] Specify a sequence of transformations that will carry a given figure onto another.
G.G-CO.B.6 : [1] Use geometric definitions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; [2] given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
G.G-CO.B.7: 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.G-CO.B.8: Explain how the criteria for triangle congruence (ASA, AAS, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G.G-CO.C.9 : 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.G-CO.C.10: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangle 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.G-CO.C.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and rectangles are parallelograms with congruent diagonals.
G.G-CO.D.12 : Make formal geometric constructions with a variety of tools and methods. Constructions include: copying segments; copying angles; bisecting segments; bisecting angles; 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.G-CO.D.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle; with a variety of tools and methods.
G.G-SRT.A.1: Verify experimentally the properties of dilations given by a center and a scale factor:
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.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.G-SRT.A.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G.G-SRT.A.3: Use the properties of similarity transformations to establish the AA, SAS, and SSS criterion for two triangles to be similar.
G.G-SRT.B.4: Prove theorems about triangles. Theorems include: an interior line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G.G-SRT.B.5: Use congruence and similarity criteria to prove relationships in geometric figures and solve problems utilizing real-world context.
G.G-SRT.C.6: 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.G-SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles.
G.G-SRT.C.8: Use trigonometric ratios (including inverse trigonometric ratios) and the Pythagorean Theorem to find unknown measurements in right triangles utilizing real-world context.
G.G-C.A.1: Prove that all circles are similar.
G.G-C.A.2: 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.G-C.A.3: Construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle.
G.G-C.B.5: [1] 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; [2] derive the formula for the area of a sector. [3] Convert between degrees and radians.
G.G-GPE.A.1: [1] Derive the equation of a circle of given center and radius using the Pythagorean Theorem; [2] complete the square to find the center and radius of a circle given by an equation.
G.G-GPE.B.4: Use coordinates to algebraically prove or disprove geometric relationships. Relationships include: proving or disproving geometric figures given specific points in the coordinate plane; and proving or disproving if a specific point lies on a given circle.
G.G-GPE.B.5: [1] Prove the slope criteria for parallel and perpendicular lines and [2] use them to solve geometric problems, including finding the equation of a line parallel or perpendicular to a given line that passes through a given point.
G.G-GPE.B.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G.G-GPE.B.7: Use coordinates to compute [1] perimeters of polygons and [2] areas of triangles and rectangles.
G.G-GMD.A.1: Analyze and verify the formulas for the volume of a cylinder, pyramid, and cone.
G.G-GMD.A.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems utilizing real-world context.
G.G-GMD.B.4: [1] Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and [2] identify three-dimensional objects generated by rotations of two-dimensional objects.
G.G-MG.A.1: Use geometric shapes, their measures, and their properties to describe objects utilizing real-world context.
G.G-MG.A.2: Apply concepts of density based on area and volume in modeling situations utilizing real-world context.
G.G-MG.A.3: Apply geometric methods to solve design problems utilizing real-world context.
A2.N-RN.A.1: Explain how the definition 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.
A2.N-RN.A.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.
A2.N-Q.A.1: [1] Use units as a way to understand problems and to guide the solution of multi-step problems; [2] choose and interpret units consistently in formulas; [3] choose and interpret the scale and the origin in graphs and data displays, include utilizing real-world context.
A2.N-Q.A.2: Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing realworld context.
A2.N-Q.A.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.
A2.N-CN.A.1: Apply the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Write complex numbers in the form (a+bi) with a and b real.
A2.N-CN.C.7: Solve quadratic equations with real coefficients that have complex solutions.
A2.A-SSE.A.2: Use structure to identify ways to rewrite polynomial and rational expressions. Focus on polynomial operations and factoring patterns.
A2.A-SSE.B.3: [1] Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Include problem-solving opportunities utilizing real-world context and focus on expressions with rational exponents. [2] Use the properties of exponents to transform expressions for exponential functions.
A2.A-SSE.B.4: 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.
A2.A-APR.B.2: Know and apply the Remainder and Factor 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).
A2.A-APR.B.3: [1] Identify zeros of polynomials when suitable factorizations are available, and [2] use the zeros to construct a rough graph of the function defined by the polynomial. Focus on quadratic, cubic, and quartic polynomials including polynomials for which factors are not provided.
A2.A-APR.C.4: [1] Prove polynomial identities and [2] use them to describe numerical relationships.
A2.A-APR.D.6: Rewrite 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.
A2.A-CED.A.1: Create equations and inequalities in one variable and use them to solve problems. Include problem-solving opportunities utilizing real-world context. Focus on equations and inequalities arising from linear, quadratic, rational, and exponential functions.
A2.A-REI.A.1: Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Extend from quadratic equations to [1] rational and [2] radical equations.
A2.A-REI.A.2: Solve [1] rational and [2] radical equations in one variable and [3] give examples showing how extraneous solutions may arise.
A2.A-REI.B.4: Fluently solve quadratic equations in one variable. 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.
A2.A-REI.C.7: Solve a system consisting of a linear equation and a quadratic equation in two variables [1] algebraically and [2] graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.
A2.A-REI.D.11: [1] 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); [2] find the solutions approximately (e.g., using technology to graph the functions, make tables of values, or find successive approximations). Include problems in real-world context. Extend from linear, quadratic, and exponential functions to cases where f(x) and/or g(x) are polynomial, rational, exponential, and logarithmic functions.
A2.F-IF.B.4: For a function that models a relationship between two quantities, [1] interpret key features of graphs and tables in terms of the quantities, and [2] sketch graphs showing key features given a verbal description of the relationship. Include problem solving opportunities utilizing a real-world context. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Functions include linear, quadratic, exponential, polynomial, logarithmic, rational, sine, cosine, tangent, square root, cube root and piecewise-defined functions.
A2.F-IF.B.6: [1] Calculate and interpret the average rate of change of a continuous function (presented symbolically or as a table) on a closed interval. [2] Estimate the rate of change from a graph. Include problem-solving opportunities utilizing real-world context. Functions include linear, quadratic, exponential, polynomial, logarithmic, rational, sine, cosine, tangent, square root, cube root and piecewise-defined functions.
A2.F-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Functions include linear, quadratic, exponential, polynomial, logarithmic, rational, sine, cosine, tangent, square root, cube root and piecewise-defined functions.
A2.F-IF.C.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the properties of exponents to interpret expressions for exponential functions and classify those functions as exponential growth or decay.
A2.F-IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions.). Functions include linear, quadratic, exponential, polynomial, logarithmic, rational, sine, cosine, tangent, square root, cube root and piecewise-defined functions.
A2.F-BF.A.1: Write a function that describes a relationship between two quantities. Functions include linear, quadratic, exponential, polynomial, logarithmic, rational, sine, cosine, tangent, square root, cube root and piecewise-defined functions. Include problem-solving opportunities utilizing real-world context.
a. [1] Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. [2] Combine function types using arithmetic operations and function composition.
A2.F-BF.A.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
A2.F-BF.B.3: [1] Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); [2] 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. Functions include linear, quadratic, exponential, polynomial, logarithmic, rational, sine, cosine, tangent, square root, cube root and piecewise-defined functions.
A2.F-BF.B.4: Find inverse functions.
a. [1] Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, recognizing that functions f and g are inverse functions if and only if f(x) = y and g(y) = x for all values of x in the domain of f and all values of y in the domain of g.
b. [2] Understand that if a function contains a point (a,b), then the graph of the inverse relation of the function contains the point (b,a).
c. [3] Interpret the meaning of and relationship between a function and its inverse utilizing real-world context.
A2.F-LE.A.4: For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithms that are not readily found by hand or observation using technology.
A2.F-LE.B.5: Interpret the parameters in an exponential function with rational exponents utilizing real-world context.
A2.F-TF.A.1: Understand radian measure of an angle as the length of the arc on any circle subtended by the angle, measured in units of the circle’s radius.
A2.F-TF.A.2: Explain how the unit circle in the coordinate plane enables the extension of sine and cosine functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
A2.F-TF.B.5: Create and interpret sine, cosine and tangent functions that model periodic phenomena with specified amplitude, frequency, and midline.
A2.F-TF.C.8: Use the Pythagorean identity sin2(θ) + cos2(θ) = 1 and the quadrant of the angle θ to find sin(θ), cos(θ), or tan(θ) given sin(θ) or cos(θ).
A2.S-ID.A.4: [1] Use the mean and standard deviation of a data set to fit it to a normal curve, and [2] use properties of the normal distribution and to estimate population percentages. [3] Recognize that there are data sets for which such a procedure is not appropriate. [4] Use calculators, spreadsheets, or tables to estimate areas under the normal curve.
A2.S-ID.B.6: [1] Represent data of two quantitative variables on a scatter plot, and [2] describe how the quantities are related. [3] Extend to polynomial and exponential models. Fit a function to the data; [4] use functions fitted to data to solve problems in the context of the data. Use given functions or chooses a function suggested by the context.
A2.S-ID.C.10: Interpret parameters of exponential models.
A2.S-IC.A.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
A2.S-IC.A.2: Explain whether a specified model is consistent with results from a given data-generating process.
A2.S-IC.B.3: Recognize the purposes of and differences between designed experiments, sample surveys and observational studies.
A2.S-IC.B.4: Use data from a sample survey to estimate a population mean or proportion; recognize that estimates are unlikely to be correct and the estimates will be more precise with larger sample sizes.
A2.S-CP.A.3: [1] Understand the conditional probability of A given B as P(A and B)/P(B), and [2] 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.
A2.S-CP.A.4: [1] 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 [2] to decide if events are independent and [3] to approximate conditional probabilities.
A2.S-CP.A.5: Recognize and explain the concepts of conditional probability and independence utilizing real-world context.
A2.S-CP.B.6: Use Bayes Rule to 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.
A2.S-CP.B.7: 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.
A2.S-CP.B.8: 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.

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