overviewkindergartenevidence:first gradeevidence:second gradeevidence: counting and cardinality know...

Overview Kindergarten Evidence: First Grade Evidence: Second Grade Evidence:

Counting and Cardinality Know number names and the count sequence. (K.CC.A)

1. Count to 100 by ones and by tens. (K.CC.A.1) (DOK 1) 2. Count forward beginning from a given number within the known sequence (instead of having

to begin at 1). (K.CC.A.2) (DOK 1,2) 3. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with

0 representing a count of no objects). (K.CC.A.3) (DOK 1)

Count to tell the number of object. (K.CC.B)

4. Understand the relationship between numbers and quantities; connect counting to cardinality.• 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.• Understand that the last number name said tells the number of objects counted. The number

of objects is the same regardless of their arrangement or the order in which they were counted.

• Understand that each successive number name refers to a quantity that is one larger. (K.CC.B.4) (DOK 2)

5. Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects. (K.CC.B.5) (DOK 2)

Compare numbers (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, e.g., by using matching and counting strategies. (K.CC.C.6) (DOK 2) Include groups with up to ten objects.

7. Compare two numbers between 1 and 10 presented as written numerals. (K.CC.C.7) (DOK 1,2)

Operations and Algebraic Thinking Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. (K.OA.A)

1. Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. (K.OA.A.1) (DOK 2) Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)

2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. (K.OA.A.2) (DOK 2)

3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). (K.OA.A.3) (DOK 2,3)

4. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation. (K.OA.A.4) (DOK 2)

5. Fluently add and subtract within 5. (K.OA.A.5) (DOK 1)

Represent and solve problems involving addition and subtraction. (1.OA.A)

1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. (1.OA.A.1) (DOK 2) See Glossary, Table 1.

2. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. (1.OA.A.2) (DOK 2)

Understand and apply properties of operations and the relationship between addition and subtraction. (1.OA.B)

3. Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) (1.OA.B.3) (DOK 2) Students need not use formal terms for these properties.

4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. (1.OA.B.4) (DOK 2)

Add and subtract within 20. (1.OA.C)

5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). (1.OA.C.5) (DOK 1,2)

6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). (1.OA.C.6) (DOK 1,2)

Work with addition and subtraction equations. (1.OA.D)

7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. (1.OA.D.7) (DOK 3)

8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = – 3, 6 + 6 = . (1.OA.D.8) (DOK 2)

Represent and solve problems involving addition and subtraction. (2.OA.A)

1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (2.OA.A.1) (DOK 2) See Glossary, Table 1.

Add and subtract within 20.

2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. (2.OA.B.2) (DOK 1) See standard 1.OA.6 for a list of mental strategies.

Work with equal groups of objects to gain foundations for multiplication. (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 2s; write an equation to express an even number as a sum of two equal addends. (2.OA.C.3) (DOK 2)

4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. (2.OA.C.4) (DOK 2)

Number and Operations in Base Ten Work with numbers 11–19 to gain foundations for place value. (K.NBT.A)

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. (K.NBT.A.1) (DOK 2)

Extend the counting sequence. (1.NBT.A)

1. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. (1.NBT.A.1) (DOK 1,2)

Understand place value. (1.NBT.B)

2. Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

• 10 can be thought of as a bundle of ten ones — called a "ten.”• The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven,

eight, or nine ones.• 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.2) (DOK 2)2. 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.B.3) (DOK 2)

Use place value understanding and properties of operations to add and subtract. (1.NBT.C)

3. Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. (1.NBT.C.4) (DOK 1,2,3)

4. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. (1.NBT.C.5) (DOK 2,3)

5. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. (1.NBT.C.6) (DOK 2,3)

Understand place value. (2.NBT.A)

1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

• 100 can be thought of as a bundle of ten tens — called a "hundred."• 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.1) (DOK 2) 2. Count within 1000; skip-count by 5s, 10s, and 100s. (2.NBT.A.2) (DOK 1) 3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded

form. (2.NBT.A.3) (DOK 1,2) 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.A.4) (DOK 2)

Use place value understanding and properties of operations to add and subtract. (2.NBT.B)

4. 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.5) (DOK 1,2)

5. Add up to four two-digit numbers using strategies based on place value and properties of operations. (2.NBT.B.6) (DOK 2)

6. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. (2.NBT.B.7) (DOK 2)

7. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900. (2.NBT.B.8) (DOK 2)

8. Explain why addition and subtraction strategies work, using place value and the properties of operations. (2.NBT.B.9) (DOK 3) Explanations may be supported by drawings or objects.

Iowa State University Extension and Outreach programs are available to all without regard to race, color, national origin, religion, sex, age, or disability. Feb. 2015

Mathematics Grades K-2 (1): in partnership with

Overview Kindergarten Evidence: First Grade Evidence: Second Grade Evidence:

Measurement and Data Describe and compare measurable attributes. (K.MD.A)

1. Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. (K.MD.A.1) (DOK 2)

2. Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter. (K.MD.A.2) (DOK 2)

Classify objects and count the number of objects in each category. (K.MD.B)

3. Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. (K.MD.B.3.) (DOK 1,2) Limit category counts to be less than or equal to 10.

Measure lengths indirectly and by iterating length units. (1.MD.A)

1. Order three objects by length; compare the lengths of two objects indirectly by using a third object. (1.MD.A.1) (DOK 2,3)

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.A.2) (DOK 1,2)

Tell and write time. (1.MD.B)

3. Tell and write time in hours and half-hours using analog and digital clocks. (1.MD.B.3) (DOK 1)

Represent and interpret data. (1.MD.C)

4. Organize, represent, and interpret data with up to three categories; 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.MD.C.4) (DOK 2,3)

Measure and estimate lengths in standard units. (2.MD.A) 1. Measure the length of an object by selecting and using appropriate tools such as rulers,

yardsticks, meter sticks, and measuring tapes. (2.MD.A.1) (DOK 1) v2. Measure the length of an object twice, using length units of different lengths for the two

measurements; describe how the two measurements relate to the size of the unit chosen. (2.MD.A.2) (DOK 2,3)

3. Estimate lengths using units of inches, feet, centimeters, and meters. (2.MD.A.3) (DOK 2) 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.A.4) (DOK 1,2)

Relate addition and subtraction to length. (2.MD.B) 5. Use addition and subtraction within 100 to solve word problems involving lengths that

are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem. (2.MD.B.5) (DOK 2)

6. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram. (2.MD.B.6) (DOK 1,2)

Work with time and money. (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.7) (DOK 1) IA.1.Describe the relationship among standard units of time: minutes, hours, days, weeks, months and years. (DOK 2,3) 8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $

and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have? (2.MD.C.8) (DOK 2)

Represent and interpret data. (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.9) (DOK 2)

IA.2Use interviews, surveys, and observations to collect data that answer questions about students' interests and/or their environment. (DOK 2,3) 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 a bar graph. (2.MD.D.10) (DOK 2) See Glossary, Table 1.

Geometry Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). (K.G.A)

1. Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to. (K.G.A.1) (DOK 1,2)

2. Correctly name shapes regardless of their orientations or overall size. (K.G.A.2) (DOK 1) 3. Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional

("solid"). (K.G.A.3) (DOK 1)

Analyze, compare, create, and compose shapes. (K.G.B)

4. Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/"corners") and other attributes (e.g., having sides of equal length). (K.G.B.4) (DOK 2,3)

5. Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes. (K.G.B.5) (DOK 2,3)

6. Compose simple shapes to form larger shapes. For example, "Can you join these two triangles with full sides touching to make a rectangle?" (K.G.B.6) (DOK 2,3)

Reason with shapes and their attributes. (1.G.A)

1. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes. (1.G.A.1) (DOK 2)

2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (1.G.A.2) (DOK 2,3) Students do not need to learn formal names such as "right rectangular prism.”

3. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. (1.G.A.3) (DOK 1,2)

Reason with shapes and their attributes. (2.G.A) 1. Recognize and draw shapes having specified attributes, such as a given number of angles

or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. (2.G.A.1) (DOK 1,2) Sizes are compared directly or visually, not compared by measuring.

2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. (2.G.A.2) (DOK 2)

3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. (2.G.A.3) (DOK 2,3)


Mathematics Grades K-2 (2): in partnership with

Mathematics Grades 3-5 (1):

Overview Third Grade Evidence: Fourth Grade Evidence: Fifth Grade Evidence:

Measurement and Data Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. (3.MD.A)

1. Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. (3.MD.A.1) (DOK 1,2)

2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. (3.MD.A.2) (DOK 1,2) Excludes compound units such as cm3 and finding the geometric volume of a container.

3. Excludes multiplicative comparison problems (problems involving notions of "times as much"; see Glossary, Table 2).

Represent and interpret data.

4. Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. (3.MD.B.3) (DOK 1,2)

5. Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters. (3.MD.B.4) (DOK 2)

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

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

• 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.

• 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.5) (DOK 1,2)

6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). (3.MD.C.6) (DOK 1,2)

7. Relate area to the operations of multiplication and addition.• 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.• Multiply side lengths to find areas of rectangles with whole-number side lengths in the

context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

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

• Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. (3.MD.C.7) (DOK 1,2)

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

8. Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. (3.MD.D.8) (DOK 1,2)

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. (4.MD.A)

1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ... (4.MD.A.1) (DOK 1)

2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. (4.MD.A.2) (DOK 1,2)

3. Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. (4.MD.A.3) (DOK 1,2)

Represent and interpret data. (4.MD.B)

4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection. (4.MD.B.4) (DOK 1,2)

Geometric measurement: understand concepts of angle and measure angles. (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:

• 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.

• An angle that turns through n one-degree angles is said to have an angle measure of n degrees. (4.MD.C.5) (DOK 1)

6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. (4.MD.C.6) (DOK 1)

7. Recognize angle measure 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. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. (4.MD.C.7) (DOK 1,2)

Convert like measurement units within a given measurement system. (5.MD.A)

1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. (5.MD.A.1) (DOK 1,2)

Represent and interpret data. (5.MD.B)

2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. 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.B.2) (DOK 1,2)

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. (5.MD.C)

3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

• 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.

• 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.3) (DOK 1)

4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. (5.MD.C.4) (DOK 1,2)

5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

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

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

• Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. (5.MD.C.5) (DOK 1,2)

Geometry Reason with shapes and their attributes. (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 of quadrilaterals that do not belong to any of these subcategories. (3.G.A.1) (DOK 1,2)

2. Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. (3.G.A.2) (DOK 1,2)

Draw and identify lines and angles, and classify shapes by properties of their lines and angles. (4.G.A)

1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. (4.G.A.1) (DOK 1)

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. Recognize right triangles as a category, and identify right triangles. (4.G.A.2) (DOK 1,2)

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. (4.G.A.3) (DOK 1)

Graph points on the coordinate plane to solve real-world and mathematical problems. (5.G.A)

1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). (5.G.A.1) (DOK 1)

2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. (5.G.A.2) (DOK 1,2)

Classify two-dimensional figures into categories based on their properties. (5.G.B)

3. Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. (5.G.B.3) (DOK 1,2)

4. Classify two-dimensional figures in a hierarchy based on properties. (5.G.B.4) (DOK 1,2)


in partnership with

Mathematics Grades 3-5 (2):Overview Third Grade Evidence: Fourth Grade Evidence: Fifth Grade Evidence:

Operations and Algebraic Thinking Represent and solve problems involving multiplication and division. (3.OA.A)

1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. (3.OA.A.1) (DOK 1,2)

2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. (3.OA.A.2) (DOK 1,2)

3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (3.OA.A.3) (DOK 1,2) See Glossary, Table 2.

4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ÷ 3, 6 × 6 = ?. (3.OA.A.4) (DOK 1,2)

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

5. Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) (3.OA.B.5) (DOK 1,2) Students need not use formal terms for these properties.

6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. (3.OA.B.6) (DOK 1,2)

Multiply and divide within 100 (3.OA.C)

7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. (3.OA.C.7) (DOK 1,2)

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

8. Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (3.OA.D.8) (DOK 1,2,3) This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. (3.OA.D.9) (DOK 1,2,3)

Use the four operations with whole numbers to solve problems. (4.OA.A)

1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. (4.OA.A.1) (DOK 1,2)

2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (4.OA.A.2) (DOK 1,2) See Glossary, Table 2.

3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (4.OA.A.3) (DOK 1,2,3)

Gain familiarity with factors and multiples. (4.OA.B)

4. Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite. (4.OA.B.4) (DOK 1)

Generate and analyze patterns. (4.OA.C)

5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. (4.OA.C.5) (DOK 1,2)

Write and interpret numerical expressions. (5.OA.A)

1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. (5.OA.A.1) (DOK 1)

2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. (5.OA.A.2) (DOK 1,2)

Analyze patterns and relationships. (5.OA.B)

3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. (5.OA.B.3) (DOK 1,2)

Number and Operations in Base Ten Use place value understanding and properties of operations to perform multi-digit arithmetic. (3.NBT.A)

1. Use place value understanding to round whole numbers to the nearest 10 or 100. (3.NBT.A.1) (DOK 1)

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.2) (DOK 1,2)

3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. (3.NBT.A.3) (DOK 1,2)

Generalize place value understanding for multi-digit whole numbers. (4.NBT.A)

1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division. (4.NBT.A.1) (DOK 1)

2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. 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.2) (DOK 1)

3. Use place value understanding to round multi-digit whole numbers to any place. (4.NBT.A.3) (DOK 1)

Use place value understanding and properties of operations to perform multi-digit arithmetic. (4.NBT.B)

4. Fluently add and subtract multi-digit whole numbers using the standard algorithm. (4.NBT.B.4) (DOK 1)

5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. (4.NBT.B.5) (DOK 1,2)

6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. (4.NBT.B.6) (DOK 1,2)

Understand the place value system. (5.NBT.A)

1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. (5.NBT.A.1) (DOK 1)

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. Use whole-number exponents to denote powers of 10. (5.NBT.A.2) (DOK 1,2)

3. Read, write, and compare decimals to thousandths.• Read and write decimals to thousandths using base-ten numerals, number names, and

expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

• Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. (5.NBT.A.3) (DOK 1)

4. Use place value understanding to round decimals to any place. (5.NBT.A.4) (DOK 1)

Perform operations with multi-digit whole numbers and with decimals to hundredths. (5.NBT.B)

5. Fluently multiply multi-digit whole numbers using the standard algorithm. (5.NBT.B.5) (DOK 1) 6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit

divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. (5.NBT.B.6) (DOK 1,2)

7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. (5.NBT.B.7) (DOK 1,2,3)

Number and Operations—Fractions Develop understanding of fractions as numbers. (3.NF.A)

1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. (3.NF.A.1) (DOK 1,2)

2. Understand a fraction as a number on the number line; represent fractions on a number line diagram.

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

• Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. (3.NF.A.2) (DOK 1,2)

3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

• Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

• Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.

• Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

• Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. (3.NF.A.3) (DOK 1,2,3)

Extend understanding of fraction equivalence and ordering. (4.NF.A)

1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. (4.NF.A.1) (DOK 1,2,3)

2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. (4.NF.A.2) (DOK 1,2,3)

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (4.NF.B)

3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.• Understand addition and subtraction of fractions as joining and separating parts referring to

the same whole.• Decompose a fraction into a sum of fractions with the same denominator in more than one

way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

• Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

• Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. (4.NF.B.3) (DOK 1,2,3)

4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

• Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

• Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

• Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. 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.B.4) (DOK 1,2)

Understand decimal notation for fractions, and compare decimal fractions. (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 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. (4.NF.C.5) (DOK 1) 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.

6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. (4.NF.C.6) (DOK 1)

7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. (4.NF.C.7) (DOK 1,2,3)

Use equivalent fractions as a strategy to add and subtract fractions. (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. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) (5.NF.A.1) (DOK 1)

2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. (5.NF.A.2) (DOK 1,2,3)

Apply and extend previous understandings of multiplication and division to multiply and divide fractions. (5.NF.B)

3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? (5.NF.B.3) (DOK 1,2)

4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

• Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

• 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.4) (DOK 1,2)

5. Interpret multiplication as scaling (resizing), by:• 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.• Explaining why multiplying a given number by a fraction greater than 1 results in a product

greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1. (5.NF.B.5) (DOK 1,2,3)

6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. (5.NF.B.6) (DOK 1,2)

7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

• Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

• Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

• Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? (5.NF.B.7) (DOK 1,2) Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.


in partnership with

Mathematics Grades 6-8 (1):Overview Sixth Grade Evidence: Seventh Grade Evidence: Eighth Grade Evidence:

Ratios and Proportional Relationships Understand ratio concepts and use ratio reasoning to solve problems. (6.RP.A)

1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes." (6.RP.A.1) (DOK 1,2)

2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger." (6.RP.A.2) (DOK 1,2) Expectations for unit rates in this grade are limited to non-complex fractions.

3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

• 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.

• Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

• Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

• Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. (6.RP.A.3) (DOK 1,2)

Analyze proportional relationships and use them to solve real-world and mathematical problems. (7.RP.A)

1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. (7.RP.A.1) (DOK 1,2)

2. Recognize and represent proportional relationships between quantities.• 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.

• Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

• 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.

• 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.2) (DOK 1,2)

3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. (7.RP.A.3) (DOK 1,2)

The Number System Apply and extend previous understandings of multiplication and division to divide fractions by fractions. (6.NS.A)

1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? (6.NS.A.1) (DOK 1,2)

Compute fluently with multi-digit numbers and find common factors and multiples. (6.NS.B)

2. Fluently divide multi-digit numbers using the standard algorithm. (6.NS.B.2) (DOK 1) 3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm

for each operation. (6.NS.B.3) (DOK 1) 4. Find the greatest common factor of two whole numbers less than or equal to 100 and the

least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). (6.NS.B.4) (DOK 1)

Apply and extend previous understandings of numbers to the system of rational numbers. (6.NS.C)

5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. (6.NS.C.5) (DOK 1,2)

6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

• Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.

• Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

• Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. (6.NS.C.6) (DOK 1)

7. Understand ordering and absolute value of rational numbers.• Interpret statements of inequality as statements about the relative position of two numbers

on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

• Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC.

• Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

• Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars. (6.NS.C.7) (DOK 1,2)

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. (6.NS.C.8) (DOK 1,2)

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. (7.NS.A)

1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

• Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

• 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). Interpret sums of rational numbers by describing real-world contexts.

• Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

• Apply properties of operations as strategies to add and subtract rational numbers. (7.NS.A.1) (DOK 1,2)

2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

• Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

• Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

• Apply properties of operations as strategies to multiply and divide rational numbers.• Convert a rational number to a decimal using long division; know that the decimal form of a

rational number terminates in 0s or eventually repeats. (7.NS.A.2) (DOK 1,2) 3. Solve real-world and mathematical problems involving the four operations with rational

numbers. (7.NS.A.3) (DOK 1,2) Computations with rational numbers extend the rules for manipulating fractions to complex fractions.

Know that there are numbers that are not rational, and approximate them by rational numbers. (8.NS.A)

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

2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. (8.NS.A.2) (DOK 1,2)

Expressions and Equations Apply and extend previous understandings of arithmetic to algebraic expressions. (6.EE.A)

1. Write and evaluate numerical expressions involving whole-number exponents. (6.EE.A.1) (DOK 1)

2. Write, read, and evaluate expressions in which letters stand for numbers.• Write expressions that record operations with numbers and with letters standing for numbers.

For example, express the calculation "Subtract y from 5" as 5 – y.• Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient,

coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

• Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2. (6.EE.A.2) (DOK 1,2)

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; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. (6.EE.A.3) (DOK 1,2)

4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. (6.EE.A.4) (DOK 1)

Reason about and solve one-variable equations and inequalities. (6.EE.B)

5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. (6.EE.B.5) (DOK 1)

6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. (6.EE.B.6) (DOK 1,2)

7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. (6.EE.B.7) (DOK 1,2)

8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. (6.EE.B.8) (DOK 1,2)

Represent and analyze quantitative relationships between dependent and independent variables. (6.EE.C)

9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. (6.EE.C.9) (DOK 1,2,3)

Use properties of operations to generate equivalent expressions. (7.EE.A)

1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. (7.EE.A.1) (DOK 1)

2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05." (7.EE.A.2) (DOK 1,2)

Solve real-life and mathematical problems using numerical and algebraic expressions and equations. (7.EE.B)

3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. (7.EE.B.3) (DOK 1,2,3)

4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

• 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. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

• Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. (7.EE.B.4) (DOK 1,2,3)

Work with radicals and integer exponents. (8.EE.A)

1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27. (8.EE.A.1) (DOK 1)

2. Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. (8.EE.A.2) (DOK 1)

3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger. (8.EE.A.3) (DOK 1,2)

4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. (8.EE.A.4) (DOK 1,2)

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

5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. 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.5) (DOK 1,2,3)

6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. (8.EE.B.6) (DOK 1,2,3)

Analyze and solve linear equations and pairs of simultaneous linear equations. (8.EE.C)

7. Solve linear equations in one variable.• Give examples of linear equations in one variable with one solution, infinitely many solutions,

or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

• Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. (8.EE.C.7) (DOK 1,2)

8. Analyze and solve pairs of simultaneous linear equations.• 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.

• Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

• Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. (8.EE.C.8) (DOK 1,2,3)


in partnership with

Mathematics Grades 6-8 (2):Overview Sixth Grade Evidence: Seventh Grade Evidence: Eighth Grade Evidence:

Functions Define, evaluate, and compare functions. (8.F.A)

1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (8.F.A.1) (DOK 1,2) Function notation is not required in Grade 8.

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.2) (DOK 1,2)

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 is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. (8.F.A.3) (DOK 1,2)

Geometry Solve real-world and mathematical problems involving area, surface area, and volume. (6.G.A)

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 in the context of solving real-world and mathematical problems. (6.G.A.1) (DOK 1,2)

2. 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. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. (6.G.A.2) (DOK 1,2)

3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. (6.G.A.3) (DOK 1,2)

4. 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 in the context of solving real-world and mathematical problems. (6.G.A.4) (DOK 1,2)

Draw, construct, and describe geometrical figures and describe the relationships between them. (7.G.A)

1. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. (7.G.A.1) (DOK 1,2)

2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. (7.G.A.2) (DOK 1,2)

3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. (7.G.A.3) (DOK 1,2)

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. (7.G.B)

4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. (7.G.B.4) (DOK 1,2)

5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. (7.G.B.5) (DOK 1,2)

6. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (7.G.B.6) (DOK 1,2)

Understand congruence and similarity using physical models, transparencies, or geometry software. (8.G.A)

1. Verify experimentally the properties of rotations, reflections, and translations:• Lines are taken to lines, and line segments 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.1) (DOK 2) 2. Understand that a two-dimensional figure is congruent to another if the second can be

obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. (8.G.A.2) (DOK 1,2)

3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (8.G.A.3) (DOK 1,2)

4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. (8.G.A.4) (DOK 1,2)

5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and 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.A.5) (DOK 1,2,3)

Understand and apply the Pythagorean Theorem. (8.G.B) Explain a proof of the Pythagorean Theorem and its converse. (8.G.B.6) (DOK 2,3) 6. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-

world and mathematical problems in two and three dimensions. (8.G.B.7) (DOK 1,2) 7. Apply the Pythagorean Theorem to find the distance between two points in a coordinate

system. (8.G.B.8) (DOK 1,2)

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. (8.G.C)

8. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. (8.G.C.9) (DOK 1,2)

Statistics and Probability Develop understanding of statistical variability. (6.SP.A)

1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students’ ages. (6.SP.A.1) (DOK 1)

2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. (6.SP.A.2) (DOK 1,2)

3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. (6.SP.A.3) (DOK 1)

Summarize and describe distributions. (6.SP.B)

4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots. (6.SP.B.4) (DOK 1,2)

5. Summarize numerical data sets in relation to their context, such as by:• Reporting the number of observations.• Describing the nature of the attribute under investigation, including how it was measured and

its units of measurement.• 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.

• Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. (6.SP.B.5) (DOK 1,2,3)

Use random sampling to draw inferences about a population. (7.SP.A)

1. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. (7.SP.A.1) (DOK 2)

2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. (7.SP.A.2) (DOK 2,3)

Draw informal comparative inferences about two populations. (7.SP.B)

3. Informally assess the degree of visual overlap of two numerical data distributions with similar variability’s, 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.3) (DOK 2,3)

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.B.4) (DOK 2,3)

Investigate chance processes and develop, use, and evaluate probability models. (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.5) (DOK 1)

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.6) (DOK 2,3)

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.

• 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.

• 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? (7.SP.C.7) (DOK 2,3)

8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

• Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

• Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.

• Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? (7.SP.C.8) (DOK 1,2,3)

Investigate patterns of association in bivariate data. (8.SP.A)

1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. (8.SP.A.1) (DOK 1,2,3) 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.2) (DOK 1,2)

2. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. (8.SP.A.3) (DOK 1,2)

3. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? (8.SP.A.4) (DOK 1,2,3)


in partnership with

Mathematics High School – Number and Quantity:

Overview Grades 9-12 Evidence:

The Real Number System Extend the properties of exponents to rational exponents. (N-RN.A)

1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. (N-RN.A.1) (DOK 1,2)

2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. (N-RN.A.2) (DOK 1)

Use properties of rational and irrational numbers. (N-RN.B)

3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. (N-RN.B.3) (DOK 1,2)

Quantities Reason quantitatively and use units to solve problems. (N-Q.A)

1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. (N-Q.A.1) (DOK 1,2)

2. Define appropriate quantities for the purpose of descriptive modeling. (N-Q.A.2) (DOK 1,2) 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. (N-Q.A.3) (DOK 1,2)

(IA) Understand and apply the mathematics of voting.

4. IA.3.Understand, analyze, apply, and evaluate some common voting and analysis methods in addition to majority and plurality, such as runoff, approval, the so-called instant-runoff voting (IRV) method, the Borda method and the Condorcet method. (DOK 1,2,3)

(IA) Understand and apply some basic mathematics of information processing and the Internet.

5. IA.4.(+) Describe the role of mathematics in information processing, particularly with respect to the Internet. (DOK 1) 6. IA.5.(+) Understand and apply elementary set theory and logic as used in simple Internet searches. (DOK 1,2) 7. IA. 6.(+) Understand and apply basic number theory, including modular arithmetic, for example, as used in keeping information secure through public-key cryptography. (DOK 1,2)

The Complex Number System Perform arithmetic operations with complex numbers. (N-CN.A)

1. Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real. (N-CN.A.1) (DOK 1) 2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. (N-CN.A.2) (DOK 1) 3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. (N-CN.A.3) (DOK 1)

Represent complex numbers and their operations on the complex plane. (N-CN.B)

4. (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. (N-CN.B.4) (DOK 1,2)

5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (–1 + √3 i)3 = 8 because (–1 + √3 i) has modulus 2 and argument 120°. (N-CN.B.5) (DOK 1,2)

6. (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. (N-CN.B.6) (DOK 1)

Use complex numbers in polynomial identities and equations. (N-CN.C)

7. Solve quadratic equations with real coefficients that have complex solutions. (N-CN.C.7) (DOK 1) 8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). (N-CN.C.8) (DOK 1,2) 9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (N-CN.C.9) (DOK 1,2)

Vector and Matrix Quantities Represent and model with vector quantities. (N-VM.A)

1. (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v). (N-VM.A.1) (DOK 1)

2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. (N-VM.A.2) (DOK 1) 3. (+) Solve problems involving velocity and other quantities that can be represented by vectors. (N-VM.A.3) (DOK 1,2)

Perform operations on vectors. (N-VM.B)

4. (+) Add and subtract vectors.a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.b. b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction

graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. (N-VM.B.4) (DOK 1,2) 5. (+) Multiply a vector by a scalar.• Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).• Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

(N-VM.B.5) (DOK 1,2)

Perform operations on matrices and use matrices in applications. (N-VM.C)

6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. (N-VM.C.6) (DOK 1,2) 7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. (N-VM.C.7) (DOK 1) 8. (+) Add, subtract, and multiply matrices of appropriate dimensions. (N-VM.C.8) (DOK 1) 9. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. (N-

VM.C.9) (DOK 1) 10. (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero

if and only if the matrix has a multiplicative inverse. (N-VM.C.10) (DOK 1) 11. (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. (N-VM.C.11) (DOK

1,2) 12. (+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. (N-VM.C.12) (DOK 1,2)


in partnership with


Seeing Structure in Expressions Interpret the structure of expressions (A-SSE.A)

1. Interpret expressions that represent a quantity in terms of its context.★• Interpret parts of an expression, such as terms, factors, and coefficients.• Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. (A-SSE.A.1) (DOK 1,2) 2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). (A-

SSE.A.2) (DOK 1,2)

Write expressions in equivalent forms to solve problems (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.★• Factor a quadratic expression to reveal the zeros of the function it defines.• Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.• Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent

monthly interest rate if the annual rate is 15%. (A-SSE.B.3) (DOK 1,2) 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. ★ (A-SSE.B.4) (DOK

1,2,3)

Arithmetic with Polynomials and Rational Expressions

Perform arithmetic operations on polynomials (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. (A-APR.A.1) (DOK 1)

Understand the relationship between zeros and factors of polynomials (A-APR.B)

2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). (A-APR.B.2) (DOK 1,2) 3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. (A-APR.B.3) (DOK 1,2)

Use polynomial identities to solve problems (A-APR.C)

4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. (A-APR.C.4) (DOK 1,2,3)

5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (A-APR.C.5) (DOK 1,2) The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.

Rewrite rational expressions (A-APR.D)

6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. (A-APR.D.6) (DOK 1,2)

7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. (A-APR.D.7) (DOK 1)

Creating Equations Create equations that describe numbers or relationships (A-CED.A)

1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. (A-CED.A.1) (DOK 1,2)

2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (A-CED.A.2) (DOK 1,2) 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example,

represent inequalities describing nutritional and cost constraints on combinations of different foods. (A-CED.A.3) (DOK 1,2,3) 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.5. (A-CED.A.4) (DOK 1)

Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning (A-REI.A)

1. Explain each step in solving a simple 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. (A-REI.A.1) (DOK 1,2,3)

2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. (A-REI.A.2) (DOK 1,2)

Solve equations and inequalities in one variable (A-REI.B)

3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (A-REI.B.3) (DOK 1) 4. Solve quadratic equations in one variable.• Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.• 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. (A-REI.B.4) (DOK 1,2,3)

Solve systems of equations (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. (A-REI.C.5) (DOK 2,3)

6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (A-REI.C.6) (DOK 1,2) 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and

the circle x2 + y2 = 3. (A-REI.C.7) (DOK 1,2) 8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. (A-REI.C.8) (DOK 1) 9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). (A-REI.C.9) (DOK 1,2)

Represent and solve equations and inequalities graphically (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). 11.(A-REI.D.10) (DOK 1) 12.Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using

technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. ★ (A-REI.D.11) (DOK 1,2,3)

13.Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. (A-REI.D.12) (DOK 1,2)


Mathematics High School – Algebra: in partnership with


Interpreting Functions Understand the concept of a function and use function notation (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). (F-IF.A.1) (DOK 1)

2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (F-IF.A.2) (DOK 1,2) 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1)

= f(n) + f(n-1) for n ≥ 1. (F-IF.A.3) (DOK 1)

Interpret functions that arise in applications in terms of the context (F-IF.B)

4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. ★ (F-IF.B.4) (DOK 1,2)

5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. ★ (F-IF.B.5) (DOK 1,2) (DOK 1,2)

6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. ★

(F-IF.B.6)

Analyze functions using different representations (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.★• Graph linear and quadratic functions and show intercepts, maxima, and minima.• Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.• Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.• (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.• Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. (F-IF.C.7) (DOK 1,2) 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.• Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.9. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10,

and classify them as representing exponential growth or decay. (F-IF.C.8) (DOK 1,2) 10. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic

function and an algebraic expression for another, say which has the larger maximum. (F-IF.C.9) (DOK 1,2)

Building Functions Build a function that models a relationship between two quantities (F-FB.A)

1. Determine an explicit expression, a recursive process, or steps for calculation from a context.2. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential,

and relate these functions to the model.3. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the

temperature at the location of the weather balloon as a function of time. (F-FB.A.1) (DOK 1,2) 4. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. ★ (F-FB.A.2) (DOK 1,2)

Build new functions from existing functions (F-FB.B)

5. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (F-FB.B.3) (DOK 1,2)

6. Find inverse functions.• Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.• (+) Verify by composition that one function is the inverse of another.• (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.• (+) Produce an invertible function from a non-invertible function by restricting the domain. (F-FB.B.4) (DOK 1,2) • +) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. (F-FB.B.5) (DOK 1,2)

Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems (F-LE.A)

1. Distinguish between situations that can be modeled with linear functions and with exponential functions.• Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.• Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.• Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. (F-LE.A.1) (DOK 1,2,3) 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a

table). (F-LE.A.2) (DOK 1,2) 3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (F-LE.A.3)

(DOK 1,2) 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 logarithm using technology. (F-LE.A.4) (DOK 1)

Interpret expressions for functions in terms of the situation they model

5. Interpret the parameters in a linear or exponential function in terms of a context. (F-LE.B.5) (DOK 1,2)

Trigonometric Functions Extend the domain of trigonometric functions using the unit circle (F-TF.A)

1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. (F-TF.A.1) (DOK 1) 2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise

around the unit circle. (F-TF.A.2) (DOK 1,2) 3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x,

and 2π–x in terms of their values for x, where x is any real number. (F-TF.A.3) (DOK 1,2) 4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. (F-TF.A.4) (DOK 2)

Model periodic phenomena with trigonometric functions (F-TF.B)

5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. ★ (F-TF.B.5) (DOK 1,2) 6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. (F-TF.B.6) (DOK 1,2) 7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. ★ (F-TF.B.7) (DOK

1,2,3)

Prove and apply trigonometric identities (F-TF.C)

8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. (F-TF.C.8) (DOK 1,2,3) 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. (F-TF.C.9) (DOK 1,2,3)


Mathematics High School – Functions: in partnership with


Congruence Experiment with transformations in the plane (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-CO.A.1) (DOK 1)

2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). (G-CO.A.2) (DOK 1,2)

3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. (G-CO.A.3) (DOK 1,2) 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (G-CO.A.4) (DOK 2) 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations

that will carry a given figure onto another. (G-CO.A.5) (DOK 1,2)

Understand congruence in terms of rigid motions (G-CO.B)

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. (G-CO.B.6) (DOK 1,2)

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-CO.B.7) (DOK 2,3)

8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (G-CO.B.8) (DOK 2,3)

Prove geometric theorems (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-CO.C.9) (DOK 3)

10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. (G-CO.C.10) (DOK 3)

11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. (G-CO.C.11) (DOK 3)

Make geometric constructions (G-CO.D)

12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. (G-CO.D.12) (DOK 2)

13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. (G-CO.D.13) (DOK 2)

Similarity, Right Triangles, and Trigonometry Understand similarity in terms of similarity transformations (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.• The dilation of a line segment is longer or shorter in the ratio given by the scale factor. (G-SRT.A.1) (DOK 2) 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-SRT.A.2) (DOK 1,2) 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (G-SRT.A.3) (DOK 2,3)

Prove theorems involving similarity (G-SRT.B)

4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. (G-SRT.B.4) (DOK 3)

5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (G-SRT.B.5) (DOK 1,2,3)

Define trigonometric ratios and solve problems involving right triangles (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-SRT.C.6) (DOK 1,2) 7. Explain and use the relationship between the sine and cosine of complementary angles. (G-SRT.C.7) (DOK 1,2) 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ (G-SRT.C.8) (DOK 1,2)

Apply trigonometry to general triangles (G-SRT.D)

9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. (G-SRT.D.9) (DOK 2,3) 10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. (G-SRT.D.10) (DOK 1,2,3) 11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). (G-SRT.D.11) (DOK

1,2)

Circles Understand and apply theorems about circles (G-C.A)

1. Prove that all circles are similar. (G-C.A.1) (DOK 3) 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-C.A.2) (DOK 1,2) 3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. (G-C.A.3) (DOK 2,3) 4. (+) Construct a tangent line from a point outside a given circle to the circle. (G-C.A.4) (DOK 2)

Find arc lengths and areas of sectors of circles (G-C.B)

5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. (G-C.B.5) (DOK 1,2,3)


Mathematics High School – Geometry (1): in partnership with


Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section (G-GPE.A)

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. (G-GPE.A.1) (DOK 1,2,3)

2. Derive the equation of a parabola given a focus and directrix. (G-GPE.A.2) (DOK 1,2) 3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. (G-GPE.A.3) (DOK 1,2)

Use coordinates to prove simple geometric theorems algebraically (G-GPE.B)

4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). (G-GPE.B.4) (DOK 3)

5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). (G-GPE.B.5) (DOK 1,2)

6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (G-GPE.B.6) (DOK 1,2) 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ★ (G-GPE.B.7) (DOK 1,2)

Geometric Measurement and Dimension Explain volume formulas and use them to solve problems (G-GMD.A)

1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. (G-GMD.A.1) (DOK 2,3)

2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. (G-GMD.A.2) (DOK 2,3)3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★ (G-GMD.A.3) (DOK 1,2)

Visualize relationships between two-dimensional and three-dimensional objects (G-GMD.B)

4. IA.7.Plot points in three-dimensions. (DOK 1) 5. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. (G-GMD.B.4) (DOK

1,2)

Modeling with Geometry Apply geometric concepts in modeling situations (G-MG.A)

1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). ★ (G-MG.A.1) (DOK 1,2)2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). ★ (G-MG.A.2) (DOK 1,2) 3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). ★

(G-MG.A.3) (DOK 2,3,4)

(IA) Use diagrams consisting of vertices and edges (vertex-edge graphs) to model and solve problems related to networks.

4. IA.8.Understand, analyze, evaluate, and apply vertex-edge graphs to model and solve problems related to paths, circuits, networks, and relationships among a finite number of elements, in real-world and abstract settings.★ (DOK 2,3)

5. IA.9.Model and solve problems using at least two of the following fundamental graph topics and models: Euler paths and circuits, Hamilton paths and circuits, the traveling salesman problem (TSP), minimum spanning trees, critical paths, vertex coloring.★ (DOK 2,3)

6. IA.10.Compare and contrast vertex-edge graph topics and models in terms of:★ (DOK 2,3)• properties• algorithms• optimization• types of problems that can be solved


Mathematics High School – Geometry (2): in partnership with


Interpreting Categorical and Quantitative Data Summarize, represent, and interpret data on a single count or measurement variable (S-ID.A)

1. Represent data with plots on the real number line (dot plots, histograms, and box plots). (S-ID.A.1) (DOK 1,2) 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. (S-ID.A.2)

(DOK 1,2) 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (S-ID.A.3) (DOK 1,2) 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not

appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. (S-ID.A.4) (DOK 1,2)

Summarize, represent, and interpret data on two categorical and quantitative variables (S-ID.B)

5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. (S-ID.B.5) (DOK 1,2)

6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.• Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic,

and exponential models.• Informally assess the fit of a function by plotting and analyzing residuals.• Fit a linear function for a scatter plot that suggests a linear association. (S-ID.B.6) (DOK 1,2)

Interpret linear models (S-ID.C)

1. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. (S-ID.C.7) (DOK 1,2) 2. Compute (using technology) and interpret the correlation coefficient of a linear fit. 3. (S-ID.C.8) (DOK 1,2) 4. Distinguish between correlation and causation. (S-ID.C.9) (DOK 1,2)

Making Inferences and Justifying Conclusions Understand and evaluate random processes underlying statistical experiments (S-IC.A)

1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population. (S-IC.A.1) (DOK 1) 2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5.

Would a result of 5 tails in a row cause you to question the model? (S-IC.A.2) (DOK 1,2)

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. (S-IC.B.3) (DOK 1,2) 4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. (S-IC.B.4) (DOK 2) 5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. (S-IC.B.5) (DOK 2,3) 6. Evaluate reports based on data. (S-IC.B.6) (DOK 2,3)

Conditional Probability and the Rules of Probability Understand independence and conditional probability and use them to interpret data (S-CP.A)

1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not") (S-CP.A.1) (DOK 1,2)

2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. (S-CP.A.2) (DOK 1)

3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. (S-CP.A.3) (DOK 1,2)

4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. (S-CP.A.4) (DOK 1,2)

5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. (S-CP.A.5) (DOK 1,2,3)

Use the rules of probability to compute probabilities of compound events in a uniform probability model (S-CP.B)

6. Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. (S-CP.B.6) (DOK 1,2) 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. (S-CP.B.7) (DOK 1,2) 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. (S-CP.B.8) (DOK 1,2) 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. (S-CP.B.9) (DOK 1,2)

Using Probability to Make Decisions Calculate expected values and use them to solve problems (S-MD.A)

1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. (S-MD.A.1) (DOK 1,2)

2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. (S-MD.A.2) (DOK 1,2) 3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical

probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. (S-MD.A.3.) (DOK 1,2,3)

4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? (S-MD.A.4.) (DOK 1,2,3)

Use probability to evaluate outcomes of decisions (S-MD.B)

5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.• Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.• Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable,

chances of having a minor or a major accident. (S-MD.B.5) (DOK 1,2,3)6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). (S-MD.B.6) (DOK 1,2) 7. Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). (S-MD.B.7) (DOK 2,3)


Mathematics High School – Statistics and Probability: in partnership with

Overview Kindergarten - Second Grade Evidence: Third – fifth Grade Evidence: Sixth – Eighth Grade Evidence: Ninth – twelth Grade Evidence:

Scientific Questioning

NGSS: 1) Asking questions/ solving problems

Asks questions about objects, organisms, and events in the environment

• Students should answer their questions by seeking information from their own observations, investigations and from reliable sources of scientific information.

Identify and generate questions that can be answered through scientific investigations

• Students ask questions that they can answer with scientific knowledge combined with their own observations.

• Students recognize that different questions lead to different types of investigations.

Identify and generate questions that can be answered through scientific investigations

• Students should develop the ability to refine and refocus broad and ill-defined questions. An important aspect of this ability consists of clarifying questions and inquiries and directing them toward objects and phenomena that can be described, explained, or predicted by scientific investigations.

• Students should develop the ability to connect their questions with scientific ideas, concepts, and quantitative relationships that guide investigations.

Identify questions and concepts that guide scientific investigations

• Students formulate a testable hypothesis and demonstrate the logical connections between the scientific concepts guiding a hypothesis and the design of an experiment. They should demonstrate appropriate procedures, a knowledge base, and conceptual understanding of scientific investigations. The key is that the student demonstrates knowledge of the scientific concepts through the investigation.

NGSS: 2) Developing using models

Scientific investigations

NGSS: 3) Planning and carrying out investigations

Plan and conduct simple investigations

• In earliest years, investigations are largely based on direct observations. As students develop, they design and conduct simple investigations to answer questions.

• It is important to follow appropriate safety procedures when conducting investigations.

Recognize that scientists perform different types of investigations

• Types of investigations include describing objects, events, and organisms; classifying them; and doing a fair test (experimenting), depending on the types of questions they want to answer.

Plan and conduct scientific investigations

• Students should engage in systematic observation, making accurate measurements, and identifying and controlling variables.

• Students understand the concept of a fair test.• Students follow appropriate safety procedures when conducting

investigations.

Design and conduct different kinds of scientific investigations

• Students understand that different kinds of questions suggest different kinds of scientific investigations.

• Students should develop general abilities such as making systematic observations, taking accurate measurements, and identifying and controlling variables.

• Students should develop the ability to clarify ideas that are influencing and guiding their inquiry, and to understand how those ideas compare with current scientific knowledge.

• Students formulate questions, design investigations, execute investigations, interpret data, use evidence to generate explanations, propose alternative explanations, and critique explanations and procedures.

• Students use appropriate safety procedures when conducting investigations.

Understand that different kinds of questions suggest different kinds of scientific investigations

• Some investigations involve observing and describing objects, organisms and events; some involve collecting specimens; some involve experiments; some involve seeking more information; some involve discovery of new objects and phenomena; and some involve making models.

Design and conduct scientific investigations

• Designing and conducting a scientific investigation requires introduction to the major concepts in the area being investigated, proper equipment, safety precautions, assistance with methodological problems, recommendations for use of technologies, clarification of ideas that guide the inquiry, and scientific knowledge obtained from sources other than the actual investigation. The investigation may also require student clarification of the question, method, controls, and variables; student organization and display of data; student revision of methods and explanations; and a public presentation of the results with a critical response from peers. Regardless of the scientific investigation performed, students must use evidence, apply logic, and construct an argument for their proposed explanations.

Tools and Techniques – Data

NGSS: 4) Analyzing and interpreting data

NGSS: 5) Using mathematics and computational thinking

Use tools to gather data and extend senses

• Students use tools such as rulers, thermometers, watches, balances, spring scales, magnifiers and microscopes to extend their senses and their abilities to gather data.

Use mathematics in scientific inquiry

• Mathematics is used to gather, organize and present data and to construct convincing explanations.

Use appropriate tools and techniques to gather, process, and analyze data

• Students enhance their skills with tools such as rulers, thermometers, balances, spring scales, magnifiers and microscopes.

• Students are introduced to the use of computers and calculators for conducting investigations.

• Students’ use of appropriate tools is guided by the questions asked and the investigations students design.

Incorporate mathematics in scientific inquiries


Select and use appropriate tools and techniques to gather, analyze and interpret data

• The use of tools and techniques, including computers, will be guided by the questions asked and the investigations students design. Students should be able to access, gather, store, retrieve, and organize data, using computer hardware and software designed for these purposes.

Incorporate mathematics in scientific inquiry


Use technology and mathematics to improve investigations and communications

• A variety of technologies, such as hand tools, measuring instruments, and calculators should be an integral component of scientific investigations. The use of computers for the collection, analysis, and display of data is also a part of this standard. Mathematics plays an essential role in all aspects of an inquiry investigation. For example, measurement is used for posing questions, formulas are used for developing explanations, and charts and graphs are used for communicating results.

Evidence and Explanations

NGSS: 6) Constructing explanations/design solutions

NGSS: 7) Engaging in argument from evidence

Use data to construct reasonable explanations

• Students should learn what constitutes evidence.• Students’ explanations should reflect the evidence they have obtained.

Use evidence to develop reasonable explanations

• Students should determine what constitutes evidence. • Students should judge the merits or strengths of the data and information

used to make explanations.• Students’ explanations should reflect the evidence they have obtained in

their investigations.• Students should check their explanations against scientific knowledge,

their own experiences, and observations of others.

Use evidence to develop descriptions, explanations, predictions, and models

• Students should base their explanations on observations and they should be able to differentiate between description and explanation.

• Developing explanations establishes connections between the content of science and the contexts in which students develop new knowledge.

• Models are often used to think about processes that happen too slowly, too quickly, or on too small a scale to observe directly, or are too vast to be changed deliberately, or are potentially dangerous.

• Different models can be used to represent the same thing.

think critically and logically to make the relationships between evidence and explanations

• Students decide what evidence should be used and develop the ability to account for anomalous data.

• Students should be able to review data from an experiment, summarize the data, and form a logical argument between cause and effect relationships.

• Students should begin to state some explanations in terms of relationships between two or more variables.

Recognize and analyze alternative explanations and predictions

• Students should develop the ability to listen to and respect the explanations proposed by other students. They should remain open to and acknowledge different ideas and explanations, be able to accept the skepticism of others, and consider alternative explanations.

Formulate and revise scientific explanations and models using logic and evidence

• Student inquiries culminate in formulating an explanation or model. Models should be physical, conceptual, and mathematical. In the process of answering the questions, the students engage in discussions and arguments that result in the revision of their explanations. These discussions should be based on scientific knowledge, the use of logic, and evidence from their investigation.

Recognize and analyze alternative explanations and models

• This aspect of the standard emphasizes the critical abilities of analyzing an argument by reviewing current scientific understanding, weighing the evidence, and examining the logic so as to decide which explanations and models are best. In other words, although there may be several plausible explanations, they do not all have equal weight. Students use scientific criteria to find the preferred explanations.

Think critically and logically to make the relationship between evidence and explanations

• Thinking critically about evidence includes deciding what evidence should be used and accounting for anomalous data. Specifically, students review data from a simple experiment, summarize the data, and form a logical argument about the cause-and-effect relationships in the experiment.

Communication

NGSS: 8) Obtaining, evaluating and communicating information

Communicate investigations and explanations

• Students should begin to develop the abilities to communicate, critique, and analyze their work and the work of other students.

• Students should communicate orally, through writing or through drawings.

Communicate scientific procedures and explanations

• Students should communicate, critique, and analyze their work and the work of other students.

• Students should share procedures and explanations through various means of communication.

Communicate and defend procedures and explanations

• Students should become competent in communicating experimental methods, describing observations and summarizing the results of investigations. Explanations can be communicated through various methods.

Communicate and defend a scientific argument

• Students in school science programs develop the abilities associated with accurate and effective communication. These include writing and following procedures, expressing concepts, reviewing information, summarizing data, using language appropriately, developing diagrams and charts, explaining statistical analysis, speaking clearly and logically, constructing a reasoned argument, and responding appropriately to critical comments.

Use mathematics in all aspects of scientific inquiry

• Mathematics is essential to asking and answering questions about the natural world. Mathematics can be used to ask questions; to gather, organize, and present data; and to structure convincing explanations.


Inquiry Science: in partnership with


Properties of Earth Materials Understand and apply knowledge of properties of earth materials. • Earth materials are solid rocks and soils, water and the gases of the

atmosphere. The varied materials have different physical and chemical properties.

• Soils have properties of color and texture, capacity to retain water, and ability to support the growth of many kinds of plants, including those in our food supply.

Understand and apply knowledge of properties and uses of earth materials.

• The different physical and chemical properties of earth materials make them useful in different ways, for example, as building materials, as sources of fuel, or for growing the plants we use as foods.

Understand and apply knowledge of energy in the earth system.

- Principles that underlie the concept and/or skill include but are not limited to:

• Internal sources of energy • External sources of energy• Plate tectonics• Energy transfer in the atmosphere and ocean• Earth systems have internal and external sources of energy, both of

which create heat. The sun is the major external source of energy. Two primary sources of internal energy are the decay of radioactive isotopes and the gravitational energy from the earth’s original formation.

• The outward transfer of Earth’s internal heat drives convection circulation in the mantle that propels the plates comprising the earth’s surface across the face of the globe.

• Heating of the earth’s surface and atmosphere by the sun drives convection within the atmosphere and oceans, producing winds and ocean currents.

• Global climate is determined by energy transfer from the sun at and near the earth’s surface. This energy transfer is influenced by dynamic processes such as cloud cover and the earth’s rotation, and static conditions such as the position of mountain ranges and oceans.

Structure of the Earth system

NGSS: Earth Systems

Understand and apply knowledge of observable information about daily and seasonal weather conditions. • Weather changes from day to day and over the seasons. • The sun provides the light and heat necessary to maintain the

temperature of the earth.

Understand and apply knowledge of events that have repeating patterns. • Seasons of the year, day and night are events that are repeated in

regular patterns.• The sun’s position in the sky can be observed and described.• The sun can only be seen during our daylight hours. We are unable to

see the sun at night because of the rotation of the earth.

Understand and apply knowledge of processes and changes on or in the earth’s land, oceans, and atmosphere.

• The surface of the earth changes. Some changes are due to slow processes, such as erosion and weathering, and some changes are due to rapid processes such as landslides, volcanic eruptions, floods and earthquakes.

Understand and apply knowledge of weather and weather patterns.

• Weather is always changing and can be described by measurable quantities such as temperature, wind direction and speed and precipitation.

• Large masses of air with certain properties move across the surface of the earth. The movement and interaction of these air masses is used to forecast the weather.

Understand and apply knowledge of the structure and processes of the earth system and the processes that change the earth and its surface.

• The solid earth consists of layers including a lithosphere; a hot, convecting mantle and a dense metallic core.

• Tectonic plates constantly move at rates of centimeters per year in response to movements in the mantle. Major geological events, such as earthquakes, volcanic eruptions, and mountain building, are results of these plate motions.

• Land forms are the result of a combination of constructive and destructive forces. Constructive forces include crustal deformation, volcanic eruption, and deposition of sediment, while destructive forces include weathering and erosion.

• Some changes in the earth can be described as the “rock cycle.” Rocks at the earth’s surface weather, forming sediments that are buried, then compacted, heated, and often re-crystallized into new rock. Eventually, those new rocks may be brought to the surface by the forces that drive plate motions, and the rock cycle continues.

• Soil consists of weathered rocks and decomposed organic matter from dead plants, animals, and bacteria. Soils are often found in layers, with each having a different chemical composition and texture.

• Living organisms have played many roles in the earth system, including affecting the composition of the atmosphere, producing some types of rocks, and contributing to the weathering of rocks.

Understand and apply knowledge of the water cycle, including consideration of events that impact groundwater quality.

• Water, which covers the majority of the earth’s surface, circulates through the crust, oceans, and atmosphere in what is known as the “water cycle.” Water evaporates from the earth’s surface, rises and cools as it rises to higher elevations, condenses as rain or snow, and falls to the surface where it collects in lakes, oceans, soil and in soil and rocks underground.

• Water is a solvent. As it passes through the water cycle, especially as it moves on the earth’s surface and underground, it dissolves minerals and gases and carries them to the oceans, rivers, and other surface water.

• Natural and human forces can contribute to contamination of surface water and groundwater.

Earth’s History

NGSS: History of Earth

Understand and apply knowledge of fossils and the evidence they provide of past life on earth.

• Fossils provide evidence of plants and animals that lived long ago and the nature of the environment at that time.

Understand and apply knowledge of earth history based on physical evidence.

• The earth processes we see today including erosion, movement of tectonic plates, and changes in atmospheric composition are similar to those that occurred in the past.

• Earth history is also influenced by occasional catastrophes such as the impact of an asteroid or a comet.

• Fossils provide important evidence of how life and environmental conditions have changed.

Geochemical Cycles

NGSS: Weather and Climate

Understand and apply knowledge of the earth’s atmospheric properties and how they influence weather and climate.

• The atmosphere is a mixture of nitrogen, oxygen, and trace gasses that include water vapor. The atmosphere has different properties at different elevations.

• Global patterns of atmospheric movement influence local weather. Oceans have a major effect on climate because water in the oceans holds a large amount of heat.

• Clouds, formed by the condensation of water vapor, affect weather and climate.

Understand and apply knowledge of Geochemical cycles. (S.9-12.ES.2)


• Elements/atoms within Earth reservoirs: Solid Earth, oceans, atmosphere, and organisms

• Movement of elements/atoms between reservoirs• The earth is a system containing essentially a fixed amount of each

stable chemical atom or element. Each element can exist in several different chemical reservoirs. Each element on Earth moves among reservoirs in the solid Earth, oceans, atmosphere, and organisms as part of geochemical cycles.

• Movement of matter between reservoirs is driven by the earth’s internal and external sources of energy. These movements are often accompanied by a change in the physical and chemical properties of the matter. Carbon, for example, occurs in carbonate rocks such as limestone, in the atmosphere as carbon dioxide gas, in water as dissolved carbon dioxide, and in all organisms as complex molecules that control the chemistry of life.

Origin and evolution of the earth system

NGSS: Space Systems

Understand and apply knowledge of the properties, movements, and locations of objects in our solar system.

• Most objects in the solar system are in regular and predictable motion. The rotation of the earth on its axis every 24 hours produces the day-and-night cycle. To people on the earth this turning of the planet makes it seem as though the sun, planets, and stars are orbiting the earth once a day.

• The sun appears to move across the sky in the same way every day. Its apparent path changes slowly across the seasons.

• The moon’s orbit around the earth once in about 28 days changes what part of the moon is lighted by the sun and how much of that part can be seen from the earth – the phases of the moon.

• Eight planets and many other objects revolve around our Sun in predictable patterns. These planets and objects are composed of varied materials.

Understand and apply knowledge of the components of our solar system. • The earth is the third planet from the sun in a system that includes the

moon, the sun, seven other planets and their moons, and smaller objects, such as asteroids and comets. The sun, an average star, is the central and largest body in the solar system.

• Gravity is the force that keeps planets in orbit around the sun and governs the rest of the motion in the solar system. Gravity alone holds us to the earth’s surface and explains the phenomena of the tides.

• The sun is the major source of energy for phenomena on the earth’s surface, such as growth of plants, winds, ocean currents, and the water cycle. Seasons result from variations in the amount of the sun’s energy hitting the surface, due to the tilt of the earth’s rotation on its axis and the length of the day.

• Most objects in the solar system are in regular and predictable motion. Those motions explain such phenomena as the day, the year, phases of the moon, and eclipses.

Understand and apply knowledge of origin and evolution of the earth system


• Formation of solar system• Geologic time• Interactions among hydrosphere, lithosphere and atmosphere• Life: origin, evolution, and effect on Earth systems• The sun, the earth, and the rest of the solar system formed from a

nebular cloud of dust and gas 10 to 15 billion years ago. The early Earth was very different from the planet on which we live today.

• Geologic time can be estimated by observing rock sequences and using fossils to correlate the sequences at various locations. Current methods for measuring geologic time include using the known decay rates of radioactive isotopes present in rocks to measure the time since the rock was formed.

• Interactions among the solid Earth, the oceans, the atmosphere, and organisms have resulted in the ongoing evolution of the earth system. We can observe some changes such as earthquakes and volcanic eruptions on a human time scale, but many processes such as mountain building and plate movements take place over hundreds of millions of years.

• Evidence for one-celled forms of life—the microbes—extends back more than 3.5 billion years. The evolution of life caused dramatic changes in the composition of the earth’s atmosphere, which did not originally contain oxygen.

Understand and apply knowledge of origin and evolution of the Universe. Principles that underlie the concept and/or skill include but are not limited to:

• Age and origin of the universe• Universe and galaxies• Star formation• The origin of the universe remains one of the greatest questions in

science. The “big bang” theory places the origin between 10 and 20 billion years ago, when the universe began in a hot dense state: According to this theory, the universe has been expanding ever since.

• Early in the history of the universe, matter—primarily the light atoms hydrogen and helium — clumped together through gravitational attraction to form countless trillions of stars. Billions of galaxies, each of which is a gravitationally bound cluster of billions of stars, now form most of the visible mass in the universe.

• Stars produce energy from nuclear reactions, primarily the fusion of hydrogen to form helium. These and other processes in stars have led to the formation of all the other elements.


Earth Science: in partnership with

Overview: Kindergarten - Second Grade Evidence: Third – fifth Grade Evidence: Sixth – Eighth Grade Evidence: Ninth – twelth Grade Evidence:

Characteristics of organisms Understand and apply knowledge of the characteristics of living things and how living things are both similar to and different from each other and from non-living things.

• Living things share some common characteristics that are both similar to and different from non-living things.

• Different species of plants and animals have different observable characteristics by which they can be classified.

Structure and function in living things Understand and apply knowledge of the basic components and functions of cells, tissues, organs, and organ systems.

• Living systems at all levels of organization demonstrate the complementary nature of structure and function. Important levels of organization for structure and function include cells, organs, tissues, organ systems, whole organisms, and ecosystems.

• All organisms are composed of cells. Most organisms are single cells; other organisms, including humans are multi-cellular.

• Cells carry on the many functions needed to sustain life. They grow and divide, thereby producing more cells. This requires that they take in nutrients, which they use to provide energy for the work that cells do and to make the materials that a cell or an organism needs.

• Specialized cells perform specialized functions in multi-cellular organisms. Groups of specialized cells cooperate to form a tissue, such as muscle. Different tissues are, in turn, grouped together to form larger functional units, called organs. Each type of cell, tissue, and organ has a distinct structure and set of functions that serve the organism as a whole.

Understand and apply knowledge of the complementary nature of structure and function and the commonalities among organisms.

• Living systems at all levels of organization demonstrate the complementary nature of structure and function. Important levels of organization for structure and function include cells, organs, tissues, organ systems, whole organisms, and ecosystems. Organisms are classified according to common characteristics.

The Cell Understand and apply knowledge of the cell.

Principles that Underlie the Concept and/or Skill:Structures and functions• Cell structures underlie functions• Cell membranes; absorption and diffusion• Basic cell processes• Cells have particular structures that underlie their functions. Every cell

is surrounded by a membrane that separates it from the outside world. Inside the cell is a concentrated mixture of thousands of different molecules which form a variety of specialized structures, notably the nucleus, mitochondria, ribosomes, chloroplasts, and the endoplasmic reticulum. Some cells have external structures facilitating movement (cilia and flagella).

Principles that Underlie the Concept and/or Skill:Functions and chemical reactions• Enzymes catalyze reactions• Food molecules (macromolecules) break down to provide molecules for

synthesis• Cell respiration breaks down complex molecules to provide energy• Most cell functions involve chemical reactions. Food molecules taken

into cells react to provide the chemical constituents needed to synthesize other molecules. Both breakdown and synthesis are made possible by protein catalysts, called enzymes.

• The chemical bonds of food molecules contain energy. Energy is released when the bonds of food molecules are broken and new compounds with lower energy bonds are formed. Cells temporarily store this energy in phosphate bonds of a small high-energy compound called ATP.

Principles that Underlie the Concept and/or Skill:• Cells grow and divide• Cells grow and divide in a cell cycle• Cell regulation allows cells to respond to their environment and to

control and coordinate cell growth and division. Environmental factors can influence cell division.

Principles that Underlie the Concept and/or Skill:Photosynthesis• Photosynthesis links sun energy to usable energy• Basic process of photosynthesis• Plant cells contain chloroplasts as sites of photosynthesis. Plants and

many microorganisms use solar energy to combine molecules of carbon dioxide and water into complex, energy rich organic compounds and release oxygen to the environment.

Behavior of organisms Understand and apply knowledge of the interdependence of the behavior of organisms.

Principles that Underlie the Concept and/or Skill:Nervous systems and behavior• Nerve cell structure and function• Nerve cell communications through neurotransmitters• Sensor organs are specialized cells detecting environmental input• Multicellular animals have nervous systems that generate behavior.

Nervous systems are formed from specialized cells that conduct signals rapidly through the long cell extensions that make up nerves. The nerve cells communicate with each other by secreting specific excitatory and inhibitory molecules. In sense organs, specialized cells detect light, sound, and specific chemicals and enable animals to monitor what is going on in the world around them.

Principles that Underlie the Concept and/or Skill:The Human Organism—Basic Functions• The human immune system protects against microscopic and foreign

substances entering the body and from cancer cells arising within.• The hormonal system exerts its influence by chemicals circulating in the

blood.• Coordinated systems (nervous, muscular and bone) are necessary for

locomotion.

Life Cycles of organisms Understand and apply knowledge of life cycles of plants and animals.

• Plants and animals have life cycles that include being born, developing into adults, reproducing, and eventually dying.

• Plants and animals closely resemble their parents.


Life Science (1): in partnership with

Overview: Kindergarten - Second Grade Evidence: Third – fifth Grade Evidence: Sixth – Eighth Grade Evidence: Ninth – twelth Grade Evidence:

Reproduction and Heredity

NGSS: Inheritance and variationUnderstand and apply knowledge of how different organisms pass on traits (heredity).

• Every organism requires a set of instructions for specifying its traits. Heredity is the passage of these instructions from one generation to another.

• Hereditary information is contained in genes, located in the chromosomes of each cell. Each gene carries a single unit of information. An inherited trait of an individual can be determined by one or by many genes, and a single gene can influence more than one trait. A human cell contains many thousands of different genes.

• The characteristics of an organism can be described in terms of a combination of traits. Some traits are inherited and others result from interactions with the environment.

Understand and apply knowledge of the molecular basis of heredity .

Principles that Underlie the Concept and/or Skill:• Genetic information in cells• DNA structure specifies genetic information in genes• Genes direct and control protein synthesis• DNA mutations• In all organisms, the instructions for specifying the characteristics of the

organism are carried in DNA, a large polymer formed from subunits of four kinds (A, G, C, and T). The chemical and structural properties of DNA explain how the genetic information that underlies heredity is both encoded in genes (as a string of molecular “letters”) and replicated (by a templating mechanism). DNA mutations occur spontaneously at low rates. Some of these changes make no difference to the organism, whereas others can change cells and organisms. Some mutations can be caused by environmental factors.

Principles that Underlie the Concept and/or Skill:• DNA, chromosomes, and sexual reproduction• DNA forms chromosomes.• Organisms have two copies of each chromosome.• Humans have 22 pairs plus two sex chromosomes.• Sex cells (sperm and egg) transmit genetic information through the

processes of meiotic cell division and fertilization.• Each DNA molecule in a cell forms a single chromosome. Most of the

cells in a human contain two copies of each of 22 different chromosomes plus two chromosomes that determine sex: a female contains two X chromosomes and a male contains one X and one Y. Transmission of genetic information to offspring occurs through meiosis that produces egg and sperm cells that contain only one representative from each chromosome pair. An egg and a sperm unite to form a new individual.

Principles that Underlie the Concept and/or Skill:Basic Inheritance Patterns• Variability occurs as a result of fertilization• Basic patterns of inheritance can be identified • The fact that an organism is formed from cells that contain two copies of

each chromosome, and therefore two copies of each gene, explains many features of heredity, such as how variations that are hidden in one generation can be expressed in the next. Different genes coding for the same feature code for it in different ways thus leading to identifiable patterns in heritable traits. These patterns of inheritance can be identified and predicted.

Humans and their environments Understand and apply knowledge of ways to help take care of the environment.

• Chapter 12 of the Iowa Administrative Code states that science instruction shall include conservation of natural resources; and environmental awareness.

• Humans depend on their natural and constructed environments.• Humans change environments in ways that can be either beneficial or

detrimental to themselves or other organisms.

Understand and apply knowledge of environmental stewardship.


• Humans change environments in ways that can be either beneficial or detrimental to themselves or other organisms.

Understand and demonstrate knowledge of the social and personal implications of environmental issues.


• The number of organisms an ecosystem can support depends on the resources available and abiotic factors, such as quantity of light and water, range of temperatures, and soil composition. Given adequate biotic and abiotic resources and no disease or predators, populations (including humans) increase at rapid rates. Lack of resources and other factors, such as predation and climate, limit the growth of populations in specific niches in the ecosystem.

Organisms and their environments

NGSS: Matter & energy in organisms & ecosystemsInterdependent Relationships in ecosystems

Understand and apply knowledge of the basic needs of plants and animals and how they interact with each other and their physical environment.

• Organisms have basic needs. For example, animals need air, water, and food; plants require air, water, nutrients, and light.

• Organisms interact with each other and their physical environment.• Organisms can survive only in environments in which their needs can be

met. • The world has many different environments, and distinct environments

support the life of different types of organisms.

Understand and apply knowledge of organisms and their environments, including: Structures, characteristics, and adaptations of organisms that allow them to function and survive within their habitats.How individual organisms are influenced by internal and external factors.The relationships among living and non-living factors in terrestrial and aquatic ecosystems

• Animals depend on plants. Some animals eat plants for food. Other animals eat animals that eat the plants.

• An organism’s patterns of behavior are related to the nature of that organism’s environment, including the kinds and numbers of other organisms present, the availability of food and resources, and the physical characteristics of the environment. When the environment changes, some plants and animals survive and reproduce, others die or move to new locations.

• All organisms cause changes in the environment in which they live. Some of these changes are detrimental to the organism or other organisms, whereas others are beneficial.

Understand and apply knowledge of the cycling of matter and energy in ecosystems & interdependency of organisms, changes in environmental conditions, and survival of individuals and species .

• All organisms must be able to obtain and use resources, grow, reproduce, and maintain stable internal conditions while living in a constantly changing external environment.

• Regulation of an organism’s internal environment involves sensing the internal environment and changing physiological activities to keep conditions within the range required to survive.

• Behavior is one kind of response an organism can make to an internal or environmental stimulus. A behavioral response requires coordination and communication on many levels, including cells, organ systems, and whole organisms. Behavioral response is a set of actions determined in part by heredity and in part from experience.

• Species acquire many of their unique characteristics through biological adaptation which involves the selection of naturally occurring variations in populations.

• Biological adaptations include changes in structures, behaviors, or physiology that enhance survival and reproductive success in a particular environment.

• For ecosystems, the major source of energy is sunlight. Energy entering ecosystems as sunlight is transferred by producers into chemical energy through photosynthesis. That energy then passes from organism to organism in food webs.

Understand and apply knowledge of the interdependence of organisms.

Principles that Underlie the Concept and/or Skill:Materials cycling• Atoms and molecules cycle (e.g., carbon, nitrogen, oxygen cycles).• The atoms and molecules on the earth cycle among the living and nonliving

components of the biosphere.

Principles that Underlie the Concept and/or Skill:• Energy transformation from producers through levels of consumers and

decomposers• Energy flows through ecosystems in one direction, from photosynthetic organisms

to herbivores to carnivores and decomposers. These tropic levels can be illustrated by food chains and food webs.

Principles that Underlie the Concept and/or Skill:Organism interrelationships• Cooperation and competition within ecosystems• Interrelationships and interdependency lead to long-term stable systems• Organisms both cooperate and compete in ecosystems. The interrelationships and

interdependencies of these organisms may generate ecosystems that are stable for hundreds or thousands of years.

Principles that Underlie the Concept and/or Skill:Humans modify ecosystems• Human modification of ecosystems• Habitat destruction threatens global stability• Human beings live within the world’s ecosystems. Increasingly, humans modify

ecosystems as a result of population growth, technology, and consumption. Human destruction of habitats through direct harvesting,

• pollution, atmospheric changes, and other factors are threatening current global stability, and if not addressed, ecosystems will be irreversibly affected.

Understand and apply knowledge of the interdependence of matter, energy, and organization of living systems.

Principles that Underlie the Concept and/or Skill:Sunlight energy conversion:• Living systems require continuous energy input.• Sunlight serves as the original energy source for life.• Plants photosynthesize, producing building blocks for making macromolecules and

storing energy in chemical bonds. • Cell respiration releases chemical bond energy stored during photosynthesis.• Living systems require a continuous input of energy, derived primarily from the sun,

to maintain their chemical and physical organization. Plants capture energy by absorbing light and using it to form strong (covalent) chemical bonds between the atoms of carbon containing (organic) molecules. These molecules can be used to assemble larger molecules (proteins, DNA, sugars, and fats). The chemical energy stored in bonds between the atoms can be used as sources of energy for life processes.

Principles that Underlie the Concept and/or Skill:Limiting factors:• Ecosystem and population limiting factors• Ecosystems have finite resources.• Environmental factors and finite resources influence ecosystem interactions.• Living organisms have the capacity to produce populations of infinite size, but

environments and resources are finite. The distribution and abundance of organisms and populations in ecosystems are limited by the availability of matter and energy and the ability of the ecosystem to recycle materials.

Principles that Underlie the Concept and/or Skill:Matter and energy flow and conservation• Living systems require continuous energy input.• Matter and energy are conserved as they flow through and between organisms.• Some energy dissipates into the environment as heat.• All matter tends toward more disorganized states. Living systems require a

continuous input of energy to maintain their chemical and physical organizations.• As matter and energy flows through different levels of organization of living systems

—cells, organs, organisms, communities—and between living systems and the physical environment, chemical elements are recombined in different ways. Each recombination results in storage and dissipation of energy into the environment as heat. Matter and energy are conserved in each change.

Biological evolution

NGSS: Natural Selection & evolution

Understand and apply knowledge of biological evolution.

Principles that Underlie the Concept and/or Skill:Species evolution• Species evolve over time.• Evolution is consequence of: population potential, genetic variability, finite

resources and environmental selection.• Species evolve over time. Evolution is the consequence of the interactions of (1) the

potential for a species to increase its numbers, (2) the genetic variability of offspring due to mutation and recombination of genes, and (3) a finite supply of the resources required for life, and (4) the ensuing selection by the environment of those offspring better able to survive and leave offspring.

Principles that Underlie the Concept and/or Skill:Natural Selection• Natural selection scientifically explains the fossil record.• Natural selection explains molecular similarity of diverse species.• Natural selection is a mechanism for evolution leading to organism diversity.• Natural selection and its evolutionary consequences provide a scientific explanation

for the fossil record of ancient life forms, as well as for the striking molecular similarities observed among the diverse species of living organisms. The great diversity of organisms is the result of more than 3.5 billion years of evolution that has filled every available niche with life forms.

Principles that Underlie the Concept and/or Skill:Relations to common ancestor:• Current diverse species are related by descent from common ancestors.• The millions of different species of plants, animals, and microorganisms that live on

Earth today are related by descent from common ancestors.

Principles that Underlie the Concept and/or Skill:Biological classification• Biological classification is based on evolutionary relationships.• Species is the most fundamental classification unit.• Biological classifications are based on how organisms are related. Organisms are

classified into a hierarchy of groups and subgroups based on similarities in development and DNA sequences which reflect their evolutionary relationships. Species is the most fundamental unit of classification.

Human health Understand and apply knowledge of basic human body structures (human body parts and their functions).

• Humans have distinct body structures for functions including but not limited to thinking, walking, holding, seeing and talking.

Understand and apply knowledge of good health habits See 21st Century Skills of the Iowa Core.

Understand and apply knowledge of basic human body systems and how they work together

• The human organism has systems which interact with one another. These systems include circulatory, respiratory, digestive, musculoskeletal, etc.

Understand and apply knowledge of personal health and wellness issues See 21st Century Skills of the Iowa Core.

Understand and apply knowledge of the functions and interconnections of the major human body systems including the breakdown in structure or function that disease causes

• The human organism has systems for digestion, respiration, reproduction, circulation, excretion, movement, control, and coordination, and for protection from disease. These systems interact with one another.

• Disease is a breakdown in structures or functions of an organism. Some diseases are the result of intrinsic failures of the system. Others are the result of damage by infection by other organisms.


Life Science (2): in partnership with


Properties of objects and materials

NGSS: Structure and property of matter

Understand and apply knowledge of observable and measurable properties of objects.

• Objects have many observable properties including size, weight, shape, color, temperature and the ability to react with other substances. Those properties can be measured using tools such as rulers, balances and thermometers.

• Objects are made of one or more materials.• Objects can be described by the properties of the materials from which

they are made. Properties can be used to separate or sort a group of objects or materials.

Understand and apply knowledge of characteristics of liquids and solids .

• Materials can exist in different states – solid, liquid, and gas. • Some common materials, such as water, can be changed from one state

to another by heating or cooling.

Understand and apply knowledge of how to describe and identify substances based on characteristic properties

• It may be necessary to use magnification to observe the component parts of some materials.

• A substance has characteristic properties. A mixture of substances often can be separated into the original substances using one or more of the characteristic properties.

• The properties of a substance can be measured using tools and technology.

• When a new material (compound) is made by chemically combining two or more materials, it has properties that are different from the original materials. For that reason, many different materials can be made from a small number of basic materials.

Understand and apply knowledge of elements, compounds, mixtures, and solutions based on the nature of their physical and chemical properties and physical and chemical changes and their relationship to the conservation of matter and energy.

• A substance has characteristic properties, such as density, a boiling point, and solubility, all of which are independent of the amount of the sample. A mixture of substances often can be separated into the original substances using one or more of the characteristic properties.

• Substances react chemically in characteristic ways with other substances to form new substances (compounds) with different characteristic properties. In chemical reactions, the total mass is conserved. Substances often are placed in categories or groups if they react in similar ways; metals is an example of such a group.

• Chemical elements do not break down during normal laboratory reactions involving such treatments as heating, exposure to electric current, or reaction with acids. There are more than 100 known elements that combine in a multitude of ways to produce compounds, which account for the living and nonliving substances that we encounter.

Understand and apply knowledge of the structure of atoms

Principles that underlie the concept and/or skill include but are not limited to:• Atomic structure• Matter is made of minute particles called atoms, and atoms are

composed of even smaller components. These components have measurable properties, such as mass and electrical charge. Each atom has a positively charged nucleus surrounded by negatively charged electrons. The electric force between the nucleus and electrons holds the atom together.

Principles that underlie the concept and/or skill include but are not limited to:• Atomic nucleus (composition and size)• Isotopes (related to relative mass) • The atom’s nucleus is composed of protons and neutrons, which are

much more massive than electrons. When an element has atoms that differ in the number of neutrons, these atoms are called different isotopes of the element.

Principles that underlie the concept and/or skill include but are not limited to:• Nuclear forces: Fission and Fusion• The nuclear forces that hold the nucleus of an atom together, at nuclear

distances, are usually stronger than the electric forces that would make it fly apart. Nuclear reactions convert a fraction of the mass of interacting particles into energy, and they can release much greater amounts of energy than atomic interactions. Fission is the splitting of a large nucleus into smaller pieces. Fusion is the joining of two nuclei at extremely high temperature and pressure, and is the process responsible for the energy of the sun and other stars.

Principles that underlie the concept and/or skill include but are not limited to:• Radioactive isotopes• Predictable rates of decay• Radioactive isotopes are unstable and undergo spontaneous nuclear

reactions, emitting particles and/or wavelike radiation. The decay of any one nucleus cannot be predicted, but a large group of identical nuclei decay at a predictable rate. This predictability can be used to estimate the age of materials that contain radioactive isotopes.

Understand and apply knowledge of the structure and properties of matter

Principles that underlie the concept and/or skill include but are not limited to:• Valence electrons• Chemical bonds• Atoms interact with one another by transferring or sharing electrons

that are the furthest from the nucleus. These outer electrons govern the chemical properties of the element.

Principles that underlie the concept and/or skill include but are not limited to:• Periodic table• Periodic trends• An element is composed of a single type of atom. When elements are

listed in order according to the number of protons (called the atomic number), repeating patterns of physical and chemical properties identify families of elements with similar properties. This “Periodic Table” is a consequence of the repeating pattern of outermost electrons and their permitted energies.

Principles that underlie the concept and/or skill include but are not limited to:• Molecular and ionic structures• Physical properties of chemical compounds• Bonds between atoms are created when electrons are paired up by

being transferred or shared. A substance composed of a single kind of atom is called an element. The atoms may be bonded together into molecules or crystalline solids. A compound is formed when two or more kinds of atoms bind together chemically.

Principles that underlie the concept and/or skill include but are not limited to:• States of matter• Relationship between pressure and volume of gasses• Solids, liquids, and gases differ in the distances and angles between

molecules or atoms and, therefore, the energy that binds them together. In solids the structure is nearly rigid; in liquids molecules or atoms move around each other but do not move apart; and in gases molecules or atoms move almost independently of each other and are mostly far apart.

Principles that underlie the concept and/or skill include but are not limited to:• Hydrocarbon compounds• Carbon atoms can bond to one another in chains, rings, and branching

networks to form a variety of structures, including synthetic polymers, oils, and the large molecules essential to life.

Properties and changes of properties in matter

NGSS: Chemical reactions

Understand and apply knowledge of states of matter and changes in states of matter.

• Materials can exist in different states – solid, liquid and gas. Some common materials can be changed from one state to another by heating or cooling.

Understand and apply knowledge of chemical reactions.

Principles that underlie the concept and/or skill include but are not limited to:• Conservation of matter• Common reactions• “Chemical reactions” is an essential concept of a world-class secondary

science curriculum. Included in “chemical reactions” is the following content:

• Chemical reactions occur all around us, for example in health care, cooking, cosmetics, and automobiles. Complex chemical reactions involving carbon-based molecules take place constantly in every cell in our bodies.

Principles that underlie the concept and/or skill include but are not limited to:• Thermochemistry• Chemical reactions may release or consume energy. Some reactions

such as the burning of fossil fuels release large amounts of energy by losing heat and by emitting light. Light can initiate many chemical reactions such as photosynthesis and the evolution of urban smog.

Principles that underlie the concept and/or skill include but are not limited to:• Types of reactions• Acids and bases• Common reactions in living systems• A large number of important reactions involve the transfer of either

electrons (oxidation/reduction reactions) or hydrogen ions (acid/base reactions) between reacting ions, molecules, or atoms. In other reactions, chemical bonds are broken by heat or light to form very reactive radicals with electrons ready to form new bonds. Radical reactions control many processes such as the presence of ozone and greenhouse gases in the atmosphere, burning and processing of fossil fuels, the formation of polymers, and explosions.

Principles that underlie the concept and/or skill include but are not limited to:• Reaction rates and equilibrium• Chemical reactions can take place in time periods ranging from the few

femtoseconds (10 – 15 seconds) required for an atom to move a fraction of a chemical bond distance to geologic time scales of billions of years. Reaction rates depend on how often the reacting atoms and molecules encounter one another, the temperature, and the properties—including shape—of the reacting elements.

Understand and apply knowledge of conservation of energy and increase in disorder

Principles that underlie the concept and/or skill include but are not limited to:• Types of energy• Energy transformations• Conservation of energy• “Conservation of energy and increase in disorder” is an essential

concept of a world-class secondary science curriculum. Included in “conservation of energy and increase in disorder” is the following content:

• The total energy of the universe is constant. Energy can be transferred by collisions in chemical and nuclear reactions, by light waves and other radiations, and in many other ways. However, it can never be destroyed. As these transfers occur, the matter involved becomes steadily less ordered.

• All energy can be considered to be either kinetic energy, which is the energy of motion; potential energy, which depends on relative position; or energy contained by a field, such as electromagnetic waves.


Physical Science (1): in partnership with

Physical Science (2):


Sounds, Light, Heat, Electricity, and Magnetism

NGSS: Waves and electromagnetic radiation

Understand and apply knowledge of the concept of conservation of mass/matter

• When something is broken into parts, the parts have the same total mass as the original item.

Understand and apply knowledge of sound, light, electricity, magnetism, and heat.

• Sound is produced when vibrations from objects travel through a medium and are received. Sound can vary in volume. The pitch of a sound can be varied by changing the rate of vibration.

• Light travels in a straight line until it strikes an object. Light can be reflected by a mirror, refracted by a lens, or absorbed by an object.

• Electricity in circuits can produce light, heat, sound, and magnetic effects. Electricity can only flow through a closed circuit.

• Magnets attract and repel each other and certain kinds of other materials.

• Heat can be produced in many ways, such as burning, rubbing, or mixing one substance with another. Heat can move from one object to another by conduction.

Transfer of Energy

NGSS: Electromagnetic radiation

Understand and apply knowledge of forms of energy and energy transfer

• Energy is a property of many substances and is associated with heat, light, electricity, mechanical motion, sound, nuclei, and the nature of a chemical. Energy is transferred in many ways.

• Heat moves in predictable ways, flowing from warmer objects to cooler ones, until both reach the same temperature.

• Light interacts with matter by transmission (including refraction), absorption, or scattering (including reflection). To see an object, light from that object- emitted by or scattered from it- must enter the eye.

• Electrical circuits provide a means of transferring electrical energy when heat, light, sound, and chemical changes are produced.

• In most chemical and nuclear reactions, energy is transferred into or out of a system. Heat, light, mechanical motion, or electricity might all be involved in such transfers.

• The sun is a major source of energy for changes on the earth’s surface. The sun loses energy by emitting light. A tiny fraction of that light reaches the earth, transferring energy form the sun to the earth. The sun’s energy arrives as light with a range of wavelengths, consisting of visible light, infrared, and ultraviolet radiation.

Understand and apply knowledge of motions and forces

Principles that underlie the concept and/or skill include but are not limited to:• Motions• Forces• Newton’s Laws• Objects change their motion only when a net force is applied. Laws of

motion are used to calculate precisely the effects of forces on the motion of objects. The magnitude of the change in motion can be calculated using the relationship F = ma, which is independent of the nature of the force. Whenever one object exerts force on another, a force equal in magnitude and opposite in direction is exerted on the first object.

Principles that underlie the concept and/or skill include but are not limited to:• Gravitation• Mass versus weight• Gravitation is a universal force that each mass exerts on any other

mass. The strength of the gravitational attractive force between two masses is proportional to the masses and is inversely proportional to the square of the distance between them.

Principles that underlie the concept and/or skill include but are not limited to:• Electric and magnetic forces• The electric force is a universal force that exists between any two

charged objects. Opposite charges attract, while like charges repel. The strength of the force is proportional to the charges, and, as with gravitation, inversely proportional to the square of the distance between them.

• Between any two charged particles, electric force is vastly greater than the gravitational force. Most observable forces such as those exerted by a coiled spring or friction may be traced to electric forces acting between atoms and molecules.

• Electricity and magnetism are two aspects of a single electromagnetic force. Moving electric charges produce magnetic forces, and moving magnets produce electric forces. These effects help students understand electric motors and generators.

Motion and Forces

NGSS: Forces and Interactions

Understand and apply knowledge of the positions and motions of objects

• The position of an object can be described by locating it relative to its background.

• An object’s motion can be described by observing and measuring its position over time.

• An object’s position or movement can be changed by pushing or pulling.

Understand and apply knowledge of how forces are related to an object’s motion.

• The motion of an object can be described by its position, direction of motion, and speed. That motion can be measured and represented on a graph.

• Changes in speed or direction of motion are caused by forces. The greater the force, the greater the change in motion. The more massive an object, the less effect a given force will have in changing its motion.

Understand and apply knowledge of motions and forces.

• The motion of an object can be described by its position, direction of motion, and speed. That motion can be measured and represented on a graph.

• An object that is not being subjected to a force will continue to move at a constant speed and in a straight line.

• If more than one force acts on an object along a straight line, then the forces will reinforce or cancel one another, depending on their direction and magnitude. Unbalanced forces will cause changes in speed or direction of an object’s motion.

Understand and apply knowledge of interactions of energy and matter

Principles that underlie the concept and/or skill include but are not limited to:• Wave phenomena• Energy and matter• Electromagnetic waves• “Interactions of energy and matter” is an essential concept of a world-

class secondary science curriculum. Included in “interactions of energy and matter” is the following content:

• Waves, including sound and seismic waves, waves on water, and light waves have energy and can transfer energy when they interact with matter.

• Electromagnetic waves result when a charged object is accelerated or decelerated. Electromagnetic waves include radio waves (the longest wavelength), microwaves, infrared radiation (radiant heat), visible light, ultraviolet radiation, X-rays, and gamma rays. The energy of electromagnetic waves is carried in packets whose magnitude is inversely proportional to the wavelength.


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