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Content Area: MathematicsGrade Level Expectations: High SchoolStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Quantitative reasoning is used to make sense of quantities and their relationships in problem situations Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Reason quantitatively and use units to solve problems (CCSS: N-Q)Use units as a way to understand problems and to guide the solution of multi-step problems. (CCSS: N-Q.1)Choose and interpret units consistently in formulas. (CCSS: N-Q.1)Choose and interpret the scale and the origin in graphs and data displays. (CCSS: N-Q.1)Define appropriate quantities for the purpose of descriptive modeling. (CCSS: N-Q.2)Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. (CCSS: N-Q.3)Describe factors affecting take-home pay and calculate the impact (PFL)Design and use a budget, including income (net take-home pay) and expenses (mortgage, car loans, and living expenses) to demonstrate how living within your means is essential for a secure financial future (PFL) Inquiry Questions: Can numbers ever be too big or too small to be useful? How much money is enough for retirement? (PFL) What is the return on investment of post-secondary educational opportunities? (PFL) Relevance & Application: The choice of the appropriate measurement tool meets the precision requirements of the measurement task. For example, using a caliper for the manufacture of brake discs or a tape measure for pant size. The reading, interpreting, and writing of numbers in scientific notation with and without technology is used extensively in the natural sciences such as representing large or small quantities such as speed of light, distance to other planets, distance between stars, the diameter of a cell, and size of a micro–organism. Fluency with computation and estimation allows individuals to analyze aspects of personal finance, such as calculating a monthly budget, estimating the amount left in a checking account, making informed purchase decisions, and computing a probable paycheck given a wage (or salary), tax tables, and other deduction schedules. Nature Of: Using mathematics to solve a problem requires choosing what mathematics to use; making simplifying assumptions, estimates, or approximations; computing; and checking to see whether the solution makes sense. Mathematicians reason abstractly and quantitatively. (MP) Mathematicians attend to precision. (MP)

Content Area: MathematicsGrade Level Expectations: PreschoolStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. Quantities can be represented and counted Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Count and represent objects including coins to 10 (PFL) Match a quantity with a numeral Inquiry Questions: What do numbers tell us? Is there a biggest number? Relevance & Application: Counting helps people to determine how many such as how big a family is, how many pets there are, such as how many members in one’s family, how many mice on the picture book page, how many counting bears in the cup. People sort things to make sense of sets of things such as sorting pencils, toys, or clothes. Nature Of: Numbers are used to count and order objects. Mathematicians reason abstractly and quantitatively. (MP) Mathematicians attend to precision. (MP)

Content Area: MathematicsGrade Level Expectations: Seventh GradeStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Linear measure, angle measure, area, and volume are fundamentally different and require different units of measure Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: State the formulas for the area and circumference of a circle and use them to solve problems. (CCSS: 7.G.4) Give an informal derivation of the relationship between the circumference and area of a circle. (CCSS: 7.G.4) Use properties of supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. (CCSS: 7.G.5) 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. (CCSS: 7.G.6) Inquiry Questions: How can geometric relationships among lines and angles be generalized, described, and quantified? How do line relationships affect angle relationships? Can two shapes have the same volume but different surface areas? Why? Can two shapes have the same surface area but different volumes? Why? How are surface area and volume like and unlike each other? What do surface area and volume tell about an object? How are one-, two-, and three-dimensional units of measure related? Why is pi an important number? Relevance & Application: The ability to find volume and surface area helps to answer important questions such as how to minimize waste by redesigning packaging, or understanding how the shape of a room affects its energy use. Nature Of: Geometric objects are abstracted and simplified versions of physical objects. Geometers describe what is true about all cases by studying the most basic and essential aspects of objects and relationships between objects. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians construct viable arguments and critique the reasoning of others. (MP)

Content Area: MathematicsGrade Level Expectations: Fifth GradeStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. Properties of multiplication and addition provide the foundation for volume an attribute of solids. Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Model and justify the formula for volume of rectangular prisms. (CCSS: 5.MD.5b)Model the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes.1 (CCSS: 5.MD.5b)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. (CCSS: 5.MD.5a)Represent threefold whole-number products as volumes to represent the associative property of multiplication. (CCSS: 5.MD.5a) Find volume of rectangular prisms using a variety of methods and use these techniques to solve real world and mathematical problems. (CCSS: 5.MD.5a)Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. (CCSS: 5.MD.4)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. (CCSS: 5.MD.5b)Use the additive nature of volume to find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts. (CCSS: 5.MD.5c) Inquiry Questions: Why do you think a unit cube is used to measure volume? Relevance & Application: The ability to find volume helps to answer important questions such as which container holds more. Nature Of: Mathematicians create visual and physical representations of problems and ideas that reveal relationships and meaning. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians model with mathematics. (MP)

1 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. (CCSS: 5.MD.3a)
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. (CCSS: 5.MD.3b)

Content Area: MathematicsGrade Level Expectations: Fourth GradeStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. Appropriate measurement tools, units, and systems are used to measure different attributes of objects and time Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. (CCSS: 4.MD)Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. (CCSS: 4.MD.1)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.1 (CCSS: 4.MD.1)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. (CCSS: 4.MD.2)Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. (CCSS: 4.MD.2)Apply the area and perimeter formulas for rectangles in real world and mathematical problems.2 (CCSS: 4.MD.3) Use concepts of angle and measure angles. (CCSS: 4.MD)Describe angles as geometric shapes that are formed wherever two rays share a common endpoint, and explain concepts of angle measurement.3 (CCSS: 4.MD.5)Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. (CCSS: 4.MD.6)Demonstrate that angle measure as additive.4 (CCSS: 4.MD.7)Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems.5 (CCSS: 4.MD.7) Inquiry Questions: How do you decide when close is close enough? How can you describe the size of geometric figures? Relevance & Application: Accurate use of measurement tools allows people to create and design projects around the home or in the community such as flower beds for a garden, fencing for the yard, wallpaper for a room, or a frame for a picture. Nature Of: People use measurement systems to specify the attributes of objects with enough precision to allow collaboration in production and trade. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians use appropriate tools strategically. (MP) Mathematicians attend to precision. (MP)

1 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), ... (CCSS: 4.MD.1)

2 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. (CCSS: 4.MD.3)

3 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. (CCSS: 4.MD.5a)
An angle that turns through n one-degree angles is said to have an angle measure of n degrees. (CCSS: 4.MD.5b)

4 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. (CCSS: 4.MD.7)

5 e.g., by using an equation with a symbol for the unknown angle measure. (CCSS: 4.MD.7)

Content Area: MathematicsGrade Level Expectations: Third GradeStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Linear and area measurement are fundamentally different and require different units of measure Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Use concepts of area and relate area to multiplication and to addition. (CCSS: 3.MD)Recognize area as an attribute of plane figures and apply concepts of area measurement.5 (CCSS: 3.MD.5)Find area of rectangles with whole number side lengths using a variety of methods6 (CCSS: 3.MD.7a)Relate area to the operations of multiplication and addition and recognize area as additive.7 (CSSS: 3.MD.7) Describe perimeter as an attribute of plane figures and distinguish between linear and area measures. (CCSS: 3.MD) Solve real world and mathematical problems involving perimeters of polygons. (CCSS: 3.MD.8)Find the perimeter given the side lengths. (CCSS: 3.MD.8)Find an unknown side length given the perimeter. (CCSS: 3.MD.8)Find rectangles with the same perimeter and different areas or with the same area and different perimeters. (CCSS: 3.MD.8) Inquiry Questions: What kinds of questions can be answered by measuring? What are the ways to describe the size of an object or shape? How does what we measure influence how we measure? What would the world be like without a common system of measurement? Relevance & Application: The use of measurement tools allows people to gather, organize, and share data with others such as sharing results from science experiments, or showing the growth rates of different types of seeds. A measurement system allows people to collaborate on building projects, mass produce goods, make replacement parts for things that break, and trade goods. Nature Of: Mathematicians use tools and techniques to accurately determine measurement. People use measurement systems to specify attributes of objects with enough precision to allow collaboration in production and trade. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians model with mathematics. (MP)

5 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. (CCSS: 3.MD.5a)
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. (CCSS: 3.MD.5b)

6 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. (CCSS: 3.MD.5a)
Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). (CCSS: 3.MD.6)
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. (CCSS: 3.MD.7a)
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. (CCSS: 3.MD.7b)

7 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. (CCSS: 3.MD.7d)
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. (CCSS: 3.MD.7c)

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 3. Time and attributes of objects can be measured with appropriate tools Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. (CCSS: 3.MD)Tell and write time to the nearest minute. (CCSS: 3.MD.1)Measure time intervals in minutes. (CCSS: 3.MD.1)Solve word problems involving addition and subtraction of time intervals in minutes8 using a number line diagram. (CCSS: 3.MD.1)Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (CCSS: 3.MD.2)Use models to add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units.9 (CCSS: 3.MD.2) Inquiry Questions: Why do we need standard units of measure? Why do we measure time? Relevance & Application: A measurement system allows people to collaborate on building projects, mass produce goods, make replacement parts for things that break, and trade goods. Nature Of: People use measurement systems to specify the attributes of objects with enough precision to allow collaboration in production and trade. Mathematicians use appropriate tools strategically. (MP) Mathematicians attend to precision. (MP)

8 e.g., by representing the problem on a number line diagram. (CCSS: 3.MD.1)

9 e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. (CCSS: 3.MD.2)

Content Area: MathematicsGrade Level Expectations: Second GradeStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Some attributes of objects are measurable and can be quantified using different tools Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Measure and estimate lengths in standard units. (CCSS: 2.MD)Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. (CCSS: 2.MD.1)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. (CCSS: 2.MD.2)Estimate lengths using units of inches, feet, centimeters, and meters. (CCSS: 2.MD.3)Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit. (CCSS: 2.MD.4) Relate addition and subtraction to length. (CCSS: 2.MD)Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units1 and equations with a symbol for the unknown number to represent the problem. (CCSS: 2.MD.5)Represent whole numbers as lengths from 0 on a number line2 diagram and represent whole-number sums and differences within 100 on a number line diagram. (CCSS: 2.MD.6) Solve problems time and money. (CCSS: 2.MD)Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. (CCSS: 2.MD.7)Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and ¢ symbols appropriately.3 (CCSS: 2.MD.8) Inquiry Questions: What are the different things we can measure? How do we decide which tool to use to measure something? What would happen if everyone created and used their own rulers? Relevance & Application: Measurement is used to understand and describe the world including sports, construction, and explaining the environment. Nature Of: Mathematicians use measurable attributes to describe countless objects with only a few words. Mathematicians use appropriate tools strategically. (MP) Mathematicians attend to precision. (MP)

1 e.g., by using drawings (such as drawings of rulers). (CCSS: 2.MD.5)

2 with equally spaced points corresponding to the numbers 0, 1, 2, ... (CCSS: 2.MD.6)

3 Example: If you have 2 dimes and 3 pennies, how many cents do you have? (CCSS: 2.MD.6)

Content Area: MathematicsGrade Level Expectations: First GradeStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Measurement is used to compare and order objects and events Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Measure lengths indirectly and by iterating length units. (CCSS: 1.MD)Order three objects by length; compare the lengths of two objects indirectly by using a third object. (CCSS: 1.MD.1)Express the length of an object as a whole number of length units.6 (CCSS: 1.MD.2) Tell and write time. (CCSS: 1.MD)Tell and write time in hours and half-hours using analog and digital clocks. (CCSS: 1.MD.3) Inquiry Questions: How can you tell when one thing is bigger than another? Why do we measure objects and time? How are length and time different? How are they the same? Relevance & Application: Time measurement is a means to organize and structure each day and our lives, and to describe tempo in music. Measurement helps to understand and describe the world such as comparing heights of friends, describing how heavy something is, or how much something holds. Nature Of: With only a few words, mathematicians use measurable attributes to describe countless objects. Mathematicians use appropriate tools strategically. (MP) Mathematicians attend to precision. (MP)

6 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. (CCSS: 1.MD.2)

Content Area: MathematicsGrade Level Expectations: KindergartenStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Measurement is used to compare and order objects Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Describe and compare measurable attributes. (CCSS: K.MD)Describe measurable attributes of objects, such as length or weight. (CCSS: K.MD.1)Describe several measurable attributes of a single object. (CCSS: K.MD.1)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.7 (CCSS: K.MD.2)Order several objects by length, height, weight, or price (PFL) Classify objects and count the number of objects in each category. (CCSS: K.MD)Classify objects into given categories. (CCSS: K.MD.3)Count the numbers of objects in each category. (CCSS: K.MD.3)Sort the categories by count. (CCSS: K.MD.3) Inquiry Questions: How can you tell when one thing is bigger than another? How is height different from length? Relevance & Application: Measurement helps to understand and describe the world such as in cooking, playing, or pretending. People compare objects to communicate and collaborate with others. For example, we describe items like the long ski, the heavy book, the expensive toy. Nature Of: A system of measurement provides a common language that everyone can use to communicate about objects. Mathematicians use appropriate tools strategically. (MP) Mathematicians attend to precision. (MP)

7 For example, directly compare the heights of two children and describe one child as taller/shorter. (CCSS: K.MD.2)

Content Area: MathematicsGrade Level Expectations: PreschoolStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Measurement is used to compare objects Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Describe the order of common events Group objects according to their size using standard and non-standard forms (height, weight, length, or color brightness) of measurement Sort coins by physical attributes such as color or size (PFL) Inquiry Questions: How do we know how big something is? How do we describe when things happened? Relevance & Application: Understanding the order of events allows people to tell a story or communicate about the events of the day. Measurements helps people communicate about the world. For example, we describe items like big and small cars, short and long lines, or heavy and light boxes. Nature Of: Mathematicians sort and organize to create patterns. Mathematicians look for patterns and regularity. The search for patterns can produce rewarding shortcuts and mathematical insights. Mathematicians reason abstractly and quantitatively. (MP) Mathematicians use appropriate tools strategically. (MP)