Pojoaque Valley Schools Math Common Core Pacing Guide Introduction The Pojoaque Valley Schools pacing guide documents are intended to guide teachers’ use of Common Core State Standards (CCSS) over the course of an instructional school year. The guides identify the focus standards by quarter. Teachers should understand that the focus standards emphasize deep instruction for that timeframe. However, because a certain quarter does not address specific standards, it should be understood that previously taught standards should be reinforced while working on the focus standards for any designated quarter. Some standards will recur across all quarters due to their importance and need to be addressed on an ongoing basis. The Math pacing guides are grounded in four key components: the key fluency expectations for each grade level, the critical areas designated in the CCSS Math Standards, the Common Core Standards for Mathematics and the integration of the Standards for Mathematical Practice. In planning instruction it is important that math teachers incorporate the 8 mathematical practices for mathematics to ensure that the Common Core standards are mastered by all students. The Math CCSS pacing guides contain the following elements: Grade Level: Identify the grade level of the intended standard Standard with code: Defines the knowledge and skills for students. The code contains the grade level, domain and standard number. Domain: Larger groups of related standards. Standards from different domains may sometimes be closely related. Cluster: Summarize groups of related standards. Skills and Knowledge: Identified as subsets of the standard and appear in one or more quarters. Define the skills and knowledge embedded in the standard to meet the full intent of the standard itself. Version 2 of the Pojoaque Valley School District Pacing guides for Reading Language Arts and Mathematics are based on the done by staff and teachers of the school district using the Kentucky model, and a synthesis of the excellent work done by Wisconsin Cooperative Educational Service Agency 7 (CESA 7) School Improvement Services, Green Bay, WI. (2010), Standards Insight project. Standards Insight was developed to give educators a tool for in depth investigation of the Common Core State Standards (CCSS). The CCSS are “unpacked” or dissected, identifying specific knowledge, skills, vocabulary, understandings, and evidence of student attainment for each standard. Standards Insight
Evidence of Student Attainment Vocabulary Knowledge Skills Common Core Essential Elements Instructional Achievement Level Descriptors Students: Given a variety of multiplication and division word problems within 100, Explain and justify solutions and solution paths using connections among a variety of representations (e.g., place value blocks, drawings, open arrays, and equations with a symbol for the unknown). See glossary for problem types (Table 2). Students know: Characteristics of multiplication and division contexts, Multiplication and division strategies. Students understand that/are able to: Represent quantities and operations (multiplication and division) physically, pictorially, or symbolically, Strategically use a variety of representations to solve multiplication and division word problems, Use informal and mathematical language to communicate the connections among multiplication and division contexts and related physical, pictorial, or symbolic representations, Accurately compute products and quotients, Use symbols to represent unknown quantities in equations. EE3.OA.3. See EE3.OA.1. for repeated addition, a foundational skill for multiplication and division. (Multiplication begins in grade 4 and division begins in grade 5). EE3.OA.3. See EE3.OA.1. for repeated addition, a foundational skill for multiplication and division. (Multiplication begins in grade 4 and division begins in grade 5).
Evidence of Student Attainment Vocabulary Knowledge Skills Common Core Essential Elements Instructional Achievement Level Descriptors Students: Given multiplication and division problems within 100, Use the properties of operations and descriptive language for the property to justify their products and quotients (e.g., If I know that 8 x 5 is 40, and two more groups of 8 would be 16, then 8 x 7 must be 40 + 16 or 56). Commutative Property of Multiplication Associative Property of Multiplication Distributive Property Students know: Commutative, Associative, Identity and Zero Properties of Multiplication and the Distributive Property, Strategies for finding products and quotients. Students understand that/are able to: Strategically and efficiently apply properties of multiplication and division in order to find products and quotients. The order in which factors are multiplied does not change the product. EE3.OA.5. N/A (Multiplication begins at grade 4). EE3.OA.5. N/A (Multiplication begins at grade 4).
Evidence of Student Attainment Vocabulary Knowledge Skills Common Core Essential Elements Instructional Achievement Level Descriptors Students: Given any single digit multiplication problem or a division problem with a single digit divisor and an unknown single digit quotient, Use an efficient strategy (e.g., recall, inverse operations, arrays, derived facts, properties of operations, etc.) to name the product or quotient. Students know: Strategies for finding products and quotients. Students understand that/are able to: Use multiplication and division strategies efficiently based on the numbers in the problems Efficient application of computation strategies are based on the numbers in the problems.. EE3.OA.7. N/A (Multiplication begins grade 4 and division begins in grade 5). EE3.OA.7. N/A (Multiplication begins grade 4 and division begins in grade 5).
Evidence of Student Attainment Vocabulary Knowledge Skills Common Core Essential Elements Instructional Achievement Level Descriptors Students: Given any number less than 1,000, Round it to the nearest 10 or 100 and justify the answer using place value vocabulary, (e.g., "Rounding 147 to the nearest 10 is 150 because 147 is between 140 and 150 and is more than half way to 150). Students know: Place value (ones, tens, hundreds), Rounding. Students understand that/are able to: Count by 10s and 100s, Determine what is halfway between two multiples of 10 or 100, Round to the nearest 10 or 100, Use place value vocabulary and logical reasoning to justify solutions to rounding problems. Rounding and place value can be used to estimate quantities by changing the original number to the closest multiple of a power of 10. EE3.NBT.1. Identify the two 10s a number comes in between on a number line (numbers 0-30). Level IV Students will: EE3.NBT.1. Identify the two 10s a number comes in between and tell which is closest (numbers 0-50). Ex. Use a color beaded number line to identify the number and round to the closest 10. Ex. Given a number line separated into tens (0-10, 10-20, etc.), stand on a number and identify the 10 that is closer. Level III Students will: EE3.NBT.1. Identify the two 10s a number comes in between on a number line (numbers 0-30). Ex. Use a color beaded number line to identify the two 10s a number falls between. Ex. Given the number 14, they would identify 10 and 20.* Level II Students will: EE3.NBT.1. Identify tens on a number line. Ex. Given a number line, circle the tens. Ex. Stand on a number chart on 10, 20, 30, etc. Level I Students will: EE3.NBT.1. Identify a number. Ex. Point to any number from one to three on a number line. Ex. Participate in a cake walk. When the music stops, look to see if they are on the number that is called out. Ex. Identify a number when point to or presented on a card. Ex. Given a number from one to three, point to the number symbol. *Refer to the Common Core Essential Elements document for diagram.
Evidence of Student Attainment Vocabulary Knowledge Skills Common Core Essential Elements Instructional Achievement Level Descriptors Students: Efficiently use strategies based on place value and properties of operations to multiply one-digit numbers by multiples of 10 (from 10-90) and justify their answers. Students know: Place value models for multiplying numbers (e.g., open arrays, place value blocks), Strategies for multiplying one-digit numbers, Strategies for mentally multiplying one-digit numbers by multiples of powers of 10. Students understand that/are able to: Use mental strategies based on an understanding of place value, properties of operations, and knowledge of one-digit multiplication to find products, Use a variety of place value models of multiplication problems to justify strategies and solutions. Patterns in the place value system and properties of operations can be used to efficiently compute products. EE3.NBT.3. Count by tens using money. Level IV Students will: EE3.NBT.3. Compare the value of money based on place value. Ex. Use money (dimes and pennies) to represent place value. Ex. Given 15 pennies, create a one group of 10 and a group of five ones. Level III Students will: EE3.NBT.3. Count by tens using money. Ex. Given three dimes, count by 10 to determine total. Ex. Given five dimes, count by 10 to determine total. Level II Students will: EE3.NBT.3. Identify whole numbers to 10. Ex. Given sets of 10 pennies, pair with numbers. Ex. Given sets of 10 pennies, pair with dimes. Level I Students will: EE3.NBT.3. Count pennies to 10.
Evidence of Student Attainment Vocabulary Knowledge Skills Common Core Essential Elements Instructional Achievement Level Descriptors Students: Tell and write time to the nearest minute using analog and digital clocks, Use strategies (e.g., watch the movement of a second or minute hand, count the changing of digits) to estimate and measure time intervals in minutes, Solve word problems involving addition and subtraction of time intervals using representations of time passage such as arrows on open number lines. Students know: Conventions for time notation, Time sequence patterns, Strategies for determining elapsed time (e.g., using number lines). Students understand that/are able to: Accurately read and write time to the nearest minute from analog and digital clocks, Measure time intervals in minutes, Strategically select and apply methods for showing elapsed time to solve word problems. Analog and digital clocks represent the time at any particular moment, Clocks show the passage of time with the movement of the hands or the changing of digits, Representations for use in solving problems are selected based on the context and numbers in the problem. EE3.MD.1. Tell time to the hour on a digital clock. Level IV Students will: EE3.MD.1. Tell time to the half hour using a digital clock. Ex. Look at a digital clock and read the time. Ex. When looking at a schedule, identify the hour. Level III Students will: EE3.MD.1. Tell time to the hour on a digital clock. Ex. Given a time written to the hour, write the digital time. Ex. Identify the time of a digital clock that is set to the hour. Ex. Given a time on a digital clock, say the time to the hour. Level II Students will: EE3.MD.1. Identify which is the hour on a digital clock. Ex. Relate the hour with the time on their daily schedule. Ex. Given cards showing digital clocks - with one clock having the hour circled and one clock with the minutes circled, indicate the clock with the hour circled. Level I Students will: EE3.MD.1. Differentiate a digital clock from other measurement tools as a tool for telling time. Ex. Given a digital clock and a measuring cup, identify the clock for telling time. Ex. Asked, “How do we know when it is time to go to lunch?” indicate a clock.
Evidence of Student Attainment Vocabulary Knowledge Skills Common Core Essential Elements Instructional Achievement Level Descriptors Students: Organize and represent data with several categories using picture graphs (pictographs) and bar graphs with scales other than 1, Reason quantitatively to answer one- and two-step "how many more?" and "how many less?" problems using information presented in the scaled pictographs and bar graph. Scaled pictograph Scaled bar graph Students know: Strategies for collecting, organizing, and recording data (including scaled pictographs and scaled bar graphs), Strategies for counting and comparing quantities, Strategies for solving addition and subtraction one and two-step problems. Students understand that/are able to: Choose and apply appropriate strategies for organizing and recording data, Read and interpret graphical representations (pictographs and bar graphs with scales other than 1) of data, Communicate and defend solutions and solutions paths. Questions concerning mathematical contexts can be answered by collecting and organizing data scaled pictographs and bar graphs, Understand that logical reasoning and connections between representations provide justifications for solutions. EE3.MD.3. Use picture or bar graph data to answer questions about data. Level IV Students will: EE3.MD.3. Interpret data to answer questions. Ex. Identify how they know there were no rainy days that week based on the chart. Ex. State two facts about the data on a graph. Level III Students will: EE3.MD.3. Use picture or bar graph data to answer questions about data. Ex. Identify from a picture or bar graph how many students in the class were identified as wearing blue shirts. Ex. State how many days were sunny as charted on a weather chart. Level II Students will: EE3.MD.3. Organize data. Ex. Take data collected from the lunch choices and place data into appropriate categories. Ex. Place data on a chart to represent the data collected. Level I Students will: EE3.MD.3. Collect data. Ex. Using two posters, one for the students with brown hair, and one for the students with “yellow” hair, place their picture on the poster board that indicates what color hair they have. Ex. Use a daily survey to collect data on different interest.
CCSS Math Pacing Guide Grade 3 Grade Level: 3rd Standard with code: 3.MD.7a Relate area to the operations of multiplication and addition. a. 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. Domain: Measurement and Data Cluster: Geometric measurement: understand concepts of area and relate area to multiplication and to addition. Quarter 1: Quarter 2: Quarter 3: Quarter 4: Find the area of a rectangle by tiling it in unit squares. Find the side lengths of a rectangle in units. Compare the area found by tiling a rectangle to the area found by multiplying the side lengths. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment Vocabulary Knowledge Skills Common Core Essential Elements Instructional Achievement Level Descriptors Students: Given a polygon that may be decomposed into 2 or more rectangles, Find the total area by decomposing the figure into non-overlapping rectangles, finding the area of each, and find Distributive Property Rectilinear figures Students know: Relationships between rectangular arrays and the corresponding multiplication problems (counting unit squares in rows and columns compared to multiplying length by width), Students understand that/are able to: Communicate the relationships between rectangular array models of areas and multiplication and addition problems including modeling the EE3.MD.5-7. N/A (Area begins at grade 6). EE3.MD.5-7. N/A (Area begins at grade 6).
the sum of the areas. Given a rectangle with whole number length sides, Find and justify the area of the rectangle by relating a tile covered model to a corresponding multiplication problem (counting unit squares in rows and columns compared to multiplying length by width). Using array cards or tiles, Create and explain rectangular models to show that the area of a rectangle with side lengths a and d (where d=b+c) is the same as the area of two smaller rectangles with area a x b and a x c. (the Distributive Property). Strategies for finding sums and products of whole numbers. Distributive Property, Model the area of rectangles using manipulatives or graph paper, Strategically and fluently choose strategies for finding sums and products, Accurately compute sums and products. The area of a plane figure is measured by the number of same-size squares that exactly cover the interior space of the figure, Multiplication is putting together equal sized groups, Rectangular arrays represent groups (rows) of equal size (number of columns), Multiplication is distributive over the addition of whole numbers.
Evidence of Student Attainment Vocabulary Knowledge Skills Common Core Essential Elements Instructional Achievement Level Descriptors Students: Given a polygon that may be decomposed into 2 or more rectangles, Find the total area by decomposing the figure into rectangles, finding the area of each, and find the sum of the areas. Given a rectangle with whole number length sides, Find and justify the area of the rectangle by relating a tile covered model to a corresponding multiplication problem (counting unit squares in rows and columns compared to multiplying length by width). Using array cards or tiles, Distributive Property Rectilinear figures Students know: Relationships between rectangular arrays and the corresponding multiplication problems (counting unit squares in rows and columns compared to multiplying length by width), Strategies for finding sums and products of whole numbers. Students understand that/are able to: Communicate the relationships between rectangular array models of areas and multiplication and addition problems including modeling the Distributive Property, Model the area of rectangles using manipulatives or graph paper, Strategically and fluently choose strategies for finding sums and products, Accurately compute sums and products. The area of a plane figure is measured by the number of squares that exactly cover the interior space of the figure, Multiplication is putting together equal sized groups, Rectangular arrays EE3.MD.5-7. N/A (Area begins at grade 6). EE3.MD.5-7. N/A (Area begins at grade 6).
CCSS Math Pacing Guide Grade 3 Grade Level: 3rd Standard with code: 3.MD.7c Relate area to the operations of multiplication and addition. c. 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. Domain: Measurement and Data Cluster: Geometric measurement: understand concepts of area and relate area to multiplication and to addition. Quarter 1: Quarter 2: Quarter 3: Quarter 4: Multiply using an area model (array). Relate area of a rectangle to multiplication and addition by modeling the distributive property. Area of a Rectangle 3 x (5+2) = 3x5 + 3x2 Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment Vocabulary Knowledge Skills Common Core Essential Elements Instructional Achievement Level Descriptors Students: Given a polygon that may be decomposed into 2 or more rectangles, Find the total area by decomposing the figure into non-overlapping rectangles, finding the area of each, and find the sum of the areas. Distributive Property Rectilinear figures Students know: Relationships between rectangular arrays and the corresponding multiplication problems (counting unit squares in rows and columns compared to multiplying length by width), Students understand that/are able to: Communicate the relationships between rectangular array models of areas and multiplication and addition problems including modeling the Distributive Property, EE3.MD.5-7. N/A (Area begins at grade 6). EE3.MD.5-7. N/A (Area begins at grade 6).
Given a rectangle with whole number length sides, Find and justify the area of the rectangle by relating a tile covered model to a corresponding multiplication problem (counting unit squares in rows and columns compared to multiplying length by width). Using array cards or tiles, Create and explain rectangular models to show that the area of a rectangle with side lengths a and d (where d=b+c) is the same as the area of two smaller rectangles with area a x b and a x c. (the Distributive Property). Strategies for finding sums and products of whole numbers. Model the area of rectangles using manipulatives or graph paper, Strategically and fluently choose strategies for finding sums and products, Accurately compute sums and products. The area of a plane figure is measured by the number of same-size squares that exactly cover the interior space of the figure, Multiplication is putting together equal sized groups, Rectangular arrays represent groups (rows) of equal size (number of columns), Multiplication is distributive over the addition of whole numbers.
Students: Given a polygon that may be decomposed into 2 or more rectangles, Find the total area by decomposing the figure into rectangles, finding the area of each, and find the sum of the areas. Given a rectangle with whole number length sides, Find and justify the area of the rectangle by relating a tile covered model to a corresponding multiplication problem (counting unit squares in rows and columns compared to multiplying length by width). Using array cards or tiles, Create and explain rectangular models to show that the area of a rectangle with side lengths a and d (where d=b+c) is the same as the area of Distributive Property Rectilinear figures Students know: Relationships between rectangular arrays and the corresponding multiplication problems (counting unit squares in rows and columns compared to multiplying length by width), Strategies for finding sums and products of whole numbers. Students understand that/are able to: Communicate the relationships between rectangular array models of areas and multiplication and addition problems including modeling the Distributive Property, Model the area of rectangles using manipulatives or graph paper, Strategically and fluently choose strategies for finding sums and products, Accurately compute sums and products. The area of a plane figure is measured by the number of same-size squares that exactly cover the interior space of the figure, Multiplication is putting together equal sized groups, Rectangular arrays represent groups (rows) of equal size (number of columns), EE3.MD.5-7. N/A (Area begins at grade 6). EE3.MD.5-7. N/A (Area begins at grade 6).
solutions to 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. Area Polygons objects, specifically perimeter and area, Strategies for modeling measurement problems involving perimeter and area, Strategies for representing and computing perimeter and area. Strategically use a variety of models and representations to solve measurement problems involving area and perimeter, Accurately compute using whole numbers, Use logical reasoning to justify solutions and solution paths by connecting models to equations and computations. Perimeter is measured in length units and is the distance around a 2-D figure, The area of a plane figure is measured by the number of same-size squares that exactly cover the interior space of the figure. CCSS Math Pacing Guide Grade 3
Students: Given squares, rectangles, or circles, Cut or draw lines to divide the shapes into equal shares and justify their divisions by reasoning about equal area, Express the area of each part as a unit fraction of the whole (e.g., 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). Area Partition Students know: Strategies for decomposing shapes into equal shares, Fraction vocabulary; halves, thirds, fourths, quarters, fifths, eighths, and tenths. Students understand that/are able to: Decompose circles, squares, and rectangles into equal shares, Communicate the size of shares using the appropriate fraction terminology, Justify equal area of congruent and shares as equal shares of the same size whole. The same fractional parts of same- size 2-D shapes have equal area but do not have to be congruent (e.g., When two same-size rectangles are cut in half vertically, horizontally, or diagonally, the pieces are all one half of the original rectangle, have equal area but are not all congruent). EE3.G.2. Recognize that shapes can be partitioned into equal areas. Level IV Students will: EE3.G.2. Given shapes with multiple lines of symmetry, will be able to identify equal areas. Ex. Complete simple tangram puzzles with tangram pieces. Ex. Identify equal areas on complex shapes (i.e., stars, rectangle cut on the diagonal)* Level III Students will: EE3.G.2. Recognize that shapes can be partitioned into equal areas. Ex. Given a shape, cut the shape into equal areas. Ex. Cut a pizza into equal areas to hand out to students in the class. Level II Students will: EE3.G.2. Create shapes. Ex. Work a pattern block puzzle that results in a shape. Ex. Given three small rectangles, rearrange them into a larger rectangle. Level I Students will: EE3.G.2. Match shapes. Ex. Match a picture of a shape, to a shape in the classroom. Ex. Match two shapes from an array of three in which one is different. *Refer to the Common Core Essential Elements document for diagram.