A Teacher s Guide to. Marching Ahead with the Mathematics Florida Standards Grade 3

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1 A Teacher s Guide to Marching Ahead with the Mathematics Florida Standards Grade 3

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3 Getting the Facts about Mathematics Florida Standards Third Grade Table of Contents I. Planning Introduction to Pacing and Sequencing Pacing and Sequencing Chart Test Item Specifications Operations and Properties Tables II. III. Standards for Mathematical Practice What Do Good Problem Solvers Do? What Constitutes a Cognitively Demanding Task? Key Ideas in Mathematics Standards for Mathematical Practice Descriptions Standards for Mathematical Practice Student Behaviors Standards for Mathematical Practice Student Friendly Language Standards for Mathematical Practice Sample Questions for Teachers to Ask Standards for Mathematical Practice in Action Standards for Mathematical Practice in 3 rd Grade Standards for Mathematical Practice Posters Getting to Know the Mathematics Florida Standards (MAFS) Breaking the Code MAFS by Grade Level at a Glance Mathematics Florida Standards Changes CCSS Domains, Clusters, and Critical Areas of Focus Domain Progressions Third Grade Domain/Cluster Descriptors and Clarifications NOTE: While some of the documents in this section were written based on Common Core Standards, they still contain information that can be used with Mathematics Florida Standards (MAFS). The changes as listed on the chart titled Mathematics Florida Standards Changes must be considered when using these documents. IV. Additional Resources Addition and Subtraction Strategies Basic Multiplication and Division Strategies Four Corners and Rhombus Math Graphic Organizer Depth of Knowledge Levels/ Cognitive Complexity of Mathematics Items

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5 Planning

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7 INTRODUCTION TO PACING AND SEQUENCING- GRADE 3 INSTRUCTION: All instruction must be standards-based. The textbook is a resource and textbook lessons must be carefully chosen and aligned with the standards targeted for instruction. It is critical that the Pacing and Sequence Chart and the FSA Test Item Specifications are used for planning and implementing lessons. The entire pacing and sequencing chart should be previewed in order to begin with the end in mind and understand how the mathematical concepts grow throughout the year. MEASUREMENT: Hands-on opportunities for students to be engaged in measurement are critical. Hands-on measurement tasks may be taught within the science and social studies curriculum VOCABULARY: Correct mathematical vocabulary MUST be used. For example, students are expected to use terms such as addend, sum, product, and so on. CONNECTIONS BETWEEN THE DOMAINS: Standards are not meant to be taught in isolation. Each standard supports other standards and will continue to be developed throughout the year. MAFS.3.MD.1.2 (volume) and MAFS.3.MD.1.1 (time) The content of these standards must be developed within short segments rather than being taught by completing the textbook topics day by day. PROBLEM-SOLVING: Emphasis should be on engaging students in deeper levels of thinking and analyzing. Students must have many opportunities to explore the content of the standards through real-world problem-solving tasks. Mathematical discourse must be an integral part of instruction. ALGORITHMS AND FORMULAS: No formal algorithms or formulas are used in 3 rd grade POST-FSA IDEAS: Students should continue to work on critical areas within the grade level standards Project-based lessons and activities are encouraged. Possible resources to use are: AIMS Solve It! Navigating Through Numbers and Operations in Grades 3-5, NCTM EnVision Math Worldscapes Literature Library The Super Source Series, ETA/Cuisenaire Teaching Student-Centered Mathematics, Vol.1, J.A.Van de Walle and L.H. Lovin Good Questions for Math Teaching, by Peter Sullivan and Pat Lilburn

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9 Standards for Mathematical Practice Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Mathematics Florida Standards (MAFS) Construct viable arguments and critique the reasoning of others. Model with mathematics. First Nine Weeks Use appropriate tools strategically. Explanations and Examples * Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. MAFS.3.MD.2.3: 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. MAFS.3.MD.1.1: 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. Students should have opportunities reading and solving problems using scaled graphs before being asked to draw one. The following graphs all use five as the scale interval, but students should experience different intervals to further develop their understanding of scaled graphs and how they relate to number facts. While exploring data concepts, students should pose a question, collect data, analyze data, and interpret data. Students should be graphing data that are relevant to their lives. This standard calls for students to solve elapsed time, including word problems. Students could use clock models or number lines to solve. On a number line, students should be given the opportunity to determine the intervals and size of jumps on their number line. Students could use predetermined number lines (e.g., intervals every 5 minutes) or open number lines (intervals determined by students). Example: James wakes up for school at 6:45 a.m. He is ready to leave for school after he showers, dresses, and eats breakfast. It takes him 5 minutes to get showered, 15 minutes to get dressed and 15 minutes to eat breakfast. What time will he be ready to leave for school? MAFS.3.MD.1.2: 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. This standard asks for students to reason about the units of mass and volume. Students need multiple opportunities weighing classroom objects and filling containers to help them develop a basic understanding of the size of a liter, and weight of a gram, and a kilogram. Milliliters may also be used to show amounts that are less than a liter. Word problems should only be one-step and include the same units. Students need practice reading scales of different intervals. Students should estimate before actually finding the actual measurement. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 1 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

10 MAFS.3.NBT.1.1: Use place value understanding to round whole numbers to the nearest 10 or 100. Number sense and computational understanding are built on a firm understanding of place value. Place value understanding extends beyond an algorithm or procedure for rounding. The expectation is that students acquire a deep understanding of place value and number sense and are able to explain and reason about the answers they get when rounding. Students need various experiences using a number line and a hundreds chart as tools to support their work with rounding. MAFS.3.NBT.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. Fluency involves accuracy (correct answer), efficiency (a reasonable amount of steps), and flexibility (using strategies that demonstrate number sense). The word algorithm refers to a procedure or a series of steps. There are other algorithms other than the standard algorithm. Third grade students should have experiences other than the standard algorithm. The standard algorithm for addition and subtraction is not introduced until 4 th grade (MAFS). Problems should include both vertical and horizontal forms, including opportunities for students to apply the commutative and associative properties. Students explain their thinking and show their work by using strategies, and verify that their answer is reasonable. Example: There are 178 fourth graders and 225 fifth graders on the playground. What is the total number of students on the playground? Student = = = = 403 students Student 2 I added 2 to 178 to get 180. I added 220 to get 400. I added the 3 left over to get 403. Student 3 I know that 75 plus 25 equals 100. I then added 1 hundred from 178 and 2 hundreds from 275. I had a total of 4 hundreds and I had 3 more left to add. So I have 4 hundreds plus 3 more which is 403. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 2 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

11 MAFS.3.OA.4.9: 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. NOTE: This standard should be developed throughout the year as new operations are introduced. This standard calls for students to examine arithmetic patterns involving both addition and multiplication. Arithmetic patterns are patterns that change by the same rate, such as adding the same number. For example, the series 2, 4, 6, 8, 10 is an arithmetic pattern that increases by 2 between each term. This standard also involves identifying patterns related to the properties of operations. Examples: Even numbers are always divisible by 2. Even numbers can always be decomposed into 2 equal addends (14 = 7 + 7). This builds on student work with doubles in K-2. Multiples of even numbers (2, 4, 6, and 8) are always even numbers. On a multiplication chart, the products in each row and column increase by the same amount (skip counting). On an addition chart, the sums in each row and column increase by the same amount. MAFS.3.OA.1.1: 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. Multiplication requires students to think in terms of groups of things or an equal amount of objects, rather than individual things. Students learn that the multiplication symbol x means groups of and problems such as 6 x 8 refer to 6 groups of 8. The terms factor and product, should be used when describing multiplication. (factor x factor = product) MAFS.3.OA.1.2: Interpret whole-number quotients of wholenumbers, 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 Students recognize the operation of division in two different types of situations. One situation requires determining how many groups (Jim purchased 48 pencils. Each package contained 6 pencils. How many packages did he buy?) and the other situation requires sharing or determining how many in each group (Jim purchased 48 pencils. They were divided equally into 8 packages. How many pencils were in each package). The terms factor and product should be used for division (product factor = factor) in order to develop understanding of these as inverse operations. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 3 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

12 MAFS.3.OA.2.6: Understand division as an unknown-factor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8. Multiplication and division are inverse operations and that understanding can be used to find the unknown. Fact family triangles demonstrate the inverse operations of multiplication and division by showing the two factors and how those factors relate to the product. Examples: 3 x 5 = 15 5 x 3 = 15 (factor x factor = product) 15 3 = = 3 (product factor = factor) Students need opportunities to see that the answer to a division problem is one of the factors. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 4 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

13 Standards for Mathematical Practice 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. Second Nine Weeks Mathematics Florida Standards (MAFS) MAFS.3.OA.1.4: 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 =? Explanations and Examples * Students explore inverse relationships between multiplication and division. Students apply their understanding of the meaning of the equal sign as the same as or balances and are able to interpret an equation with an unknown. 3x 18 5 x 6 x 5 or Equations in the form of a x b = c and c = a x b should be used interchangeably, with the unknown in different positions. Example: Rachel has 3 bags of marbles. She has a total of 12 marbles. If each bag contains the same amount of marbles, how many marbles are in each bag? 3 x = = Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 5 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

14 MAFS.3.OA.1.3: 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. Students use a variety of representations for creating and solving one-step word problems. They use multiplication and division of whole numbers up to 10x10. Word problems should be represented in multiple ways including but not limited to: Equations: 3 x 4 =?, 4 x 3 =?, 12 4 =? and 12 3 =? Array: Equal groups Repeated addition: or repeated subtraction Three equal jumps forward from 0 on the number line to 12 or three equal jumps backwards from 12 to 0 Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 6 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

15 MAFS.3.OA.2.5: 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) 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) = = 56. (Distributive property) Students represent equations using various objects, pictures, words and symbols in order to develop their understanding of properties. They must apply these properties flexibly and fluently. Students need not use formal terms for these properties. Models help build understanding of the commutative property: 3 x 6 = 6 x 3 is the same quantity as 4 x 3 = 3 x 4 An array explicitly demonstrates the concept of the commutative property. 4 rows of 3 or 4 x 3 3 rows of 4 or 3 x 4 Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 7 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

16 MAFS.3.OA.2.5: (Continued) 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) 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) = = 56. (Distributive property) Using the associative property, students are able to create simpler calculations that are easier to do mentally. Students are introduced to the distributive property of multiplication over addition as a strategy for using products they know to solve products they don t know. Example: 3 x 5 x 2 = 3 x (5 x 2) = 3 x 10 = 30 3 x 5 x 2 = (3 x 5) x 2 = 15 x 2 = 30 If students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at or 56. Students should learn that they can decompose either of the factors. It is important to note that the students may record their thinking in different ways = 56 MAFS.3.NBT.1.3: Multiply one-digit whole numbers by multiples of 10 in the range (e.g., 9 80, 5 60) using strategies based on place value and properties of operations. This standard extends students work in multiplication by having them apply their understanding of place value. Students use base ten blocks, diagrams, or hundreds charts to multiply one-digit numbers by multiples of 10 from Examples: They can interpret 2 x 40 as 2 groups of 4 tens or 8 groups of 10. They understand that 5 x 60 is 5 groups of 6 tens or 30 tens, and know that 30 tens is 300. After developing this understanding they begin to recognize the patterns in multiplying one-digit numbers by multiples of 10. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 8 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

17 MAFS.3.OA.4.8: 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. Examples: Jerry earned 231 points at school last week. This week he earned 79 points. If he uses 60 points to earn free time on a computer, how many points will he have left? A student may use the number line above to describe his/her thinking, = 240 so now I need to add 70 more. 240, 250 (10 more), 260 (20 more), 270, 280, 290, 300, 310 (70 more). Now I need to count back , 300 (back 10), 290 (back 20), 280, 270, 260, 250 (back 60). On Monday Mike ran 5 miles. On Tuesday he ran 6 miles. His goal is to run 30 miles by the end of the week. How many miles does Mike have left to run in order to meet his goal? Write an equation and find the solution. (5+6+b=30) When students solve word problems, they use various estimation skills which include identifying when estimation is appropriate, determining the level of accuracy needed, selecting the appropriate method of estimation, and verifying solutions or determining the reasonableness of solutions. Estimation strategies include, but are not limited to: using benchmark numbers that are easy to compute (students select close whole numbers to determine an estimate). using friendly or compatible numbers (students seek to fit numbers together, e.g., rounding to factors and grouping numbers together that have round sums like 100 or 1000). Here are some typical estimation strategies for the problem: On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34 miles on the third day. About how many miles did they travel? Student 1 I first thought about 267 and 34. I noticed that their sum is about 300. Then I knew that 194 is close to 200. When I put 300 and 200 together, I get 500. Student 2 I first thought about 194. It is really close to 200. I also have 2 hundreds in 267. That gives me a total of 4 hundreds. Then I have 67 in 267 and the 34. When I put 67 and 34 together that is really close to 100. When I add that hundred to the 4 hundreds that I already had I end up with 500. Student 3 I rounded 267 to 300. I rounded 194 to 200. I rounded 34 to 30. When I added 300, 200 and 30, I know my answer will be about 530. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 9 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

18 MAFS.3.OA.3.7: 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. Fluency involves accuracy (correct answer), efficiency (a reasonable amount of steps) and flexibility (using strategies that demonstrate number sense. Students demonstrate fluency with multiplication facts through 10 and the related division facts. Multiplying and dividing fluently refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently. Strategies students may use to attain fluency include Multiplication Properties (Zero, Identity, Commutative, Associative, and Distributive Doubles (2s facts), Doubling twice (4s), Doubling three times (8s) Tens facts (relating to place value, 5 x 10 is 5 tens or 50) Five facts (half of tens) Skip counting (counting groups of and knowing how many groups have been counted) Square numbers (ex: 3 x 3) Nines (10 groups less one group, e.g., 9 x 3 is 10 groups of 3 minus one group of 3) Decomposing into known facts (6 x 7 is 6 x 6 plus one more group of 6) Students should have exposure to multiplication and division problems presented in both vertical and horizontal forms. MAFS.3.MD.3.5: Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. b. 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. MAFS.3.MD.3.6: Measure areas by counting unit squares (square cm, square m, square in., square ft., and improvised units). Students explore the concept of covering a region with unit squares to measure area. a. One unit square can be used as a measuring tool to find the area of a shape. b. The unit squares cover the shape and then are added together to determine the area of the shape. Students should find the area by counting in metric, customary or non-standard square units. Using different sized graph paper, students can explore the area measured in square centimeters and square inches. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 10 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

19 MAFS.3.MD.3.7: Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with wholenumber side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. c. Use tiling to show in a concrete case that the area of a rectangle with wholenumber side lengths a and b + c is the sum of a x b and a x c. Use area models to represent the distributive property in mathematical reasoning. d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the nonoverlapping parts, applying this technique to solve real world problems. Students tile areas of rectangles, determine the area, record the length and width of the rectangle, investigate the patterns in the numbers, and discover that the area is the length times the width. For shapes made of composite rectangles (rectilinear figures), teachers can use a hands-on activity to have students physically cut the shape into component rectangles before finding the area of each rectangle. Example: Joe and John made a poster that was 4 by 3. Mary and Amir made a poster that was 4 by 2. They placed their posters on the wall sideby-side so that that there was no space between them. How much area will the two posters cover? Students use pictures, words, and numbers to explain their understanding of the distributive property in this context. A rectilinear figure is a polygon composed of squares and rectangles x x 2 = 20 square feet 4 (3 + 2) = 20 square feet 4 x 5 = 20 square feet Students can decompose a rectilinear figure into different rectangles or squares. They find the area of the figure by adding the areas of each of the quadrilaterals together Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 11 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications. 10

20 Standards for Mathematical Practice 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. Third Nine Weeks Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Mathematics Florida Standards (MAFS) Explanations and Examples * MAFS.3.G.1.2: 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 4 1 of the area of the shape. MAFS.3.NF.1.1: Understand a fraction b 1 as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction b a as the quantity formed by a parts of size b 1. Given a shape, students partition it into equal parts recognizing that these parts all have the same area. They identify the fractional name of each part and are able to partition a shape into parts with equal areas in several different ways. Fractional models in third grade include area (parts of a whole) and number lines. Denominators are limited to 2,3,4,6,and 8. Set models (parts of a group) are not introduced in Grade 3. Students should focus on the concept that a fraction is composed of many pieces of a unit fraction, which has a numerator of 1. For example, the fraction 6 3 is composed of 3 pieces that each have a size of 6 1. Some important concepts related to developing understanding of fractions include: Fractional parts must be equal-sized The number of equal parts tell how many make a whole As the number of equal pieces in the whole increases, the size of the fractional pieces decrease The size of the fractional part is relative to the whole When a whole is cut into equal parts, the denominator represents the number of equal parts The numerator of a fraction is the count of the number of equal parts Students need many opportunities with concrete models to develop understanding of fractions. They also need many opportunities to solve word problems that require parts of a whole. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 12 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

21 MAFS.3.NF.1.2: Understand a fraction as a number on the number line; represent fractions on a number line diagram. On a number line from 0 to 1, students can partition (divide) it into equal parts and recognize that each segmented part represents the same length. a. Represent a fraction b 1 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 b 1 and that the endpoint of the part based at 0 locates the number b 1 on the number line. Students label each fractional part based on how far it is from zero to the endpoint. a b. Represent a fraction on a number b line diagram by marking off a lengths 1 from 0. Recognize that the resulting b a interval has size and that its b a endpoint locates the number on the b number line. Denominators are limited to 2,3,4,6, and 8. Number lines may extend beyond 1. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 13 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

22 MAFS.3.NF.1.3: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions, e.g., =, = Explain why the fractions are 3 equivalent, e.g., by using a visual fraction model. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 1 3 ; recognize that 1 6 = 6; locate 4 4 and 1 at the same point of a number line diagram. d. 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.1.3a and 3.NF.1.3b: These standards call for students to use visual fraction models (area models) and number lines (linear models) to explore the idea of equivalent fractions. Students should only explore equivalent fractions using models and reasoning, rather than using algorithms or procedures. 3.NF.1.3c: This standard includes writing whole numbers as fractions. 3.NF.1.3d: This standard involves comparing fractions with or without visual fraction models including number lines. An important concept when comparing fractions is to look at the size of the parts and the number of the parts. Experiences should encourage students to reason about the size of pieces, the fact that 3 1 of a cake is larger than 4 1 of the same cake. Since the same cake (the whole) is split into equal pieces, thirds are larger than fourths. In this standard, students should also reason that comparisons are only valid if the wholes are identical. For example, 2 1 large pizza is a different amount than is larger and why? 2 Denominators are limited to 2,3,4,6, and 8. Items may not use the term simplify or lowest terms in directives. of a small pizza. Students should be given opportunities to discuss and reason about which of a Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 14 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

23 MAFS.3.MD.2.4: 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. It s important to review with students how to read and use a standard ruler including details about halves and quarter marks on the ruler. Students should connect their understanding of fractions to measuring to one-half and one-quarter inch. Third graders need many opportunities measuring the length of various objects in their environment. Some important ideas related to measuring with a ruler are: The starting point of where one places a ruler to begin measuring Measuring is approximate. Items that students measure will not always measure exactly 4 1, 2 1 or one whole inch. Students will 1 1 need to measure to the nearest or. 4 2 Making paper rulers and folding to find the half and quarter marks may help students develop a stronger understanding of measuring length. Students generate data by measuring and then creating a line plot to display their findings. An example of a line plot is shown below: Objects Measured Measurement in Inches Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 15 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

24 MAFS.3.MD.4.8: 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 Examples: Students should be able to use geoboards, tiles, and graph paper to find the perimeter of a polygon. Students should also be able to find the perimeter of all the possible rectangles that have a given perimeter (e.g., find the rectangles with a perimeter of 14 cm). Students use geoboards, tiles, graph paper, or technology to find all the possible rectangles with a given area (e.g., find the rectangles that have an area of 12 square units). Example: Same Perimeter/Different Area Same Area/Different Perimeter Perimeter Length Width Area Area Length Width Perimeter 14 in. 4 in. 3 in. 12 sq. in 12 sq. in. 1 in. 12 in. 26 in. 14 in. 5 in. 2 in. 10 sq. in 12 sq. in. 2 in. 6 in. 16 in. 14 in. 6 in. 1 in. 6 sq. in 12 sq. in 3 in. 4 in. 14 in. 14 in. 3 in. 4 in. 12 sq. in 12 sq. in 4 in. 3 in. 14 in. 14 in. 2 in. 5 in. 10 sq. in 12 sq. in 6 in. 2 in. 16 in. 14 in. 1 in. 6 in. 6 sq. in 12 sq. in 12 in. 1 in. 26 in. Charts can allow the students to identify the factors of 12, connect the results to the commutative property, and discuss the differences in perimeter within the same area. This chart can also be used to investigate rectangles with the same perimeter. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 16 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

25 Standards for Mathematical Practice 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. Fourth Nine Weeks Mathematics Florida Standards (MAFS) Explanations and Examples * MAFS.3.G.1.1: 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. Students recognize shapes that are and are not quadrilaterals by examining the properties of the geometric figures. They conceptualize that a quadrilateral must be a closed figure with four straight sides and begin to notice characteristics of the angles and the relationship between opposite sides. Students should be encouraged to provide details and use proper vocabulary when describing the properties of quadrilaterals. They sort geometric figures and identify squares, rectangles, t r a p e z o i d, parallelograms and rhombuses as quadrilaterals. Students should classify shapes by attributes and draw shapes that fit specific categories. For example, parallelograms include: squares, rectangles, parallelograms and rhombuses. Also, the broad category quadrilaterals include all types of parallelograms, trapezoids and other four-sided figures. *The terms parallel and perpendicular lines are not assessed until fourth grade. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 17 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

26 In Grade 3, instructional time should focus on four critical areas: THIRD GRADE CRITICAL AREAS OF FOCUS (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes. (1) Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division. (2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 2 1 of the paint in a small bucket could be less paint than 3 1 of the paint in a larger bucket, but 3 1 of a ribbon is longer than 5 1 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators. (3) Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle. (4) Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole. Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 18 of 18, Brevard Public Schools, *Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

27 DRAFT Grade 3 Mathematics Item Specifications

28 Grade 3 Mathematics Item Specifications Florida Standards Assessments The draft Florida Standards Assessments (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as provided in CPALMs. The Specifications are a resource that defines the content and format of the test and test items for item writers and reviewers. Each grade level and course Specifications document indicates the alignment of items with the Florida Standards. It also serves to provide all stakeholders with information about the scope and function of the FSA. Item Specifications Definitions Also assesses refers to standard(s) closely related to the primary standard statement. Clarification statements explain what students are expected to do when responding to the question. Assessment limits define the range of content knowledge and degree of difficulty that should be assessed in the assessment items for the standard. Item types describe the characteristics of the question. Context defines types of stimulus materials that can be used in the assessment items. Context Allowable refers to items that may but are not required to have context. Context No context refers to items that should not have context. Context Required refers to items that must have context. 2 P age May 2016

29 Grade 3 Mathematics Item Specifications Florida Standards Assessments Technology Enhanced Item Descriptions: The Florida Standards Assessments (FSA) are composed of test items that include traditional multiple choice items, items that require students to type or write a response, and technology enhanced items (TEI). Technology enhanced items are computer delivered items that require students to interact with test content to select, construct, and/or support their answers. Currently, there are nine types of TEIs that may appear on computer based assessments for FSA Mathematics. For students with an IEP or 504 plan that specifies a paper based accommodation, TEIs will be modified or replaced with test items that can be scanned and scored electronically. For samples of each of the item types described below, see the FSA Training Tests. Technology Enhanced Item Types Mathematics 1. Editing Task Choice The student clicks a highlighted word or phrase, which reveals a drop down menu containing options for correcting an error as well as the highlighted word or phrase as it is shown in the sentence to indicate that no correction is needed. The student then selects the correct word or phrase from the drop down menu. For paper based assessments, the item is modified so that it can be scanned and scored electronically. The student fills in a circle to indicate the correct word or phrase. 2. Editing Task The student clicks on a highlighted word or phrase that may be incorrect, which reveals a text box. The directions in the text box direct the student to replace the highlighted word or phrase with the correct word or phrase. For paper based assessments, this item type may be replaced with another item type that assesses the same standard and can be scanned and scored electronically. 3. Hot Text a. Selectable Hot Text Excerpted sentences from the text are presented in this item type. When the student hovers over certain words, phrases, or sentences, the options highlight. This indicates that the text is selectable ( hot ). The student can then click on an option to select it. For paper based assessments, a selectable hot text item is modified so that it can be scanned and scored electronically. In this version, the student fills in a circle to indicate a selection. 3 P age May 2016

30 Grade 3 Mathematics Item Specifications Florida Standards Assessments b. Drag and Drop Hot Text Certain numbers, words, phrases, or sentences may be designated draggable in this item type. When the student hovers over these areas, the text highlights. The student can then click on the option, hold down the mouse button, and drag it to a graphic or other format. For paper based assessments, drag and drop hot text items will be replaced with another item type that assesses the same standard and can be scanned and scored electronically. 4. Open Response The student uses the keyboard to enter a response into a text field. These items can usually be answered in a sentence or two. For paper based assessments, this item type may be replaced with another item type that assesses the same standard and can be scanned and scored electronically. 5. Multiselect The student is directed to select all of the correct answers from among a number of options. These items are different from multiple choice items, which allow the student to select only one correct answer. These items appear in the online and paper based assessments. 6. Graphic Response Item Display (GRID) The student selects numbers, words, phrases, or images and uses the drag and drop feature to place them into a graphic. This item type may also require the student to use the point, line, or arrow tools to create a response on a graph. For paper based assessments, this item type may be replaced with another item type that assesses the same standard and can be scanned and scored electronically. 7. Equation Editor The student is presented with a toolbar that includes a variety of mathematical symbols that can be used to create a response. Responses may be in the form of a number, variable, expression, or equation, as appropriate to the test item. For paper based assessments, this item type may be replaced with a modified version of the item that can be scanned and scored electronically or replaced with another item type that assesses the same standard and can be scanned and scored electronically. 8. Matching Item The student checks a box to indicate if information from a column header matches information from a row. For paper based assessments, this item type may be replaced with another item type that assesses the same standard and can be scanned and scored electronically. 9. Table Item The student types numeric values into a given table. The student may complete the entire table or portions of the table depending on what is being asked. For paper based assessments, this item type may be replaced with another item type that assesses the same standard and can be scanned and scored electronically. 4 P age May 2016

31 Grade 3 Mathematics Item Specifications Florida Standards Assessments Mathematical Practices: The Mathematical Practices are a part of each course description for Grades 3 8, Algebra 1, Geometry, and Algebra 2. These practices are an important part of the curriculum. The Mathematical Practices will be assessed throughout. Make sense of problems and persevere in solving them. MAFS.K12.MP.1.1: MAFS.K12.MP.2.1: Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. 5 P age May 2016

32 Grade 3 Mathematics Item Specifications Florida Standards Assessments Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Construct viable arguments and critique the reasoning of others. MAFS.K12.MP.3.1: MAFS.K12.MP.4.1: Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their 6 P age May 2016

33 Grade 3 Mathematics Item Specifications Florida Standards Assessments mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Use appropriate tools strategically. MAFS.K12.MP.5.1: Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Attend to precision. MAFS.K12.MP.6.1: Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7 P age May 2016

34 Grade 3 Mathematics Item Specifications Florida Standards Assessments Look for and make use of structure. MAFS.K12.MP.7.1: MAFS.K12.MP.8.1: Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x² + x + 1), and (x 1)(x³ + x² + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 8 P age May 2016

35 Grade 3 Mathematics Item Specifications Florida Standards Assessments Reference Sheets: Reference sheets and z tables will be available as online references (in a pop up window). A paper version will be available for paper based tests. Reference sheets with conversions will be provided for FSA Mathematics assessments in Grades 4 8 and EOC Mathematics assessments. There is no reference sheet for Grade 3. For Grades 4, 6, and 7, Geometry, and Algebra 2, some formulas will be provided on the reference sheet. For Grade 5 and Algebra 1, some formulas may be included with the test item if needed to meet the intent of the standard being assessed. For Grade 8, no formulas will be provided; however, conversions will be available on a reference sheet. For Algebra 2, a z table will be available. Grade Conversions Some Formulas z table 3 No No No 4 On Reference Sheet On Reference Sheet No 5 On Reference Sheet With Item No 6 On Reference Sheet On Reference Sheet No 7 On Reference Sheet On Reference Sheet No 8 On Reference Sheet No No Algebra 1 On Reference Sheet With Item No Algebra 2 On Reference Sheet On Reference Sheet Yes Geometry On Reference Sheet On Reference Sheet No 9 P age May 2016

36 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.1 Represent and solve problems involving multiplication and division. MAFS.3.OA.1.1 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. Assessment Limits Whole number factors may not exceed 10 x 10. Students may not be required to write an equation to represent a product of whole numbers. Calculator No Item Types Equation Editor Multiple Choice Multiselect Open Response Table Item Context Allowable Sample Item Item Type Tom told Mary he planted 4 x 5 flowers. How might Mary describe the Open Response arrangement of flowers in Tom s rectangular shaped garden? Tom told Mary he planted 48 flowers in the rectangular shaped garden. Which sentence could Mary use to describe how the flowers were planted? Multiple Choice A. Tom planted 24 rows of 24 flowers. B. Tom planted 4 rows of 24 flowers. C. Tom planted 40 rows of 8 flowers. D. Tom planted 8 rows of 6 flowers. See Appendix for the practice test item aligned to this standard. 10 P age May 2016

37 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.1 Represent and solve problems involving multiplication and division. MAFS.3.OA.1.2 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 Assessment Limits Whole number quotients and divisors may not exceed 10. Items may not require students to write an equation to represent a quotient of whole numbers. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Open Response Context Allowable Sample Item Item Type Heidi has 12 apples and 6 bags. She places an equal number of apples in each bag. GRID Drag apples to show how many apples are in each bag. See Appendix for the practice test item aligned to this standard. 11 P age May 2016

38 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.1 Represent and solve problems involving multiplication and division. MAFS.3.OA.1.3 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. Assessment Limits All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Items may not contain more than one unknown per equation. Items may not contain the words times as much/many. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Context Required Sample Item Item Type Craig has 72 grapes. He separates the grapes into 9 equal groups. How many Equation Editor grapes are in each group? See Appendix for the practice test item aligned to this standard. 12 P age May 2016

39 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.1 Represent and solve problems involving multiplication and division. MAFS.3.OA.1.4 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 =? Assessment Limits All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Items must provide the equation. Students may not be required to create the equation. Calculator No Item Types Equation Editor Multiple Choice Multiselect Context No context Sample Item Item Type A division problem is shown. Equation Editor 9 = 3 What is the value of the unknown number? What is the value of the unknown number in the equation 72 = 9? Equation Editor See Appendix for the practice test item aligned to this standard. 13 P age May 2016

40 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.2 Understand properties of multiplication and the relationship between multiplication and division. MAFS.3.OA.2.5 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.) 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) = = 56. (Distributive property.) Assessment Limit All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Items may contain no more than two properties in an equation (e.g., a x (b + c) = (a x b) + (c x a)). Calculator No Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect Context No context Sample Item Item Type An equation is shown. Multiple Choice 4 x 9 = 9 x What is the missing value? A. 4 B. 5 C. 9 D. 13 Drag numbers to the boxes to create a different expression that is equal to (3 + 4) + 5. GRID 14 P age May 2016

41 Grade 3 Mathematics Item Specifications Florida Standards Assessments Sample Item Select all the expressions that could be used to find 6 x 10. Item Type Multiselect 10 x 6 6 x (2 x 5) 6 + (2 x 5) (6 x 2) x 5 (6 x 8) x (6 x 2) See Appendix for the practice test item aligned to this standard. 15 P age May 2016

42 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.2 Understand properties of multiplication and the relationship between multiplication and division. MAFS.3.OA.2.6 Understand division as an unknown factor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8. Assessment Limit All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Context No context Sample Item Item Type Create a multiplication equation that could be used to solve 21 3 =. Equation Editor See Appendix for the practice test item aligned to this standard. 16 P age May 2016

43 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.3 Multiply and divide within 100. MAFS.3.OA.3.7 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. Assessment Limit All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Calculator No Item Types Equation Editor Multiple Choice Multiselect Table Item Context No context Sample Item Item Type Multiply: 8 x 2 Equation Editor See Appendix for the practice test item aligned to this standard. 17 P age May 2016

44 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.4 Solve problems involving the four operations, and identify and explain patterns in arithmetic. MAFS.3.OA.4.8 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. Assessment Limits Adding and subtracting is limited to whole numbers within 1,000. All values in multiplication or division situations may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Students may not be required to perform rounding in isolation. Equations may be provided in items. Calculator No Item Types Editing Task Choice Equation Editor Hot Text Multiple Choice Multiselect Open Response Context Required Sample Item Item Type A bookstore has 4 boxes of books. Each box contains 20 books. On Monday, the Equation Editor bookstore sold 16 books. How many books remain to be sold? On Monday, a bookstore sold 75 books. On Tuesday, the bookstore sold 125 books. The bookstore must sell 500 books by Friday. Create an equation that can be used to find how many more books, b, the bookstore must sell by Friday. Equation Editor See Appendix for the practice test item aligned to this standard. 18 P age May 2016

45 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.4 Solve problems involving the four operations, and identify and explain patterns in arithmetic. MAFS.3.OA.4.9 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. Assessment Limits Adding and subtracting is limited to whole numbers within 1,000. All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Calculator No Item Types Editing Task Choice Equation Editor GRID Hot Text Multiple Choice Multiselect Table Item Context No context Sample Item Item Type See Appendix for the practice test item aligned to this standard. 19 P age May 2016

46 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.NBT Number and Operations in Base Ten MAFS.3.NBT.1 Use place value understanding and properties of operations to perform multi digit arithmetic. MAFS.3.NBT.1.1 Use place value understanding to round whole numbers to the nearest 10 or 100. Assessment Limit Items may contain whole numbers up to 1,000. Calculator No Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect Table Item Context No context Sample Item What value is 846 rounded to the nearest 100? Item Type Equation Editor A. Round 846 to the nearest hundred. B. Round 846 to the nearest ten. Select all the numbers that will equal 800 when rounded to the nearest hundred An incomplete table is shown. Complete the table by filling in the missing original numbers with possible values. Equation Editor Multiselect Table Item Original Number Rounded to Nearest Ten Plot points on the number line to represent all whole number values that round to 500 when rounded to the nearest hundred and to 450 when rounded to the nearest ten. GRID See Appendix for the practice test item aligned to this standard. 20 P age May 2016

47 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.NBT Number & Operations in Base Ten MAFS.3.NBT.1 Use place value understanding and properties of operations to perform multi digit arithmetic. MAFS.3.NBT.1.2 Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. Assessment Limits Addends and sums are less than or equal to 1,000. Minuends, subtrahends, and differences are less than or equal to 1,000. Items may not require students to name specific properties. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Table Item Context No context Sample Item Item Type What is the sum of 153, 121, and 178? Equation Editor See Appendix for the practice test item aligned to this standard. 21 P age May 2016

48 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.NBT Number & Operations in Base Ten MAFS.3.NBT.1 Use place value understanding and properties of operations to perform multi digit arithmetic. MAFS.3.NBT.1.3 Multiply one digit whole numbers by multiples of 10 in the range (e.g., 9 80, 5 60) using strategies based on place value and properties of operations. Assessment Limit Items may not require students to name specific properties. Calculator No Item Types Equation Editor Matching Item Multiple Choice Multiselect Context Allowable Sample Item What is the product of 7 and 50? Item Type Equation Editor Select all expressions that have a product of x 90 4 x 80 5 x 60 8 x 40 9 x 30 Mr. Engle has 10 tables in his classroom. There are 3 students at each table. Each student has 6 glue sticks. Multiselect Equation Editor A. How many glue sticks are at each table? B. How many glue sticks do all of Mr. Engle s students have combined? See Appendix for the practice test item aligned to this standard. 22 P age May 2016

49 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.NF Number and Operations Fractions MAFS.3.NF.1 Develop understanding of fractions as numbers. MAFS.3.NF.1.1 Understand a fraction as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction as the quantity formed by a parts of size. Also Assesses: MAFS.3.G Geometry MAFS.3.G.1 Reason with shapes and their attributes. MAFS.3.G.1.2 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 of the area of the shape. Assessment Limits Denominators are limited to 2, 3, 4, 6, and 8. Items are limited to combining or putting together unit fractions rather than formal addition or subtraction of fractions. Maintain concept of a whole as one entity that can be equally partitioned in various ways when working with unit fractions. Fractions a/b can be fractions greater than 1. Items may not use the term simplify or lowest terms in directives. Items may not use number lines. Shapes may include: quadrilateral, equilateral triangle, isosceles triangle, regular hexagon, regular octagon, and circle. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Table Item Context Allowable for 3.NF.1.1; no context for 3.G P age May 2016

50 Grade 3 Mathematics Item Specifications Florida Standards Assessments Sample Item Each model shown has been shaded to represent a fraction. Which model shows shaded? Item Type Multiple Choice A. B. C. D. Each model shown has been shaded to represent a fraction. Which model shows shaded? Multiple Choice A. B. C. D. A figure is shown. Part of the figure is shaded. Equation Editor Which fraction of the total area of the figure does the shaded part represent? A figure is shown. Part of the figure is shaded. Equation Editor Which fraction of the total area of the figure does the shaded part represent? 24 P age May 2016

51 Grade 3 Mathematics Item Specifications Florida Standards Assessments Sample Item A half of a shape is shown. Item Type GRID Click squares to complete the whole shape. A sixth of a shape is shown. GRID Click squares to complete the whole shape. Each shape shown represents of a whole. Drag the shapes into the box to show. GRID Each shape shown represents of a whole. Equation Editor How many shapes should be put together to make? See Appendix for the practice test item aligned to a standard in this group. 25 P age May 2016

52 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.NF Number and Operations Fractions MAFS.3.NF.1 Develop understanding of fractions as numbers. MAFS.3.NF.1.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. MAFS.3.NF.1.2a Represent a fraction 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 and that the endpoint of the part based at 0 locates the number on the number line. MAFS.3.NF.1.2b Represent a fraction on a number line diagram by marking off a lengths from 0. Recognize that the resulting interval has size and that its endpoint locates the number on the number line. Assessment Limits Denominators are limited to 2, 3, 4, 6, and 8. Number lines in MAFS.3.NF.1.2b items may extend beyond 1. Only whole number marks may be labeled on number lines. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Context No context Sample Item Which number line is divided into thirds? Item Type Multiple Choice A. B. C. D. 26 P age May 2016

53 Grade 3 Mathematics Item Specifications Florida Standards Assessments Sample Item What fraction is represented by the total length marked on the number line shown? Item Type Equation Editor What fraction is represented by the length marked on the number line shown? Equation Editor See Appendix for the practice test item aligned to a standard in this group. 27 P age May 2016

54 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.NF Number and Operations Fractions MAFS.3.NF.1 Develop understanding of fractions as numbers. MAFS.3.NF.1.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. MAFS.3.NF.1.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. MAFS.3.NF.1.3b Recognize and generate simple equivalent fractions, e.g.,,. Explain why the fractions are equivalent, e.g., by using a visual fraction model. MAFS.3.NF.1.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = ; recognize that = 6; locate and 1 at the same point of a number line diagram. MAFS.3.NF.1.3d 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. Assessment Limits Denominators are limited to 2, 3, 4, 6, and 8. Fractions must reference the same whole entity that can be equally partitioned, unless item is assessing MAFS.3.NF.1.3d. Items may not use the term simplify or lowest terms in directives. Visual models may include number lines and area models. Only whole number marks may be labeled on number lines. Calculator No Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect Table Item Context Allowable 28 P age May 2016

55 Grade 3 Mathematics Item Specifications Florida Standards Assessments Sample Item Jenni and Jimmy s equal sized pizzas are each cut into 8 pieces. Jenni eats 2 slices of her pizza, and Jimmy eats 3 slices of his pizza. Click on Jenni s pizza to show how much she ate. Click on Jimmy s pizza to show how much he ate. Drag <, >, or = to the box to make a true statement. Item Type GRID Jenni s and Jimmy s equal sized pizzas are each cut into 8 slices. Jenni eats 2 slices of her pizza, and Jimmy eats 3 slices of his pizza. GRID Complete the comparison of Jenni s pizza to Jimmy s pizza. 29 P age May 2016

56 Grade 3 Mathematics Item Specifications Florida Standards Assessments Sample Item Mary has two models, each divided into equal sized sections. The first model has been shaded to represent a fraction. Item Type GRID Click to shade sections on the second model to show a fraction equivalent to the one in the first model. Create a true comparison of the 2 fractions. See Appendix for the practice test item aligned to a standard in this group. 30 P age May 2016

57 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.1 Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. Assessment Limits Calculator Item Types MAFS.3.MD.1.1 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. Clocks may be analog or digital. Digital clocks may not be used for items that require telling or writing time in isolation. No Equation Editor GRID Multiple Choice Multiselect Table Item Allowable Context Sample Item Alex arrives at the grocery store at 5:15 p.m. He leaves the grocery store 75 minutes later. Place an arrow on the number line to show the time he left the grocery store. Item Type GRID Alex arrives at the grocery store at 5:17 p.m. He leaves at 5:59 p.m. How many minutes was he in the grocery store? Equation Editor Alex has chores every day. The length of time, in minutes, of each chore is shown. He starts at 9:00 a.m. Complete the table to show what time he will start and finish each chore. Chore Time Needed to Complete the Chore Start Time End Time Watering flowers 12 minutes 9:00 : Sweeping kitchen 7 minutes : : Dusting all rooms 14 minutes : : Table Item See Appendix for the practice test item aligned to this standard. 31 P age May 2016

58 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.1 Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. Assessment Limits MAFS.3.MD.1.2 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. Items may not contain compound units such as cubic centimeters (cm 3 ) or finding the geometric volume of a container. Items may not require multiplicative comparison (e.g., times as much/many ). Unit conversions are not allowed. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Context Allowable Sample Item How many liters (L) of water are in the following container? Item Type Equation Editor 32 P age May 2016

59 Grade 3 Mathematics Item Specifications Florida Standards Assessments Sample Item Gina and Maurice have same sized containers filled with different amounts of water, as shown. Item Type Equation Editor Gina s container has 4 liters (L) of water. About how much water, in liters (L), does Maurice s container have? Gina and Maurice have the containers shown. Equation Editor Gina does not know how much water is in her container. Maurice s container is the same size as Gina s container. About how much less water, in liters (L), does Gina have than Maurice? See Appendix for the practice test item aligned to this standard. 33 P age May 2016

60 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.2 Represent and interpret data. MAFS.3.MD.2.3 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. Assessment Limits The number of data categories are six or fewer. Items must provide appropriate scale and/or key unless item is assessing that feature. Only whole number marks may be labeled on number lines. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Table Item Context Required Sample Item Item Type John surveys his classmates about their favorite foods, as shown in the table. GRID Favorite Food Hamburger 2 Salad 5 Pizza 8 Click on the graph to complete the bar graph. 34 P age May 2016

61 Grade 3 Mathematics Item Specifications Florida Standards Assessments Sample Item John surveys his classmates about their favorite foods, as shown in the bar graph. Item Type Equation Editor How many more classmates prefer pizza over salad? John surveys his classmates about their favorite foods, as shown in the table. GRID Favorite Food Hot Dogs 5 Pizza 9 Salad 6 Chicken 3 Fish 8 Click on the graph to create a bar graph that represents the data. See Appendix for the practice test item aligned to this standard. 35 P age May 2016

62 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.2 Represent and interpret data. Assessment Limits Calculator Item Types Context Sample Item A pencil is shown. MAFS.3.MD.2.4 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. Standard rulers may not be used; only special rulers that are marked off in halves or quarters are allowed. Measurements are limited to inches. No Equation Editor GRID Matching Item Multiple Choice Multiselect Allowable Item Type Equation Editor What is the length of the pencil to the nearest whole inch? A pencil is shown. Equation Editor What is the length of the pencil to the nearest half inch? A pencil is shown. Equation Editor What is the length of the pencil to the nearest quarter inch? See Appendix for the practice test item aligned to this standard. 36 P age May 2016

63 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.3 Geometric measurement: understand concepts of area and relate area to multiplication and addition. MAFS.3.MD.3.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. MAFS.3.MD.3.5a 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. MAFS.3.MD.3.5b 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. Also Assesses: MAFS.3.MD.3.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Assessment Limits Items may include plane figures that can be covered by unit squares. Items may not include exponential notation for unit abbreviations (e.g., cm 2 ). Calculator No Item Types Equation Editor Multiple Choice Multiselect Context Allowable Sample Item Item Type Alex put the tiles shown on his floor. Equation Editor What is the area, in square feet, of Alex s floor? 37 P age May 2016

64 Grade 3 Mathematics Item Specifications Florida Standards Assessments Sample Item The area of Alex s floor is 30 square feet. Select all the floors that could be Alex s. Item Type Multiselect See Appendix for the practice test item aligned to a standard in this group. 38 P age May 2016

65 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.3 Geometric measurement: understand concepts of area and relate area to multiplication and addition. MAFS.3.MD.3.7 Relate area to the operations of multiplication and addition. MAFS.3.MD.3.7a 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. MAFS.3.MD.3.7b Multiply side lengths to find areas of rectangles with wholenumber side lengths in the context of solving real world and mathematical problems, and represent whole number products as rectangular areas in mathematical reasoning. MAFS.3.MD.3.7c 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. Assessment Limits Calculator Item Types Context MAFS.3.MD.3.7d 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. Figures are limited to rectangles and shapes that can be decomposed into rectangles. Dimensions of figures are limited to whole numbers. All values in items may not exceed whole number multiplication facts of 10 x 10. No Equation Editor GRID Multiple Choice Multiselect Allowable 39 P age May 2016

66 Grade 3 Mathematics Item Specifications Florida Standards Assessments Sample Item A park is in the shape of the rectangle shown. Item Type Equation Editor What is the area, in square miles, of the park? A park is shown. Equation Editor What is the area, in square miles, of the park? See Appendix for the practice test item aligned to a standard in this group. 40 P age May 2016

67 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.4 Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. Assessment Limits Calculator Item Types MAFS.3.MD.4.8 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. For items involving area, only polygons that can be tiled with square units are allowable. Dimensions of figures are limited to whole numbers. All values in items may not exceed whole number multiplication facts of 10 x 10. Items are not required to have a graphic, but sufficient dimension information must be given. No Equation Editor GRID Multiple Choice Multiselect Required Context Sample Item Ben is planning a garden. Which measurement describes the perimeter of his garden? A. the length of fence he will need B. the amount of soil he will need C. the number of seeds he will buy D. the length of the garden multiplied by the width Item Type Multiple Choice Ben s garden has a perimeter of 32 feet. Draw a rectangle that could represent the garden. GRID Ben has a rectangular garden with side lengths of 2 feet and 5 feet. What is the perimeter, in feet, of Ben s garden? Equation Editor Ben wants to create a rectangular garden with an area less than 40 square feet. He has 30 feet of fencing. Draw a rectangle that could represent Ben s garden. GRID See Appendix for the practice test item aligned to this standard. 41 P age May 2016

68 Grade 3 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.3.G Geometry MAFS.3.G.1 Reason with shapes and their attributes. MAFS.3.G.1.1 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. Assessment Limits Shapes may include two dimensional shapes and the following quadrilaterals: rhombus, rectangle, square, parallelogram, and trapezoid. Items may reference and/or rely on the following attributes: number of sides, number of angles, whether the shape has a right angle, whether the sides are the same length, and whether the sides are straight lines. Items may not use the terms parallel or perpendicular. Items that include trapezoids must consider both the inclusive and exclusive definitions. Items may not use the term "kite" but may include the figure. Calculator No Item Types Editing Task Choice GRID Hot Text Matching Item Multiple Choice Multiselect Open Response Context No context Sample Item Item Types A square and a trapezoid are shown below. Multiselect Which attributes do these shapes always have in common? number of sides side lengths angle measures right angles number of angles Select the shapes that are always quadrilaterals and not rectangles. rhombus parallelogram triangle trapezoid square Multiselect 42 P age May 2016

69 Grade 3 Mathematics Item Specifications Florida Standards Assessments Sample Item Draw a quadrilateral that is not a rectangle. Item Type GRID What is the name of a shape that is a quadrilateral but not a rectangle? Multiple Choice A. hexagon B. parallelogram C. square D. triangle See Appendix for the practice test item aligned to this standard. 43 P age May 2016

70 Grade 3 Mathematics Item Specifications Florida Standards Assessments Appendix A The chart below contains information about the standard alignment for the items in the Grade 3 Mathematics FSA Computer Based Practice Test at andfamilies/practice tests/. Content Standard Item Type Computer Based Practice Test Item Number MAFS.3.OA.1.1 Table Item 10 MAFS.3.OA.1.2 Multiselect 4 MAFS.3.OA.1.3 Equation Editor 17 MAFS.3.OA.1.4 Multiple Choice 1 MAFS.3.OA.2.5 Multiple Choice 23 MAFS.3.OA.2.6 GRID 14 MAFS.3.OA.3.7 Table Item 6 MAFS.3.OA.4.8 Multiple Choice 21 MAFS.3.OA.4.9 Multiple Choice 12 MAFS.3.NBT.1.1 Matching Item 2 MAFS.3.NBT.1.2 Multiselect 15 MAFS.3.NBT.1.3 Equation Editor 22 MAFS.3.NF.1.1 GRID 19 MAFS.3.NF.1.2b GRID 5 MAFS.3.NF.1.3c Multiselect 9 MAFS.3.MD.1.1 Multiple Choice 13 MAFS.3.MD.1.2 Equation Editor 3 MAFS.3.MD.2.3 GRID 11 MAFS.3.MD.2.4 GRID 16 MAFS.3.MD.3.6 Multiple Choice 20 MAFS.3.MD.3.7d Equation Editor 8 MAFS.3.MD.4.8 GRID 18 MAFS.3.G.1.1 Open Response 7 44 P age May 2016

71 Grade 3 Mathematics Item Specifications Florida Standards Assessments Appendix B: Revisions Page(s) Revision Date 13 Item types and sample items revised. May Assessment limits revised. May Sample item revised. May Sample items revised. May Item types revised. May Item types revised. May Item types revised. May Item types revised. May Item types revised. May Assessment limits revised. May Assessment limits revised. May Assessment limits and sample items revised. May Item types and sample items revised. May Assessment limits, item types, and sample items revised. May Assessment limits and item types revised. May Item types revised. May Sample items revised. May Assessment limits revised. May Assessment limits, item types, and sample items revised. May Appendix A added to show Practice Test information. May P age May 2016

72 Table 1. Common addition and subtraction situations. 6 Result Unknown Change Unknown Start Unknown Add to Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? =? Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? 2 +? = 5 Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before?? + 3 = 5 Take from Five apples were on the table. I ate two apples. How many apples are on the table now? 5 2 =? Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5? = 3 Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before?? 2 = 3 Put together/take apart 2 Total Unknown Three red apples and two green apples are on the table. How many apples are on the table? =? Addend Unknown Five apples are on the table. Three are red and the rest are green. How many apples are green? 3 +? = 5, 5 3 =? Both Addends Unknown 1 Grandma has five flowers. How many can she put in her red vase and how many in her blue vase? 5 = = = = = = difference Unknown Bigger Unknown Smaller Unknown Difference Unknown Bigger Unknown Smaller Unknown Compare 3 ( How many more? version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy? (Version with more ): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have? (Version with more ): Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have? ( How man fewer? version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie? (Version with fewer ): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have? (Version with fewer ): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have? 2 +? = =? =? =? 5 3 =?? + 3 = 5 1These take apart situations can be used to show all the decompositions of a given number. The associated equations, which have the total on the left of the equal sign, help children understand that the = sign does not always mean makes or results in but always does mean is the same number as. 2Either addend can be unknown, so there are three variations of these problem situations. Both Addends Unknown is a productive extension of this basic situation, especially for small numbers less than or equal to 10. 3For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version using more for the bigger unknown and using less for the smaller unknown). The other versions are more difficult. 6 Adapted from Box 2-4 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33). Common Operation Situations and Properties, page 1 of 3, Brevard Public Schools,

73 Table 2. Common multiplication and division situations. 7 Unknown Product 3 6 =? Group Size Unknown ( How many in each group? Division) 3? = 18, and 18 3 =? Number of Groups Unknown ( How many groups? Division)? 6 = 18, and 18 6 =? Equal Groups There are 3 bags with 6 plums in each bag. How many plums are there in all? Measurement example: You need 3 lengths of string, each 6 inches long. How much string will you need altogether? If 18 plums are shared equally into 3 bags, then how many plums will be in each bag? Measurement example: You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be? If 18 plums are to be packed 6 to a bag, then how many bags are needed? Measurement example: You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have? Arrays 4, Area 5 There are 3 rows of apples with 6 apples in each row. How many apples are there? Area example: What is the area of a 3 cm by 6 cm rectangle? If 18 apples are arranged into 3 equal rows, how many apples will be in each row? Area example: A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it? If 18 apples are arranged into equal rows of 6 apples, how many rows will there be? Area example: A rectangle has area 18 square centimeters. If one side is 6 cm long, how long is a side next to it? Compare 3 A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost? Measurement example: A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long? A red hat costs $18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost? Measurement example: A rubber band is stretched to be18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first? A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat? Measurement example: A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first? General a b =? a? = p, and p a =?? b = p, and p b =? 4 The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable. 5 Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations. 7 The first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement examples. Common Operation Situations and Properties, page 2 of 3, Brevard Public Schools,

74 TABLE 3. THE PROPERTIES OF OPERATIONS. Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, a nd the complex number system. Associative property of addition Commutative property of addition Additive identity property of 0 Existence of additive inverses Associative property of multiplication Commutative property of multiplication Multiplicative identity property of 1 Existence of multiplicative inverses Distributive property of multiplication over addition (a + b) + c = a + (b + c) a + b = b + a a + 0 = 0 + a = a For every a there exists a so that a + ( a) = ( a) + a = 0 (a b) c = a (b c) a b = b a a 1 = 1 a = a For every a 0 there exists 1/a so that a 1/a = 1/a a = 1 a (b + c) = a b + a c TABLE 4. THE PROPERTIES OF EQUALITY. Here a, b and c stand for arbitrary numbers in the rational, real, or complex number systems. Reflexive property of equality Symmetric property of equality Transitive property of equality Addition property of equality Subtraction property of equality Multiplication property of equality Division property of equality Substitution property of equality a = a If a = b, then b = a. If a = b and b = c, then a = c. If a = b, then a + c = b + c. If a = b, then a c = b c. If a = b, then a c = b c. If a = b and c 0, then a c = b c. If a = b, then b may be substituted for a in any expression containing a. TABLE 5. THE PROPERTIES OF INEQUALITY. Here a, b and c stand for arbitrary numbers in the rational or real number systems. Exactly one of the following is true: a < b, a = b, a > b. If a > b and b > c then a > c. If a > b, then b < a. If a > b, then a < b. If a > b, then a ± c > b ± c. If a > b and c > 0, then a c > b c. If a > b and c < 0, then a c < b c. If a > b and c > 0, then a c > b c. If a > b and c < 0, then a c < b c Common Operation Situations and Properties, page 3 of 3, Brevard Public Schools,

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76 Standards for Mathematical Practice

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78 Do what makes sense and be persistent Use number sense when representing a problem Look for and use patterns and connections Use math to describe a real situation or problem Make conjectures and prove or disprove them What do good problem solvers do? Look for and create efficient strategies Be precise with words, numbers, and symbols Use tools and technology strategically

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80 What Constitutes a Cognitively Demanding Task? Lower-level demands (memorization) Involve either reproducing previously learned facts, rules, formulas, or definitions or committing facts, rules, formulas or definitions to memory. Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure Are not ambiguous. Such tasks involve the exact reproduction of previously seen material, and what is to be reproduced is clearly and directly stated. Have no connection to the concepts or meaning that underlie the facts, rules, formulas, or definitions being learned or reproduced. Lower-level demands (procedures without connections to meaning) Are algorithmic. Use of the procedure either is specifically called for or is evident from prior instruction, experience, or placement of the task. Require limited cognitive demand for successful completion. Little ambiguity exists about what needs to be done and how to do it. Have no connection to concepts or meaning that underlie the procedure being used. Are focused on producing correct answers instead of on developing mathematical understanding. Require no explanation or explanations that focus solely on describing the procedure that was used. Higher-level demands (procedures with connections to meaning) Focus students attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas. Suggest explicitly or implicitly pathways to follow that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts. Usually are represented in multiple ways, such as visual diagrams, manipulatives, symbols, and problem situations. Making connections among multiple representations helps develop meaning. Require some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with conceptual ideas that underlie the procedures to complete the task successfully and that develop understanding. Higher-level demands (doing mathematics) Require complex and non-algorithmic thinking - a predictable, well-rehearsed approach or pathway is not explicitly suggested by the task, task instructions, or a worked-out example. Require students to explore and understand the nature of mathematical concepts, processes, or relationships. Demand self-monitoring or self-regulation of one s own cognitive processes. Require students to access relevant knowledge and experiences and make appropriate use of them in working through the task. Require considerable cognitive effort and may involve some level of anxiety for the student because of the unpredictable nature of the solution process required. Arbaugh, F., & Brown, C.A. (2005). Analyzing mathematical tasks: a catalyst for change? Journal of Mathematics Teacher Education, 8, p. 530.

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82 Key Ideas in the Mathematics Florida Standards (MAFS) Focus: Greater focus on fewer topics Focus deeply on the standards for mastery and the ability to transfer skills. Focus deeply on the major work of each grade as follows: In grades K-2: Concepts, skills, problem solving related to addition and subtraction. In grades 3-5: Concepts, skills, and problem solving related to multiplication and division of whole numbers and fractions. In grade 6: Ratios and proportional relationships, and early algebraic expressions and equations. This focus will enable students to gain strong foundations, including a solid understanding of concepts, and the ability to apply the math they know to solve problems inside and outside the classroom. Coherence: Linking topics and thinking across grades Coherence is about making math make sense. Mathematics is a coherent body of knowledge made up of interconnected concepts. The standards are designed around coherent progressions from grade to grade. Learning is carefully connected across grades so that students can build new understanding onto foundations built in previous years. Each standard is not a new event, but an extension of previous learning. It is critical to think across grade levels and examine the progressions to see how major content is developed across grades. Rigor: Calls for a balance of tasks that require conceptual understanding, procedural skills and fluency, and application of mathematics to solve problems Rigor refers to deep, authentic command of mathematical concepts. The following three aspects of rigor must be pursued with equal intensity to help students meet the standards: Conceptual understanding: The standards call for conceptual understanding of key concepts. Students must be able to access concepts from a number of perspectives. This will allow them to see math as more than a set of mnemonics or discrete procedures. Procedural skills and fluency: The standards call for speed and accuracy in calculation with a balance of practice and understanding. Students must practice simple calculations such as single-digit multiplication with meaning, in order to have access to more complex concepts and procedures. Application: The standards call for students to have solid conceptual understanding and procedural fluency. They are expected to apply their understanding and procedural skills in mathematics to problem solving situations. -Adapted from

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84 Standards for Mathematical Practice The Standards for Mathematical Practice describe behaviors that all students will develop in the Common Core Standards. These practices rest on important processes and proficiencies including problem solving, reasoning and proof, communication, representation, and making connections. These practices will allow students to understand and apply mathematics with confidence. When given a problem, I can make a plan to solve it and check my answer. 1. Make sense of problems and persevere in solving them. Find meaning in problems Analyze, predict, and plan solution pathways Verify answers Ask them the question: Does this make sense? 2. Reason abstractly and quantitatively. Make sense of quantities and their relationships in problems Create coherent representations of problems I can explain my thinking and consider the mathematical thinking of others. I can use numbers and words to help me make sense of problems. 3. Construct viable arguments and critique the reasoning of others. Understand and use information to construct arguments Make and explore the truth of conjectures Justify conclusions and respond to arguments of others 4. Model with mathematics. Apply mathematics to problems in everyday life Identify quantities in a practical situation Interpret results in the context of the situation and reflect on whether the results make sense I can recognize math in everyday life and use math I know to solve problems.

85 I can use math tools to help me explore and understand math in my world. 5. Use appropriate tools strategically. Consider the available tools when solving problems Be familiar with tools appropriate for their grade or course (pencil and paper, concrete models, ruler, protractor, calculator, spreadsheet, computer programs, digital content located on a website, and other technological tools) 6. Be precise. Communicate precisely to others Use clear definitions, state the meaning of symbols and be careful about specifying units of measure and labeling axes Calculate accurately and efficiently I can see and understand how numbers and shapes are put together as parts and wholes. I can be careful when I use math and clear when I share my ideas. 7. Look for and make use of structure. Recognize patterns and structures Step back for an overview and shift perspective See complicated things as single objects or as being composed of several objects 8. Look for and identify ways to create shortcuts when doing problems. When calculations are repeated, look for general methods, patterns and shortcuts Be able to evaluate whether an answer makes sense I can notice when calculations are repeated.

86 Standard for Mathematical Practice 1. Make sense of problems and persevere in solving them. Student Friendly Language I can try many times to understand and solve a math problem. 2. Reason abstractly and quantitatively. I can think about the math problem in my head, first. 3. Construct viable arguments and critique the reasoning of others. I can make a plan, called a strategy, to solve the problem and discuss other students strategies too. 4. Model with mathematics. I can use math symbols and numbers to solve the problem. 5. Use appropriate tools strategically. I can use math tools, pictures, drawings, and objects to solve the problem. 6. Attend to precision. I can check to see if my strategy and calculations are correct. 7. Look for and make use of structure I can use what I already know about math to solve the problem. 8. Look for and express regularity in repeated reasoning. I can use a strategy that I used to solve another math problem. Carroll County Public Schools,

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88 Make sense of problems and persevere in solving them Teachers ask: What is this problem asking? How could you start this problem? How could you make this problem easier to solve? How is s way of solving the problem like/different from yours? Does your plan make sense? Why or why not? What tools/manipulatives might help you? What are you having trouble with? How can you check this? Use appropriate tools strategically Teachers ask: How could you use manipulatives or a drawing to show your thinking? Which tool/manipulative would be best for this problem? What other resources could help you solve this problem? Florida State Standards Standards for Mathematical Practice Sample Questions for Teachers to Ask Reason abstractly and quantitatively Teachers ask: What does the number represent in the problem? How can you represent the problem with symbols and numbers? Create a representation of the problem. Attend to precision Teachers ask: What does the word mean? Explain what you did to solve the problem. Compare your answer to s answer What labels could you use? How do you know your answer is accurate? Did you use the most efficient way to solve the problem? Construct viable arguments and critique the reasoning of others Teachers ask: How is your answer different than s? How can you prove that your answer is correct? What math language will help you prove your answer? What examples could prove or disprove your argument? What do you think about s argument What is wrong with s thinking? What questions do you have for? *it is important that the teacher implements tasks that involve discourse and critiquing of reasoning Look for and make use of structure Teachers ask: Why does this happen? How is related to? Why is this important to the problem? What do you know about that you can apply to this situation? How can you use what you know to explain why this works? What patterns do you see? *deductive reasoning (moving from general to specific) Model with mathematics Teachers ask: Write a number sentence to describe this situation What do you already know about solving this problem? What connections do you see? Why do the results make sense? Is this working or do you need to change your model? *It is important that the teacher poses tasks that involve real world situations Look for and express regularity in repeated reasoning Teachers ask: What generalizations can you make? Can you find a shortcut to solve the problem? How would your shortcut make the problem easier? How could this problem help you solve another problem? *inductive reasoning (moving from specific to general)

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90 Standards for Mathematical Practice in Action Practice Sample Student Evidence Sample Teacher Actions 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Display sense-making behaviors Show patience and listen to others Turn and talk for first steps and/or generate solution plan Analyze information in problems Use and recall multiple strategies Self-evaluate and redirect Assess reasonableness of process and answer Represent abstract and contextual situations symbolically Interpret problems logically in context Estimate for reasonableness Make connections including real life situations Create and use multiple representations Visualize problems Put symbolic problems into context Questions others Use examples and non-examples Support beliefs and challenges with mathematical evidence Forms logical arguments with conjectures and counterexamples Use multiple representations for evidence Listen and respond to others well Uses precise mathematical vocabulary Connect math (numbers and symbols) to real-life situations Symbolize real-world problems with math Make sense of mathematics Apply prior knowledge to solve problems Choose and apply representations, manipulatives and other models to solve problems Use strategies to make problems simpler Use estimation and logic to check reasonableness of an answer Choose appropriate tool(s) for a given problem Use technology to deepen understanding Identify and locate resources Defend mathematically choice of tool Communicate (oral and written) with precise vocabulary Carefully formulate questions and explanations (not retelling steps) Decode and interpret meaning of symbols Pay attention to units, labeling, scale, etc. Calculate accurately and effectively Express answers within context when appropriate Look for, identify, and interpret patterns and structures Make connections to skills and strategies previously learned to solve new problems and tasks Breakdown complex problems into simpler and more manageable chunks Use multiple representations for quantities View complicated quantities as both a single object or a composition of objects Design and state shortcuts Generate rules from repeated reasoning or practice (e.g. integer operations) Evaluate the reasonableness of intermediate steps Make generalizations Provide open-ended problems Ask probing questions Probe student responses Promote and value discourse Promote collaboration Model and accept multiple approaches Model context to symbol and symbol to context Create problems such as what word problem will this equation solve? Give real world situations Offer authentic performance tasks Place less emphasis on the answer Value invented strategies Think Aloud Create a safe and collaborative environment Model respectful discourse behaviors Find the error problems Promote student to student discourse (do not mediate discussion) Plan effective questions or Socratic formats Provide time and value discourse Model reasoning skills Provide meaningful, real world, authentic performance-based tasks Make appropriate tools available Model various modeling techniques Accept and value multiple approaches and representations Provide a toolbox at all times with all available tools students then choose as needed Model tool use, especially technology for understanding Model problem solving strategies Give explicit and precise instruction Ask probing questions Use ELA strategies of decoding, comprehending, and text-to-self connections for interpretation of symbolic and contextual math problems Guided inquiry Let students explore and explain patterns Use open-ended questioning Prompt students to make connections and choose problems that foster connections Ask for multiple interpretations of quantities Provide tasks that allow students to generalize Don t teach steps or rules, but allow students to explore and generalize in order to discover and formalize Ask deliberate questions Create strategic and purposeful check-in points

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92 STANDARDS FOR MATHEMATICAL PRACTICE IN THIRD GRADE The Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students Grades K-12. Below are a few examples of how these Practices may be integrated into tasks that students complete. Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. Explanation and Example Mathematically proficient students in third grade know that doing mathematics involves solving problems and discussing how they solved them. Students make sense of a problem by determining and explaining its meaning and look for ways to solve it. Third grade students may use concrete objects or pictures to help themselves conceptualize and solve problems. They may check their thinking by asking themselves, Does this make sense?. They listen to the strategies of others and will try different approaches. They often will use another method to check their answers. Mathematically proficient students in third grade should recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities. Mathematically proficient students in third grade may construct arguments using concrete referents, such as objects, pictures, and drawings. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like How did you get that? and Why is that true?. They explain their thinking to others and respond to others thinking. Mathematically proficient students in third grade experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart, list, or graph, creating equations, etc. Students need opportunities to connect these different representations and explain the connections. They should be able to use all of these representations as needed. Third grade students should evaluate their results in the context of the situation and reflect on whether or not the results make sense. Mathematically proficient students in third grade consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use graph paper to find all the possible rectangles that have a given perimeter. They compile the possibilities into an organized list or a table and determine whether they have all the possible rectangles. Standards for Mathematical Practice, (from North Carolina Department of Education, Third Grade, page 1 of 2,

93 Practice 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Explanation and Example Mathematically proficient students in third grade develop their mathematical communication skills. They try to use clear and precise language in their discussions with others and in their own reasoning. They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, when figuring out the area of a rectangle, they record their answers in square units. Mathematically proficient students in third grade look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to multiply and divide (commutative and distributive properties). Mathematically proficient students in third grade should notice repetitive actions in computation and look for more shortcut methods. For example, students may use the distributive property as a strategy for using products they know to solve products that they don t know. If students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at or 56. In addition, third grade students continually evaluate their work by asking themselves, Does this make sense?. Standards for Mathematical Practice, (from North Carolina Department of Education, Third Grade, page 2 of 2,

94 Make sense of problems and persevere in solving them. Mathematical Practice 1 When given a problem, I can make a plan, carry out my plan, and check my answer. BEFORE... DURING... AFTER... Think about the problem. Ask myself, "Which strategy will I use?" Make a plan to solve the problem. Stick to it! Ask myself, "Does this make sense?" Change my plan if it isn't working out. CHECK my work. Ask myself, "Is there another way to solve the problem?" Clip art licensed from the Clip Art Gallery on DiscoverySchool.com Jordan School District 2012, Grades 2-3

95 Reason abstractly and quantitatively. Mathematical Practice 2 I can use numbers and words to help me make sense of problems. Numbers to Words Words to Numbers = 53 There are 26 boys and 27 girls on the playground. How many children are on the playground? There are 26 boys and 27 girls on the playground. How many children are on the playground? = = = = = = = 53 Clip art licensed from the Clip Art Gallery on DiscoverySchool.com Jordan School District 2012, Grades 2-3

96 Construct viable arguments and critique the I can explain my strategy using reasoning of others. Mathematical Practice 3 I can explain my thinking and respond to the mathematical thinking of others. I can compare strategies with others by objects, drawings, and actions examples and non-examples contexts listening asking useful questions understanding mathematical connections between strategies Clip art licensed from the Clip Art Gallery on DiscoverySchool.com Jordan School District 2012, Grades 2-3

97 I can use. Model with mathematics. Mathematical Practice 4 I can recognize math in everyday life and use math I know to solve problems. I can use take-away to find the difference between the number of crayons Jill and Rob have (Symbols) (Words) Rob has 23 crayons. Jill has 46 crayons. How many more crayons does Jill have than Rob? (Pictures) Difference of 23 crayons (Objects) Rob's Crayons Jill's Crayons to solve everyday problems. Clip art licensed from the Clip Art Gallery on DiscoverySchool.com Jordan School District 2012, Grades 2-3

98 Use appropriate tools strategically. Mathematical Practice 5 I can use certain tools to help me explore and deepen my math understanding. I have a math toolbox I know HOW and WHEN to use math tools I can reason: Did the tool I used give me an answer that makes sense? Clip art licensed from the Clip Art Gallery on DiscoverySchool.com Jordan School District 2012, Grades 2-3 Toolbox

99 Attend to precision. Mathematical practice 6 I can be precise when solving problems and clear when I share my ideas. Careful and clear mathematicians use PLUS: join symbols EQUAL: the same as = 75 units of measure: CENTS math vocabulary symbols that have meaning context labels units of measure calculations that are accurate and efficient Clip art licensed from the Clip Art Gallery on DiscoverySchool.com Jordan School District 2012, Grades 2-3

100 For example: Look for and make use of structure. Mathematical Practice 7 Numbers I can see and understand how numbers and shapes are organized and put together as parts and wholes. For example: Shapes These shapes have three sides hundred, 2 tens, and 3 ones These are the same! These shapes have four right angles Base Ten System Orientation Attributes Clip art licensed from the Clip Art Gallery on DiscoverySchool.com Jordan School District 2012, Grades 2-3

101 Look for and express regularity in repeated reasoning. Mathematical Practice 8 I can notice when calculations are repeated = Clip art licensed from the Clip Art Gallery on DiscoverySchool.com Jordan School District 2012, Grades 2-3