Grade 3 Mathematics, Quarter 2, Unit 2.1 Fractions as ... 3 Mathematics, Quarter 2, Unit 2.2...

Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin

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Grade 3 Mathematics, Quarter 2, Unit 2.1

Fractions as Parts of a Whole

Overview Number of Instructional Days: 5 (1 day = 45–60 minutes)

Content to be Learned Mathematical Practices to Be Integrated • Partition shapes into parts with equal areas and

express each equal part as a unit fraction (1/b) when the whole has b equal parts. (Each part is 1/3 when there are 3 equal parts.)

• Understand a fraction a/b as the quantity formed by a parts of size 1/b. (2/3 is two 1/3s)

• Represent fractional parts by using the standard fraction notation (1/2; 3/4)—limited to fractions with denominators of 2, 3, 4, 6, and 8.

Use appropriate tools strategically.

• Use concrete models (number line, fraction bars, fraction strips, fraction circles) to develop conceptual understanding of fraction quantities.

• Choose appropriate models to demonstrate understanding of fraction concepts.

Essential Questions • How do you identify and record the fraction of

a given shape?

• How do you partition this shape so the fraction ____ is represented?

• What does the numerator tell you about a fraction?

• What does the denominator tell you about a fraction?

• How you can represent a unit fraction using a variety of materials?

• If a shape is divided into ____(4) equal pieces, what is the size of each piece? How many pieces are needed to show ____ (3/4)?

Grade 8 Mathematics, Quarter 2, Unit 2.1 Fractions as Parts of a Whole (5 days)


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Written Curriculum

Common Core State Standards for Mathematical Content

Geometry 3.G

Reason with shapes and their attributes.

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

Number and Operations—Fractions5 3.NF 5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Develop understanding of fractions as numbers.

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

Common Core Standards for Mathematical Practice

5 Use appropriate tools strategically.

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.

Clarifying the Standards

Prior Learning

Students in grade 2 partitioned circles and rectangles into two, three, or four equal shares and identified the numbers of halves, thirds, and fourths using pictures and words only. Standard fraction notation was not introduced.



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Current Learning

Developing an understanding of unit fractions as numbers is a critical area. In this unit, students partition shapes into 2, 3, 4, 6, and 8 equal parts with equal areas and express each part as a unit fraction (1/b) when the whole has b equal parts. They are now using the standard fraction notation (1/4; 1/2) and developing the understanding that the size of a fractional part is relative to the size of the whole. They are identifying a fraction a/b as the quantity formed by a parts of size 1/b. Students recognize that two 1/3s are the same amount as 2/3. They understand and use the terms numerator and denominator.

In Units 2.2 and 3.1 they use their understanding of fractions of an area to develop and reinforce their understanding of fractions as numbers on a number line. They develop an understanding of equivalence of simple fractions and whole numbers as fractions. Students compare two fractions with the same numerator or the same denominator by reasoning about their size.

Future Learning

In grade 4, students will further develop their understanding of fraction equivalence by comparing two fractions with different numerators and different denominators. They will apply knowledge of composing and decomposing fractions and operations on whole numbers to add and subtract fractions with like denominators and multiply fractions by a whole number.

Additional Findings

According to the Progressions 3–5 Number and Operations—Fractions, “Grade 3 students start with unit fractions (fractions with numerator 1), which are formed by partitioning a whole into equal parts and taking one part, e.g., if a whole is partitioned into 4 equal parts then each part is 1/4 of the whole, and 4 copies of that part make the whole. Next, students build fractions from unit fractions, seeing the numerator 3 of 3/4 as saying that 3/4 is the quantity you get by putting 3 of the 1/4s together. They read any fraction this way, and in particular there is no need to introduce ‘proper fractions’ and ‘improper fractions’ initially; 5/3 is the quantity you get by combining 5 parts together when the whole is divided into 3 equal parts. Two important aspects of fractions provide opportunities for the mathematical practice of attending to precision (MP6): Specifying the whole. Explaining what is meant by ‘equal parts’ (p. 2).



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Fractions as Numbers


Content to be Learned Mathematical Practices to Be Integrated • Understand a fraction as a number on the

number line.

• Represent fractions on a number line diagram in which they have defined the interval from 0–1 as the whole.

• Express whole numbers as fractions.

• Recognize fractions that are equivalent to whole numbers (4/4; 6/1).

• Generate measurement data by using rulers marked with halves and fourths of an inch.

• Show the data on a line plot where the horizontal scale is marked off in whole numbers, halves, or quarters.

Use appropriate tools strategically.

• Consider available tools when solving a mathematical problem (e.g., concrete models such as fraction bars, fraction strips, fraction circles, number line, ruler).

• Makes sound decisions about when each tool might be helpful.

• Detect possible errors by strategically using estimation and other mathematical knowledge.

Essential Questions • How do you represent a whole number as a

fraction?

• How would you show equivalent fractions on a number line diagram?

• How do you know if two fractions are equivalent? How do you know if they are not equivalent?

• Where would the following fractions be located on a number line diagram?

• How do you represent your data on a labeled line plot diagram?

• What steps must you take when deciding where to place a fraction on a number line diagram?

• How long is this item to the nearest whole number, 1/2 or 1/4 of an inch?

Grade 3 Mathematics, Quarter 2, Unit 2.2 Fractions as Numbers (10 days)


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Written Curriculum


Number and Operations—Fractions5 3.NF 5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Develop understanding of fractions as numbers.

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

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

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

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

Measurement and Data 3.MD

Represent and interpret data.

3.MD.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.


5 Use appropriate tools strategically.

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.



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Prior Learning

Students in grade 2 partitioned circles and rectangles into two, three, or four equal shares and identified the numbers of halves, thirds, and fourths using pictures and words only. Standard fraction notation was not introduced. Second grade students have used appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes to estimate and measure length. Students used units of inches, feet, centimeters, and meters. Students represented measurement data on a line plot involving whole numbers.

Current Learning

In the previous unit, students develop an understanding of unit fractions. They partition shapes into 2, 3, 4, 6, and 8 equal parts with equal areas and expressed each as a unit fraction (1/b) when the whole has b equal parts. They use the standard fraction notation (1/2, 3/4). Students identify a fraction a/b as the quantity formed by a parts of size 1/b. (3/8 = 3 parts of 1/8 each). They use the terms numerator and denominator.

In Unit 2.1, students develop an understanding of fractions within an area model. In this unit, they extend their understanding of fractions as a distance on a number line. The context of linear measurement supports this development. Students generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. They show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.

In this unit, students continue to develop their understanding of a fraction as a number by using the number line model to explore unit fractions. This is a critical area of focus for students this year. They understand the fraction represents the distance from 0 on a number line. They define the interval from 0 to 1 as the whole and partition it into b equal parts while recognizing that each part has size 1/b. (For example, 2/3 is ‘two’ intervals of ‘1/3’ from zero.) Students explore whole numbers and their fractional equivalence. (They see 2 as 2/1 or 4/2 or 6/3 and 1 as 2/2 or 3/3.)

Best practice indicates referring to fractional quantities as an amount or a distance from zero, not as a part of a set or “three out of four” for 3/4.

Later in grade 3, students use their understanding of fractions on a number line to compare fractions by reasoning about their size. They generate simple equivalent fractions (1/2 = 2/4; 2/3 = 4/6).

Future Learning

In grade 4, students will further develop their understanding of fraction equivalence and use their understanding of how to compose fractions from unit fractions and decompose fractions into unit fractions. They will solve problems involving addition and subtraction of fractions with like denominators and multiply fractions by a whole number.



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Additional Findings

According to Progressions 3–5 Number and Operations—Fractions, “As students move towards thinking of fractions as points on the number line, they develop an understanding of order in terms of position. Given two fractions—thus two points on the number line—the one to the left is said to be smaller and the one to right is said to be larger. This understanding of order as position will become important in Grade 6 when students start working with negative numbers.” (p. 4)

According to PARCC Model Content Frameworks Mathematics Grades 3–11, “Students in grade 3 begin to enlarge their concept of numbers by developing an understanding of fractions as numbers. This work will continue in grades 3–6, preparing the way for work with the complete rational number system in grades 6–7.” (p. 15)

“It is critical that students at this grade are able to place fractions on a number line diagram and understand them as a related component of their ever expanding number system.” (p. 16)


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Multiplication and Division Within 100


Content to be Learned Mathematical Practices to Be Integrated • Multiply and divide within 100.

• Develop and use strategies based on the relationship between multiplication and division and the commutative and distributive properties of multiplication to solve problems.

• Solve one-step word problems using the four operations.

• Represent problems using equations.

• Apply strategies of rounding and mental computation to check for reasonableness.

Construct viable arguments and critique the reasoning of others.

• Justify conclusions and explain thinking by communicating them to others.

• Compare strategies for effectiveness.

• Support arguments using object, pictures, and/or drawings.

Attend to precision.

• Use the equal sign appropriately.

• Develop efficient strategies and calculate accurately.

• Check their work.

Look for and make use of structure.

• Look for patterns in multiplication and division.

Essential Questions • How can multiplication help you solve division

problems?

• What strategies can be used to find products and/or quotients?

• How can you use the array model to help you solve multiplication problems?

• What number sentences could be used to solve this problem?

• How can simpler multiplication facts help you solve a more difficult fact?

• How do you know that your equation accurately represents this word problem?

• How do you know your answer is reasonable?

Grade 3 Mathematics, Quarter 2, Unit 2.3 Multiplication and Division Within 100 (10 days)


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Written Curriculum


Operations and Algebraic Thinking 3.OA

Multiply and divide within 100.

3.OA.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.

Understand properties of multiplication and the relationship between multiplication and division.

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

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

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


3 Construct viable arguments and critique the reasoning of others.

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.



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6 Attend to precision.

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 Look for and make use of structure.

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 7 × 5 + 7 × 3, 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 2 + 7. 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.


Prior Learning

In grade 2, students used additive thinking to skip count by 5s, 10s, and 100s, and used additive thinking to find the total number of objects arranged in arrays up to 5 row and 5 columns.

Current Learning

In this unit, students continue to develop their conceptual understanding of multiplication. In unit 1.3, they use area as a context for developing multiplicative thinking and to interpret products of whole numbers as the total number of objects grouped equally. In Unit 1.4, they study patterns and relationships in multiplication facts and the commutative strategy of multiplication to begin multiplying within 100. Students build on previous learned strategies to multiply and divide within 100 and solve single-step word problems of various types (see CCSS table 1, pp. 88, 89) with whole numbers using all four operations. They develop an understanding of the relationship between multiplication and division as they apply the commutative property.

Throughout third grade, students continue to develop their understanding of multiplication and division. In Unit 3.3, they develop more strategies for multiplying and dividing within 100 and solve two-step problems with a letter standing for the unknown quantity. In Unit 4.4, students solve two-step problems using all four operations and represent these problems using equations with a letter standing for the unknown quantity. By the end of grade 3, they will know from memory all products of two one-digit numbers.



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Future Learning

In grade 4, they will solve multi-step word problems including those in which remainders must be interpreted. They will multiply up to 4-digit by 1-digit whole numbers and two 2-digit numbers and find whole number quotients and remainders with up to 4-digit dividends and 1-digit divisors.

Additional Findings

According to the PARCC Model Content Frameworks Mathematics Grades 3–11, “Students must begin multiplication and division at or near the very start of the year in order to allow time for understanding and fluency to develop” (p. 15)

According to the Progressions Counting and Cardinality, K–5, Operations and Algebraic Thinking, “Note that mastering this material, and reaching fluency in single- digit multiplications and related divisions with understanding, 3.OA.7 may be quite time consuming because there are no general strategies for multiplying or dividing all single-digit numbers as there are for addition and subtraction. Instead, there are many patterns and strategies dependent upon specific numbers. So it’s imperative that extra time and support be provided if needed.” (p. 22)


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Application of Area Concepts


Content to be Learned Mathematical Practices to Be Integrated • Find areas of rectangles by tiling.

• Relate finding area to multiplication.

• Find perimeter of polygons by determining the length of an unknown side.

• Recognize attributes that quadrilaterals share.

Make sense of problems and persevere in solving them.

• Use concrete objects to conceptualize area and perimeter.

• Justify and communicate solutions using pictures, words, numbers, etc.

Look for and express regularity in repeated reasoning.

• Notice repetitive actions in computation and look for efficient methods.

Model with mathematics.

• Represent problems in multiple ways.

• Connect different representations and explain these representations.

Essential Questions • Do rectangles with the same area always have

the same perimeter?

• Do rectangle with the same perimeter always have the same area?

• How would you explain the process for finding the area of a rectangle?

• What attributes do all quadrilaterals share?

• Given the perimeter and the length of one side of a rectangle, how can you determine the length of the other side?

Grade 3 Mathematics, Quarter 2, Unit 2.4 Application of Area Concepts (15 days)


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Written Curriculum


Measurement and Data 3.MD

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

3.MD.7 Relate area to the operations of multiplication and addition.

a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

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.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

3.MD.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.

Geometry 3.G

Reason with shapes and their attributes.

3.G.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.


1 Make sense of problems and persevere in solving them.

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



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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.

4 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 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.

8 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.


Prior Learning

In grade 2, students described and analyzed shapes by examining their sides and angles. They developed the foundation for understanding area. They recognized and drew shapes having specified attributes.

Current Learning

In this unit, students build on their understanding of perimeter developed in Unit 1.2. In this unit, students continue to develop an understanding of the concept of perimeter by using real-world and classroom contexts (i.e., walking around the perimeter of a room; using elastics on a geo-board). They use addition to find perimeter and recognize the patterns that exist when finding the sum of the length and width of rectangles. Given the perimeter and a length or width, students use objects or pictures to find the missing length or width.

Students in grade 3 tile areas of rectangles with whole number sides, determine the area, record the length and width of the rectangle, investigate the pattern in the numbers and discover that the area is length times width.



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Students recognize shapes that are and are not quadrilaterals by examining the properties of the geometric figures.

Later this year in Unit 3.2, students find the area of rectilinear figures (polygons having all right angles) by decomposing them into nonoverlapping rectangles and adding the areas of the nonoverlapping parts. They will use concrete area models to reinforce their understanding of the distributive property of multiplication. They will apply their understanding of area and perimeter to explore relationships between area and perimeter.

Future Learning

In grade 4, they will apply the area and perimeter formulas for rectangles in real-world and mathematical problems and will classify 2-D figures based on attribute of their lines and angles.

Additional Findings

According to the PARCC Model Content Frameworks, “Area is a major concept within measurement, and area models must function as a support for multiplicative reasoning in grade 3 and beyond” (p. 60).

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