All the resources on this page come from the GADOE Framework 3rd Grade Unit 4 Overview. Reviewing this information before or during planning can help ensure your instruction is aligned with the standards and is at the depth expected.

Unit 4 Overview

In this unit students will:

• Further develop understandings of geometric figures by focusing on identification and descriptions of plane figures based on geometric properties.

• Identifies examples and non-examples of plane figures based on geometric properties.

• Identify differences among quadrilaterals.

• Understand that shapes in different categories may share attributes and those attributes can define a larger category (example: rhombuses, rectangles, and others have four sides and are all called quadrilaterals).

• Expand the ability to see geometry in the real world.

• Can draw plane figure shapes based on attributes. • Further develop understanding of partitioning shapes into parts with equal areas.

• Partitions shapes in several different ways into equal parts of halves, thirds, fourths, sixths, and eighths and recognizes the partitioned parts have the same area. • Use data collected to make bar and picture graphs.

• Interpret line plots.

• Find the perimeter of polygons; use addition to find perimeters; solve for an unknown length and recognize the patterns that exist when finding the sum of the lengths and widths of rectangles.

Third grade students will 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. Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language.

Mathematically proficient students communicate clearly by engaging in discussion about their reasoning, using appropriate mathematical language. 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 square units required to cover the shape without gaps or overlaps. 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.

In second grade, students identify and draw triangles, quadrilaterals, pentagons, and hexagons. Third graders build on this experience and further investigate quadrilaterals (technology may be used during this exploration). 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 (see examples below) and identify squares, rectangles, and rhombuses as quadrilaterals.

Students should classify shapes by attributes and by drawing shapes that fit specific categories. For example, parallelograms include: squares, rectangles, rhombi, or other shapes that have two pairs of parallel sides. Also, the broad category, quadrilaterals, includes all types of parallelograms, trapezoids and other four-sided figures.

Students should also use this standard to help build on their understanding of fractions and area. Students are responsible for partitioning shapes into halves, thirds, fourths, sixths and eighths. 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.

As an ongoing process throughout all third-grade units, students should continue to develop understanding of representing and interpreting data using picture and bar graphs. They should also continue their work in generating measurement data by measuring lengths with rulers marked with halves and fourths of an inch. In second grade, students measured length in whole units using both metric and U.S. customary systems. It is important to review with students how to read and use a standard ruler including details about half 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. This standard provides a context for students to work with fractions by measuring objects to a quarter of an inch.

With geometry, many student misconceptions might occur. The four content goals for geometry include shapes and properties, transformation, location, and visualization (see Van de Walle, page 205.) Students often have a difficult time recognizing shapes if the shape has been transformed by a translation, reflection, or rotation. Students may also identify a square as a “non-rectangle” or a “non-rhombus” based on limited images they see. They do not recognize that a square is a rectangle because it has all of the properties of a rectangle. They may list properties of each shape separately, but not see the interrelationships between the shapes. For example, students do not look at the properties of a square that are characteristic of other figures as well. Using straws to make four congruent figures have students change the angles to see the relationships between a rhombus and a square. As students develop definitions for these shapes, relationships between the properties will be understood.

Third grade should prepare students to be able to easily transition into fourth grade geometry. In fourth grade students will be required to draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. Students in fourth grade will classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category and identify right triangles. Recognizing a line of symmetry for a two- dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts will also be a part of fourth grade. http://www.learner.org/courses/learningmath/geometry/session10/index35.html - Lesson for the teacher to learn more about teaching Geometry to elementary students. The lesson includes ideas for lessons, video of classrooms, etc.

STANDARDS FOR MATHEMATICAL PRACTICES (SMP)

This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion. Students are expected to:

1. Make sense of problems and persevere in solving them. Students make sense of problems involving the attributes of shapes.

2. Reason abstractly and quantitatively. Students demonstrate abstract reasoning about perimeters when considering the values of these numbers in relation to distance.

3. Construct viable arguments and critique the reasoning of others. Students construct and critique arguments regarding shape attributes and perimeters.

4. Model with mathematics. Students use inch tiles and charts to solve real world perimeter problems.

5. Use appropriate tools strategically. Students select and use tools such as inch counters, TanGrams, and geometric shapes to represent attributes and perimeter.

6. Attend to precision. Students use clear and precise language when discussing the attributes of shapes.

7. Look for and make use of structure. Students look closely to discover a pattern or structure when sorting shapes based on common attributes.

8. Look for and express regularity in repeated reasoning. Students demonstrate repeated reasoning by showing the relationship between partitioning shapes and perimeter.

***Mathematical Practices 1 and 6 should be evident in EVERY lesson***

CONTENT STANDARDS

Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.

Reason with shapes and their attributes.

MGSE3.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.

MGSE3.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.

Represent and Interpret Data

MGSE3.MD.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.

MGSE3.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. Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

MGSE3.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.

c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

MGSE3.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.

For more detailed information about unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the Grade Level Overview.

BIG IDEAS

• Identify and describe properties of two-dimensional shapes using properties that are shared between the shapes.

• Generalize that shapes fit into a particular classification.

• Compare and classify shapes by their sides and angles and connect these with definitions of shapes.

• Geometric figures can be classified according to their properties.

• Quadrilaterals can be classified according to the lengths of their sides.

• Recognize shapes that are and are not quadrilaterals by examining the properties of the geometric figures.

• 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

• Provided details and use proper vocabulary when describing the properties of quadrilaterals.

• Sort geometric figures and identify squares, rectangles, and rhombuses as quadrilaterals.

• Classify shapes by attributes and by drawing shapes that fit specific categories. (e.g.; parallelograms include: squares, rectangles, rhombi, or other shapes that have two pairs of parallel sides.

• The broad category “Quadrilaterals” includes all types of parallelograms, trapezoids and other four-sided figures.

• Relate fraction work to geometry by expressing the area of a shape as a unit fraction of the whole.

• Shapes can be partitioned with equal areas in a variety of ways to show halves, thirds, fourths, sixths, and eighths.

• The length around a polygon can be calculated by adding the lengths of its sides.

• The space inside a rectangle or square can be measured in square units.

ESSENTIAL QUESTIONS

• How do the attributes help us identify the different quadrilaterals/shapes?

• How it is possible to have a shape that has fits into more than one category?

• What does it mean to partition a shape into parts?

• What is the relationship between perimeter and area?

CONCEPTS/SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

• Represent and solve problems involving multiplication and division

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

• Multiply and divide within 100

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

• Use place value

• Recognize basic geometric figures and spatial relationships of triangle, quadrilateral (squares, rectangles, and trapezoids), pentagon, hexagon, cube, trapezoid, half/quarter circle, circle, cone, cylinder, sphere

Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.

Deep Understanding: Teachers teach more than simply “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives. Therefore, students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.

Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency and automaticity. Fluency implies a much richer kind of mathematical knowledge and experience.

Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context.

Fluent students:

• flexibly use a combination of deep understanding, number sense, and memorization.

• are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.

• are able to articulate their reasoning.

• find solutions through a number of different paths.

For more about fluency, see: http://www.youcubed.org/wpcontent/uploads/2015/03/FluencyWithoutFear-2015.pdf https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/nctm-timed-tests.pdf

STRATEGIES FOR TEACHING AND LEARNING:

In earlier grades, students have experiences with informal reasoning about particular shapes through sorting and classifying using their geometric attributes. Students have built and drawn shapes given the number of faces, number of angles and number of sides. The focus now is on identifying and describing properties of two-dimensional shapes in more precise ways using properties that are shared rather than the appearances of individual shapes.

These properties allow for generalizations of all shapes that fit a particular classification. Development in focusing on the identification and description of shapes’ properties should include examples and non-examples, as well as examples and non-examples drawn by students of shapes in a particular category. For example, students could start with identifying shapes with right angles. An explanation as to why the remaining shapes do not fit this category should be discussed. Students should determine common characteristics of the remaining shapes.

In Grade 2, students partitioned rectangles into two, three or four equal shares, recognizing that the equal shares need not have the same shape. They described the shares using words such as, halves, thirds, half of, a third of, etc., and described the whole as two halves, three thirds or four fourths. In Grade 3 students will partition shapes into parts with equal areas (the spaces in the whole of the shape). These equal areas need to be expressed as unit fractions of the whole shape, i.e., describe each part of a shape partitioned into four parts as ¼ of the area of the shape. Have students draw different shapes and see how many ways they can partition the shapes into parts with equal areas.

http://www.learner.org/courses/learningmath/geometry/pdfs/session9/vand.pdf - Geometric Thinking from John Van de Walle

Also in the unit, students will use what they have learned in second grade about representing the length of several objects by making a line plot. In second grade, students would have rounded their lengths to the nearest whole unit. A line plot shows data on a number line with an X or other mark to show frequency.

Examples of Line Plot

•The line plot below shows the test scores of 26 students.

The count of cross marks above each score represents the number of students who obtained the respective score. For students in second and third grade, they will use the data from measuring with rulers to create line plots.

Area and Perimeter…

• Students can cover rectangular shapes with tiles and count the number of units (tiles) to begin developing the idea that area is a measure of covering. Area describes the size of an object that is two-dimensional. The formulas should not be introduced before students discover the meaning of area.

• The area of a rectangle can be determined by having students lay out unit squares and count how many square units it takes to completely cover the rectangle completely without overlaps or gaps. Students need to develop the meaning for computing the area of a rectangle. A connection needs to be made between the number of squares it takes to cover the rectangle and the dimensions of the rectangle.

Ask questions such as: o What does the length of a rectangle describe about the squares covering it? o What does the width of a rectangle describe about the squares covering it?

• The concept of multiplication can be related to the area of rectangles using arrays. Students need to discover that the length of one dimension of a rectangle tells how many squares are in each row of an array and the length of the other dimension of the rectangle tells how many squares are in each column. Ask questions about the dimensions if students do not make these discoveries. For example:

o How do the squares covering a rectangle compare to an array?

o How is multiplication used to count the number of objects in an array?

• Students should also make the connection of the area of a rectangle to the area model used to represent multiplication. This connection justifies the formula for the area of a rectangle.

• Provide students with the area of a rectangle (i.e., 42 square inches) and have them determine possible lengths and widths of the rectangle. Expect different lengths and widths such as, 6 inches by 7 inches or 3 inches by 14 inches.

**For additional assistance see the Unit Webinar:

https://www.georgiastandards.org/Archives/Pages/default.aspx

SELECTED TERMS AND SYMBOLS

The following terms are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them. These terms are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real-life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers. Mathematics Glossary • 2-dimensional • 3-dimensional • acute angle • attributes • closed figure • congruent • cubes, cones, cylinders and rectangular prisms (as subcategories of 3-dimensional figures) • polygon • line plot • obtuse angle • open figure • parallel • parallelogram • partition • polygon • properties • quadrilateral • rectangle • rhombi, rectangles, and squares (as subcategories of quadrilaterals) • rhombus/rhombi • right angle • square • three-sided • unit fraction • area • overlap • plane figure • side length • square centimeter • square foot • square inch • square meter • square unit • tiling

Due to the preponderance of advantages, inclusive definitions are used. For example, the inclusive definition of trapezoid specifies that it is a quadrilateral with at least one pair of parallel sides. Additional resources for finding definitions for common geometry terms: http://www.amathsdictionaryforkids.com/dictionary.html