How To Understand And Understand Mathematics

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1 Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) Common Core and Research from the CCSS Progression Documents Geometry Students learn to analyze and relate categories of two-dimensional and three-dimensional shapes explicitly based on their properties.5.g.4 Based on analysis of properties, they classify two-dimensional figures in hierarchies. For example, they conclude that all rectangles are parallelograms, because they are all quadrilaterals with two pairs of opposite, parallel, equal-length sides (MP3). In this way, they relate certain categories of shapes as subclasses of other categories.5.g.3 This leads to understanding propagation of properties; for example, students understand that squares possess all properties of rhombuses and of rectangles. Therefore, if they then show that rhombuses diagonals are perpendicular bisectors of one another, they infer that squares diagonals are perpendicular bisectors of one another as well. Note that in the U.S., that the term trapezoid has two different meanings: T(E)exclusive and T(I) inclusive T(E): a trapezoid is a quadrilateral with exactly one pair of parallel sides T(I): a trapezoid is a quadrilateral with at least one pair of parallel sides. These different meanings result in different classifications. According to T(E), a parallelogram is not a trapezoid; according to T(I), a parallelogram is a trapezoid. Both definitions are legitimate; however; when teaching trapezoids to 4 th and 5 th graders, we will use the inclusive definition since most college bound textbooks and PARCC use the inclusive definition. A note about research The ability to describe, use, and visualize the effects of composing and decomposing geometric regions is significant in that the concepts and actions of creating and then iterating units and higher-order units in the context of constructing patterns, measurin g, and computing are established bases for mathematical understanding and analysis. Additionally, there is suggestive evidence that this type of composition corresponds with, and may support, children s ability to compose and decompose numbers. Coordinate Planes Students extend their Grade 4 pattern work by working briefly with two numerical patterns that can be related and examining these relationships within sequences of ordered pairs and in the graphs in the first quadrant of the coordinate plane. 5.OA.3 This work prepares students for studying proportional relationships and functions in middle school. Fifth grade students extend their knowledge of the coordinate plane, understanding the continuous nature of two-dimensional space and the role of fractions in specifying locations in that space. Thus, spatial structuring underlies coordinates for the plane as well, and students learn both to apply it and to distinguish the objects that are structured. For example, they learn to interpret the components of a rectangular grid structure as line segments or lines (rather than regions) and understand the precision of location that these lines require, rather than treating them as fuzzy boundaries or indicators of intervals. Students learn to reconstruct the levels of counting and quantification that they had already constructed in the domain of discrete objects to the coordination of (at first) two continuous linear measures. That is, they learn to apply their knowledge of number and length to the order and distance relationships of a coordinate grid and to coordinate this across two dimensions. 5.G.1 1

2 Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) Although students can often locate a point, these understandings are beyond simple skills. For example, initially, students often fail to distinguish between two different ways of viewing the point (2, 3), say, as instructions: right 2, up 3 ; and as the point defined by being a distance 2 from the x-axis and a distance 3 from the y-axis. In these two descriptions the 2 is first associated with the x-axis, then with the y-axis. They connect ordered pairs of (whole number) coordinates to points on the grid, so that these coordinate pairs constitute numerical objects and ultimately can be operated upon as single mathematical entities. Students solve mathematical and real-world problems using coordinates. For example, they plan to draw a symmetric figure using computer software in which students input coordinates that are then connected by line segments. 5.G.2 Progressions for the Common Core State Standards in Mathematics (draft) Commoncoretools.wordpress.com 2

3 Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) The chart below highlights the key understandings of this cluster along with important questions that teachers should pose to promote these understandings. The chart also includes key vocabulary that should be modeled by teachers and used by students to show precision of language when communicating mathematically. Enduring Understandings Essential Questions Key Vocabulary Two-Dimensional figures are classified by their properties. Two-Dimensional figures can fit into more than one category. Identify and describe properties of two-dimensional figures more precisely. Graphical representations can be used to make predications and interpretations about real world situations. How can plane figures be categorized and classified? What is a quadrilateral? What are the properties of quadrilaterals? How can you classify different types of quadrilaterals? How are quadrilaterals alike and different? How can angle and side measures help us to create and classify triangles? Why are some quadrilaterals classified as parallelograms? Why is a square always a rectangle? What are ways to classify triangles? Where is geometry found in your everyday world? How does the coordinate system work? How do coordinate grids help you organize information? The terms below 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. Geometry acute angle acute triangle circle congruence/congruent equilateral triangle half circle/semicircle hexagon irregular polygon isosceles triangle kite obtuse angle parallel lines parallelogram pentagon perpendicular lines plane figure polygon quadrilateral rectangle regular polygon right angle right triangle rhombus/rhombi scalene triangle square triangle trapezoid two-dimensional vertex Coordinate Planes Coordinate system Coordinate plane First quadrant Points Lines Axis/Axes x-axis y-axis Horizontal Vertical Intersection of lines Origin Ordered Pairs Coordinates x-coordinate y-coordinate 3

4 Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) Throughout this cluster, students will develop their use of the 8 Mathematical Practices while learning the instructional standards. Specific connections to this cluster and instructional strategies are provided in the following chart. Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them Cluster Connections and Instructional Strategies Students make sense of problems involving two-dimensional figures based on their geometric properties. Students make sense of solving real world problems involving points on the coordinate plane. 2. Reason abstractly and quantitatively Students demonstrate abstract reasoning about rational relationships among geometric properties. Students go beyond simple recognition to an analysis of the properties and how they interrelate. Students demonstrate abstract reasoning about ordered pairs with their visual representations. 3. Construct viable arguments and critique the reasoning of others Students construct and critiques arguments regarding their knowledge of triangles and the ability to belong to one or more of the categories and sub-categories. Students construct and critique arguments regarding patterns and relationship of ordered pairs as they are plotted on a coordinate plane to represent real-world contexts. 4. Model with mathematics Students use tables to identify and draw all three types of triangles comparing their attributes in mathematical and real-world contexts. Students use the coordinate plane to compare two numbers in mathematical and real-world contexts. 5. Use appropriate tools strategically Students select and use tools such as tables and the quadrilateral hierarchy to represent situations involving the categories and sub-categories of two-dimensional figures. Students select and use tools such as number line models and the coordinate plane to represent situations involving positive numbers. 6. Attend to precision Students attend to the geometric precision when classifying two-dimensional figures in the hierarchy. Students attend to the language of real-world situations to determine how far to travel from the origin and the given direction of the coordinates being represented. 7. Look for and make use of structure Students relate the attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. Students relate the structure of number lines to values of positive integers as they use the coordinate plane. 8. Look for and express regularity in repeated reasoning Students relate new experiences to experiences with similar contexts when studying the hierarchy of polygons based on properties. Students relate new experiences to experiences with similar contexts when studying positive representations of distance and quantity. 4

5 Classify two-dimensional figures into categories based on their properties. Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) Maryland College and Career- Ready Standards 5.G.3 Understand that attributes belonging to a category of twodimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. SMP.3 Construct viable arguments and critique the reasoning of others SMP.7 Look for and make use of structure SMP.5 Use appropriate tools strategically Instructional Strategies and Resource Support This standard calls for students to reason about the attributes (properties) of shapes. Student should have experiences discussing the property of shapes and reasoning. Example: Examine whether all quadrilaterals have right angles. Give examples and non-examples. Example: If the opposite sides on a parallelogram are parallel and congruent, then rectangles are parallelograms A sample of questions that might be posed to students include: A parallelogram has 4 sides with both sets of opposite sides parallel. What types of quadrilaterals are parallelograms? Regular polygons have all of their sides and angles congruent. Name or draw some regular polygons. All rectangles have 4 right angles. Squares have 4 right angles so they are also rectangles. True or False? A trapezoid has 2 sides parallel so it must be a parallelogram. True or False? Use tangrams to have students make shapes using a designated number of tangram pieces. For example: make a square, rectangle, trapezoid and a parallelogram by using 3 tangram pieces for each. Which of the figures can you make using only 2 tangram pieces? Make quadrilaterals using Geoboards. Record several on dot paper. Have students bring the dot paper quadrilaterals to small group instruction and use them to sort the quadrilaterals by different attributes or properties. Teaching Student-Centered Mathematics (Grades 3-5) pg. 212 Shape Sorts - Activity 8.1 (BLMs 20-26) pg. 213 What's My Shape - Activity 8.2 pg. 214 Constructing & Dissecting Shapes (BLM 27) pg. 217 *Can You Make It - Activity 8.5 pg. 221 Categories of Two-Dimensional Shapes - Table 8.1 Students make use of structure to build a logical progression of statements and explore hierarchical relationships among 2-dimensional shapes (SMP.3, SMP.7) Sample Formatives What property do they have in common? Name each shape. Explain how the two shapes are alike and how they are different. Use the Geoboards to draw two different quadrilaterals. Name each of the quadrilaterals. Compare the properties of the quadrilaterals you drew. Complete the table by placing Xs in each column to indicate properties of each figure. 5

6 Classify two-dimensional figures into categories based on their properties. Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) Maryland College and Career- Ready Standards 5.G.4. Classify twodimensional figures in a hierarchy based on properties. SMP.3 Construct viable arguments and critique the reasoning of others Instructional Strategies and Resource Support Figures from previous grades: polygon, rhombus/rhombi, rectangle, square, triangle, quadrilateral, pentagon, hexagon, cube, trapezoid, half/quarter circle, circle Example: Create a Hierarchy Diagram using the following terms: polygons a closed plane figure formed from line segments that meet only at their endpoints. quadrilaterals - a four-sided polygon. parallelogram a quadrilateral with two pairs of parallel and congruent sides. rectangle a quadrilateral with two pairs of congruent, parallel sides and four right angles. rhombus a parallelogram with all four sides equal in length. square a parallelogram with four congruent sides and four right angles. trapezoid-a quadrilateral with at least one pair of parallel sides. Sample Formatives Put the labels from the word box in the appropriate box to make the diagram true. SMP.7 Look for and make use of structure SMP.5 Use appropriate tools strategically Student should be able to reason about the attribute/properties of shapes by examining: What are ways to classify triangles? Which quadrilaterals have opposite angles congruent and why is this true of certain quadrilaterals? How many lines of symmetry does a regular polygon have? Why can t all rectangles be classified as squares? Teaching Student-Centered Mathematics (Grades 3-5) pg. 222 Figure 8.11 (Bottom figure) pg. 225 Triangle Sort - Activity 8.7 (BLM 29) pg. 226 Property Lists for Quadrilaterals -Activity 8.8 (BLM 30-33) pg. 227 Diagonal Strips - Activity 8.9, *Figure 8.16 pg. 230 Minimal Defining Lists - Activity 8.11 pg. 231 True or False - Activity 8.12 Students make use of structure to build a logical progression of statements and explore hierarchical relationships among 2-dimensional shapes (MP.3, MP.7) 6

7 Graph points on the coordinate plane to solve real-world and mathematical problems. Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) Maryland College and Career-Ready Standards Instructional Strategies and Resource Support Sample Formatives 5.G.1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x- axis and x-coordinate, y-axis and y- coordinate). TSCM-page G.1 and 5.G.2 These standards deal with only the first quadrant (positive numbers) in the coordinate plane. Although students can often locate a point, these understandings are beyond simple skills. For example, initially, students often fail to distinguish between two different ways of viewing the point (2, 3), say, as instructions: right 2, up 3 ; and as the point defined by being a distance 2 from the y-axis and a distance 3 from the x-axis. In these two descriptions the 2 is first associated with the x-axis, then with the y-axis. Plot the ordered pairs on the coordinate plane. Label the points as indicated. 5.G.2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. SMP.6 Attend to precision SMP.4 Model with mathematics SMP.2 Reason abstractly and quantitatively SMP.5 Use appropriate tools strategically Students precisely describe the coordinate of points and the relationship of the coordinate plane to the number line (SMP.6). Students both generate and identify relationship in numerical patterns, using the coordinate plane as a way of representing these relationship and patterns (SMP.4) 7

8 Graph points on the coordinate plane to solve real-world and mathematical problems. Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) Maryland College and Career-Ready Standards 5.G.2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. SMP.6 Attend to precision SMP.4 Model with mathematics SMP.5 Use appropriate tools strategically SMP.1 Make sense of problems and persevere in solving them Instructional Strategies and Resource Support This standard references real-world and mathematical problems, including the traveling from one point to another and identifying the coordinates of missing points in geometric figures, such as squares, rectangles, and parallelograms. Example 1: Students precisely describe the coordinate of points and the relationship of the coordinate plane to the number line (SMP.6). Students both generate and identify relationship in numerical patterns, using the coordinate plane as a way of representing these relationship and patterns (SMP.4) Sample Formatives Plot the following points on the coordinate grid. Plot a fourth point on the grid to form a rectangle with all 4 point are connected. Write the coordinates for the point you plotted. 8