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Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Maryland College and Career-Ready Standards In this unit, students will build their understanding of geometry by making constructions, identifying attributes within geometric figures, such as perpendicular and parallel lines, obtuse, right, and acute angles, and classify figures based on those attributes. Students will also understand angles as the space between intersecting rays or line segments. They will see angles as parts of a circle and will be able to measure angles in degrees using information about the angle in relation to a circle or a protractor. Students will recognize patterns within a group of shapes and be able to make a conjecture based on the pattern. Patterns will also be found when studying symmetry of shapes. Students will look at repeating shape patterns and make algebraic conjectures by studying the terms in the pattern and noticing patterns in the multiples matched with each term. Research Students, by 4th grade, should be ready to make conjectures about relationships they notice within geometry. This helps student learn to analyze and interpret the world around them and gives students tools for solving problems. Geometry instruction should focus on the relationship among geometric figures and the properties of those figures, not just naming shapes or figures. Geometric thinking extends students mathematical thinking within other concepts. Helping Children Learn Mathematics Reys, 2004 “As students sort, build, draw, model, trace, measure, and construct, their capacity to visualize geometric relationships will develop” (p. 165) Students can learn to be special and visual thinkers, they need rich experiences provided for them to develop this skill. Navigations through Geometry NCTM, 2000 As students learn geometric concepts, they pass through five sequential levels of development, called the Van Hiele levels. The levels are not based on age/grade, instead they describe how we think and what types of geometric ideas we think about. A student is likely to be at different levels at any one time. Geometric experience is the greatest single factor influencing advancement through the levels. It is important to be aware of these levels, so that instruction can be designed to help students move through this hierarchy or thinking. “When instruction or language is at a level higher than that of the students, there will be a lack of communication. Students required to wrestle with objects of thought that have not been constructed at the earlier level may be forced into rote learning and achieve only temporary and superficial success.” (p208) The first three levels (0-2) of thinking will most likely be developed during elementary school and are briefly described below. See TSCM pages 206-210 for more detailed information. Level 0: Visualization: Students identify shapes/figures only on the bases of appearance. Instruction needs to include examples of a variety of each type of shape shown in various orientations. This encourages students to make generalizations about common features. Level 1: Analysis: Students begin to analyze shapes and figures based on properties and attributes. Students make generalizations about how different shapes relate to one another by spending time comparing and classifying properties of shapes. Level 2: Informal Deduction: Students can generalize relationships between various shapes. Students are able to justify their generalizations (“if-then” reasoning) by providing examples and non-examples. (not full proofs) Level 3: Formal Deduction: This is the level of the traditional high school geometry course. Level 4: Rigor: This is the level of college mathematics. Teaching Student Centered Mathematics Van de Walle, 2006 1 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days The chart below highlights the key understandings of Unit 7 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 Two-dimensional figures can be analyzed and classified by their properties. Parallel sides, perpendicular sides, angle measures, and symmetry can be used to classify geometric figures. Some two-dimensional figures have one or more lines of symmetry. Angles are measured in degrees using protractors. An angle can be decomposed into two or more smaller angles. The original angle measure is equal to the sum of the smaller angle measures. Essential Questions How can two-dimensional figures be classified? How can parallelism, perpendicularity, and angle measure be used to analyze quadrilaterals and triangles? How are quadrilaterals similar and different? How are triangles alike and different? How do I know when a figure is symmetrical? How do you measure an angle? How do you construct an angle of a given measure? How can an angle be divided? Key Vocabulary acute angle/triangle angle circle decompose degree endpoint figure geometric shape intersect line line segment parallel perpendicular point protractor obtuse angle/triangle one-degree angle ray right angle/triangle straight unknown vertex equilateral triangle isosceles triangle scalene triangle line of symmetry Background Reading Focus in Grade 4: Teaching with Curriculum Focal Points Teaching StudentCentered Mathematics – Grades 3-5 Putting the Practices into Action: Implementing the Common Core Standards for Mathematical Practice K-8 2 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Throughout this unit, students will develop their use of the 8 Mathematical Practices while learning the instructional standards. The mathematical practices in the shaded boxes should be emphasized during instruction this unit due the how well they connect with the content standards in this unit. Standards for Mathematical 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 Connections to this Cluster Solve problems in which the solution is not immediately evident. To determine and articulate what the problem is asking: Ask students to restate the problem in their own words. Have students turn to a partner to state the problem. Discuss familiar problems (When have we seen something like this before? What did we do?) To self-monitor progress and change directions when necessary: Have students talk or write about how they got “stuck” and then “unstuck” when solving a problem. Think aloud to show students how to change course when needed. To demonstrate perseverance in problem-solving and identify different ways to solve a problem: Make a classroom list of possible strategies. Acknowledge those who modify their thinking and persevere to get to the solution and have students show and talk about how they solved problems. Encourage students to show at least two ways to solve a problem. Write an equation for a situation and be able to explain how the equation relates to the situation presented. Solve the equation outside of the context of the problem, and then connect the solution back to the situation presented. Represent how to find unknown angle measures using understanding of complementary and supplementary angles. To make sense of quantities and their relationships in problem situations: Represent a given problem situation with an equation using a variable for the unknown. Write a situation problem that matches a specific equation. Justify conclusions, communicate them to others, and respond to the arguments of others. Ask useful questions to clarify or improve the arguments. Examine the steps taken that produce an incorrect solution and provide a viable argument as to why the process produced an incorrect response. Discuss strategies for categorizing two-dimensional figures, finding lines of symmetry, finding unknown angle measures. -Ask students to restate and try peer methods for solving problems. -Help students make mathematical connections between different representations/solution methods. -Evaluate the efficiency of the strategies based on the probability of an error occurring. Identify models that are most efficient for solving specific problems or representing specific geometry ideas. Construct visual models using concrete or virtual geometric manipulatives, pictures, or equations to justify thinking and display the solution. Represent real-world geometry situations. Use tools effectively when classifying, measuring, and constructing geometric figures. Tiles, ruler, straight edges for constructing two-dimensional figures and drawing lines of symmetry 3 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days 6. Attend to precision Index cards or corners of paper for right angles Protractors for measuring and constructing angles Graphic organizers for classifying two-dimensional shapes Create accurate drawings and representations of mathematical situations. Use specific math vocabulary to communicate mathematical ideas. Develop and display anchor charts with precise math vocabulary, such as parallel, perpendicular, etc. Orally rephrase student explanations using appropriate vocabulary. Label geometric components and angles accurately. 7. Look for and make use of structure Look closely to discover a pattern or structure. 8. Look for and express regularity in repeated reasoning Analyze properties of geometric figures and classify two-dimensional figures based on properties. Model the composing and decomposing of angles to build understanding of the additive quality of angles. Notice repetitive actions in mathematics and try to make sense of it. Make generalizations about rules and “shortcuts” to get to answers more quickly. 4 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Maryland College and CareerReady Standards 4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures. SMP 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure Instructional Strategies and Resource Support This standard asks students to identify and construct foundational geometric figures including points, lines, line segments, rays, and angles. They build on this understanding in this standard and throughout the unit as they: describe, compare and classify two-dimensional shapes by their properties relate parallelism and perpendicularity measure and classify angles and triangles It is vital that students construct figures throughout the unit. Drawing shapes and their geometric components help to deepen students’ understanding of two-dimensional figures and help them make meaning of abstract concepts. These drawings provide a basis for a student to connect visual images with vocabulary and enable him to analyze and communicate precisely. Geoboards, grid paper (square and triangle grids), popsicle sticks, and straws are useful tools to have available as students explore twodimensional figures. Provide criteria for students to use to construct figures: Draw a polygon with at least 2 pairs of parallel sides. Construct a trapezoid with 2 right angles. Sketch a triangle with an obtuse angle. Additionally, have students identify properties of given shapes: What types of angles does this figure have? Are there parallel sides? Perpendicular sides? Study Jams PPT – Types of Lines http://studyjams.scholastic.com/studyjams/jams/math/geometry/types-oflines.htm Formative Assessments FA.4.G.A.1.a Seth uses a ruler to construct a twodimensional figure on his paper. He follows these steps: Draw a line segment on your paper with point A and point B as endpoints. Draw another line segment that is perpendicular to the first and shares endpoint A. Label the other endpoint C. FA.4.G.A.1.b Adam, Brian, and Charlie all live in the same neighborhood. The points below show the location of their homes. Each point matches the first letter in each boy’s name. A B C FA.4.G.A.1.c Look at each of the figures below. Use a highlighter to trace one pair of perpendicular sides in each figure. Use a highlighter to trace one pair of parallel sides in each figure. 5 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Maryland College and CareerReady Standards 4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures. Instructional Strategies and Resource Support Geometric Term point line This is just a visual. A point actually lacks dimension. Definition a location in space a straight path that continues in both directions with an infinite number of points but no endpoints part of a line with 2 endpoints line segment ray SMP 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure Figure angle acute angle right angle obtuse angle straight angle parallel perpendicular Formative Assessments part of a line with 1 endpoint and extends forever in one direction the amount of space between 2 intersecting lines, line segments, or rays FA.4.G.A.1.d On each object, trace one pair of lines that are parallel using a blue colored pencil. How do you know if lines are parallel? an angle that measures less than 90° a 90° angle an angle that measures 91° 179° An angle that forms a straight line - 180° lines that are always the same distance apart intersecting lines that form right angles 6 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Maryland College and Career-Ready Standards 4.G.A.2 Classify twodimensional 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. SMP 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure Instructional Strategies and Resource Support Two-dimensional shapes are classified based on relationships of sides and angles. Multiple opportunities in comparing and contrasting shapes allow students to gain a deeper understanding about shapes and their properties. http://www.illustrativemathematics.org/illustrations/1275 Formative Assessments FA.4.G.A.2.a Use the Geoboards and a ruler to draw each twodimensional figure. Quadrilaterals: A quadrilateral is a 4-sided polygon. In 3rd grade, students learn to identify quadrilaterals and recognized parallel lines within them. In 4th grade, you can build on that by exploring quadrilaterals in-depth. Students can determine if the sides are parallel or perpendicular, analyze angles and classify accordingly. Characteristics of rectangles (including squares) can be used to strengthen the concept of parallel and perpendicular lines. Quadrilateral Figure . rectangle square rhombus parallelogram trapezoid kite Properties 4 right angles (perpendicular sides) 2 sets of parallel sides Opposite sides are congruent 4 right angles (perpendicular sides) 4 congruent sides 2 sets of parallel sides FA.4.G.A.2.b Julia sorted the figures into a Venn diagram as shown. 4 congruent sides 2 sets of parallel sides 2 sets of parallel sides Opposite sides are congruent At least 1 set of parallel sides 2 pairs of congruent sides adjacent (next to each other) FA.4.G.A.2.c Jeff classified triangles into the following groups: Students can try to write the shortest definition of a quadrilateral. For a parallelogram do you need to say 2 sets of parallel sides and opposite sides are equal? Why or why not? 7 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Maryland College and CareerReady Standards 4.G.A.2 Classify twodimensional 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. Instructional Strategies and Resource Support Informal understanding of the characteristics of triangles is developed through angle measures and side length relationships. Triangles are named according to their angle measures (right, acute or obtuse) and side lengths (scalene, isosceles or equilateral). Triangles Classified by Angles: Name Figure Properties . right 1 right angle (measures 90º) 3 angles that are acute (measure less than 90º) acute 1 angle that is obtuse (measures greater than 90º) obtuse SMP 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure Formative Assessments Triangles Classified by Side Length: Name equilateral isosceles scalene Figure Properties . 3 equal sides 2 equal sides no sides equal 8 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Maryland College and CareerReady Standards 4.G.A.2 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. SMP 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure Instructional Strategies and Resource Support Formative Assessments Students can also explore parallelism, perpendicularity, and angle size when analyzing qualities of other polygons including pentagons, hexagons, octagons in addition to irregular polygons (polygons that do not have all sides equal and all angles equal). Students can create and analyze sorts based on properties. Example: At least 1 set of parallel sides At least 1 obtuse angle Additionally, students should have opportunities to prove/disprove conjectures (with counter-examples). Pose statements such as the ones below for students to test validity: A parallelogram can have exactly one right angle. A trapezoid can have 2 right angles. A right triangle can be isosceles. Draw and name a figure that has 2 right angles and has 1 pair of parallel sides. A quadrilateral must be a parallelogram. Geoboards, dot paper, virtual geoboards, wiki sticks, and pipe cleaners are useful tools to create 2-dimensional figures. 9 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Maryland College and CareerReady Standards 4.G.A.3 Recognize a line of symmetry for a twodimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. SMP 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure Instructional Strategies and Resource Support In this standard, students explore symmetry in relation to two-dimensional figures. A line of symmetry divides a figure into 2 congruent halves. It can be considered a “folding line” so that if the figure is folded along the line, the 2 parts match. Students need opportunities to work with symmetrical figures and nonsymmetrical figures including regular and irregular polygons. Through work with paper cut-outs, students can identify a shape’s line(s) of symmetry and begin to investigate symmetry patterns with questions such as: Do all triangles have 3 lines of symmetry? How many lines of symmetry does a circle have? Is the number of lines of symmetry related to the number of angles/sides of a regular polygon? Students need to be able to apply their understanding of symmetry to drawings of two-dimensional figures as well. On dot paper, have students draw polygons based on given characteristics and identify lines of symmetry. Example: Draw a regular hexagon and draw all lines of symmetry. What do you notice? What about a regular pentagon? Formative Assessments FA.4.G.A.3.a Draw all of the lines of symmetry for each of shapes below: Rectangle A has ____ lines of symmetry. Rectangle B has ____ lines of symmetry. FA.4.G.A.3.b A square has 4 lines of symmetry. Draw all of the lines of symmetry for each of the quadrilaterals below. Write the number of lines on the blank under each figure. FA.4.G.A.3.c Half of each figure is shown. The dotted line represents the line of symmetry. Use this to draw the other half of the figure. Additionally, provide one-half of a figure and the line of symmetry. Students should draw the other half of the shape. FA.4.G.A.3.d http://illuminations.nctm.org/lesson.aspx?id=1800 Draw all of the lines of symmetry for each of the polygons. Write the number of lines on each blank. 10 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Maryland College and CareerReady Standards Geometric measurement: understand concepts of angle and measure angles. 4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: 4.MD.C.5.a An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles. 4.MD.C.5.b An angle that turns through n one-degree angles is said to have an angle measure of n degrees. Instructional Strategies and Resource Support This standard connects angles and circular measurement (360 degrees). An angle is the union of two rays with a shared endpoint called the vertex. The rays can be called the sides of the angles. The rays can rotate one from the other about the vertex. This rotation determines the size of the angle between the rays. Angles are measured in reference to a circle with its center at the vertex of the rays. An angle that turns through 1/360 of a circle is called a “onedegree angle,” and degrees are the unit used to measure angles. A complete rotation is 360º. Real world connections can be seen in the movement of hands on a clock, a rotating water sprinkler, or the opening of a door (the hinge is the vertex), or degrees of a compass. Formative Assessments FA.4.MD.C.5.a.a Harry is standing on the court with a basketball in his hands. He pivots to find a teammate so that he can pass the ball. Harry turns 60°with each pivot. How many times will Harry pivot by the time he makes a complete circle? FA.4.MD.C.5.a.b Scott is hiking through the woods. He is holding a compass to help him with direction. FA.4.MD.C.5.c Jason’s farm uses a large sprinkler that rotates in a circle. The sprinkler rotates 23 degrees in 10 seconds. How far will it rotate in 30 seconds? Show your thinking: 11 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Maryland College and CareerReady Standards Geometric measurement: understand concepts of angle and measure angles. 4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: Instructional Strategies and Resource Support Formative Assessments Students should use their understanding of the degrees in a circle to explore questions such as: If each circle was divided into equal parts, what is the measure of each angle? 4.MD.C.5.a An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “onedegree angle,” and can be used to measure angles. 4.MD.C.5.b (360º ÷ 4 = 90º) (360º ÷ 3 = 120º) (360º ÷ 6 = 60º) Students should also explore the concept that an angle measurement is not related to the area between 2 rays. For example, the visual below shows that while the area between the 2 rays is different in each of the circles; the angle measure is the same. An angle that turns through n one-degree angles is said to have an angle measure of n degrees. 12 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Geometric measurement: understand concepts of angle and measure angles. Maryland College and CareerReady Standards 4.MD.C.6 Measure angles in wholenumber degrees using a protractor. Sketch angles of specified measure. SMP 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure Instructional Strategies and Resource Support Prior to using a protractor, students need to have a strong understanding of benchmark angles. Through their earlier work in the unit, they have explored a circle as having 360º, a straight angle (line) measuring 180º, a right angle as 90º, an acute angle as less than 90º and an obtuse angle measuring from 91º to 179º. Formative Assessments FA.4.MD.C.6.a Classify (acute, obtuse, right) and measure each angle. Students need to be able to both measure and draw angles of a given value using a protractor. Measuring an angle: Place the vertex at the center point of the protractor. Align one ray along the horizontal edge of the protractor. The angle is measured by finding where the second ray intersects the arc of the protractor. Drawing an angle: Draw a ray and place the center of the protractor on the endpoint of the ray. This will become the vertex. Place a point on the paper that aligns with the angle measure on the protractor. Use the straightedge of the protractor to connect point to the vertex. To avoid misconceptions in measuring or drawing angles, students should: Classify the type of angle to determine an appropriate estimate of the angle measure. Have experience with many different angles (not just those that are in a standard position with one ray that is horizontal). Identify angle measures within figures. Use your protractor and ruler to draw the following angles. Identify each as acute, obtuse, right, or straight. 45º 135º 90º 60º FA.4.MD.C.6.b Use your protractor to measure the angles. Label the degrees in each angle. 13 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Geometric measurement: understand concepts of angle and measure angles. Maryland College and CareerReady Standards 4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. SMP 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure Instructional Strategies and Resource Support Angle measure is additive. An angle can be divided into two smaller angles. The combined measures of the two smaller angles equal the original angle. B BAC = BAD + DAC D A C If a right angle is decomposed (broken apart) into two adjacent angles, the sum is 90º, thus they are complementary. Two adjacent angles that compose a “straight angle” of 180º are supplementary. You can find an unknown angle measure, if you know the measure of the original angle and one of the smaller angles. Formative Assessments FA.4.MD.C.7.a Stacey’s bedroom door is opened 32 degrees. Her mom wants it to be opened the full 90 degrees. How many degrees does Stacey need to open her door? ________________ Draw an angle sketch to show what Stacey’s door looked like at 32 degrees. FA.4.MD.C.7.b Vikki was ice skating and attempted to complete a 180 degree turn. She turned some and then tripped. She needed to turn 42 degrees more to complete her turn. How many degrees did Vikki turn before she tripped? FA.4.MD.C.7.c Find the missing angle measurements: (Progressions for the CCSSM, Geometric Measurement, CCSS Writing Team, June 2012, page 23) 14 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Geometric measurement: understand concepts of angle and measure angles. Maryland College and CareerReady Standards 4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into nonoverlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. SMP 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure Instructional Strategies and Resource Support Formative Assessments Students apply the additive quality of angles in order to find unknown angles. Examples: 1. A revolving door rotates 65º, pauses, and then rotates 45º more. What is the total angle measure of the door so far? (110º) How many more degrees will the door need to turn to make a full 360º rotation? (360º - 110º = 250º) 45º 65º 2. There are 360º in a complete circle. If the hands of a clock are exactly on 12 and 1, the angle created measures 30º. What is the angle measure if the hands are exactly on 12 and 4? (120º); 8 and 11? (90º); 1 and 7? (180º) 3. If the 2 rays are perpendicular, what is the value of n? n = 90º - (30º + 35º) 30º n = 90º - 65º = 25º 35º n 15 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Maryland College and CareerReady Standards Generate and analyze patterns. 4.OA.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Instructional Strategies and Resource Support In unit 1, students understood this standard in terms of generating, identifying, and extending number patterns. In this unit, students will make conjectures and explore patterns using two-dimensional figures, or properties of two-dimensional figures. Symmetry Patterns Students make and test conjectures about the number of lines of symmetry in regular polygons (polygons having equal sides and equal angles). Identify and record the number of sides and the number of lines of symmetry in each figure. SMP 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure Number of Sides 3 4 5 6 Number of Lines of Symmetry 3 4 5 ? Formative Assessments FA.4.OA.C.5.a A new high school is under construction, and the materials for the tile floor are being ordered. The design of the main hallway floor is shown below: The school needs to order a total of 224 shape tiles for the hallway design. How many square tiles should the school order? Show your thinking. How many lines of symmetry will a hexagon have? An octagon? What pattern is created by where the lines of symmetry are located for an even number of sides versus an odd number of sides? 16 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Maryland College and CareerReady Standards Generate and analyze patterns. 4.OA.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Instructional Strategies and Resource Support Students can solve real world pattern problems related to angle measurement and properties of polygons. They can use a table to find a rule, extend elements of a pattern, and answer questions based on the pattern. Angle Measure Example At the carnival, the Ferris wheel has 12 seats, evenly spaced. At the beginning of the ride, the people get on the ride as follows: Seat Number Number of Passengers Number of Degrees Wheel Moved SMP 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure Formative Assessments 1 2 3 2 4 6 30º 60º 90º 4 5 6 How many passengers will be on the Ferris wheel when the wheel has rotated 240º? When 20 people are on the ride, how many degrees has the wheel rotated? Polygon Example David is designing a toothpick bridge. He makes a drawing of the bridge before constructing it. Each line segment is 1 toothpick. ? 1 section 2 sections 3 sections 4 sections ? 5 sections Draw how 4 sections and 5 sections of the bridge would look. Create a table to record the number of toothpicks used for each section size. Extend your table to find the number of toothpicks needed for 10 sections. How many toothpicks are needed for 20 sections? 100 sections? What rule connects the number of sections to the number of toothpicks needed? 17 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days Triangle Sort Cut out the triangles. Sort the triangles into categories. (Each triangle MUST fit into one of the categories). Describe your sort. Then, mix the triangles and create a new sort. How did your groups change? A B C D E F G H I J K L 18 Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4) Unit 7: Geometry and Patterns – 26 Days 19