Download Trapezoid defini-ons

Trapezoid defini-ons Everyday Math Expressions Sco4 Foresman A quadrilateral that has exactly one pair of parallel sides (K & 1st) (3rd & 4th) A quadrilateral with only one pair of parallel sides. (5th) A quadrilateral that has exactly one pair of parallel sides A quadrilateral with at least one pair of parallel sides. (5th) A quadrilateral with one pair of parallel sides. Maybe by Middle School? CMP Glencoe Holt A quadrilateral with at least one pair of opposite sides parallel. (6th) A quadrilateral with one pair of opposite sides parallel. (7th) A quadrilateral with one pair of parallel sides. (8th) A quadrilateral with exactly one pair of parallel opposite sides. A quadrilateral with exactly one pair of parallel sides. Parallelogram Defini-ons Everyday Math Expressions Sco4 Foresman • A quadrilateral with two pairs of parallel sides. • Opposite sides of a parallelogram are congruent. • Opposite angles in a parallelogram have the same measure. (K-­‐2nd) A quadrilateral in which both pairs of opposite sides are parallel and opposite angles are congruent. (3rd-­‐5th) A quadrilateral with both pairs of opposite sides parallel. (3rd & 4th) A quadrilateral in which opposite sides are parallel. (5th) A quadrilateral with both pairs of opposite sides parallel. Progressions •  Read through Grade 3 Geometry and Geometric Measurement and jot down a few ideas to share. • 
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Think about what you read. Note important ideas. Share ideas with a partner Share ideas with the whole group. Geometry Learning Progression 3
Area DefiniAons FracAons 4
Parallel, Perpendicular, Right, Acute, Obtuse angles, Line segments, Ray, Symmetry 5
Volume Coordinate System Categorize Shapes Grade 3 •  Four Cri-cal Areas –  Developing understanding of •  Mul-plica-on/division and strategies within 100 •  Frac-ons – especially unit frac-ons •  THE STRUCTURE OF RECTANGULAR ARRAYS AND OF AREA •  TWO-­‐DIMENSIONAL SHAPES Reason with shapes and their a]ributes. 3.G.1 •  Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share a]ributes (e.g., having four sides), and that the shared a]ributes 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. rd
Geometry in 3 Grade •  Grade 3 students work on analyzing, comparing, categorizing, and drawing shapes using their proper-es. •  Building on previous work, Grade 3 students should par-cipate in many hands on experiences which involve a strong language component to lead them to understanding and correctly using geometric vocabulary for shapes and their a]ributes. MathemaAcally proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathemaAcal language From previous grades: •  triangle, quadrilateral, pentagon, hexagon, cube, trapezoid, half/quarter circle, circle, cone, cylinder, sphere Grade 3 •  properAes, a4ributes, features, , quadrilateral, open figure, closed figure , three-­‐sided, 2-­‐dimensional, rhombi, rectangles, and squares are subcategories of quadrilaterals, polygon, rhombus/rhombi, rectangle, square, parAAon, unit fracAon, kite •  1The term “property” in these standards is reserved for those a]ributes that indicate a rela-onship between components of shapes. Thus, “having parallel sides” or “having all sides of equal lengths” are proper-es. •  “A4ributes” and “features” are used interchangeably to indicate any characteris-c of a shape, including proper-es, and other defining characteris-cs (e.g., straight sides) and non-­‐
defining characteris-cs (e.g., “right-­‐side up”). •  (Progressions for the CCSSM, Geometry, CCSS Wri-ng Team, June 2012, page 3 footnote) Property Criteria
Property Criteria
• all right angles
• at least one right angle
• both pairs of opposite sides congruent
• both pairs of opposite sides parallel
• both pairs of opposite angles congruent
• congruent diagonals
• diagonals bisect each other
• at least one pair of parallel sides
• exactly one pair of parallel sides
But there is a problem for us! Third graders are moving toward iden-fying shapes by defini-on, or at least by parts of their defini-ons. The problem is that even the “experts” do not agree on what those defini-ons are in elementary school math text books. Trapezoid defini-ons Everyday Math Expressions Sco4 Foresman A quadrilateral that has exactly one pair of parallel sides. (Exclusive definiAon.) (K & 1st) A quadrilateral with at least one pair of parallel sides. (Inclusive definiAon.) (5th) A quadrilateral with one pair of parallel sides. (Exclusive definiAon) (3rd & 4th) A quadrilateral with only one pair of parallel sides. (Exclusive definiAon) (5th) A quadrilateral that has exactly one pair of parallel sides (Exclusive definiAon) Sort the quadrilaterals according to the inclusive and exclusive defini5ons. What is the problem? Maybe by Middle School? CMP Glencoe Holt A quadrilateral with at least one pair of opposite sides parallel. (Inclusive definiAon) (6th) A quadrilateral with one pair of opposite sides Parallel. (Exclusive) (7th) A quadrilateral with one pair of parallel sides. (8th) A quadrilateral with exactly one pair of parallel opposite sides. A quadrilateral with exactly one pair of parallel sides. (Exclusive definiAon.) From CC. Progression Doc: “Usiskin et al. conclude, ‘The preponderance of advantages to the inclusive definiDon of trapezoid has caused all the arDcles we could find on the subject, and most college-­‐bound geometry books, to favor the inclusive definiDon.’” Parallelogram Definitions
Everyday Math
Expressions
• A quadrilateral with (K-2nd)
two pairs of parallel A quadrilateral in
sides.
which both pairs of
opposite sides are
• Opposite sides of a parallel and
parallelogram are
opposite angles are
congruent.
congruent.
• Opposite angles in
a parallelogram
have the same
measure.
(3rd-5th)
A quadrilateral
with both pairs of
opposite sides
parallel.
Scott Foresman
(3rd & 4th)
A quadrilateral in
which opposite
sides are parallel.
(5th)
A quadrilateral with
both pairs of
opposite sides
parallel.
What is a Trapezoid?
•  Some authors choose to define
trapezoid as a quadrilateral with at least
one pair of parallel sides.
•  That definition is more inclusive and
leads to the conclusion that all
parallelograms are trapezoids.
True or False
A square is
•  a special kind of rectangle.
•  It is a rectangle in which all four sides
are the same length
A parallelogram is
•  a special kind of trapezoid.
•  It is a trapezoid with two pairs of
parallel sides.
True or False
A rhombus
•  is a special kind of kite.
•  It is a kite in which all four sides are the
same length.
Sor-ng Quadrilaterals Using geoboard paper, draw 4 different types of quadrilaterals, including one that is neither a trapezoid nor a parallelogram. Cut your geoboards apart and put them in the center of the table. How many different ways can you sort these quadrilaterals? As you work, have one person record the ways. Discuss any geometric vocabulary that is used coming up with common defini-ons before you sort. Jus-fy your sor-ng decisions. Reason with shapes and their a]ributes 3.G.2 •  Par--on shapes into parts with equal areas. Express the area of each part as a unit frac-on of the whole. For example, parDDon a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. Students need to noAce and see why equal parts can have different shapes. How might a 3rd grader jus-fy that all the parts of the two shapes at the bo]om are equal? Folding paper to explore this standard. 1. 
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Fold a square of paper side edge to side edge. Open and trace the fold line. What is each part called? Now open the square and fold it in half the other way. Trace the new line. Now, how many parts do you have? What is each part called? 5.  Open the square and fold it corner to corner, two -mes. Trace the new lines. How many parts do you have? What is each part called? •  Can you fold a square in different ways to make halves, fourths and eighths? •  Challenge: Try a triangle. Can you fold it into halves, fourths, and eighths? Was that easier or more difficult? The large rectangle above is divided into a series of smaller quadrilaterals and triangles. Each of the shapes is a frac-onal part of the large rectangle. Can you untangle what frac-onal part is represented by each of the ten numbered shapes? Write down your answers for each number and share with your table what kind of thinking you did to solve this problem. Solve problems involving measurement and es-ma-on. 3. MD. 1 •  Tell and write -me to the nearest minute and measure -me intervals in minutes. Solve word problems involving addi-on and subtrac-on of -me intervals in minutes, e.g., by represenDng the problem on a number line diagram. •  Solve problems involving measurement and esAmaAon of intervals of Ame, liquid volumes, and masses of objects. •  TERMS •  esAmate, Ame, Ame intervals, minute, hour, elapsed Ame, measure, liquid volume, mass, standard units, metric, gram (g), kilogram (kg), liter (L) 3.MD.1 Solve •  Mrs. Ford’s math class starts at 8:15. They do 3 fluency ac-vi-es that each last 4 minutes. Just when they finish all of the fluency, the fire alarm goes off. •  When they return to the room amer the drill, it is 8:46. How many minutes did the fire drill last? Solve problems involving measurement and es-ma-on. 3. MD. 2 •  Measure and es-mate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).1 Add, subtract, mul-ply, or divide to solve one-­‐step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem 3.MD.2 •  Students iden-fy 5 things that weigh about one gram. They record their findings with words and pictures. (Students can repeat this for 5 grams and 10 grams.) This ac-vity helps develop gram benchmarks. •  One large paperclip weighs about one gram. A box of large paperclips (100 clips) weighs about 100 grams, so 10 boxes would weigh one kilogram. Founda-onal understandings to help with measure concepts: •  Understand that larger units can be subdivided into equivalent units (par--on). •  Understand that the same unit can be repeated to determine the measure (itera-on). •  Understand the rela-onship between the size of a unit and the number of units needed (compensatory principal). Typical problem for 3MD.2. •  Baggage handlers lim heavy luggage into the plane. The weight of one bag is shown on the scale at the top right. •  a. One baggage handler lims 3 bags of the same weight. Es-mate what you think the total weight he lims will be? Now calculate the exact weight. •  b. Another baggage handler lims luggage that weighs a total of 200 kilograms. Write and solve an equa-on to show how much more weight he lims than the first handler in Part (a). •  A tank has a capacity of 1,500 liters. If there are already 950 liters of water in the tank, how much more water is needed to fill it up completely? •  The volume of a bo]le of water is 750 milliliters. What is the volume of 6 bo]les? •  John weighs 100 kilograms. He is 4 -mes heavier than Mike. What’s Mike’s weight? More Challenging: Oranges and Lemons On the table there is a pile of oranges and lemons that weighs exactly one kilogram. The oranges all weigh 130 grams. The lemons are also all the same weight, which is less than 2/3 of the weight of an orange. There are twice as many lemons as oranges in the pile. How many lemons are there and how much does each one weigh? Is there more than one answer? (from h]p://nrich.maths.org ) Represent and interpret data. 3. 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 informa-on presented in scaled bar graphs •  While exploring data concepts, students should •  Pose a ques-on, Collect data, Analyze data, and Interpret data (PCAI). Students should be graphing data that is relevant to their lives •  Graphs should include a -tle, categories, category label, key, and data. 3.MD.3 Solve •  The following chart shows the number of -mes an insect’s wings vibrate each second. Use the following clues to complete the unknowns in the chart. •  The beetle’s number of wing vibra-ons is the same as the difference between the fly and honeybee’s. •  The mosquito’s number of wing vibra-ons is the same as 50 less than the beetle and fly’s combined. 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. Understand concepts of area and relate area to mul-plica-on and to addi-on 3. MD. 5 •  Recognize area as an a]ribute of plane figures and understand concepts of area measurement. •  A. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area •  B. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 3. MD. 6 •  Measure areas by coun-ng unit squares (square cm, square m, square in, square m, and improvised units). Geometric measurement 3. MD. 7 •  Relate area to the operaAons of mulAplicaAon and addiAon. • 
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A. Find the area of a rectangle with whole-­‐number side lengths by -ling it, and show that the area is the same as would be found by mul-plying the side lengths. B. Mul-ply side lengths to find areas of rectangles with whole-­‐number side lengths in the context of solving real world and mathema-cal problems, and represent whole-­‐number products as rectangular areas in mathema-cal reasoning. C. Use -ling 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 distribu-ve property in mathema-cal reasoning. D. Recognize area as addi-ve. Find areas of rec-linear figures by decomposing them into non-­‐
overlapping rectangles and adding the areas of the non-­‐overlapping parts, applying this technique to solve real world problems. E. Solve real world and mathema-cal problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibi-ng rectangles with the same perimeter and different areas or with the same area and different perimeters. Tiling, adding and mul-plying to find area. Mapping Arrays with Specific Dimensions to find area d. Recognize area as addi-ve. Find areas of rec-linear figures by decomposing them into non-­‐overlapping rectangles and adding the areas of the non-­‐overlapping parts, applying this technique to solve real world problems. . Par--oning Arrays to Solve for Area Using Distribu-ve Property C. Use -ling 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 distribu-ve property in mathema-cal reasoning. Erika was wondering how to arrange 20 pieces of fencing to make a rectangular dog run. Task 1: •  Draw a rectangle with 20 toothpicks (fencing pieces) •  Sketch, label dimensions and find area. •  Display all rectangles on chart paper. •  Label which arrangement has the largest area and which has the smallest. Erika has 20 square pieces of sod (grass) for the dog run. Which rectangular arrangement of sod would take the most fencing? The least fencing? Task 2: •  Build a rectangle with 20 -les •  Sketch, label and find the perimeter •  Display all rectangles on chart paper •  Label which requires the most fencing and which requires the least fencing Solve •  Mario plans to completely cover his 8-­‐inch by 6-­‐inch cardboard with square-­‐inch -les. He has 42 square-­‐inch -les. How many more square-­‐inch -les does Mario need to cover the cardboard without any gaps or overlap? Explain your answer. Problem Solving •  3.MD.8 •  Solve real world and mathema-cal problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibi-ng rectangles with the same perimeter and different areas or with the same area and different perimeters. Jaya drew a shaded rectangle on a grid, as shown below •  . •  What is the area, in square units, of Jaya’s shaded rectangle? __________ Show or explain how you got your answer. •  Marc also drew a rectangle.Marc’s rectangle has the same area as Jaya’s rectangle.Marc’s rectangle has a width of 3 units. Corey put together the two triangles shown below to make a new . shape
•  The rules he used to make his new shape are below. –  The sides labeled a are next to each other. –  The triangles should touch but not overlap •  What new shape did Corey make? Discuss •  A rectangle has a perimeter of 64 inches. What are possible areas for this rectangle? •  How would you tackle this problem? What strategies would you use? Discuss. The teacher is helping students understand area, perimeter, and their relaDonship. Solve •  Can you find rectangles where the value of the area is the same as the value of the perimeter? •  Do you think there is any pa]ern or mathema-cal rela-onship among rectangles that have equal perimeters and area? Read Geometry and Measurement/
Data Progression Overview Grade 4
Jot down a few ideas to share. Grade 4 •  Three Cri-cal Areas –  Developing understanding of •  GEOMETRIC FIGURES –  CLASSIFYING BY PROPERTIES •  Mul--­‐digit mul-plica-on/division –  Fluency •  Frac-ons –  Equivalence –  +/-­‐ of unlike frac-ons –  Mul-plica-on of whole numbers by frac-ons Draw and iden-fy lines and angles, and classify shapes by proper-es of their lines and angles 4.G.1 •  Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Iden-fy these in two-­‐
dimensional figures. Draw and iden-fy lines and angles, and classify shapes by proper-es of their lines and angles 4.G.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. . A regular octagon is shown below
Iden-fy which of these shapes have perpendicular or parallel sides and jus-fy your selec-on. Draw and iden-fy lines and angles, and classify shapes by proper-es of their lines and angles 4.G.3 •  Lines of symmetry? Recognize 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. Iden-fy line-­‐symmetric figures and draw lines of symmetry. How many lines of symmetry does the le]er below have? Symmetry? Below is half of a line-symmetric figure and its line of symmetry. Use a ruler to
complete Mike’s drawing.
How many line segments are needed to make the le]er A? How many angles are there? Are they acute, obtuse, or right angles? Are any of the line segments perpendicular? Are any of the line segments parallel? We can build all of these le]ers from line segments and arcs of circles. Build all of the capital le]ers with the smallest number of "pieces," where each piece is either a line segment or an arc of a circle. Which le]ers have perpendicular line segments? Which le]ers have parallel line segments? Which le]ers have no line segments? Do any le]ers contain both parallel and perpendicular lines? What makes the lower case le]ers "i" and "j" different than all of the capital le]ers? Le]ers can be thought of as geometric figures. •  measure, metric, customary, convert/
conversion, relaAve size, liquid volume, mass, length, distance, kilometer (km), meter (m), cenAmeter (cm), millimeter (mm), kilogram (kg), gram (g), liter (L), milliliter (mL), cup (c) Ame, hour, minute, second, equivalent, operaAons, add, subtract, mulAply, divide, fracAons, decimals, area, perimeter Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit 4.MD.1 •  Know rela-ve sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. •  Record measurement equivalents in a two-­‐column table. For example, know that 1 R is 12 Dmes as long as 1 in. •  Express the length of a 4 m snake as 48 in. •  Generate a conversion table for feet and inches lis-ng the number pairs (1, 12), (2, 24), (3, 36).... •  Billy has been training for a half-­‐marathon. He has a strict gym rou-ne that he follows six -mes a week. For the problems below, use tape diagrams, numbers, and words to explain each answer. •  •  Each day Billy runs on the treadmill for 5 kilometers and runs on the outdoor track for 6,000 meters. In all, how many kilometers does Billy run each day? •  •  Since Billy has started training, he has also been drinking more water. On Saturday, he drank 2 L 755 mL of water. On Sunday, he drank some more. If Billy drank a total of 4 L 255 mL of water on Saturday and Sunday, how many milliliters of water did Billy drink on Sunday? •  Since exercising so much for his half-­‐marathon, Billy has been losing weight. In his first week of training, he lost 2 kg 530 g of weight. In the following two weeks of training, he lost 1 kg 855 g per week. Billy now weighs 61 kg 760 g. What was Billy’s weight, in grams, before he started training? Explain your thinking Problems 4.MD.2 •  Use the four opera-ons to solve word problems involving distances, intervals of -me, liquid volumes, masses of objects, and money, including problems involving simple frac-ons or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. •  Represent measurement quan--es using diagrams such as number line diagrams that feature a measurement scale . Number line Diagrams Problems Area and Perimeter 4.MD.3 •  Apply the area and perimeter formulas for rectangles in real world and mathema-cal problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a mulDplicaDon equaDon with an unknown factor. Problem Solving •  Ms. McCrary wants to make a rabbit pen in a sec-on of her lawn. Her plan for the rabbit pen includes the following: •  It will be in the shape of a rectangle. •  It will take 24 feet of fence material to make. •  Each side will be longer than 1 foot. •  The length and width will measure whole feet. Part A •  Draw 3 different rectangles that can each represent Ms. McCrary’s rabbit pen. Be sure to use all 24 feet of fence material for each pen. •  Use the grid below. Part B to have more than 60 Ms. McCrary wants her rabbit square feet of ground area inside the pen. She finds that if she uses the side of her house as one of the sides of the rabbit pen, she can make the rabbit pen larger. •  Draw another rectangular rabbit pen. •  Use all 24 feet of fencing for 3 sides of the pen. •  Use one side of the house for the other side of the pen. •  Make sure the ground area inside the pen is greater than 60 square feet. Students should be challenged to solve mul-step problems. •  Example: A plan for a house includes rectangular room with an area of 60 square meters and a perimeter of 32 meters. What are the length and the width of the room? Represent and interpret data. •  4.MD.4 •  Make a line plot to display a data set of measurements in frac-ons of a unit (1/2, 1/4, 1/8). •  Solve problems involving addi-on and subtrac-on of frac-ons by using informa-on presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collecDon. Ten students in Room 31 measured their pencils at the end of the day. . They recorded their results on the line plot below
•  What is the difference in length from the longest to the shortest pencil? •  If you were to line up all the pencils, what would the total length be? •  If the 5 1/8-­‐inch pencils are placed end to end, what would be their total length? Geometric measurement: understand concepts of angle and measure angles. •  The terms students should learn to use with increasing precision with this cluster are: measure, point, end point, geometric shapes, ray, angle, circle, fracAon, intersect, one-­‐
degree angle, protractor, decomposed, addiAon, subtracAon, unknown Geometric measurement: understand concepts of angle and measure angles. •  4.MD.5 •  Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: •  a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the frac-on of the circular arc between the points where the two rays intersect the circle. An angle that turns through of a circle is called a "one-­‐
degree angle," and can be used to measure angles. •  b. An angle that turns through n one-­‐degree angles is said to have an angle measure of n degrees. •  This standard brings up a connec-on between angles and circular measurement (360 degrees). •  Angle measure is a “turning point” in the study of geometry. Students omen find angles and angle measure to be difficult concepts to learn, but that learning allows them to engage in interes-ng and important mathema-cs. •  An angle is the union of two rays, a and b, with the same ini-al point P. The rays can be made to coincide by rota-ng one to the other about P; this rota-on determines the size of the angle between a and b. The rays are some-mes called the sides of the angles. The diagram below will help students understand that an angle measurement is not related to an area since the area between the 2 rays is different for both circles yet the angle measure is the same. Geometric measurement: 4.MD.6 •  Measure angles in whole-­‐number degrees using a protractor. •  Sketch angles of specified measure For each part below, explain how the measure of the unknown angle can be found without using a protractor. Problems •  Joey knows that when a clock’s hands are exactly on 12 and 1, the angle formed by the clock’s hands measures 30°. •  What is the measure of the angle formed when a clock’s hands are exactly on the 12 and 4? Problem solving •  List -mes on the clock in which the angle between the hour and minute hands is 90°. Use a student clock, watch, or real clock. Verify your work using a protractor. Geometric measurement: 4.MD.7 •  Recognize angle measure as addi-ve. 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 addi-on and subtrac-on problems to find unknown angles on a diagram in real world and mathema-cal problems, e.g., by using an equa-on with a symbol for the unknown angle measure. •  Example: A lawn water sprinkler rotates 65 degress and then pauses. It then rotates an addi-onal 25 degrees. What is the total degree of the water sprinkler rota-on? To cover a full 360 degrees how many -mes will the water sprinkler need to be moved? •  If the water sprinkler rotates a total of 25 degrees then pauses. How many 25 degree cycles will it go through for the rota-on to reach at least 90 degrees? If the two rays are perpendicular, what is the value of ? Write equa-ons using variables to represent the unknown angle measurements. Find the unknown •  . angle measurements numerically. Grade 5 •  Three Cri-cal Areas –  Extend division to 2-­‐digit divisors, integrate decimal frac-ons, understand decimals to hundredths, develop fluency with whole numbers and decimals –  UNDERSTANDING OF VOLUME –  Frac-ons •  Fluency with +/-­‐ •  Understanding of mul-plica-on •  Division in limited cases Graph points on the coordinate plane to solve real-­‐
world and mathema-cal problems. •  5.G.1 – •  Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersec-on 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 direc-on of one axis, and the second number indicates how far to travel in the direc-on of the second axis, with the conven-on that the names of the two axes and the coordinates correspond (e.g., x-­‐axis and x-­‐coordinate, y-­‐axis and y-­‐ coordinate). Problem •  Edna drew a square that had an area of 36 square units using the grid shown below. She started drawing her square at (7,10). Draw the square that Edna could have made in the grid below. Be sure to label the coordinates of each vertex of the square. Problem •  (a) Plot the following coordinate pairs on a grid: A (2,4) B (2,8) C (8,4) D(8,8) •  (b) If you connected point A to point B. Then connect point D to point •  C. Describe the rela-onship between the line segments. •  (c) If you connect point A to point B, point B to point D, point D to point C, and point C to point A, what geometric figure have you drawn? •  (d) List two proper-es of this geometric figure. Problem solving 5.G.2 •  Represent real world and mathema-cal problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situa-on. Sample •  Sara has saved $20. She earns $8 for each hour she works. ♣ If Sara saves all of her money, how much will she have amer working 3 hours? 5 hours? 10 hours? ♣ Create a graph that shows the rela-onship between the hours Sara worked and the amount of money she has saved. •  ♣ What other informa-on do you know from analyzing the graph? Classify two-­‐dimensional figures into categories based on their proper-es. 5.G.3 •  Understand that a]ributes belonging to a category of two-­‐dimensional 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. Sample •  If the opposite sides on a parallelogram are parallel and congruent, then rectangles are parallelograms. •  ♣ 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? Classify two-­‐dimensional figures into categories based on their proper-es •  5.G.4 – Classify two-­‐dimensional figures in a hierarchy based on proper-es. Proper-es of figure may include: • 
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♣ Proper-es of sides – parallel, perpendicular, congruent, number of sides ♣ Proper-es of angles – types of angles, congruent • Examples: ♣ A right triangle can be both scalene and isosceles, but not equilateral. ♣ A scalene triangle can be right, acute, and obtuse. • Triangles can be classified by: ♣ Angles σ Right – the triangle has one angle that measures . σ Acute: The triangle has exactly three angles that measure between σ Obtuse: The triangle has exactly one angle that measures greater than ♣ Sides σ Equilateral: All sides of the triangle are the same length. σ Isosceles: At least two sides of the triangle are the same length. σ Scalene: No sides of the triangle are the same length. Convert like measurement units within a given measurement system. •  5.MD.1 – •  Convert among different-­‐sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving mul--­‐step, real world problems. Represent and interpret data. 5.MD.2 – Make a line plot to display a data set of measurements in frac-ons of a unit (1/2, 1/4, 1/8). Use opera-ons on frac-ons for this grade to solve problems involving informa-on presented in line plots. For example, given different measurements of liquid in idenDcal beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. Ten beakers, measured in liters, are filled with a liquid. •  The above line plot shows the amount of liquid in liters in 10 beakers. If the liquid is redistributed equally, how much liquid would each beaker have? Geometric measurement: understand concepts of volume and relate volume to mul-plica-on and to addi-on. 5.MD.3 – •  Recognize volume as an a]ribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. •  b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. Geometric measurement 5.MD.4 – •  Measure volumes by coun-ng unit cubes, using cubic cm, cubic in, cubic m, and improvised units. Geometric Measurement 5.MD.5 – Relate volume to the opera-ons of mul-plica-on and addi-on and solve real world and mathema-cal problems involving volume. • 
Find the volume of a right rectangular prism with whole-­‐number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by mul-plying the edge lengths, equivalently by mul-plying the height by the area of the base. • 
Represent threefold whole-­‐number products as volumes, e.g., to represent the associa-ve property of mul-plica-on. • 
Apply the formulas V = l w h and V = b h for rectangular prisms to find volumes of right rectangular prisms with whole-­‐number edge lengths in the context of solving real world and mathema-cal problems. • 
Recognize volume as addi-ve. Find volumes of solid figures composed of two non-­‐
overlapping right rectangular prisms by adding the volumes of the non-­‐
overlapping parts, applying this technique to solve real world problems. Determine the volume of concrete needed to build the steps in the diagram below. Aaron says more informa-on is needed to find the volume of the prisms. Is Aeron mistaken? Can you calculate the volume of the prisms ? . The tank, shaped like a rectangular prism, is filled to the top with water
•  Will the beaker hold all the water in the box? If yes, how much more will the beaker hold? If not, how much more will the cube hold than the beaker? Explain how you know. Problem •  A rectangular tank with a base area of 24 cm2 is filled with water and oil to a depth of 9 cm. The oil and water separate into two layers when the oil rises to the top. If the thickness of the oil layer is 4 cm, what is the volume of the water? Grade 6 •  Four Cri-cal Areas –  Connect ra-o and rate to whole number mul-plica-on and division –  Division of frac-ons and extending to ra-onal number system including nega-ves –  Wri-ng, interpre-ng, and using expressions and equa-ons –  Develop understanding of sta-s-cal thinking Solve real-­‐world and mathema-cal problems involving area, surface area, and volume. 6.G.1 •  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; •  apply these techniques in the context of solving real-­‐world and mathema-cal problems. Problems Find the area of a triangle with a base length of three units and a height of four units. Find the area of the trapezoid shown below using the formulas for rectangles and triangles. Problems •  A rectangle measures 3 inches by 4 inches. If the lengths of each side double, what is the effect on the area? •  The area of the rectangular school garden is 24 square units. The length of the garden is 8 units. What is the length of the fence needed to enclose the en-re garden? Problems The 6th grade class at Hernandez School is building a giant wooden H for their school. The H will be 10 feet tall and 10 feet wide and the thickness of the block le]er will be 2.5 feet. How large will the H be if measured in square feet? The truck that will be used to bring the wood from the lumberyard to the school can only hold a piece of wood that is 60 inches by 60 inches. What pieces of wood (how many pieces and what dimensions) are needed to complete the project? The 6th grade class at Hernandez School is building a giant wooden H for their school. The H will be 10 feet tall and 10 feet wide and the thickness of the block le]er will be 2.5 feet. How large will the H be if measured in square feet? The truck that will be used to bring the wood from the lumberyard to the school can only hold a piece of wood that is 60 inches by 60 inches. What pieces of wood (how many pieces and what dimensions) are needed to complete the project? Solve real-­‐world and mathema-cal problems involving area, surface area, and volume 6.G.2 Find the volume of a right rectangular prism with frac-onal edge lengths by packing it with unit cubes of the appropriate unit frac-on edge lengths, and show that the volume is the same as would be found by mul-plying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with frac-onal edge lengths in the context of solving real-­‐world and mathema-cal problems Kelly wants to wrap 20 golf balls, each in a cube-­‐shaped box, together in one larger box. Which arrangement will use the least wrapping paper? •  Build a box with 20 cubes •  Sketch each box, label dimensions, find area of each face and the total surface area •  Display all boxes on chart paper •  Label which arrangement has the largest surface area and which has the smallest. •  Post Wait a minute… •  How can the boxes have the same volume of 20 cubes and have different surface areas? •  Discuss with your table group how students in 5th grade may respond to the above ques-on. Problem •  Cube-­‐shaped boxes will be loaded into the cargo hold of a truck. The cargo hold of the truck is in the shape of a rectangular prism. The edges of each box measure 2.50 feet and the dimensions of the cargo hold are 7.50 feet by 15.00 feet by 7.50 feet, as shown below. •  What is the volume, in cubic feet, of each box? Determine the number of boxes that will completely fill the cargo hold of the truck. Use words and/or numbers to show how you determined your answer. •  What is the volume, in cubic feet, of each box? Determine the number of boxes that will completely fill the cargo hold of the truck. Use words and/or numbers to show how you determined you answer. Problem •  Determine the volume of a rectangular prism length and width are in a ra-o of 3:1. •  The width and height are in a ra-o of 2: 3. The length of the rectangular prism is 5 m. Solve real-­‐world and mathema-cal problems involving area, surface area, and volume 6.G.3 – Draw polygons in the coordinate plane given coordinates for the ver-ces; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-­‐world and mathema-cal problems. Problem •  On a map, the library is located at , the city hall building is located at , and the high school is located at . Represent the loca-ons as points on a coordinate grid with a unit of 1 mile. •  ♣ What is the distance from the library to the city hall building? The distance from the city hall building to the high school? How do you know? •  ♣ What shape is formed by connec-ng the three loca-ons? The city council is planning to place a city park in this area. How large is the area of the planned park? Solve real-­‐world and mathema-cal problems involving area, surface area, and volume •  6.G.4 •  Represent three-­‐dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. •  Apply these techniques in the context of solving real-­‐world and mathema-cal problems Examples •  Describe the shapes of the faces needed to construct a rectangular pyramid. Cut out the shapes and create a model. Did your faces work? Why or why not? •  Create the net for a given prism or pyramid and then use the net to calculate the surface area. Classify each net as represen-ng a rectangular prism, a triangular prism, or a pyramid. Reset bu]on. What three-­‐dimensional figure will the net create? •  What is the volume of the figure? •  What is the surface area of the figure? 6.EE.9 •  6.EE.9 – •  Use variables to represent two quan--es in a real-­‐
world problem that change in rela-onship to one another; write an equa-on to express one quan-ty, thought of as the dependent variable, in terms of the other quan-ty, thought of as the independent variable. •  Analyze the rela-onship between the dependent and independent variables using graphs and tables, and relate these to the equa-on. Problem •  Plot four unique points on the coordinate grid that are each 5 units from the point (1, 2). Each point must contain coordinates with integer values. Problem •  On a map, the theater is located at (-­‐4,-­‐4), the courthouse is located at (2, 9), and the train sta-on is located at (2,-­‐4). Represent the loca-ons as points on a coordinate grid with a unit of 1 km. •  (a) What is the distance from the the theater to the train sta-on? The distance from the train sta-on to the courthouse? How do you know? •  (b) What shape does connec-ng the three loca-ons form? •  (c) The mayor is planning on construc-ng a public plaza in this area. What is the area of the planned plaza? Grade 7 •  Four Cri-cal Areas –  Develop understanding of and applying propor-onal rela-onships –  Develop understanding of opera-ons with ra-onal numbers and working with expressions and linear equa-ons –  Draw inferences based on popula-on samples –  SOLVE PROBLEMS INVOLVING •  SCALE DRAWINGS •  INFORMAL GEOMETRIC CONSTRUCTIONS •  AREA, SURFACE AREA, VOLUME Draw, construct, and describe geometrical figures and describe . the rela-onships between them
7.G.1 •  Solve problems involving scale drawings of geometric figures, such as compu-ng actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. • 
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Students determine the dimensions of figures when given a scale and iden-fy the impact of a scale on actual length (one-­‐dimension) and area (two-­‐ dimensions). Students iden-fy the scale factor given two figures. Using a given scale drawing, students reproduce the drawing at a different scale. Students understand that the lengths will change by a factor equal to the product of the magnitude of the two size transforma-ons. • Scale drawings of geometric figures connect understandings of propor-onality to geometry and lead to future work in similarity and congruence. As an introduc-on to scale drawings in geometry, students should be given the opportunity to explore scale factor as the number of -me you mul-ple the measure of one object to obtain the measure of a similar object. It is important that students first experience this concept concretely progressing to abstract contextual situa-ons. • Provide opportuni-es for students to use scale drawings of geometric figures with a given scale that requires them to draw and label the dimensions of the new shape. Ini-ally, measurements should be in whole numbers, progressing to measurements expressed with ra-onal numbers. This will challenge students to apply their understanding of frac-ons and decimals. • 
• Students should move on to drawing scaled figures on grid paper with proper figure labels, scale, and dimensions. Provide word problems that require finding missing side lengths, perimeters or areas. For example, if a 4 by 4.5 cm rectangle is enlarged by a scale of 3, what will be the new perimeter? What is the new area? Or, if the scale is 6, what will the new side length look like? Or, suppose the area of one triangle is 16 sq units and the scale factor between this triangle and a new triangle is 2.5. What is the area of the new triangle? • 
• Reading scales on maps and determining the actual distance (length) is an appropriate contextual situa-on. •  Julie showed you the scale drawing of her room. If each 2 cm on the scale drawing equals 5 m, what are the actual dimensions of Julie's room? Reproduce the drawing at 3 -mes its current size. Draw, construct, and describe geometrical figures and describe the rela-onships between them. 7.G.2 •  Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given condi-ons. Focus on construc-ng triangles from three measures of angles or sides, no-cing when the condi-ons determine a unique triangle, more than one triangle, or no triangle condi-ons •  Students understand that three given lengths determine a triangle, provided the largest length is less than the sum of the other two lengths; otherwise, no triangle can be formed. •  Students understand that if two side lengths of a triangle are given, then the third side length must be between the difference and the sum of the first two side lengths. •  Students understand that two angle measurements determine many triangles, provided the angle sum is less than 180°; otherwise, no triangle can be formed Problems •  Is it possible to draw a triangle with a angle and one leg that is 4 inches long and one leg that is 3 inches long? If so, draw one. Is there more than one such triangle? •  Draw a triangle with angles that are 60 degrees. Is this a unique triangle? Why or why not? •  Draw an isosceles triangle with only one 80° angle. Is this the only possibility or can you draw another triangle that will also meet these condi-ons? •  Can you draw a triangle with sides that are 13 cm, 5 cm, and 6 cm? •  Draw a quadrilateral with one set of parallel sides and no right angles. Draw, construct, and describe geometrical figures and describe the rela-onships between them 7.G.3 •  Describe the two-­‐dimensional figures that result from slicing three-­‐dimensional figures, as in plane sec-ons of right rectangular prisms and right rectangular pyramids. Example •  Using a clay model of a rectangular prism, describe the shapes that are created when planar cuts are made diagonally, perpendicularly, and parallel to the base. •  John and Joyce are sharing a piece of cake with the dimensions shown in the diagram. John is about to cut the cake at the mark indicated by the do]ed lines. Joyce says this cut will make one of the pieces three -mes as big as the other. Is she right? Jus-fy your response. •  Three ver-cal slices •  Rectangular pyramid perpendicular to the base of the right rectangular pyramid are to be made at the marked loca-ons: (1) through 𝐴𝐵, , (2) through 𝐶𝐷, , and (3) through vertex 𝐸. •  Based on the rela-ve loca-ons of the slices on the pyramid, make a reasonable sketch of each slice. Include the appropriate nota-on to indicate measures of equal length. solu-on solu-on •  VolumeTrapezoidal Prism = ,,1-­‐2.. (5+2.5)(6)(10) = 225 cm3 •  VolumeTriangular Prism = ,,1-­‐2.. (2.5)(6)(10) = 75 cm3 •  Joyce is right, the current cut would give 225 cm3 of cake for the trapezoidal prism piece, and 75 cm3 of cake for the triangular prism piece, making the larger piece, ,,22.5-­‐7.5.. = 3 -mes the size of the smaller piece. Solve real-­‐life and mathema-cal problems involving angle measure, area, surface area, and volume. 7.G.4 – •  Know the formulas for the area and circumference of a circle and solve problems; •  give an informal deriva-on of the rela-onship between the circumference and area of a circle. Note:“Know the formula” does not mean memoriza-on of the formula. To “know” means to have an understanding of why the formula works and how the formula relates to the measure (area and circumference) and the figure. This understanding should be for all students. •  The illustra-on shows the rela-onship between the circumference and area. If a circle is cut into wedges and laid out as shown, a parallelogram results. Half of an end wedge can be moved to the other end a rectangle results. The height of the rectangle is the same as the radius of the circle. •  The base length is the circumference . The area of the rectangle (and therefore the circle) is found by the following calcula-ons Problem •  The seventh grade class is building a mini golf game for the school carnival. The end of the pu}ng green will be a circle. If the circle is 10 feet in diameter, how many square feet of grass carpet will they need to buy to cover the circle? •  How might you communicate this informa-on to the salesperson to make sure you receive a piece of carpet that is the correct size? Example •  Students measure the circumference and diameter of several circular objects in the room (clock, trashcan, doorknob, wheel, etc.). •  Students organize their informa-on and discover the rela-onship between circumference and diameter by no-cing the pa]ern in the ra-o of the measures. •  Students write an expression that could be used to find the circumference of a circle with any diameter and check their expression on other circles •  The side of square 𝐷𝐸𝐹𝐺, , 𝐸𝐹=2 =2 cm, is also the radius of circle ,𝐶-­‐𝐹.. .. What is the area of the en-re shaded region? Provide all evidence of your calcula-ons. Solve real-­‐life and mathema-cal problems involving angle measure, area, surface area, and volume •  7.G.5 – Use facts about supplementary, complementary, ver-cal, and adjacent angles in a mul--­‐step problem to write and solve simple equa-ons for an unknown angle in a figure. In the triangle, what is the degree measure of angle ABC? Solve real-­‐life and mathema-cal problems involving angle measure, area, surface area, and volume 7.G.6 – Solve real-­‐world and mathema-cal problems involving area, volume and surface area of two-­‐ and three-­‐dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Problem •  Jim wants to know how much his family spends on water for showers. Water costs $1.50 for 1,000 gallons. His family averages 4 showers per day. The average length of a shower is 10 minutes. •  He places a bucket in his shower and turns on the water. Amer one minute, the bucket has 2.5 gallons of water. •  About how much money does his family spend on water for showers in a 30-­‐day month? The diagram below represents a solid of uniform cross-­‐sec-on. All the lines of the figure meet at right angles. The dimensions are marked in the drawing in terms of x Write simple formulas in terms of x for each of the following: (a)  the volume of the solid; (b) the surface area you would have to cover in order to paint this solid (c) the length of decoraDve cord you would need if you wanted to outline all the edges of this solid. Grade 8 •  Three Cri-cal Areas –  Formula-ng and reasoning about expressions and equa-ons, solving linear equa-ons, solving systems of linear equa-ons –  Concept of func-on and using func-ons to describe quan-ta-ve rela-onships –  ANALYZING 2 AND 3-­‐D SPACE AND FIGURES • 
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DISTANCE ANGLE SIMILARITY CONGRUENCE PYTHAGOREAN THEOREM Understand congruence and similarity using physical models, . transparencies, or geometry somware
8.G.1 •  Verify experimentally the proper-es of rota-ons, reflec-ons, and transla-ons: •  a. Lines are taken to lines, and line segments to line segments of the same length. •  b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. •  Aaron is drawing some designs for gree-ng cards. He divides a grid into 4 quadrants and starts by drawing a shape in one quadrant. He then reflects, rotates, or translates the shape into the other three quadrants. •  ♣ Finish Aaron’s first design by reflec-ng the gray shape over the ver-cal line. Then, reflect both of the shapes over the horizontal line. This will . make a design in all four quadrants
Understand congruence and similarity using physical models, transparencies, or geometry somware 8.G.2 •  Understand that a two-­‐dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rota-ons, reflec-ons, and transla-ons; given two congruent figures, •  describe a sequence that exhibits the congruence between them Is Figure A congruent to Figure A’? Explain how you know. Describe the sequence of transforma-ons that results in the transforma-on of Figure A to Figure A’. Understand congruence and similarity using physical models, transparencies, or geometry somware 8.G.3 – •  Describe the effect of dila-ons, transla-ons, rota-ons, and reflec-ons on two-­‐dimensional figures using coordinates. Transla-on Reflec-on: Rota-on: Understand congruence and similarity using physical models, transparencies, or geometry somware 8.G.4 Understand that a two-­‐dimensional figure is similar to another if the second can be obtained from the first by a sequence of rota-ons, reflec-ons, transla-ons, and dila-ons; given two similar two-­‐dimensional figures, describe a sequence that exhibits the similarity between them. Is Figure A similar to Figure A’? Explain how you know Describe the sequence of transforma-ons that results in the transforma-on of Figure A to Figure A’. Understand congruence and similarity using physical models, transparencies, or geometry somware •  8.G.5 – •  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-­‐angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so Understand and apply the Pythagorean Theorem. 8.G.5 •  Explain a proof of the Pythagorean Theorem and its converse. ATTEMPT 1 ATTEMPT 2 Pythagorean Theorem •  8.G.7 – •  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-­‐
world and mathema-cal problems in two and three dimensions. •  Jane is hoping to buy a large new television for her den, but she is not sure what size screen will be suitable for her wall. The is because television screens are measured by their diagonal line. The following 42-­‐inch screen measures 32 inches along the base. –  ♣ What is the height of the screen? Show how you know. –  ♣ What is the area of the screen in square inches? –  ♣ Jane would like to have a screen 40-­‐inches wide and 32-­‐inches high. What screen size, in inches, will she need to buy? Show your work •  What is the ra-o of the areas of the two squares? Show your work. •  ♣ If a second circle is inscribed inside the smaller square, what is the ra-o of the areas of the two circles? •  Explain your reasoning. Pythagorean Theorem •  8.G.8 •  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system Find the distance between the two points on the coordinate plane. Solve real-­‐world and mathema-cal problems involving volume of cylinders, cones, and spheres. •  8.G.9 •  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-­‐world and mathema-cal problems James wanted to fill his container with water. He wondered how much water he would need to fill it. Use the measurements in the diagram below to determine the container’s volume.