# Unit 5 Area. What Is Area?

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Trainer/Instructor Notes: Area What Is Area? Unit 5 Area What Is Area? Overview: Objective: Participants determine the area of a rectangle by counting the number of square units needed to cover the region. Group discussion deepens participants understanding of area (number of square units needed to cover a given region) and connects the formula for the area of a rectangle to the underlying array structure. TExES Mathematics Competencies III.013.D. The beginning teacher computes the perimeter, area, and volume of figures and shapes created by subdividing and combining other figures and shapes (e.g., arc length, area of sectors). V.018.D. The beginning teacher uses formal and informal reasoning to justify mathematical ideas. Geometry TEKS b.3.d. The student uses inductive reasoning to formulate a conjecture. e.1.a. The student finds area of regular polygons and composite figures. Background: Materials: New Terms: No prerequisite knowledge is necessary for this activity. index cards, patty paper, straightedge area Procedures: Background information: In order to accurately count the units of a given space, it is necessary to mentally organize the space in a structured manner. However, research referenced in Schifter, Bastable, and Russell (2002), on how children learn the concept of area supports the theory that the structure of the rectangular array is not intuitively obvious to children. When asked to cover a rectangular region, children progress from incomplete or unsystematic coverings to individually drawn units to the use of a row or column iteration. Gradually they will rely less on drawing and move towards multiplication or repeated addition. Covering a rectangular region with unit squares helps children understand area measure, but they must ultimately be able to formally connect area, linear measurement, and multiplication in order to truly understand the area formula A= b h. According to Schifter, Bastable, and Russell (2002), the drawing, filling, and counting that children use in this developmental process are both motor and mental actions that coordinate to organize spatial structuring. What implications does this research have for secondary teachers? Many of our students come to us with an understanding of area at the Visual Level of the van Hiele model of Geometry Module 5-1

2 Trainer/Instructor Notes: Area What Is Area? geometric development. They recognize figures by their shape and understand area as the blank space within the boundaries. Others have moved to the Descriptive Level, which implies that they can mentally see the rectangular array which overlays the shape. This activity asks participants to draw the grid, rather than giving them a pre-structured grid in order for them to experience the type of activity necessary to move a student from the Visual to the Descriptive Level. Secondary students functioning on the Relational Level are able to compare linear dimensions with grid areas and apply formulas with understanding. Teachers must be aware that although students may become proficient at rote application of formulas, they may not be functioning at the Relational Level. If they have not been given sufficient opportunity to understand the principles behind the formulas, they will have difficulty modifying a procedure to fit a particular situation as is necessary to find areas of composite figures and shaded regions. Distribute index cards to participants and ask them to write a response to the question What is area?. Indicate that they will have an opportunity to share and revise their responses at the conclusion of this activity. Then, allow time for participants, working independently or in pairs, to complete the activity using the patty paper or straight edge to determine the number of square units needed to completely cover the rectangular region. Note that the given square units are not convenient measures, such as 1 cm 2 or 1 in. 2. Consequently, participants will be less likely to simply measure the rectangle and use the area formula without having the experience of drawing the units. Whether participants mark off the units on two adjacent sides of the rectangle and multiply or actually draw in one or more rows and columns of units, they will be counting the units by considering how many rows of squares are needed to cover the region. 1. Determine the number of square units needed to cover this rectangular region. 1 square unit The rectangle measures 6 8 square units. Therefore, it will take 48 square units to cover the rectangle. Geometry Module 5-2

3 Trainer/Instructor Notes: Area What Is Area? 2. Determine the number of square units needed to cover this rectangular region. (Same rectangle, different square unit) 1 square unit The rectangle measures 9 12 square units. Therefore, it will take 108 square units to cover the rectangle. Was it necessary to draw all 108 square units to determine that it would take 108 units to cover the rectangular region? No. After drawing one row and one column of square units the total number of squares can be obtained by considering how many rows and columns of squares will be needed to cover the entire region. Has the activity caused you to reconsider your definition of area? Some participants may have responded to the question What is area? by stating that area is the amount of space covered by a particular region. It is important to make the distinction that area is the number of square units needed to completely cover a particular region. If the same figure is measured in different units, the number representing the area of the region will be different, but the area will remain constant. A more abstract definition of area, provided by Michael Serra (Serra, 2003) states that area is a function that assigns to each two-dimensional geometric shape a nonnegative real number so that (1) the area of every point is zero, (2) the areas of congruent figures are equal, and (3) if a shape is partitioned into sub regions, then the sum of the areas of those sub regions equals the area of the shape. If a figure is rotated so that a different side is considered the base, will the area formula necessarily give the same result? Yes. Surprisingly, the answer to this question is not evident to all students. According to the work of Clements and Battista referenced in Schifter, Bastable, and Russell (2002), orientation, the position of objects in space in relation to an external frame of reference, is for some children a part of their definition of a particular shape. If secondary students have not had adequate experience manipulating by rotating, flipping or sliding shapes, they may be working at the Visual Level with an inadequate understanding of shape. In developing an understanding of area, students should observe that rotations, reflections and translations preserve area while dilations do not. Geometry Module 5-3

4 Activity Page: Area What is Area? What Is Area? 1. Determine the number of square units needed to cover this rectangular region. 1 square unit Geometry Module 5-4

5 Activity Page: Area What is Area? 2. Determine the number of square units needed to cover this rectangular region. (Same rectangle, different square unit) 1 square unit Geometry Module 5-5

6 Trainer/Instructor Notes: Area Investigating Area Formulas Investigating Area Formulas Overview: Objective: Participants cut and rearrange two triangles, a parallelogram, and a kite to form rectangles with the same areas. Examination of the points at which figures must be cut will lead to a deeper understanding of the formula for the area of each figure. TExES Mathematics Competencies III.011.A. The beginning teacher applies dimensional analysis to derive units and formulas in a variety of situations (e.g., rates of change of one variable with respect to another and to find and evaluate solutions to problems. III.013.C. The beginning teacher uses geometric patterns and properties (e.g., similarity, congruence) to make generalizations about two- and three-dimensional figures and shapes (e.g., relationships of sides, angles). V.018.D. The beginning teacher uses formal and informal reasoning to justify mathematical ideas. Geometry TEKS b.2.a. The student uses constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. b.3.c. The student demonstrates what it means to prove mathematically that statements are true. d.2.c. The student develops and uses formulas including distance and midpoint. Background: Materials: Participants should know the formula for the area of a rectangle and be able to identify the base and altitude of a triangle, the base and altitude of a parallelogram, and the diagonals of a kite. transparency sheets, colored pencils, glue or tape, patty paper, scissors New Terms: Procedures: Participants sit in groups of 3-4 to work collaboratively. However, each participant should cut and glue his/her own figures. Can any parallelogram, triangle or kite be cut and rearranged to form a rectangle of the same area? Allow time for discussion within groups and then ask two or three groups to share their responses with the entire group. It is fairly obvious that any parallelogram can be cut to Geometry Module 5-6

7 Trainer/Instructor Notes: Area Investigating Area Formulas form a rectangle of equal area, but the group may or may not be able to reach a consensus on the triangle and kite. Briefly describe the activity. Each of the figures on the activity sheet has the same area as the rectangle. Using a colored pencil, trace the parallelogram on patty paper. Then, using the least number of cuts possible, cut the parallelogram and rearrange the pieces to form a rectangle of equal area. The rectangle will help you determine where to cut. Lay the patty paper tracing over the rectangle and slide it around to decide where to cut. Using a different colored pencil, draw the cut line on the patty paper figure and then cut. Assemble the pieces to form a rectangle and glue it next to the original parallelogram. Repeat the process for each of the figures. While the groups are working, assign each of the figures to a different participant to draw on a transparency for use during the group discussion. When most participants have completed the task, reconvene as a large group for discussion. Parallelogram: b h Can we express the dimensions of the rectangle in terms of the dimensions of the parallelogram? Yes. The base of the rectangle is the base of the parallelogram. The height of the rectangle is the height of the parallelogram. What does this tell us about the formula for the area of the parallelogram? Since the area of the parallelogram is equal to the area of the rectangle, the area of the parallelogram is b h. Geometry Module 5-7

8 Trainer/Instructor Notes: Area Investigating Area Formulas Obtuse Triangle: b= m h How would you describe the location of the cut lines on the obtuse triangle? The cuts must pass through the midpoints of the sides of the triangle as shown. What do we call the segment that connects the midpoints of the sides of a triangle? The midsegment What do we know about the midsegment of a triangle? The midsegment is parallel to the base and one half the length of the base of the triangle. Can we express the dimensions of the rectangle in terms of the dimensions of the triangle? Yes. The length of the base of the rectangle is equal to the length of the midsegment of the triangle, m. The height of the rectangle, h, is the height of the triangle. Can we use this information to derive the formula for the area of a triangle? Since we know Area of triangle = Area of rectangle = b h By substitution, Area of triangle = m h By the definition of a midsegment, Area of triangle = 1 (2 b) 2 h 2b = 1 (base of triangle) (height) 2 Geometry Module 5-8

9 Trainer/Instructor Notes: Area Investigating Area Formulas Acute Triangle: 2h h How would you describe the location of the cut lines on the acute triangle? One cut line goes through the midsegment of the triangle and one cut line is the altitude joining the midsegment to the opposite vertex of the triangle. b Can we use this information to derive the formula for the area of a triangle? Since we know Area of triangle = Area of rectangle = b h 1 = b ( 2 h) 2 = 1 (base of triangle) (height of triangle) 2 Geometry Module 5-9

10 Trainer/Instructor Notes: Area Investigating Area Formulas Kite: h d 2 d 1 b How can we use the formula we have derived for the area of a triangle to derive the formula for the area of the kite? d 1 lies on the line of symmetry for the kite. The two triangles formed by the line of symmetry, d 1, are congruent. 1 2 (d 2) is the length of the altitude of each triangle, and 1 ( d 2) = h 2 1 Area of one triangle = ( d 1) h = (d 1 1) 2 (d 2) Area of kite = 2( 1 )( d )( 1 1 )( d2) = 1 ( d1 d2) = 1 d 1 d 2 2 Success in this activity indicates that participants are working at the Relational Level because they must discover the relationship between the area rule for a rectangle and the area rule for a parallelogram, triangle, or kite. While a participant at the Descriptive Level will be able to cut the figures to form the rectangle of the same area, he/she will need prompting to explain why it works using informal deductive arguments. Geometry Module 5-10

11 Activity Page: Area Investigating Area Formulas Investigating Area Formulas Trace the parallelogram, triangles, and kite on patty paper. Then cut and arrange the pieces of each figure to form a rectangle congruent to the given rectangle. Glue each new figure next to the original figure from which it was made. Label the dimensions of the rectangle b and h. Determine the area for each figure in terms of b and h. Geometry Module 5-11

12 Activity Page: Area Investigating Area Formulas Geometry Module 5-12

13 Trainer/ Instructor Notes: Area Area of Trapezoids Area of Trapezoids Overview: Objective: Participants find many different ways to derive the formula for the area of a trapezoid. TExES Mathematics Competencies III.011.A. The beginning teacher applies dimensional analysis to derive units and formulas in a variety of situations (e.g., rates of change of one variable with respect to another and to find and evaluate solutions to problems. III.013.C. The beginning teacher uses geometric patterns and properties (e.g., similarity, congruence) to make generalizations about two- and three-dimensional figures and shapes (e.g., relationships of sides, angles). V.018.D. The beginning teacher uses formal and informal reasoning to justify mathematical ideas. Geometry TEKS b.2.a. The student uses constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. b.3.c. The student demonstrates what it means to prove mathematically that statements are true. d.2.c. The student develops and uses formulas including distance and midpoint. Background: Materials: Participants need to know the formulas for areas of triangles, rectangles, and parallelograms. A clear understanding of the midsegment of a trapezoid is a prerequisite. patty paper, scissors New Terms: Procedures: Distribute the activity sheet and allow about 10 minutes for participants to work on it individually so that each will have an opportunity for independent discovery. Within their group of four, allow another 15 minutes for participants to share all of the methods for deriving the formula for the area of a trapezoid that they have discovered. Then jigsaw the groups and allow about 20 minutes for the new groups to share their work. During the final 15 minutes of the activity, participants return to their original groups and share any new method they have seen during the jigsaw. Scaffolding questions to help participants who are having difficulty finding multiple solutions: Geometry Module 5-13

14 Trainer/ Instructor Notes: Area Area of Trapezoids Can you use a technique from the previous activity on areas of triangles and parallelograms? In the previous activity we saw that the midsegment could be used to cut and rearrange a triangle into a rectangle of the same area. Can a trapezoid be divided into figures whose area formulas are known? Yes, it can be divided into a rectangle and two triangles, three triangles, or four triangles. Can you create a larger figure in which the trapezoid is one of the composite parts? Yes, the trapezoid can be copied and rotated 180 o to create a parallelogram whose area is equal to twice the area of the original trapezoid. A triangle can be drawn such that a rectangle is formed from the triangle and the original trapezoid. Answers will vary. Seven derivations are given below. 1. Base of rectangle = Midsegment of trapezoid Height of rectangle = Height of trapezoid Area of trapezoid = Area of rectangle = Midsegment of trapezoid height of trapezoid m 2. Area of triangle 1 = 1 2 ah b Area of triangle 2 = 1 2 ch Area of rectangle = bh Area of trapezoid = 1 2 ah ch + h h = h 1 a 1 c b 2 2 a b c = 1 2 h (a + c + 2b) = 1 h[(a + c + b) + b] 2 = 1 2 (height) (sum of bases) Geometry Module 5-14

15 Trainer/ Instructor Notes: Area Area of Trapezoids 3. Area of trapezoid = Area of parallelogram + area of triangle = h b h (b 1 b 2 ) = h b h b h b 2 b 2 = = = 1 2 h (2b 2 + b 1 b 2 ) 1 2 h (b 1 + b 2 ) 1 (height) (sum of bases) 2 b 1 h 1 4. Area of trapezoid = 2 h a h b hc 1 = 2 h (a + b 1 + c) 1 = 2 h (b 1 + b 2 ) 1 = (height) (sum of bases) 2 a h b 1 h h c b Area of trapezoid = 2 h 1 1 b h 1 b = 2 h 1 (b 1 + b 2 ) + h 1 m = h 1 m + h 1 m = 2hm 1 2 h 1 m h 1 m b 2 h h 1 m h 1 h 1 h 1 b 1 Geometry Module 5-15

16 Trainer/ Instructor Notes: Area Area of Trapezoids 6. Area of trapezoid = Area of parallelogram = 2 m h 1 = 2m 1 2 h = mh b 2 h m m h 1 b 1 b Area of trapezoid = Area of parallelogram 2 1 = 2 h (b 1 + b 2 ) = (midsegment of trapezoid) (height of trapezoid) b 2 b 1 h b 1 b 2 Success in this activity indicates that participants are working at the Relational Level because they must show the relationships among area rules and give informal deductive arguments to justify the rule determining the area of a trapezoid. Geometry Module 5-16

17 Activity Page: Area Area of Trapezoids Area of Trapezoids How many ways can you derive the formula for the area of a trapezoid? Geometry Module 5-17

18 Trainer/ Instructor Notes: Area Area of Circles Area of Circles Overview: Objective: Participants develop the concept of area of circles by drawing a circle and then cutting it into sectors. The sectors are rearranged to form a shape that resembles a parallelogram. TExES Mathematics Competencies III.011.B. The beginning teacher applies formulas for perimeter, area, surface area, and volume of geometric figures and shapes (e.g., polygons, pyramids, prisms, cylinders, cones, spheres) to solve problems. III.013.D. The beginning teacher computes the perimeter, area, and volume of figures and shapes created by subdividing and combining other figures and shapes (e.g., arc, length, area of sectors). V.018.E. The beginning teacher understands the problem-solving process (i.e., recognizing that a mathematical problem can be solved in a variety of ways, selecting an appropriate strategy, evaluating the reasonableness of a solution). V.019.C. The beginning teacher translates mathematical ideas between verbal and symbolic forms. V.019.D. The beginning teacher communicates mathematical ideas using a variety of representations (e.g., numeric, verbal, graphical, pictorial, symbolic, concrete). Geometry TEKS b.3.c. The student demonstrates what it means to prove mathematically that statements are true. b.4. The student selects an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. d.2.c. The student develops and uses formulas including distance and midpoint. e.1.b. The student finds areas of sectors and arc lengths of circles using proportional reasoning. Background: Materials: New Terms: Participants need to understand area as a covering of a space, pi, and the circumference of a circle. cups (preferably large plastic cups), glue or tape, graphing calculator, colored markers, patty paper, scissors circumference, sector Procedures: Participants work in pairs for both parts of this activity. Geometry Module 5-18

19 Trainer/ Instructor Notes: Area Area of Circles Part 1: Developing the Formula for the Area of a Circle Using a piece of patty paper, trace the circular rim of the plastic cup. Fold the patty paper to locate the center of the circle. Fold the circle in half four additional times to create sixteenths. Firmly crease the folds. Using a marker, color half of the circle in one color and the other half of the circle in a different color. Cut out the 16 sectors of the circle. Paste the sectors in one color upside down on a sheet of paper. Paste the sectors in the other color between the sectors in an interlocking manner, forming a parallelogram shaped figure. Possible scaffolding questions to help participants develop the formula of area: How is the sum of the sectors related to the area of the circle? The sum of the areas of the sectors is equal to the area of the circle. Identify the length of the parallelogram. The length is half of the circumference of the circle. Because the circumference is C = 2π r, and the length of the parallelogram is half of the circle, the length is π r. Label the lengthπ r. Identify the width of the parallelogram. The width is the same as the radius of the circle. Label it r. Geometry Module 5-19

20 Trainer/ Instructor Notes: Area Area of Circles What is the area of the parallelogram-shaped figure? 2 The area is ( π r)( r) = π r. Is the area of a circle a function of the radius? Yes; the area depends on the length of the radius. Part 2: Applying the Formula for the Area of a Circle 1. The circumference of a circle and the perimeter of a square are each 20 in. Which has the greater area, the circle or the square? How much greater? C=2πr P=20in 20 = 2πr P=4s 10 r 20 = 4s π 2 5=s A=πr A=5 =25in 10 A=π π 2 A 31.8 in The circle has the greater area. The approximate difference is in 25in = 6.8 in. 2. A rice farmer uses a rotating irrigation system in which 8 sections of 165-foot pipe are connected end to end. The resulting length of sprinkler pipe travels in a circle around a well, which is located in the center of the circle. a. How many acres of the square field are irrigated? (one acre is 43,560 ft 2 ) There are 8 sections of 165 feet pipe, or 1320 feet of pipe, which is the radius of the 2 irrigated field. The area is π 1320 or π ft 2 or ft 2. To find the number of acres, divide the total square feet by the 43,560 ft 2 in one acre acres b. Broccoli is often planted in the corners of the field, which are not irrigated. How many acres of broccoli are planted in this field? Area of square Area of circle = Area of corners Geometry Module 5-20

21 Trainer/ Instructor Notes: Area Area of Circles The diameter of the circle is the length of the side of the square. The diameter is 2640 ft. The area of the square is or is ft 2. Area of square Area of circle = Area of corners Using Substitution, = ft 2 Changing the area to acres, acres 3. Ron opened a new restaurant. He asked his wait staff to determine the greatest number of cylindrical glasses that would fit on a 24 in. by 12 in. tray. Each glass has a radius of 1.5 inches. Being a math geek, he began to wonder how much area was not being used on the tray. Make a drawing of the arrangement that fits the most glasses on the tray. Make a prediction of how many glasses will fit on the tray. What is the area on the tray not used by the cylindrical glasses? Area of glass = π ( 1.5) 7.1 in 2 2 Area of 32 glasses = 32( 7.1) in Area of tray = 24( 12) = 288 in Area of difference = 288 in in = 60.8 in When Ruth s family moved during her sophomore year of high school, they stayed in a motel for 3 weeks. Their dog, Frosty, was tied at the corner of the motel on a 50 foot rope. If the motel were a 40 ft by 30 ft rectangle, how many square feet could Frosty wander? (Give answer in terms of π.) The dog could run in a three quarters circle having a radius of 50 feet. At both ends of the building his path forms a quarter circle, one with a radius of 20 feet, and the other with a radius of 10 feet. The dog can wander 2000π square feet. 30 feet 10 feet 40 feet Motel 20 feet 5. Circle Pizzeria is changing the size of its circular pizza from 12 inches to 16 inches and increasing the number of slices per pizza from 8 to 10. What is the percent increase of the size of each new slice? (Round answer to the nearest tenth of a percent.) Geometry Module 5-21

22 Trainer/ Instructor Notes: Area Area of Circles r = 6 r = 8 12 inch pizza Area of 12 inch pizza = π 6 2 = 36 π 16 inch pizza Area of 16 inch pizza = π 8 2 = 64 π 36π 8slices 2 64π = 4.5π in. per slice = 6.4π in. 2 per slice 10 slices 6.4 π 4.5 π = 1.9 π difference 1.9π π % increase 6. Circles of radius 4 with centers at (4,0) and (0,4) overlap in the shaded region shown in the figure. Find the area of the shaded region in terms of π. One way to find the area of the shaded region is to draw the diagonal of the square. 1 Area of shaded region = Area of quarter circle Area of triangle 2 Area of sector: πr bh π (4π 8) (0,4) (4,0) 8π 16 units 2 Geometry Module 5-22

23 Trainer/ Instructor Notes: Area Area of Circles 7. In the figure shown, all arcs are semicircles, and those that appear to be congruent are. What is the area of the shaded region? (Give answer in terms of π.) 1 2 2(Area midsized semi-circle) = 2 π 2 = 4π 2 4(Area small semi-circle) = π 1 2π 2 = Area midsized semi-circles. Area small semi-circles = 4π - 2π = 2π Remind participants to add the new terms circumference and sector to the glossaries. Part 1: Success in this part of the activity indicates that participants are working at the Relational Level because they must discover the relationship between the formula for the area of a parallelogram and the formula for the area of a circle. Part 2: Success in this part of the activity indicates that participants are working at the Relational Level because they must solve geometric problems by using known properties of figures and insightful approaches. Geometry Module 5-23

24 Activity Page: Area Area of Circles Area of Circles Solve the following problems. 1. The circumference of a circle and the perimeter of a square are each 20 in. Which has the greater area, the circle or the square? How much greater? 2. A rice farmer uses a rotating irrigation system in which 8 sections of 165 foot pipe are connected end to end. The resulting length of sprinkler pipe travels in a circle around a well, which is located in the center of the circle. a. How many acres of the square field are irrigated? (One acre is 43,560 ft 2.) b. Broccoli is often planted in the corners of the field, which are not irrigated. How many acres of broccoli are planted in this field? Geometry Module 5-24

25 Activity Page: Area Area of Circles 3. Ron opened a new restaurant. He asked the wait staff how many cylindrical glasses would fit on a 24 in. by 12 in. tray. Each glass has a radius of 1.5 inches. Being a math geek, he began to wonder how much area was not being used on the tray. Make a drawing of the arrangement that fits the most glasses on the tray. Make a prediction of how many glasses will fit on the tray. What is the area on the tray not used by the cylindrical glasses? 4. When Ruth s family moved during her sophomore year of high school, they stayed in a motel for 3 weeks. Their dog, Frosty, was tied at the corner of the motel on a 50 foot rope. If the motel was a 40 ft by 30 ft rectangle, how many square feet could Frosty wander? (Give answer in terms of π.) 40 feet 30 feet Motel 50 feet 5. Circle Pizzeria is changing the size of its circular pizza from 12 inches to 16 inches and increasing the number of slices per pizza from 8 to 10. What is the percent of increase of the size of each new slice? (Round answer to nearest tenth of a percent.) Geometry Module 5-25

26 Activity Page: Area Area of Circles 6. Circles of radius 4 with centers at (4,0) and (0,4) overlap in the shaded region shown in the figure. Find the area of the shaded region in terms of π. (0,4) (4,0) 7. In the figure shown, all arcs are semicircles, and those that appear to be congruent are. What is the area of the shaded region? (Give answer in terms of π.) Geometry Module 5-26

27 Trainer/Instructor Notes: Area Applying Area Formulas Applying Area Formulas Overview: Objective: Participants use the problem solving process to find the area of composite figures (composite of triangles, quadrilaterals and circles). TExES Mathematics Competencies III.011.B. The beginning teacher applies formulas for perimeter, area, surface area, and volume of geometric figures and shapes (e.g., polygons, pyramids, prisms, cylinders, cones, spheres) to solve problems). III.013.D. The beginning teacher computes the perimeter, area, and volume of figures and shapes created by subdividing and combining other figures and shapes (e.g., arc length, area of sectors). V.018.E. The beginning teacher understands the problem-solving process (i.e., recognizing that a mathematical problem can be solved in a variety of ways, selecting an appropriate strategy, evaluating the reasonableness of a solution). V.019.C. The beginning teacher translates mathematical ideas between verbal and symbolic forms. V.019.D. The beginning teacher communicates mathematical ideas using a variety of representations (e.g., numerical, verbal, graphical, pictorial, symbolic, concrete). Geometry TEKS b.4. The student selects an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. d.2.c. The student develops and uses formulas including distance and midpoint. e.1.a. The student finds the area of regular polygons, and composite figures. Background: Materials: New Terms: Participants need to know the area formulas and how to connect the formulas to models of composite figures (composite of triangles, quadrilaterals and circles). calculator composite figures Procedures: Write the word composite on the overhead and brainstorm some meanings of this word. The Merriam-Webster Collegiate Dictionary, from defines composite, when used as an adjective, as consisting of separate interconnected parts. Remind the participants to add the new term to their glossaries. Geometry Module 5-27

### Unit 6 Pythagoras. Sides of Squares

Sides of Squares Unit 6 Pythagoras Sides of Squares Overview: Objective: Participants discover the Pythagorean Theorem inductively by finding the areas of squares. TExES Mathematics Competencies II.004.A.

### Middle Grades Mathematics 5 9

Middle Grades Mathematics 5 9 Section 25 1 Knowledge of mathematics through problem solving 1. Identify appropriate mathematical problems from real-world situations. 2. Apply problem-solving strategies

### Perfume Packaging. Ch 5 1. Chapter 5: Solids and Nets. Chapter 5: Solids and Nets 279. The Charles A. Dana Center. Geometry Assessments Through

Perfume Packaging Gina would like to package her newest fragrance, Persuasive, in an eyecatching yet cost-efficient box. The Persuasive perfume bottle is in the shape of a regular hexagonal prism 10 centimeters

### *1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles.

Students: 1. Students understand and compute volumes and areas of simple objects. *1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles. Review

### Chapter 1: Essentials of Geometry

Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,

### 12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### NEW MEXICO Grade 6 MATHEMATICS STANDARDS

PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical

### Area of Parallelograms, Triangles, and Trapezoids (pages 314 318)

Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base

### Geometry Course Summary Department: Math. Semester 1

Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give

### Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

### Geometry Vocabulary Booklet

Geometry Vocabulary Booklet Geometry Vocabulary Word Everyday Expression Example Acute An angle less than 90 degrees. Adjacent Lying next to each other. Array Numbers, letter or shapes arranged in a rectangular

COURSE OVERVIEW The geometry course is centered on the beliefs that The ability to construct a valid argument is the basis of logical communication, in both mathematics and the real-world. There is a need

### Overview Mathematical Practices Congruence

Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason

### GEOMETRY CONCEPT MAP. Suggested Sequence:

CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### Honors Geometry Final Exam Study Guide

2011-2012 Honors Geometry Final Exam Study Guide Multiple Choice Identify the choice that best completes the statement or answers the question. 1. In each pair of triangles, parts are congruent as marked.

### 17.1 Cross Sections and Solids of Rotation

Name Class Date 17.1 Cross Sections and Solids of Rotation Essential Question: What tools can you use to visualize solid figures accurately? Explore G.10.A Identify the shapes of two-dimensional cross-sections

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### MATHEMATICS Grade 6 Standard: Number, Number Sense and Operations

Standard: Number, Number Sense and Operations Number and Number C. Develop meaning for percents including percents greater than 1. Describe what it means to find a specific percent of a number, Systems

E XPLORING QUADRILATERALS E 1 Geometry State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Statement of Purpose: The activities in this

### Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

### Geometry. Higher Mathematics Courses 69. Geometry

The fundamental purpose of the course is to formalize and extend students geometric experiences from the middle grades. This course includes standards from the conceptual categories of and Statistics and

### Week 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test

Thinkwell s Homeschool Geometry Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Geometry! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan

### Unit 2 - Triangles. Equilateral Triangles

Equilateral Triangles Unit 2 - Triangles Equilateral Triangles Overview: Objective: In this activity participants discover properties of equilateral triangles using properties of symmetry. TExES Mathematics

### Grade 3 Math Expressions Vocabulary Words

Grade 3 Math Expressions Vocabulary Words Unit 1, Book 1 Place Value and Multi-Digit Addition and Subtraction OSPI words not used in this unit: add, addition, number, more than, subtract, subtraction,

### Archdiocese of Washington Catholic Schools Academic Standards Mathematics

5 th GRADE Archdiocese of Washington Catholic Schools Standard 1 - Number Sense Students compute with whole numbers*, decimals, and fractions and understand the relationship among decimals, fractions,

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### Scope & Sequence MIDDLE SCHOOL

Math in Focus is a registered trademark of Times Publishing Limited. Houghton Mifflin Harcourt Publishing Company. All rights reserved. Printed in the U.S.A. 06/13 MS77941n Scope & Sequence MIDDLE SCHOOL

### Area of Parallelograms (pages 546 549)

A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular

### Geometry Enduring Understandings Students will understand 1. that all circles are similar.

High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### Conjectures. Chapter 2. Chapter 3

Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

### II. Geometry and Measurement

II. Geometry and Measurement The Praxis II Middle School Content Examination emphasizes your ability to apply mathematical procedures and algorithms to solve a variety of problems that span multiple mathematics

### Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees

Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees Apex in a pyramid or cone, the vertex opposite the base; in

### Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms.

Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game

### Discovering Math: Exploring Geometry Teacher s Guide

Teacher s Guide Grade Level: 6 8 Curriculum Focus: Mathematics Lesson Duration: Three class periods Program Description Discovering Math: Exploring Geometry From methods of geometric construction and threedimensional

### Surface Area of Rectangular & Right Prisms Surface Area of Pyramids. Geometry

Surface Area of Rectangular & Right Prisms Surface Area of Pyramids Geometry Finding the surface area of a prism A prism is a rectangular solid with two congruent faces, called bases, that lie in parallel

### FCAT Math Vocabulary

FCAT Math Vocabulary The terms defined in this glossary pertain to the Sunshine State Standards in mathematics for grades 3 5 and the content assessed on FCAT in mathematics. acute angle an angle that

### Name: Date: Geometry Honors Solid Geometry. Name: Teacher: Pd:

Name: Date: Geometry Honors 2013-2014 Solid Geometry Name: Teacher: Pd: Table of Contents DAY 1: SWBAT: Calculate the Volume of Prisms and Cylinders Pgs: 1-6 HW: Pgs: 7-10 DAY 2: SWBAT: Calculate the Volume

### Such As Statements, Kindergarten Grade 8

Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential

### Su.a Supported: Identify Determine if polygons. polygons with all sides have all sides and. and angles equal angles equal (regular)

MA.912.G.2 Geometry: Standard 2: Polygons - Students identify and describe polygons (triangles, quadrilaterals, pentagons, hexagons, etc.), using terms such as regular, convex, and concave. They find measures

### Geometry Essential Curriculum

Geometry Essential Curriculum Unit I: Fundamental Concepts and Patterns in Geometry Goal: The student will demonstrate the ability to use the fundamental concepts of geometry including the definitions

### Chapter 4: Area, Perimeter, and Volume. Geometry Assessments

Chapter 4: Area, Perimeter, and Volume Geometry Assessments Area, Perimeter, and Volume Introduction The performance tasks in this chapter focus on applying the properties of triangles and polygons to

### Teacher Page Key. Geometry / Day # 13 Composite Figures 45 Min.

Teacher Page Key Geometry / Day # 13 Composite Figures 45 Min. 9-1.G.1. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles

### CARMEL CLAY SCHOOLS MATHEMATICS CURRICULUM

GRADE 4 Standard 1 Number Sense Students understand the place value of whole numbers and decimals to two decimal places and how whole numbers 1 and decimals relate to simple fractions. 4.1.1 Read and write

### Integrated Algebra: Geometry

Integrated Algebra: Geometry Topics of Study: o Perimeter and Circumference o Area Shaded Area Composite Area o Volume o Surface Area o Relative Error Links to Useful Websites & Videos: o Perimeter and

### CONJECTURES - Discovering Geometry. Chapter 2

CONJECTURES - Discovering Geometry Chapter C-1 Linear Pair Conjecture - If two angles form a linear pair, then the measures of the angles add up to 180. C- Vertical Angles Conjecture - If two angles are

### SURFACE AREA AND VOLUME

SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has

### Begin recognition in EYFS Age related expectation at Y1 (secure use of language)

For more information - http://www.mathsisfun.com/geometry Begin recognition in EYFS Age related expectation at Y1 (secure use of language) shape, flat, curved, straight, round, hollow, solid, vertexvertices

### Angles that are between parallel lines, but on opposite sides of a transversal.

GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

### 12 Surface Area and Volume

12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids

### Coordinate Coplanar Distance Formula Midpoint Formula

G.(2) Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the oneand two-dimensional coordinate systems to

### Surface Area and Volume

1 Area Surface Area and Volume 8 th Grade 10 days by Jackie Gerwitz-Dunn and Linda Kelly 2 What do you want the students to understand at the end of this lesson? The students should be able to distinguish

### Line Segments, Rays, and Lines

HOME LINK Line Segments, Rays, and Lines Family Note Help your child match each name below with the correct drawing of a line, ray, or line segment. Then observe as your child uses a straightedge to draw

### BASIC GEOMETRY GLOSSARY

BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that

### 3 rd 5 th Grade Math Core Curriculum Anna McDonald School

3 rd 5 th Grade Math Core Curriculum Anna McDonald School Our core math curriculum is only as strong and reliable as its implementation. Ensuring the goals of our curriculum are met in each classroom,

### New York State Student Learning Objective: Regents Geometry

New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students

### Geometry. Geometry is the study of shapes and sizes. The next few pages will review some basic geometry facts. Enjoy the short lesson on geometry.

Geometry Introduction: We live in a world of shapes and figures. Objects around us have length, width and height. They also occupy space. On the job, many times people make decision about what they know

### Chapter 8 Geometry We will discuss following concepts in this chapter.

Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles

### Conjectures for Geometry for Math 70 By I. L. Tse

Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:

### Unit 6 Grade 7 Geometry

Unit 6 Grade 7 Geometry Lesson Outline BIG PICTURE Students will: investigate geometric properties of triangles, quadrilaterals, and prisms; develop an understanding of similarity and congruence. Day Lesson

### 1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 mile (mi) = 5280 feet (ft) Replace 1 with 1 ft/12 in. 1ft

2 MODULE 6. GEOMETRY AND UNIT CONVERSION 6a Applications The most common units of length in the American system are inch, foot, yard, and mile. Converting from one unit of length to another is a requisite

### 3. If AC = 12, CD = 9 and BE = 3, find the area of trapezoid BCDE. (Mathcounts Handbooks)

EXERCISES: Triangles 1 1. The perimeter of an equilateral triangle is units. How many units are in the length 27 of one side? (Mathcounts Competitions) 2. In the figure shown, AC = 4, CE = 5, DE = 3, and

### A Different Look at Trapezoid Area Prerequisite Knowledge

Prerequisite Knowledge Conditional statement an if-then statement (If A, then B) Converse the two parts of the conditional statement are reversed (If B, then A) Parallel lines are lines in the same plane

### Geometry Unit 6 Areas and Perimeters

Geometry Unit 6 Areas and Perimeters Name Lesson 8.1: Areas of Rectangle (and Square) and Parallelograms How do we measure areas? Area is measured in square units. The type of the square unit you choose

### Geometry Honors: Extending 2 Dimensions into 3 Dimensions. Unit Overview. Student Focus. Semester 2, Unit 5: Activity 30. Resources: Online Resources:

Geometry Honors: Extending 2 Dimensions into 3 Dimensions Semester 2, Unit 5: Activity 30 Resources: SpringBoard- Geometry Online Resources: Geometry Springboard Text Unit Overview In this unit students

### Session 9 Solids. congruent regular polygon vertex. cross section edge face net Platonic solid polyhedron

Key Terms for This Session Session 9 Solids Previously Introduced congruent regular polygon vertex New in This Session cross section edge face net Platonic solid polyhedron Introduction In this session,

### Perimeter and area formulas for common geometric figures:

Lesson 10.1 10.: Perimeter and Area of Common Geometric Figures Focused Learning Target: I will be able to Solve problems involving perimeter and area of common geometric figures. Compute areas of rectangles,

### Surface Area and Volume

UNIT 7 Surface Area and Volume Managers of companies that produce food products must decide how to package their goods, which is not as simple as you might think. Many factors play into the decision of

### Shape Dictionary YR to Y6

Shape Dictionary YR to Y6 Guidance Notes The terms in this dictionary are taken from the booklet Mathematical Vocabulary produced by the National Numeracy Strategy. Children need to understand and use

### Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade

Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade Standards/Content Padrões / Conteúdo Learning Objectives Objetivos de Aprendizado Vocabulary Vocabulário Assessments Avaliações Resources

### Wallingford Public Schools - HIGH SCHOOL COURSE OUTLINE

Wallingford Public Schools - HIGH SCHOOL COURSE OUTLINE Course Title: Geometry Course Number: A 1223, G1224 Department: Mathematics Grade(s): 10-11 Level(s): Academic and General Objectives that have an

### Grade 9 Mathematics Unit 3: Shape and Space Sub Unit #1: Surface Area. Determine the area of various shapes Circumference

1 P a g e Grade 9 Mathematics Unit 3: Shape and Space Sub Unit #1: Surface Area Lesson Topic I Can 1 Area, Perimeter, and Determine the area of various shapes Circumference Determine the perimeter of various

### Shape and Space. General Curriculum Outcome E: Students will demonstrate spatial sense and apply geometric concepts, properties and relationships.

Shape and Space General Curriculum Outcome E: Students will demonstrate spatial sense and apply geometric concepts, properties and relationships. Elaboration Instructional Strategies/Suggestions KSCO:

### 2 feet Opposite sides of a rectangle are equal. All sides of a square are equal. 2 X 3 = 6 meters = 18 meters

GEOMETRY Vocabulary 1. Adjacent: Next to each other. Side by side. 2. Angle: A figure formed by two straight line sides that have a common end point. A. Acute angle: Angle that is less than 90 degree.

### Name: Date: Geometry Solid Geometry. Name: Teacher: Pd:

Name: Date: Geometry 2012-2013 Solid Geometry Name: Teacher: Pd: Table of Contents DAY 1: SWBAT: Calculate the Volume of Prisms and Cylinders Pgs: 1-7 HW: Pgs: 8-10 DAY 2: SWBAT: Calculate the Volume of

### Areas of Rectangles and Parallelograms

CONDENSED LESSON 8.1 Areas of Rectangles and Parallelograms In this lesson you will Review the formula for the area of a rectangle Use the area formula for rectangles to find areas of other shapes Discover

### Everyday Mathematics CCSS EDITION CCSS EDITION. Content Strand: Number and Numeration

CCSS EDITION Overview of -6 Grade-Level Goals CCSS EDITION Content Strand: Number and Numeration Program Goal: Understand the Meanings, Uses, and Representations of Numbers Content Thread: Rote Counting

### Overview. Essential Questions. Grade 8 Mathematics, Quarter 4, Unit 4.3 Finding Volume of Cones, Cylinders, and Spheres

Cylinders, and Spheres Number of instruction days: 6 8 Overview Content to Be Learned Evaluate the cube root of small perfect cubes. Simplify problems using the formulas for the volumes of cones, cylinders,

### Mathematics Scope and Sequence, K-8

Standard 1: Number and Operation Goal 1.1: Understands and uses numbers (number sense) Mathematics Scope and Sequence, K-8 Grade Counting Read, Write, Order, Compare Place Value Money Number Theory K Count

### Illinois State Standards Alignments Grades Three through Eleven

Illinois State Standards Alignments Grades Three through Eleven Trademark of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other

### Open-Ended Problem-Solving Projections

MATHEMATICS Open-Ended Problem-Solving Projections Organized by TEKS Categories TEKSING TOWARD STAAR 2014 GRADE 7 PROJECTION MASTERS for PROBLEM-SOLVING OVERVIEW The Projection Masters for Problem-Solving

### Activity Set 3. Trainer Guide

Geometry and Measurement of Plane Figures Activity Set 3 Trainer Guide GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development

### Implementing the 6 th Grade GPS via Folding Geometric Shapes. Presented by Judy O Neal

Implementing the 6 th Grade GPS via Folding Geometric Shapes Presented by Judy O Neal (joneal@ngcsu.edu) Topics Addressed Nets Prisms Pyramids Cylinders Cones Surface Area of Cylinders Nets A net is a

### Glencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 3-3, 5-8 8-4, 8-7 1-6, 4-9

Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 6-8 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,

### Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, June 19, :15 a.m. to 12:15 p.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 19, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### 10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres. 10.4 Day 1 Warm-up

10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres 10.4 Day 1 Warm-up 1. Which identifies the figure? A rectangular pyramid B rectangular prism C cube D square pyramid 3. A polyhedron

### Grade 4 - Module 4: Angle Measure and Plane Figures

Grade 4 - Module 4: Angle Measure and Plane Figures Acute angle (angle with a measure of less than 90 degrees) Angle (union of two different rays sharing a common vertex) Complementary angles (two angles

### Grade Level Expectations for the Sunshine State Standards

for the Sunshine State Standards Mathematics Grades 6-8 FLORIDA DEPARTMENT OF EDUCATION http://www.myfloridaeducation.com/ Strand A: Number Sense, Concepts, and Operations Standard 1: The student understands

### Overview of PreK-6 Grade-Level Goals. Program Goal: Understand the Meanings, Uses, and Representations of Numbers Content Thread: Rote Counting

Overview of -6 Grade-Level Goals Content Strand: Number and Numeration Program Goal: Understand the Meanings, Uses, and Representations of Numbers Content Thread: Rote Counting Goal 1: Verbally count in

### PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES NCERT

UNIT 12 PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES (A) Main Concepts and Results Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines,

### Everyday Mathematics GOALS

Copyright Wright Group/McGraw-Hill GOALS The following tables list the Grade-Level Goals organized by Content Strand and Program Goal. Content Strand: NUMBER AND NUMERATION Program Goal: Understand the

### PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

### TargetStrategies Aligned Mathematics Strategies Arkansas Student Learning Expectations Geometry

TargetStrategies Aligned Mathematics Strategies Arkansas Student Learning Expectations Geometry ASLE Expectation: Focus Objective: Level: Strand: AR04MGE040408 R.4.G.8 Analyze characteristics and properties

### Solids. Objective A: Volume of a Solids

Solids Math00 Objective A: Volume of a Solids Geometric solids are figures in space. Five common geometric solids are the rectangular solid, the sphere, the cylinder, the cone and the pyramid. A rectangular