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TEKS 4.5(C), 4.5(D) – Readiness

Unit 25 Standards

(Student pages 199–206)

Reporting Category

3 Geometry and Measurement The student will demonstrate an understanding of how to represent and apply geometry and measurement concepts.

Domain TEKS

Student Expectation

Algebraic Reasoning 4.5 The student applies mathematical process standards to develop concepts of expressions and equations. 4.5(C) This student expectation is not identified as readiness or supporting. Use models to determine the formulas for the perimeter of a rectangle (l + w + l + w or 2l + 2w), including the special form for perimeter of a square (4s) and the area of a rectangle (l x w). 4.5(D) – Readiness Standard Solve problems related to perimeter and area of rectangles where dimensions are whole numbers.

Mathematical Process TEKS Addressed in This Unit The student uses mathematical processes to acquire and demonstrate mathematical understanding. 4.1(A) 4.1(B)

4.1(C)

4.1(F) 4.1(G)

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Apply mathematics to problems arising in everyday life, society, and the workplace. Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. Analyze mathematical relationships to connect and communicate mathematical ideas. Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

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Unit 25

Use Formulas and Models to Solve Problems with Perimeter and Area

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teacher sample Use Formulas and Models to Solve Problems with Perimeter and Area

TEKS 4.5(C), 4.5(D) – Readiness

Unpacking the Standards In grade 3, students found the perimeters of polygons or found a missing side length when given the perimeter and the remaining side lengths. While students did not formally work with formulas to calculate perimeter or area, they used multiplication of whole numbers related to the number of rows times the number of square units in each row to find the areas of rectangles and reported areas in square units. They also used an additive model for area as they decomposed figures formed by up to three non-overlapping rectangles and understood that the total area of the composite figure could be calculated by finding the sum of the areas of the smaller rectangles. In grade 4, students investigate formulas for perimeter and area using models and tools (e.g., geoboards, tiles, grid paper) and record the findings in whole number measures. Students reason about the relationship between side lengths of rectangles and their perimeters and areas. They will use P = l + w + l + w or P = 2l + 2w for perimeter. Students will also use the formula P = 4s for the perimeter of a square and A = l x w to find the area of a rectangle. Students show their understandings of finding perimeter and area by applying derived formulas to real-world problems. Problems may require students to find a missing side length before computing area, or use a given area and the measure of length or width to determine the missing measure of a side in order to compute perimeter. All calculations to find area and perimeter in grade 4 are limited to whole numbers.

Getting Started Introduction Activity The teacher reads the book Spaghetti and Meatballs for All by Marilyn Burns. Students or student pairs use Color Tiles® to represent tables and paper clips to represent chairs. As the story is read, students use the tiles and clips to model the described seating arrangements, recording the arrangements as sketches in math journals or on centimeter grid paper. Next, the teacher leads students to informally define area as the number of tables and perimeter as the number of chairs placed around the tables. The teacher may opt to introduce these ideas as questions, as students may remember the concepts from third grade. As an extension, students brainstorm methods of finding the areas and perimeters of tables without counting all the tiles or paper clips. As a class, students work together to develop the formulas for area and perimeter: A = l x w, P = 2 x (l + w), P = (2 x l) + (2 x w), or P = 2l = 2w. (DOK: 3, Bloom’s/RBT: Application/Apply, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I, (c)3.D, (c)3.E, (c)5.B)

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Unit 25

motivationmath

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TEKS 4.5(C), 4.5(D) – Readiness

Suggested Formative Assessment The teacher asks probing questions to determine students’ understanding of area and perimeter. • When Mrs. Comfort’s relatives push the tables together, what happens to the total number of people who can eat at the tables? Why? • Is it possible for two rectangles to have the same area but different perimeters? Justify your answer. • In the story, what table arrangement seated the greatest number of people? Why? • How can you prove that the formula A = l x w will help calculate the area of a table? • What is a general rule you can use for finding the perimeter of a rectangle? How can you express this rule using numbers and symbols, in which l represents length and w represents width? • Suppose you know that the width of a table is 7 units and the area is 42 square units. How can you find the length? • Suppose a table is square and its area is 25 square units. How can you find its length and width? (DOK: 2, Bloom’s/RBT: Analysis/Analyze, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.C, (c)2.E, (c)2.I, (c)3.D, (c)3.E, (c)3.F, (c)3.H)

Children’s Literature Connections Chickens on the Move – Pam Pollack and Meg Belviso Perimeter and Area at the Amusement Park – Dianne Irving Perimeter, Area, and Volume: A Monster Book of Dimensions – David A. Adler Racing Around: Perimeter – Stuart J. Murphy Sam’s Sneaker Squares – Nat Gabriel Sir Cumference and the Isle of Immeter – Cindy Neuschwander Spaghetti and Meatballs for All!: A Mathematical Story – Marilyn Burns

Vocabulary Focus The following are essential vocabulary terms for this unit. area (A)

length (l)

rectangle

square unit

dimension

model

side

width (w)

formula

perimeter (P)

square

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Unit 25

Use Formulas and Models to Solve Problems with Perimeter and Area

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teacher sample Use Formulas and Models to Solve Problems with Perimeter and Area

TEKS 4.5(C), 4.5(D) – Readiness

Vocabulary Activity Word Web The teacher provides a topic for the middle space of a word web. Students complete the webs connecting vocabulary words to everyday activities. An example follows. installing baseboards in a room

placing a border around a bulletin board

completing a home run in baseball

Activities that Involve Perimeter installing a fence around a garden

using a weedeater around the edges of a yard

decorating the edges of a cake with frosting

Other topics for word webs might include: • Activities that Involve Area • Activities in Which Dimensions Must be Measured • Objects Measured in Square Units (DOK: 3, Bloom’s/RBT: Analysis/Analyze, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I, (c)5.B, (c)5.G)

Suggested Formative Vocabulary Assessment Students write short stories about situations in which they would need to determine the area or perimeter of an object or space. Students use a minimum of four vocabulary words in the stories. The teacher gathers and evaluates evidence of understanding demonstrated by student stories and plans additional vocabulary activities as needed. (DOK: 2, Bloom’s/RBT: Application/Apply, ELPS: (c)1.E, (c)1.H, (c)5.B, (c)5.G)

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Unit 25

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TEKS 4.5(C), 4.5(D) – Readiness

Suggested Instructional Activities 1. Students use painter’s tape or other colored tape to outline rectangles and squares found on tile floors or walls of the classroom or hallway. The teacher guides students as they identify the lengths and widths of the rectangles in units. Then, the teacher identifies the perimeter of an outlined rectangle and students work together to discover the formula for finding the perimeter of a rectangle. Next, the teacher identifies the area of the same outlined rectangle. The teacher and students work together, as a class, to discover the formula used to calculate the area of a rectangle. The teacher records the formulas on the board and students use them to find the areas and perimeters of the other outlined rectangles and squares. (DOK: 2, Bloom’s/RBT: Analysis/Analyze, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I, (c)3.D, (c)3.E, (c)4.C)

2. The teacher provides students with geoboards and rubber bands. Students enclose the smallest square possible. The teacher defines the distance from one peg to the next (vertically or horizontally) as 1 unit and counts the rim of the rectangle to find a perimeter of 4 units. The teacher further defines the amount of space enclosed by the rubber band as one square unit, identifying this as area. Then, the teacher repeats the process with a larger rectangle such as the one shown in the example. The teacher asks probing questions, leading students to understand the formulas for finding perimeter. Questions might include the following.

Example

• What is the number of units on one long side of the rectangle? • What is the number of units on one short side of the rectangle? • How many long sides are on the rectangle? The longer sides are called length. Do both lengths have the same measure? • How many short sides are on the rectangle? • The shorter sides are called width. Do both widths have the same measure? • Using what you know about perimeter, what is one way to find the perimeter of this rectangle using an addition equation? • Since we added two 4s and two 3s, what is another way to write this equation? • If 2 represents length, w represents width, and P represents the perimeter, what formula could be used to find the perimeter of a rectangle? • From our knowledge of the Distributive Property of Multiplication, how else can we write this formula? Next, students count squares to determine the area of the rectangle. Students count the number of squares in each row and the number of squares in one column. In the example above, students count four square units horizontally and three square units vertically. The teacher again uses probing questions to help students generalize the formula for finding the area of a rectangle. Questions might include the following. • Look at the rectangle enclosed by the rubber band. What is another mathematical term for a rectangular figure made of rows and columns? • How many square units are in the top row of the array? How many square units are in the second row? How many square units are in the bottom row? • Do all rows contain the same number of square units? • What addition and multiplication equations could be used to find the total number of square units? ©2014 mentoringminds.com

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Unit 25

Use Formulas and Models to Solve Problems with Perimeter and Area

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TEKS 4.5(C), 4.5(D) – Readiness

• If l represents the length, w represents the width, and A represents the total area, what formula could be used to find the area of a rectangle? Students apply the generalizations to other rectangles on the geoboard. (DOK: 2, Bloom’s/RBT: Analysis/Analyze, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I, (c)3.D, (c)3.E, (c)3.F, (c)3.H, (c)4.C)

3. Given a perimeter, area, and one dimension (e.g., perimeter = 30 units, area = 50 square units, and length = 10 units), small groups work to find and record a rectangle that meets all given specifications. Students use the information to show how the formulas for area and perimeter can be used in reverse to determine the missing dimensions (e.g., A = l x w, so A ÷ l = w). Students then write real-world scenarios that could be applied to the given parameters. Groups exchange and solve problems, using the appropriate formulas. (DOK: 2, Bloom’s/RBT: Application/Apply, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I, (c)3.D, (c)3.E, (c)3.G, (c)3.H, (c)4.F, (c)4.J, (c)5.B, (c)5.G)

Suggested Formative Assessment The teacher uses agreement circles to assess student understanding of applying the area and perimeter formulas for rectangles. The teacher develops three or four statements about this concept. Students form a circle in the classroom. The teacher reads the statements, allows think time, and students then move toward the center of the circle to show agreement with the statement or remain on the circumference of the circle to show disagreement. The teacher then groups students into small groups and provides time for students to discuss and solidify understandings. The teacher uses the results from this assessment to plan additional instruction and/or provide interventions. (DOK: 2, Bloom’s/RBT: Analysis/Analyze, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.C, (c)2.E, (c)2.I, (c)3.D, (c)3.E, (c)3.G)

Suggested Reflection/Closure Activity Students draw rectangles and squares on centimeter grid paper. The teacher displays a four-column table on the board with the headings Length, Width, Perimeter, and Area. Student volunteers describe the rectangles by length, width, perimeter, and area. The teacher records students’ measurements on the table. As a class, discuss patterns between the columns. (DOK: 2, Bloom’s/RBT: Application/Apply, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I, (c)3.D)

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Unit 25

motivationmath

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TEKS 4.5(C), 4.5(D) – Readiness

Suggested Formative Assessment Students fold sheets of centimeter grid paper in half and then in half again to create four columns. Students draw and shade a rectangle on the top half of the paper. On the bottom half of the paper, students title the columns Length, Width, Perimeter, and Area and fill in the table with the measurements for the shaded rectangle. Students write the formulas they used to determine the perimeter and area in the corresponding columns and show their work. The teacher reviews student tables, noting misconceptions about perimeter and area and makes instructional adjustments to meet identified student needs. (DOK: 2, Bloom’s/RBT: Application/Apply, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I, (c)3.D, (c)5.B)

Interventions 1. The teacher provides pairs or small groups of students with an assortment of Color Tiles® and assigns each group an area, for example, 36 square units. Each group uses the tiles to make as many squares and rectangles as possible with the given area. Repeat the activity with perimeter. The teacher asks students follow-up questions. • • • •

How many different squares and rectangles were you able to make? Why does the perimeter of each shape change? Why doesn’t the area of each shape change? How would this activity change if you were given a perimeter instead of area? Explain your answer.

(DOK: 2, Bloom’s/RBT: Analysis/Analyze, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I, (c)3.D, (c)3.E)

2. The teacher provides students with Color Tiles® and grid paper to build the rectangles described below. • Rectangle 1 has a perimeter between 10 and 15 units and an area between 10 and 15 square units. • Rectangle 2 has a perimeter between 20 and 24 units and an area between 30 and 36 square units. Next, students draw the two rectangles on the grid paper. Students color, cut out, and display the rectangles on black construction paper. Students record the perimeters and areas for the rectangles on the black paper. The teacher displays students’ rectangles around the room. Students discuss the similarities and differences among the rectangles. (DOK: 2, Bloom’s/RBT: Application/Apply, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I, (c)3.D, (c)3.E, (c)3.F)

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Unit 25

Use Formulas and Models to Solve Problems with Perimeter and Area

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TEKS 4.5(C), 4.5(D) – Readiness

3. The teacher wraps a length of string around the rim of a rectangle to measure the perimeter. The teacher cuts the length of the string to match the perimeter. Then, the teacher straightens the string and measures its length with a ruler. The teacher emphasizes that perimeter is a linear measure. Students use strings to find the perimeters of rectangles on an activity sheet. The teacher emphasizes the connection between the string activity and the formulas for the perimeters of rectangles and squares. Next, the teacher uses Color Tiles® to cover as much of the surface of a book in rows and columns as possible, and then counts the number of tiles needed to cover the surface. The teacher emphasizes that area is a measure of the number of square units contained in a space. Students use Color Tiles® to find the areas of rectangles on an activity sheet. The teacher emphasizes the connection between the tiles and formulas for the areas of rectangles and squares. (DOK: 1, Bloom’s/RBT: Application/Apply, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I)

Suggested Formative Assessment The teacher gives each student three note cards. On the first note card, students draw a rectangle or square and label its length and width or show the length and width using centimeter squares. On the second note card, students write an equation to show how to determine the perimeter of the drawn shape. On the third card, students write an equation to show how to determine the area of the drawn shape. The teacher collects, shuffles, and distributes three cards to each student. Students trade cards with classmates until they have a set of three matched cards. The teacher observes students as they match cards and plans additional instruction and/or interventions as needed.

5 P=5+5+3+3

3

A=5×3

(DOK: 2, Bloom’s/RBT: Analysis/Analyze, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I, (c)3.D, (c)3.E, (c)3.F, (c)4.C, (c)4.F, (c)5.B)

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TEKS 4.5(C), 4.5(D) – Readiness

Extending Student Thinking Students investigate the relationship of the formula for finding the area of a rectangle to the formula for finding the area of a triangle. Students use tangrams to form rectangles. Using their knowledge of area, students generalize and justify a formula for finding the area of a triangle. Students create models using grid paper and/or geoboards to support their generalizations and share their results with the class. (DOK: 3, Bloom’s/RBT: Synthesis/Create, ELPS: (c)1.C, (c)1.E, (c)1.H, (c)2.E, (c)2.I, (c)3.D, (c)3.E, (c)3.G, (c)3.H)

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Unit 25

Use Formulas and Models to Solve Problems with Perimeter and Area

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teacher sample Use Formulas and Models to Solve Problems with Perimeter and Area

TEKS 4.5(C), 4.5(D) – Readiness

Answer Codings (Student pages 199–201)

Page Question

Process TEKS

Answer

Bloom’s Original/ Revised

DOK Level

4.1(A)

Application/Apply

1

(c)1.C, (c)1.E, (c)1.H, (c)4.G

4.1(A)

Comprehension/Understand

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G, (c)5.B

4.1(A) 4.1(G)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G, (c)5.G

4

96 centimeters Answers may vary. Students may explain that they first divided the perimeter of the small square, 48, by 4 to find the length of each side. They then multiplied 12 x 2 = 24 to find the length of 1 side of the original sheet of paper. They could then use the formula for finding the perimeter of a square, P = 4s to multiply P = 4 x 24. The perimeter is 96 cm. A = 24 x 24 = 576 square centimeters

4.1(B) 4.1(G)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G, (c)5.G

1

C

4.1(A)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

2

F

4.1(A)

Comprehension/Understand

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

3

B

4.1(A)

Analysis/Analyze

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

4

G

4.1(A)

Application/Apply

1

(c)1.C, (c)1.E, (c)1.H, (c)4.G

1

D

4.1(F)

Analysis/Analyze

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

2

J

4.1(B)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

3

C

4.1(B)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

4

25

4.1(B)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

5

B

4.1(A)

Application/Apply

1

(c)1.C, (c)1.E, (c)1.H, (c)4.G

1

2

3 199

16 feet; 15 square feet Answers may vary. Students should explain that carpet covers the area of a space. To find the area of a rectangle multiply l x w, therefore 14 x 10 is used to find the area of the room. 10 feet Answers may vary. Students should explain that if the perimeter of a rectangle is 48 feet, then the measure of one length and one width is half the perimeter, or 24 feet. Since the length is 14 feet, the width must be 24 – 14 = 10 feet.

200

201

298

ELPS

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GRADE LEVEL 4 teacher edition

Unit 25

motivationmath

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TEKS 4.5(C), 4.5(D) – Readiness

Answer Codings

(Student pages 202–205)

Page Question

202

Process TEKS

Answer

Bloom’s Original/ Revised

DOK Level

ELPS

1

B

4.1(A)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

2

256

4.1(B)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

3

A

4.1(A)

Application/Apply

1

(c)1.C, (c)1.E, (c)1.H, (c)4.G

4

J

4.1(C)

Application/Apply

1

(c)1.C, (c)1.E, (c)1.H, (c)4.G

5

B

4.1(B)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

4.1(B)

Analysis/Analyze

3

(c)1.C, (c)1.E, (c)1.H, (c)4.G

4.1(B)

Analysis/Analyze

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

4.1(F) 4.1(G)

Analysis/Analyze

3

(c)1.C, (c)1.E, (c)1.H, (c)4.G, (c)5.B, (c)5.G

4.1(D)

Comprehension/Understand

2

(c)1.C, (c)1.E, (c)1.H, (c)3.E, (c)4.G

Comprehension/Understand

2

(c)1.C, (c)1.E, (c)1.H, (c)3.E, (c)4.G

Bedroom Dimensions Area (square feet)

Length (feet)

Width (feet)

Dell

120

12

10

Anna Mason

132

12

11

121 128

11 16

11 8

Child

1

Jenna

Answers may vary. One possible arrangement is shown. Accept all responses with a perimeter of 32 units.

203

2

36

1

P = 74

18 2

P = 40

9

12 3

P = 30

4

P = 26

6

Journal 204

205

6

P = 24

Answers will vary. Students should explain that the more spread out a rectangle is, the greater its perimeter; the more compact (closer to square) a rectangle is, the smaller its perimeter. Vocabulary Activity

Answers may vary.

Motivation Station

Results may vary.

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Unit 25

Use Formulas and Models to Solve Problems with Perimeter and Area

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teacher sample Use Formulas and Models to Solve Problems with Perimeter and Area

TEKS 4.5(C), 4.5(D) – Readiness

Answer Codings (Student page 206)

Answer

Process TEKS

72 feet; no Based on the dimensions of Carmen’s garden, the area is 288 square feet. The bag only covers 250 square feet. Carmen will not have enough fertilizer.

4.1(B) 4.1(G)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G, (c)5.B

4.1(A)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

4.1(B)

Application/Apply

2

(c)1.C, (c)1.E, (c)1.H, (c)4.G

Page Question

1

Bloom’s Original/ Revised

DOK Level

ELPS

206 2

3

300

Sport

I

W

P

A

Basketball

28

15

86

420

Volleyball

18

9

54

162

Badminton

13

6

38

78

90 square meters

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