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Chapter
7
Multiplication in Geometry
Chapter Overview 7-1
Local Standards
The Area Model for Multiplication A Find the area of a triangle given appropriate dimensions. B Find the area of a trapezoid (including special types) given appropriate dimensions. E Recognize and use the Distributive Property and the Commutative and Associative Properties of Multiplication. F Recognize the differences between perimeter and area. G Find areas of rectangles and the number of elements in rectangular arrays in applied situations. K Picture multiplication using arrays or area.
Pacing (in days) Average
Advanced
Block
1.5
1
0.5
7-2
Multiplication of Fractions C Multiply fractions. K Picture multiplication using arrays or area.
1
1
0.5
7-3
The Distributive Property E Recognize and use the Distributive Property and the Commutative and Associative Properties of Multiplication. L Represent the Distributive Property with areas of rectangles.
1
1
0.5
0.5
0.5
0.25
QUIZ 1 7-4
The Area of a Triangle A Find the area of a triangle given appropriate dimensions. H Find areas of triangles or trapezoids in real situations.
1
1
0.5
7-5
The Area of a Trapezoid B Find the area of a trapezoid (including special types) given appropriate dimensions. H Find areas of triangles or trapezoids in real situations.
1
1
0.5
Circles D Find the area and circumference of a circle. F Recognize the differences between perimeter and area. I Find the area and circumference of a circle in real-world situations.
1.5
1
0.5
QUIZ 2
0.5
0.5
0.25
1
1
0.5
Self-Test
1
1
0.5
Chapter Review
2
1
1
Test
1
1
0.5
13
11
6
7-6
7-7
The Size-Change Model for Multiplication J Apply the Size-Change Model for Multiplication in real-world situations. M Perform expansions or contractions on a coordinate graph.
Technology Resources Teacher’s Assessment Assistant, Ch. 7 Electronic Teacher’s Edition, Ch. 7
428A Chapter 7
TOTAL
Chapter 7 Overview
Differentiated Options
7-1 7-2 7-3 7-4 7-5 7-6 7-7
Universal Access
Accommodating the Learner
Vocabulary Development
Ongoing Assessment
pp. 432, 434 pp. 440, 441 pp. 446, 447 pp. 452, 453 pp. 458, 459 pp. 465, 467 p. 474
p. 432 p. 440 p. 446 p. 452 p. 458 p. 465 p. 473
small groups, p. 436 small groups, p. 442 individual, p. 449 small groups, p. 454 small groups, p. 462 individual, p. 469 individual, p. 476
Materials rulers
grid paper grid paper, scissors, ruler ruler
Objectives S kills A Find the area of a triangle given appropriate dimensions. B Find the area of a trapezoid (including special types) given appropriate dimensions. C Multiply fractions. D Find the area and circumference of a circle.
Lessons 7-1, 7-4
Self-Test Questions 9, 20
Chapter Review Questions 1–4
7-1, 7-5 7-2
12 6, 7, 16
5–9 10–17
7-6
11
18–20
7-1, 7-3
2, 3, 4
21–33
7-1, 7-6
13
34–39
7-1
5, 14
40–43
7-4, 7-5 7-6
17 1
44–45 46–49
7-7
10
50–55
7-1, 7-2
8
56–59
7-3
15
60–63
7-7
18, 19
64–69
2-1
26
55–60
P roperties E Recognize and use the Distributive Property and the Commutative and Associative Properties of Multiplication. F Recognize the differences between perimeter and area.
U ses G Find areas of rectangles and the number of elements in rectangular arrays in applied situations. H Find areas of triangles or trapezoids in real situations. I Find the area and circumference of a circle in real-word situations. J Apply the Size-Change Model for Multiplication in real-world situations.
R epresentations K Picture multiplication using arrays or area. L Represent the Distributive Property with areas of rectangles. M Perform expansions or contractions on a coordinate graph. L Represent a relationship between two variables using a table.
Chapter 7 Overview 428B
Chapter 7 Resource Masters
Resource Masters Chapter 7 Resource Master 1, Graph Paper (page 2), can be used with Lesson 7-7. Resource Master 2, Four-Quadrant Graph Paper (page 3), can be used with Lessons 7-1 and 7-7.
Resource Master for Lesson 7-1
Resource Master for Lesson 7-1
Resource Master for Lesson 7-2
Resource Master for Lesson 7-3
Resource Master for Lesson 7-4
Resource Master for Lesson 7-5
428C Chapter 7
Chapter 7 Overview
Resource Master for Lesson 7-6
Resource Master for Lesson 7-7
Chapter 7 Overview 428D
Chapter
7 Pacing Each lesson in this chapter is designed to be covered in 1 to 2 days. At the end of the chapter, you should plan to spend 1 day to review the Self-Test, 1–2 days for the Chapter Review, and 1 day for a test. Consequently, this chapter should take about 13–16 days.
FPO
Overview Geometry plays two roles in arithmetic. One role is to picture or represent the arithmetic. Students have encountered this role when using a number line, thinking of slides to add positive and negative numbers, and graphing ordered pairs. In this chapter, we represent multiplication by area. The second role of geometry is as a context in which to apply the arithmetic. We add lengths to find perimeter. We use subtraction to find distances between points on a number line. And in this chapter, we multiply to find area. It can be argued that the importance of multiplication as an arithmetic operation can be traced historically to its applications in the calculation of area. Knowing how much land a person controlled or owned was as important in ancient times as it is today, for at least two of the same reasons—taxes and inheritance. Ancient civilizations knew that calculations of area were associated with multiplication.
Chapter 7 Overview This chapter is a link between the concrete geometry activities involving area and volume, which students should have studied in earlier grades, with the formal study of area and volume that should occur in high school. We make this link in three ways: (1) Even those students who have seen some of these ideas before often do not have them internalized, and we provide some concrete activities for them and for students whose backgrounds are weaker. (2) Whereas in earlier grades students may have associated the area of a rectangle with the arithmetic of multiplication,
428 Chapter 7
here we put some emphasis on algebraic formulas. (3) The notion of a sequence of formulas, working from simpler figures to more complex ones, paralleling the sequence in formal proof, is introduced. Lessons 7‑1 to 7‑3 provide the arithmetic and algebraic basis for the chapter. Lesson 7‑1 reviews the area formulas for a rectangle and a right triangle and the basic properties of multiplication, material that students should have encountered in previous years. Lesson 7‑2, covering the multiplication of fractions,
Using Pages 428–429 We strongly suggest that you probe students to find out what they know about the area and volume of figures before beginning this chapter. Depending on the curricula students have studied, there could be vast differences in earlier opportunities to learn about area and volume. Some teachers very much enjoy this content and dwell on it, while others do not like the content and may even skip it entirely.
FPO
Ask students to look at the drawings on page 429. Without reading the text, can they identify any or all of the figures? You might write down the names with arrows as shown: square → rectangle → right triangle → triangle → polygon → parallelogram → circle. This order almost mirrors the sequence in this chapter, which is square → rectangle → right triangle → triangle → trapezoid → parallelogram → circle.
Chapter 7 Projects At this time you might want to have students look over the projects on page 477.
Chapter 7 Overview This chapter is a link between the concrete geometry activities involving area and volume, which students should have studied in earlier grades, with the formal study of area and volume that should occur in high school. We make this link in three ways: (1) Even those students who have seen some of these ideas before often do not have them internalized, and we provide some concrete activities for them and for students whose backgrounds are weaker. (2) Whereas in earlier grades students may have associated the area of a
rectangle with the arithmetic of multiplication,
here we put some emphasis on algebraic formulas. (3) The notion of a sequence of formulas, working from simpler figures to more complex ones, paralleling the sequence in formal proof, is introduced. Lessons 7‑1 to 7‑3 provide the arithmetic and algebraic basis for the chapter. Lesson 7‑1 reviews the area formulas for a rectangle and a right triangle and the basic properties of multiplication, material that students should
Chapter 7 Opener 429
7
Multiplication in Geometry
Chapter Overview 7-1
Local Standards
The Area Model for Multiplication A Find the area of a triangle given appropriate dimensions. B Find the area of a trapezoid (including special types) given appropriate dimensions. E Recognize and use the Distributive Property and the Commutative and Associative Properties of Multiplication. F Recognize the differences between perimeter and area. G Find areas of rectangles and the number of elements in rectangular arrays in applied situations. K Picture multiplication using arrays or area.
Pacing (in days) Average
Advanced
Block
1.5
1
0.5
7-2
Multiplication of Fractions C Multiply fractions. K Picture multiplication using arrays or area.
1
1
0.5
7-3
The Distributive Property E Recognize and use the Distributive Property and the Commutative and Associative Properties of Multiplication. L Represent the Distributive Property with areas of rectangles.
1
1
0.5
0.5
0.5
0.25
QUIZ 1 7-4
The Area of a Triangle A Find the area of a triangle given appropriate dimensions. H Find areas of triangles or trapezoids in real situations.
1
1
0.5
7-5
The Area of a Trapezoid B Find the area of a trapezoid (including special types) given appropriate dimensions. H Find areas of triangles or trapezoids in real situations.
1
1
0.5
Circles D Find the area and circumference of a circle. F Recognize the differences between perimeter and area. I Find the area and circumference of a circle in real-world situations.
1.5
1
0.5
QUIZ 2
0.5
0.5
0.25
1
1
0.5
Self-Test
1
1
0.5
Chapter Review
2
1
1
Test
1
1
0.5
13
11
6
7-6
7-7
The Size-Change Model for Multiplication J Apply the Size-Change Model for Multiplication in real-world situations. M Perform expansions or contractions on a coordinate graph.
Technology Resources Teacher’s Assessment Assistant, Ch. 7 Electronic Teacher’s Edition, Ch. 7
428A Chapter 7
TOTAL
Chapter 7 Overview
Differentiated Options
7-1 7-2 7-3 7-4 7-5 7-6 7-7
Universal Access
Accommodating the Learner
Vocabulary Development
Ongoing Assessment
pp. 432, 434 pp. 440, 441 pp. 446, 447 pp. 452, 453 pp. 458, 459 pp. 465, 467 p. 474
p. 432 p. 440 p. 446 p. 452 p. 458 p. 465 p. 473
small groups, p. 436 small groups, p. 442 individual, p. 449 small groups, p. 454 small groups, p. 462 individual, p. 469 individual, p. 476
Materials rulers
grid paper grid paper, scissors, ruler ruler
Objectives S kills A Find the area of a triangle given appropriate dimensions. B Find the area of a trapezoid (including special types) given appropriate dimensions. C Multiply fractions. D Find the area and circumference of a circle.
Lessons 7-1, 7-4
Self-Test Questions 9, 20
Chapter Review Questions 1–4
7-1, 7-5 7-2
12 6, 7, 16
5–9 10–17
7-6
11
18–20
7-1, 7-3
2, 3, 4
21–33
7-1, 7-6
13
34–39
7-1
5, 14
40–43
7-4, 7-5 7-6
17 1
44–45 46–49
7-7
10
50–55
7-1, 7-2
8
56–59
7-3
15
60–63
7-7
18, 19
64–69
2-1
26
55–60
P roperties E Recognize and use the Distributive Property and the Commutative and Associative Properties of Multiplication. F Recognize the differences between perimeter and area.
U ses G Find areas of rectangles and the number of elements in rectangular arrays in applied situations. H Find areas of triangles or trapezoids in real situations. I Find the area and circumference of a circle in real-word situations. J Apply the Size-Change Model for Multiplication in real-world situations.
R epresentations K Picture multiplication using arrays or area. L Represent the Distributive Property with areas of rectangles. M Perform expansions or contractions on a coordinate graph. L Represent a relationship between two variables using a table.
Chapter 7 Overview 428B
Chapter 7 Resource Masters
Resource Masters Chapter 7 Resource Master 1, Graph Paper (page 2), can be used with Lesson 7-7. Resource Master 2, Four-Quadrant Graph Paper (page 3), can be used with Lessons 7-1 and 7-7.
Resource Master for Lesson 7-1
Resource Master for Lesson 7-1
Resource Master for Lesson 7-2
Resource Master for Lesson 7-3
Resource Master for Lesson 7-4
Resource Master for Lesson 7-5
428C Chapter 7
Chapter 7 Overview
Resource Master for Lesson 7-6
Resource Master for Lesson 7-7
Chapter 7 Overview 428D
Chapter
7 Pacing Each lesson in this chapter is designed to be covered in 1 to 2 days. At the end of the chapter, you should plan to spend 1 day to review the Self-Test, 1–2 days for the Chapter Review, and 1 day for a test. Consequently, this chapter should take about 13–16 days.
FPO
Overview Geometry plays two roles in arithmetic. One role is to picture or represent the arithmetic. Students have encountered this role when using a number line, thinking of slides to add positive and negative numbers, and graphing ordered pairs. In this chapter, we represent multiplication by area. The second role of geometry is as a context in which to apply the arithmetic. We add lengths to find perimeter. We use subtraction to find distances between points on a number line. And in this chapter, we multiply to find area. It can be argued that the importance of multiplication as an arithmetic operation can be traced historically to its applications in the calculation of area. Knowing how much land a person controlled or owned was as important in ancient times as it is today, for at least two of the same reasons—taxes and inheritance. Ancient civilizations knew that calculations of area were associated with multiplication.
Chapter 7 Overview This chapter is a link between the concrete geometry activities involving area and volume, which students should have studied in earlier grades, with the formal study of area and volume that should occur in high school. We make this link in three ways: (1) Even those students who have seen some of these ideas before often do not have them internalized, and we provide some concrete activities for them and for students whose backgrounds are weaker. (2) Whereas in earlier grades students may have associated the area of a rectangle with the arithmetic of multiplication,
428 Chapter 7
here we put some emphasis on algebraic formulas. (3) The notion of a sequence of formulas, working from simpler figures to more complex ones, paralleling the sequence in formal proof, is introduced. Lessons 7‑1 to 7‑3 provide the arithmetic and algebraic basis for the chapter. Lesson 7‑1 reviews the area formulas for a rectangle and a right triangle and the basic properties of multiplication, material that students should have encountered in previous years. Lesson 7‑2, covering the multiplication of fractions,
Using Pages 428–429 We strongly suggest that you probe students to find out what they know about the area and volume of figures before beginning this chapter. Depending on the curricula students have studied, there could be vast differences in earlier opportunities to learn about area and volume. Some teachers very much enjoy this content and dwell on it, while others do not like the content and may even skip it entirely.
FPO
Ask students to look at the drawings on page 429. Without reading the text, can they identify any or all of the figures? You might write down the names with arrows as shown: square → rectangle → right triangle → triangle → polygon → parallelogram → circle. This order almost mirrors the sequence in this chapter, which is square → rectangle → right triangle → triangle → trapezoid → parallelogram → circle.
Chapter 7 Projects At this time you might want to have students look over the projects on page 477.
Chapter 7 Overview This chapter is a link between the concrete geometry activities involving area and volume, which students should have studied in earlier grades, with the formal study of area and volume that should occur in high school. We make this link in three ways: (1) Even those students who have seen some of these ideas before often do not have them internalized, and we provide some concrete activities for them and for students whose backgrounds are weaker. (2) Whereas in earlier grades students may have associated the area of a
rectangle with the arithmetic of multiplication,
here we put some emphasis on algebraic formulas. (3) The notion of a sequence of formulas, working from simpler figures to more complex ones, paralleling the sequence in formal proof, is introduced. Lessons 7‑1 to 7‑3 provide the arithmetic and algebraic basis for the chapter. Lesson 7‑1 reviews the area formulas for a rectangle and a right triangle and the basic properties of multiplication, material that students should
Chapter 7 Opener 429
