Download 7.1 Notes - crunchy math

Changes for
nd
2
Semester:
1. Two separate interactive notebooks
(Notes & Scholar Work)
2. No intervention/reteach week
3. Retakes will be taken after school on
determined dates. All assignments &
notes must be complete
4. Crunchymath.weebly.com
Opening Activity
1. A tight end scored 6 touchdowns in 14
games. Find the ratio of touchdowns per game.
2. In a schedule of 6 classes, Marta has 2
elective classes. What is the ratio of elective to
non-elective classes in Marta’s schedule.
3. An artist in Portland, Oregon, makes bronze
sculptures of dogs. The ratio of the height of a
sculpture to the actual height of the dog is 2:3.
If the height of the sculpture is 14 inches, find
the height of the dog.
D. N. A.
Solve the following
equations.
1) 7 x  63
3
2)
x9
4
6a
3)
 15
5
2
4)
y46
3
Slides
Skills Practice
Practice
1, 2, 7, 8, 9
1-5
1-3
11, 12, 13
6-11
4-9
15
12-15
10-12
17
13-15
Geometry Chapter 7:
Proportions and Similarity
Chapter 7 Test on Friday 1/18
Retake 2/7 or 2/8
Chapter 7-1:
Proportions
• Write ratios.
• ratio
• Use properties of proportions. • proportion
• cross products
• extremes
• means
Reinforcement of CA Standard 6NS1.3 Use
proportions to solve problems (e.g., determine the value
of N if 4/7 = N/21, find the length of a side of a polygon
similar to a known polygon). Use cross-multiplication as
a method for solving such problems, understanding it as
the multiplication of both sides of an equation by a
multiplicative inverse. (Key)
Prerequisite Algebra Review
 Solve the following equation:
3
2) x  9
4
Multiply both sides by the reciprocal of the fraction.
 4   3   9  4 
   x    
 3   4   1  3 
12
36
x
12
3
x  12
Prerequisite Algebra Review (cont.)
 Solve the following equation:
6a
3)
 15
5
5
 5   6a   15   5 
     
6 5   1 6
2
25
a
2
Prerequisite Algebra Review (cont.)
 Solve the following equation:
2
4)
y46
3
2
y  10
3
5
 3   2   10   3 
   y    
 2   3   1  2 
a  15
Ratios
A comparison of two quantities using division.
a
or a: b
b
Example: The ratio of 5 and 7 can be written as 5:7
or as the fraction 5 and we say the ratio is “five to
seven”.
7
Example # 2:
• Gary has a bag with 4 marbles, 3 books, 5
pencils, and 2 erasers.
a. What is the ratio of pencils to books?
5:3
b. What is the ratio of marbles to the total
number of items in the bag?
4:14  2:7 (Must be reduced!)
Write a Ratio
SCHOOL The total number of students who
participate in sports programs at Central High School
is 500. The total number of students in the school is
2000. Find the athlete-to-student ratio to the nearest
tenth.
To find this ratio, divide the number of athletes by the
total number of students.
# of athletes
500 1

 or 1 : 4
total # of students 2000 4
Answer: The athlete-to-student ratio is 1:4.
1. A tight end scored 6 touchdowns in 14 games.
Find the ratio of touchdowns per game.
2. In a schedule of 6 classes, Marta has 2 elective
classes. What is the ratio of elective to nonelective classes in Marta’s schedule.
3. An artist in Portland, Oregon, makes bronze
sculptures of dogs. The ratio of the height of a
sculpture to the actual height of the dog is 2:3.
If the height of the sculpture is 14 inches, find
the height of the dog.
Proportions
• If two ratios are equal, they can be written
as a proportion.
a c

b d
Extremes
Means
Proportion Practice
• Which proportions are not correct?
4
8
6 12
4 12
a) 
b) 
c) 
6 12
4
8
8
6
48 = 48 
48 = 48 
24  96 
6 4
4 6
8 12
d )  e)

f) 
12 8
12 8
4 6
48 = 48 
32  72 
48 = 48 
Proportion Practice
• Solve the following proportions
4 5
28

4  7  5x x 
x 7
5
3
2
3 y  2( y  2)

y2 y
3y  2y  4
y  4 Check your answer!
Solve each proportion.
2 x
6) 
5 40
5x 35
9)

4
8
7 21
7)

10 x
x 1 7
10)

3
2
20 4 x
8)

5
6
15 x  3
11)

3
5
Using Ratios Example #1
• The Perimeter of a rectangle is 60 cm. The ratio of
AB:BC is 3:2. Find the length and width of the
A
B
rectangle.
3:2 is in lowest terms.
D
AB:BC could be
3:2, 6:4, 9:6, 12:8, etc.
AB = 3x Perimeter = l + w+ l + w
BC = 2x 60 = 3x + 2x + 3x + 2x
60 = 10x
L = 3(6) = 18
x=6
W = 2(6) = 12
C
Find the measures of the sides of each triangle.
12. The ratio of the measures of the sides of a
triangle is 3:5:7, and its perimeter is 450
centimeters.
13. The ratio of the measures of the sides of a
triangle is 5:6:9, and its perimeter is 220 meters.
14. The ratio of the measures of the sides of a
triangle is 4:6:8, and its perimeter is 126 feet.
Find the measures of the angles in each triangle.
15) The ratio of the measures of the angles is 4:5:6.
Using Ratios Example #2
• The angle measures in ABC are in the extended
ratio of 2:3:4. Find the measure of the three angles.
mA+ mB+ mC = 180o
Triangle Sum Thm.
2x + 3x + 4x = 180o
9x = 180o
B
4x
x = 20o
mA =
40o
mB = 60o
mC = 80o
A
2x
3x
C
Using Ratios Example #3
• The ratio of the measures of the three side lengths of
1 1 1
a ABC are 2 : 4 : 5 , and the perimeter is 19 m.
Find the measure of each side of the triangle.
x  x  x  19
1
1
1
2
4
5
Change the fractions into common denominators?
10
5
4
20
20
20
Multiply everything by the common denominator.
x
x
x  19
20  x   20  x   20  x   20 19  1 (20)  10
2
10
20
5
20
10 x  5 x  4 x  380
19 x  380
x  20
4
20
1
4
(20)  5
1
5
(20)  4
Extended Ratios in Triangles
In a triangle, the ratio of the measures of three sides
is 5:12:13, and the perimeter is 90 centimeters. Find
the measure of the shortest side of the triangle.
A 13 cm
B 15 cm
C 38 cm
D 39 cm
The shortest side is 15 centimeters. The answer is B.
Check Add the lengths of the sides to make sure that
the perimeter is 90.
In a triangle, the ratio of the measures of three sides
is 3:4:5, and the perimeter is 42 feet. Find the
measure of the longest side of the triangle.
A. 10.5 ft
B. 14 ft
C. 17.5 ft
D. 37 ft
Proportion Properties
(The names are not important, the ideas are!!!)
Cross Product Property—The product of the
means equals the product of the extremes.
a c
if 
b d
then ad  bc
Solve Proportions by Using Cross
Products
B.
Cross products
Simplify.
Add 30 to each side.
Divide each side by 24.
Answer:
–2
A.
A. 0.65
B. 4.5
C. –14.5
D. 147
B.
A. 9
B. 8.9
C. 3
D. 1.8
1.
2.
3.
4.
A
B
C
D
Solve Problems Using Proportions
TRAINS A boxcar on a train has a length of
40 feet and a width of 9 feet. A scale model
is made with a length of 16 inches. Find the
width of the model.
Substitution
Cross products
Multiply.
Divide each side by 40.
Answer: The width of the model is 3.6 inches.
Solve Problems Using Proportions
Substitution
Cross products
Multiply.
Divide each side by
40.
Answer: The width of the model is 3.6 inches.
Proportion Practice #2
• A picture of a tree is shown, the actual tree
1
is 84 in. tall. How wide is the tree?
1 in
5
pic width 
4
7
1
pic height 
3 in
2
2
tree height  84 in tree width  x
5
5
7
(84)  x
4  x
4
2
7
84
5(84 )  14 x 420  14 x
2
4
x  30
Two large cylindrical containers are in proportion.
The height of the larger container is 25 meters with
a diameter of 8 meters. The height of the smaller
container is 7 meters. Find the diameter of the
smaller container.
A. 0.6 m
B. 2.24 m
C. 2.52 m
D. 28.57 m
Homework
Chapter 8.1
Pg 383:
2 – 9, 12 – 26,
56 – 61, 63, 64