name angle pairs - wsdblog.wsd.net/keperkins/files/2015/01/may-4-skills-15.pdf · ... you used the...
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MAY 4, 2015www.scholastic.com/math
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Geometry
Angle PAirsIn “Shapes in the Sand” on page 4, you used the degree formula to find unknown angle measures in a given polygon. You can also use the properties of different angle pairs to solve for missing angle measures.
Complementary angles: Angles whose measures have a sum of 90° (a right angle)
Supplementary angles: Angles whose measures have a sum of 180° (a straight line)
Vertical angles: Angles opposite one another, formed when two lines intersect; these angles are always congruent (they have the same degree measure)
Adjacent angles: Angles that have a common side and a common vertex (corner point) and do not overlap
Types of Angle Pairs
Use these definitions to classify angle pairs and find missing angle measures in the questions below.
1 A. What is the sum of the angle measures above?
B. Are the angles complementary, supplementary, vertical, and/or adjacent?
2 A. What is the sum of the angle measures above?
B. Are the angles complementary, supplementary, vertical, and/or adjacent?
3A. Are angles a and c complementary, supplementary, vertical, and/or adjacent?
B. Angle a is 116°. What is the measure of angle c?
C. Are angles a and b complementary, supplementary, vertical, and/or adjacent?
D. What is the measure of angle b?
E. What is the measure of angle d?
F. What is the sum of the measures of angles a, b, c, and d?
75° 105°
a
cb
d
24°66°
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inTeriOr AnglesIn “Shapes in the Sand” on page 4, you used the formula for degrees in a polygon to find the sum of the interior angles of a given shape. Use what you learned and the formula below to answer five more questions about interior-angle sums.
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Geometry
1 A. How many sides does the polygon pictured above
have?
B. Based on the degree formula, what will be the sum of its interior angles?
2What is the sum of the interior angles of the polygon above?
3What is the sum of the interior angles of the polygon above?
4The sum of the interior angles of a particular polygon is 900°. How many sides does the figure have?
5The sum of the interior angles of a particular polygon is 1,800°. How many sides does the figure have?
Formula for Degrees in a Polygon:
180° 5 (number of sides – 2)
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COMBining liKe TerMsIn “Battle of the Bots” on page 6, you practiced writing and evaluating algebraic expressions to represent word problems. To simplify algebraic expressions, you can combine like terms: terms that contain the same power of the same variable.
EXAMPLE: Simplify this algebraic expression: 3xy + 4x 2 + 7y 3 – 8x – 5y 3.
The expression above has 5 terms. Each term is associated with the addition or subtraction sign that comes before it: 3xy, + 4x 2, + 7y 3, – 8x, – 5y 3
The terms “+ 7y 3” and “– 5y 3” have the same power of the same variable: y 3. That means these terms can be combined. To combine them, add the coefficients (the integers in front of those terms). Keep the power of the variable the same: +7y 3 – 5y 3 = +2y 3
Rewrite the simplified expression using the combined like terms and the remaining terms: 3xy + 4x 2 + 2y 3 – 8x
Solve the following questions by combining like terms to simplify the given algebraic expressions.
MAY 4, 2015
Algebraic Expressions
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1 7g – h + 2f + 3f – g – f
2 4x3 + 5x2 – x2 + 7x –2x + 8
3 3p2 + 4n – 2p2 + p3 – 2n
4 5ab + 8a4 + b2 – 6a2 + 3b2 + a – 3ab
5A. During the morning shift, a coffee shop sold 25 cups of coffee and 18 muffins. During the afternoon
shift, the shop sold 16 cups of coffee and 13 muffins. Write an expression with four terms that describes the shop’s total coffee and muffin sales that day, where c represents the cost of a cup of coffee, and m represents the cost of a muffin.
B. Simplify the expression by combining like terms.
C. If a cup of coffee costs $1.75 and a muffin costs $1.95, how much revenue did the coffee shop make?
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BUilDing BOTsIn “Battle of the Bots” on page 6, you wrote and evaluated algebraic expressions to find unknown quantities in mathematical scenarios involving a robotics challenge. Use what you learned to write five more algebraic expressions and solve for their variables.
1A. To build a dog-shaped robot to enter into a robotics competition, you will need four robot legs, two internal
computers, and one stabilizing electronic “tail.” Write an expression for the total cost of your dog-bot where l represents the cost of each leg, c represents the cost of an internal computer, and t represents the cost of an electronic tail.
B. How much will it cost to build your dog-bot if each leg costs $250, an internal computer costs $200, and a tail costs $50?
2A. Your finished dog-bot can trot at a pace of 10 miles per hour. Write an expression for how far the
dog-bot can travel in x hours.
B. How far can your dog-bot travel in 15 minutes?
3A. A Rubik’s Cube-solving robot can unscramble a Rubik’s Cube in 5 seconds flat. Write an expression for
how long it would take the robot to solve a series of cubes, where n represents the total number of cubes solved.
B. How many cubes could the Rubik’s Cube-solving robot solve in 2 minutes?
4 An updated model of the Rubik’s Cube robot can unscramble a cube in 3 seconds. Write an expression
for how much longer it would take the older robot to solve n cubes than it would take the newer robot.
5How many Rubik’s Cubes can the newer model solve in the time it takes the older model to solve 12 cubes?
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MAY 4, 2015
Scatter Plots
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ClUsTers & OUTliersIn “Manatees on the Move” on page 8, you interpreted scatter plot data by drawing and analyzing trend lines. When interpreting the meaning of data shown on a scatter plot, it is also helpful to identify clusters and outliers in the data.
Clusters are distinct groups of points that are close together within a scatter plot. These points may overlap or merely cluster near one another.
Outliers are points that do not seem to follow the overall pattern of the data. These isolated points will be located far apart from other points on the plot.
Analyze the scatter plot below to identify outliers and clusters.
1What two variables are being compared in this scatter plot?
2Circle any clusters you see in the data. What do these clusters indicate?
3Circle any outliers you see in the data. What do the outliers indicate?
4Based on the clusters and outliers in the scatter plot, how would you characterize the relationship between
dog height and weight? Use the data to support your answer.
Dog Weights by Height for Select Dogs
10 15 20 25 30
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Dog HEigHT (iNcHES)
Do
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MAY 4, 2015
Scatter Plots
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MOre MAnATees!In “Manatees on the Move” on page 8, you analyzed scatter plots by identifying and drawing trend lines through data points. Use what you learned to answer five more questions based on the scatter plot below, which shows the combined totals of east and west Florida manatee counts.
1Draw a trend line through the data in the scatter plot above.
2Does the trend line indicate a positive or a negative relationship between the two variables?
3In which year did the number of manatees sighted increase the most compared with the prior year?
4In which year did the number of manatees sighted decrease the most compared with the prior year?
5Look back at the scatter plots on page 11 of your May issue. In what ways does the scatter plot of combined
totals differ from the individual east and west scatter plots? In what ways are the plots similar?
MA
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6,000
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YEAR
Manatee counts in East and West Florida*
sourCe: fLoriDA fisH AnD WiLDLife ConserVATion Commission*Gaps in data indicate years in which manatee counts were not conducted.
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MUlTiPlYing & DiViDing DeCiMAls
In “Super Skills” on page 14, you practiced solving word problems involving rational numbers. A rational number is a number that can be expressed as the quotient of two integers, where the divisor is not zero. Rational numbers can be expressed as integers, fractions, or decimals. When multiplying or dividing decimals in a problem, you can use an algorithm.
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Mixed Skills
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1 1.936 5 2.7
2 94.17 ÷ 7.3
3 369 ÷ 1.5
4 72.15 5 0.03
5 261.6 ÷ 1.2
EXAMPLE: 3.25 5 4.8
STEP 1: Rewrite the expression vertically, leaving out the decimal points.
STEP 2: Multiply as you would with whole numbers.
STEP 3: Count the total number of decimal places in the factors. This will be the total number of decimal places in the product.
3.25 5 4.8 ➔ 15.600
So 3.25 5 4.8 = 15.6
EXAMPLE: 30.155 ÷ 3.7
STEP 1: Rewrite the expression using a division bracket. The number outside the bracket is the divisor; the number inside the bracket is the dividend.
STEP 2: Move the decimal point in the divisor to the right until the divisor is a whole number. Move the decimal point in the dividend to the right the same number of times.
STEP 3: Divide as you would with whole numbers. Place the decimal point in the quotient (your answer) directly above the decimal point in the dividend.
So 30.155 ÷ 3.7 = 8.15
325485
32548
26001300015600
5
+
three decimal places in the factors
three decimal places in the product
30.1553.7)
301.5537.)
301.55296
0
5537185185
37.)–
–
–
8.15
Use the algorithms above to solve the following questions. Use another piece of paper to show your work.
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MigHTY MeAsUresIn “Super Skills” on page 14, you used mixed skills to uncover fun facts about the Avengers’ incredible powers and abilities. Use what you learned to answer five more questions about these superheroes’ superstrengths!
1Bruce Banner weighs 128 pounds. The Hulk weighs 1,400 pounds. By what factor does Banner’s weight
increase when he transforms into the Hulk? Round to the nearest whole number.
2The Hulk can lift 200,000 pounds. A common American field ant can lift 5,000 times its weight.
Based on the Hulk’s weight from question 1, who can lift more in proportion to its weight: the Hulk or the ant?
3Hawkeye’s vision is 8 times better than normal human vision. If he can spot a target from a distance of 56
feet, how close to the target would a person with normal vision have to be to see it?
4Thor’s bones are 4 times denser than human bone, which means they can withstand more weight before
they break. If Thor’s bones can handle a load of 76,000 pounds, how many pounds can normal human bone withstand?
5 Iron Man’s suit is not made of iron. This metal is very heavy and rusts easily. Instead, Tony Stark uses a mix
of metals, also called an alloy. Iron’s density is 0.284 pounds per cubic inch. If the alloy’s density is 0.233 pounds per cubic inch, what percent less dense is the alloy compared with iron? Round to the nearest percent.
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Mixed Skills
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PYTHAgOreAn TriPlesIn “Triple Threat!”on page 16, you used the Pythagorean theorem, a2 + b2 = c2, to find the length of an unknown side in a right triangle. When all three sides of a right triangle are integers, the set of these numbers is known as a Pythagorean triple. Use what you learned and the right-triangle diagram below to find the missing side in each of the following sets of Pythagorean triples.
1 a = 11, b = 60, c = _____
2 a = 12, b = 35, c = _____
3 a = 20, b = 21, c = _____
4 a = 4, b = _____, c = 5
5 a = _____, b = 15, c = 17
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Geometry
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a
b
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Name
Practice Test
MAY 4, 2015
1The superhero Quicksilver travels at breakneck speeds, moving even faster than the speed of sound! If he travels
675,000 meters in 30 minutes, what is his speed in meters per second?
2Bruce Banner’s height is 5'11". As the Hulk, he shoots up to 8 feet tall. What is Banner’s percent increase in
height when he transforms into the Hulk? Round to the nearest whole percent.
3Tickets to a robotics competition are sold at three different price points. Floor seats cost $20, upper-level
seats cost $10, and standing room costs $5. Write an expression for the total cost of tickets purchased in a batch, where f represents the number of floor-seat tickets, l represents the number of upper-level tickets, and s represents the number of standing-room tickets.
4Use the expression from question 3 to find the total cost of tickets if you were to purchase 10 floor tickets,
7 upper-level tickets, and 8 standing-room tickets.
5If a trend line drawn on a scatter plot has a slope of 2, what type of relationship must the plotted values have?
6If you are unable to draw a trend line, what does this tell you about the variables on the plot?
7What is the sum of the interior angles in the figure to the right?
Hint: Use the formula for degrees in a polygon: 180° 5 (number of sides – 2).
8What is the sum of the interior angles in the figure to the right?
9Use the Pythagorean theorem to find the length of the hypotenuse c for a right triangle with legs a = 5
inches and b = 4 inches. Round to the nearest tenth.
10 Find the length of the hypotenuse c for a right triangle with legs a = 7 inches and b = 9 inches.
Round to the nearest tenth.
ISSUE SKILLS REVIEW
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