Math combined
8oz= ______cup
1 cup
Which of the following lists all the factors of a composite number? 1, 7, 21 1, 11 2, 3, 6 1, 5, 25
1, 5, 25 EXPLANATION: When listing ALL of the factors of a composite number the number itself and 1 must always be included. Remember also that a composite number is a positive integer that has a positive divisor other than one or itself. The factor pairs for the composite number 25 are: 1 and 25; 5 and 5. Notice that 5 x 5 causes the 5 to be a repeated factor. So when listing the factors of the composite number 25 in order, we have: 1, 5, and 25.
There is a 15% increase in tuition at UT for next fall. If the current tuition is $3500 per semester, which equation could be used to find x, the new tuition for the fall? 0.15 * 3500 = x 1.15 * 3500 = x (15/100)= (x/3500) .85 * 3500 = x
1.15 * 3500 = x EXPLANATION:. With a 15% increase, the tuition of 3500 will increase: 100% + 15% = 1 + 0.15 = 1.15. So, 1.15 • 3500 = x would yield the new tuition rate. .85 * 3500 = x
In the image below, the area of the triangle is 1 unit2. What is the total area of the figure?
11 unit2 Dissecting the figure, it becomes apparent that each square consists of two triangles. The hexagon consists of 6 triangles. Therefore, the total area is 2 + 2 + 6 + 1 = 11 units2
Julie brings and eats 1/3 of a sandwich every day for lunch. If she made 11 sandwiches this month, how many lunches did she bring? Which of the following expressions would be a correct computation for the answer to the problem above?
11 ÷ 1/3 This expression would equal 11 x 3/1, which equals 33, the correct amount of lunches.
The images below are similar triangles. Solve for the missing value, x.
12 Since the triangles are similar, this question is best solved by setting up a proportion. 15/20 = x/16. Solving leads to 20x = 240. This reduces to x = 12.
What percent is represented by the shaded area of the decimal square?
24% EXPLANATION: There are 24 small squares shaded of the 100 total squares; therefore 24% is the correct answer.
What is the volume of a ball that is 12 cm in diameter?
288π cm3 The volume of a sphere is equal to 4/3 × π × r3. Since the diameter of the ball is 12 cm, the radius of the ball is 6 cm. Therefore the volume of the sphere is equal to 4/3 × π × 63, which equals 288π cm3.
What is the perimeter of the figure provided?
29 cm The perimeter of the figure is 2 × (l + w) = 2 × (5.5 + 9) = 29 cm.
The state sales tax is 7.5%. Which number could also represent 7.5%?
3/40 EXPLANATION: There is always the option of arriving at the answer by eliminating incorrect answer choices, but it is always a good idea to double-check the final choice. In the case of this question, 3/40 = 3 ÷ 40 = 0.075 = 75/1000=7.5/100 = 7.5%
Which of the following fractions has the smallest value?
3/7 3/7 is less than half because half of 7 is 3.5. The decimal equivalent is about 0.43.
Provided is a diagram of Tanya's new swimming pool. What is the perimeter of Tanya's pool in yards?
30 yds The units of measure in the drawing are not the same units as in the choices, so the easiest thing to do would be to change the measurements in the drawing from feet to yards. Recall that there are 3 feet in each yard. 15 feet is equivalent to 5 yards and 30 feet is equivalent to 10 yards. Thus, the perimeter is: P = 5 + 10 + 5 + 10 = 30 yds.
What is the volume of an ice cream cone that is 6cm wide at the top and 12cm tall?
36π cm3 The volume of a cone is equal to ⅓ × π × r2 × h. Since the diameter of the base of the cone is 6 cm, the radius of the base of the cone is 3 cm. Therefore the volume of the cone is equal to ⅓ × π × 32 × 12, which equals 36π cm3.
The ratio of Antone's height to arm span is the same as the ratio of his brother's height to arm span. About how tall is Antone?
39" EXPLANATION: This is the correct choice, 39", is found by solving the proportion: (56/60)=(36/x). In this proportion, x is Antone's height. In the first ratio, 56 is Antone's brother's arm span and 60 is his height. In the second ratio, 36 is Antone's arm span, and x, his height. Proportions are solved by cross- multiplying. When cross multiplication is performed on this proportion, we get: 56•x = 60•36. When simplified we get: 56x = 2160; then you divide each side by 56 resulting in x = 2160/56=38.57 but because the question says "about" you would use 38.57 ≈ 39".
12π/ 9 is approximately equivalent to:
4 EXPLANATION: 4 is the correct answer. Recall that a close estimation for pi is 3. So 12π = 36 and 36/9 = 4.
It took Julie ¾ of an hour to run 3½ miles. What is her average speed in miles per hour?
4 ⅔ miles per hour 3 ½ needs to be converted to an improper fraction: \frac{7}{2}27. Then divide \frac{7}{2}27 by \frac{3}{4}43. To divide by a fraction, multiply by the reciprocal: \frac{7}{2} \times \frac{4}{3} = \frac{28}{6} = \frac{14}{3} = 4 ⅔27×34=628=314=4⅔.
If each square in the decimal square has a value of 0.1, then which of the following is the decimal numeric representation of the shaded area?
4.8 EXPLANATION: If each square has a value of 0.1, then each column of 10 squares would be equivalent to 10 • 0.1 = 1. There are 4 columns shaded for a total of 4 and 8 additional small squares for 0.1 • 8 = 0.8. The total value of the shaded area is 4.8.
If the number 888 is written as a product of its prime factors in the form a3bc, what is the numerical value of a + b + c?
42 The prime factorization of 888 is 23 \times× 3 \times× 37. 42 is the correct answer because 2 + 3 + 37 is 42.
What is the value of the "7" in the number 432.0769?
7/100 In a base 10 system, each place location for a number has a value that is a power of 10. In the opposite direction of the ones place, there is the tenths place from 10⁻¹ = 0.1 or 1/10, the hundredths place from 10⁻² = 0.01 or 1/100, the thousandths place from 10⁻³= 0.001 or 1/1,000, etc. When a digit is in a specific position, its value is equal to the product of that digit and the power of 10 that is assigned to its position. Therefore, in the number 432.0769, the 7 represents 7 \times 10^{-2} = 7 × 0.01 = 0.07 = \frac{7}{100}7×10−2=7×0.01=0.07= 1007.
Jim enrolled in a new gym membership this year which required a $50 membership fee and a $70 monthly fee. Jim wrote an equation in the form y=mx+by=mx+b to find the total amount he has spent on his gym membership this year. What is the meaning of the $70 monthly fee in his equation?
70 is the slope of the line The recurring monthly fee in this scenario is the slope.
Larry started the following number pattern: 900, 888, 876, 864, ... Which number could not be a part of Larry's pattern? 736 816 756 624
736 EXPLANATION: As you look at the sequence of numbers, you should see that the numbers in the sequence decrease by 12. 736 cannot be reached by subtracting 12 any number of times. Subtracting 12 thirteen times will get 744, and fourteen times will result in 732
Which of the following word problems is the correct question to be written and solved with the given equation? 8 - 5 + 3 = 6
8 birds were sitting on a fence. 5 birds flew away and 3 more birds landed on the fence. How many birds are on the fence now?
How many 1/2 ounces are in 8 ounces?
8 divided by half = 16 ounces
What is the digit in the hundreds place in the product of 63 × 31?
9 63 × 31 = 1953. The digit that occupies the hundreds place is 9.
The Trout family just purchased a large table in the shape of a perfect circle. It is 600 cm across. John helps set one side of the table for dinner and walks exactly halfway around the table. Which of the following is closest to how far has he walked?
950 cm Since John walked halfway around the table, we are solving for half of the circumference, C = 𝜋d. Since the table is 600 cm across, d = 600 cm and therefore P = 600𝜋cm. John walked halfway around, so John walked \frac{1}{2}(600)\pi21(600)π which is about 950 cm.
Which symbol most accurately reflects the relationship between the two numbers below? 0.8 ☐ 4/3
< Since the numerator (4) is more than the denominator (3) of the fraction, it is more than 1.0.
Which of the following relations represent a function? Select all answers that apply.
A In order for a graph to represent a function, it must pass the vertical line test. A vertical line pass through any part of the graph must only intersect the graph at one point. This graph passes the vertical line test, and so it is a function. B A relation is a function if every input has exactly one output. In an (x, y) ordered pair, the x is the input and the y is the output. None of the x values repeat in the set of ordered pairs, so each x value corresponds with only one y value, and so the relation is a function.
Mr. Miller is teaching his students about the volume of rectangular prisms. He writes the formula volume = length × width × height on the board and tells his students to get to work. He notices two of his students arguing over which leg represents length and which represents width. What should he do? Select all answers that apply.
A Remind students that multiplication is commutative so it does not matter which leg they select to represent each variable. This is a good way to tie current learning back to essential properties of mathematics. Allow each student to select which means length and which represents width on their own and then do math and compare the volume they compute. The students will realize their choice does not matter when they arrive at the same solution.
Fact Family
A fact family consists three numbers which are used in two addition problems with the addends reversed (a + b = c and b + a = c) and two subtraction problems (c - b = a and c - a = b).
Linear Equation
A linear equation is an equation whose graph is a line.
A student asks the teacher, "Why is the area of a triangle formula ½bh?" Which of the following would be the most appropriate answer for the teacher to provide?
A parallelogram is the combination of two congruent triangles. Since the area of a parallelogram is bh, one half of the area of a parallelogram equals the area of a triangle. Every parallelogram is made up of two congruent triangles. The area for a parallelogram is base × height (bh), so the area of each of the two congruent triangles is one half of the parallelogram, or (1/2bh).
What should the teacher NOT consider with respect to remediation? A. Students need more work reducing fractions. B. Students need more work finding common denominators. C. Students need more practice working with equivalent fractions at the concrete level. D. Students need more practice finding equivalent factors using scale factors.
A. Students need more work reducing fractions. The student reduced the answer correctly, but the answer was not accurate.
Which of the manipulative materials below would be most suitable for teaching decimal notation to the hundredths place? Select all answers that apply. A. decimal squares B. geoboards C. tangrams D. pattern blocks E. base ten blocks
A. decimal squares Decimal squares are tag-board pictures of 10x10 grids that have portions of the 100 smaller squares shaded. E. base ten blocks
Tom wants to mentally calculate a 20% tip on his bill of $40. Which of the following is the best for Tom to use in the mental calculation of the tip? A. 40 x (200/1000) B. 40 x .02 C. 40 x (20/100) D. 40 x .1 x 2
D. 40 x .1 x 2 Tom can quickly find 10% of 40 and then double it. In this case the answer is $8 because 10% of 40 is 4 and 4 × 2 is 8.
Who is credited with creating much of what we consider geometry? A. the Babylonians B. the Indians C. the Americans D. the Greeks
D. the Greeks The ancient Greeks are responsible for much of what is studied in Geometry, most famously the Pythagorean Theorem by Pythagoras.
Mr. King gives his students this figure and asks students to determine its perimeter. About 80% of the students give the correct response, but he receives several responses of 100. How should he address the issue?
During group work, pull aside the students who go the answer incorrect and review the difference between perimeter and area. This allows students to correct their misunderstanding without delaying the progress of the entire class.
Which of the following equations are linear? Select all answers that apply.
F y=-3x-5y=−3x−5 This equation will create a straight line so it is linear. D y=x-2y=x−2 This equation will create a straight line so it is linear. Select all answers that apply. A y=- \left( 1/3 \right) x- \left( 5/3 \right)y=−(1/3)x−(5/3) This equation will create a straight line so it is linear. B y=3y=3 Since this equation only contains a constant, it is linear. Note that the equation could be written y=3+0xy=3+0x.
A fifth-grade teacher wants to assess students' ability to calculate the area of nonstandard polygons. Which of the following figures would assess a student's ability to find the area of non-standard polygons?
Figure 5 Figure 5 is a non-standard polygon because it does not have a standard geometric structure, such as a triangle, square, or rectangle do.
A teacher wants to introduce her students to three dimensional figures. Which of the following is the best first activity to do?
Give students models of various three dimensional figures and have them write what they observe about the figures. Starting with something concrete that students can interact with will help them understand what three dimensional objects are.
A teacher is introducing the concept of volume to his fifth-grade class. Which of the follow is the best initial activity?
Give students several hollow objects and ask them which they think can hold more water and why they think that. Then allow them to determine the volume of water that fits inside each object. This is an engaging activity at the concrete level of learning.
Mrs. Stallings wants her students to learn the divisibility rules before committing them to memory. Using Bloom's Taxonomy levels, how can she elevate her lesson from "Remembering"?
Have them explain how to use the rules for large numbers. This allows them to analyze the rules which is a higher level of Bloom's Taxonomy.
When considering the addition problem 1/3 + 3/8, which of the following statements is true? I. The LCD = 24 II. 3 and 8 are relatively prime
I and II EXPLANATION: Two numbers are relatively prime if they have no common factors besides 1. Because 3 and 8 are relatively prime, there is no whole number, other than 1, that will divide both 3 and 8 evenly without a remainder. Because 3 and 8 are relatively prime their least common denominator (LCD) can be found using the formula 3 • 8 = 24. Therefore 24 is the LCD.
Which of the following situations might require the use of a common denominator? I. Addition of Fractions II. Multiplication of fractions III. Subtraction of Fractions IV. Division of fractions
I and III EXPLANATION: Unless fractions have like denominators, you must always find a common denominator before you can add or subtract. Finding a common denominator requires finding a common multiple of the two (or more) denominators. It is important to note that you do not have to find the LCM or lowest common denominator, any common multiple will work. However, if the LCM is found and used, there will be considerably less simplifying to do in order to reduce to the lowest terms. Multiplication and division never require finding a common denominator.
I. Y=X II. Y=X2 III. Y=X-2 IV. Y=-2X2+3 V. Y=-3X-5 VI. Y=-1/3X-5/3 Which of the following lists ALL the equations that are linear? I, III, and IV I, II, III, and IV I, II, and IV I, III, V, and VI
I, III, V, and VI EXPLANATION: A linear equation is an equation whose graph is a line. The x and y variables contained in a linear equation are always first degree, meaning that they have no exponents - or they have exponents of 1; which are not usually written. So, x and x^1 are equivalent expressions. Of the equations I - VI, only I, III, V, and VI are linear. Equations II and IV are quadratic equations. All quadratic equations contain one second degree variable: a variable with an exponent of 2. When quadratic equations are plotted, their graphs are parabolas.
Sixteen teachers placed a book order for books to be used in their classrooms. The bill, totaling $350, is to be shared equally. How much will each teacher pay?
III EXPLANATION: This situation involves money. We are looking for an answer that can be "translated" into money. You can use this option by taking the exact answer of 21.875 and rounding it to $21.88.
A field trip to the planetarium is being planned. If each bus will hold 16 people, how many busses will be needed to transport 350 children and teachers? Which answer above is the most appropriate for this situation?
IV EXPLANATION: This is an example of a situation in which an overestimation is required. A bus can carry only 16 people. So 21 busses will not allow for everyone to have a ride and it does not make sense to take a fraction of a bus. Therefore, 22 busses must be taken.
Mr. Harris assigns his students to create a factor tree of the products of a number. One student turns in the product factor tree of 32 shown here. Which of the following best describes the error in this factor tree?
The factor tree identifies the sums and not the product factors of many of numbers. The third level of factors is the sum of the second level numbers (4 + 4 = 8). The correct factors of eight would be either (4 and 2) or (1 and 8).
As Kate runs more miles per week, her time per mile improves steadily. She begins running 4 miles per week and it takes her more than 8 minutes to complete each mile. On the graph above, where would the line begin and what direction would it travel?
The line would begin in the top left and move towards the bottom right. The mile time would be the slowest when she is running the least. The line would start at the top left and move towards the bottom right.
Which of the following best describes the polygon shown?
a convex pentagon A pentagon has 5 sides. A convex polygon has no angles greater than 180°. Another way to think of how to identify a convex pentagon is that it has no angles pointing inward.
Quadratic Equations
contain one second degree variable: a variable with an exponent of 2. When quadratic equations are plotted, their graphs are parabolas.
Median
found by first putting all the numbers in order from least to greatest the median can be either the middle number that has an odd amount of numbers or when there is an even set of numbers you find the average of the middle two numbers
Composite number
is a positive integer that has a positive divisor other than one or itself
Miguel is playing with his model cars. Which transformation is represented in the picture of his cars?
rotation This is a rotation because the top car has been rotated 180° about the point at the center, which is the transformation between the two cars, leaving you with this position and orientation of the bottom car.
Mean
the mean is found by adding all of the numbers together and dividing that sum by the number of numbers. Mean is also known as the average
Mode
the most frequent number
A student is instructed to draw a four-pointed geometric shape on an xy-plane. After the shape is drawn, the student is instructed to add 5 to each x-coordinate and add 3 to each y-coordinate. Which of the following did the student perform?
translation A translation is simply moving the object from one point on a plane to another point on the plane. The shape of the object remains the same; the object is simply moved along the plane.
If 15 ml is equivalent to ½ oz, which equation could be used to find x, the number of ml in 1 cup? x = 15 / 1/2 + 8 x = 15 * 1/2 * 8 x = 15 / 8 * ½ x = 15 * 8 / ½
x = 15 * 8 / ½ EXPLANATION: There are 8 oz in one cup, but the problem references ½ oz. So, how many ½ ounces are in 8 oz? 8 ÷ 1/2 = 16. If each 1/2 ounce is equivalent to 15 ml, then 15 • 16 would give us the number of ml in 8 ounces or 1 cup = 240 ml. x = 15 • 8 ÷ ½ = 120 ÷ ½ = 240 ml.
Which expression best represents y in terms of x? y = x - 2 y = 2x - 3 y = 1 - 2x y = -(x + 4)
y = 1 - 2x EXPLANATION: The expression y = 1 - 2x is the only expression that satisfies ALL of the possibilities on the chart. By plugging the value of X, found in the chart, into the equation, the result will equal the value of Y indicated in the chart.
Mr. Harris wants to divide a long string of gummy candy into pieces for his class. He folds the gummy candy in half and makes one cut along the end of the folded string, so that he now has three pieces. He asks his students to create an equation so he can find out how many cuts he should make in the candy to have enough for all the students. Which of the following equations represents the data in the table? Number of Cuts Number of Pieces 0 1 1 3 2 5 3 7
y = 2x + 1 The data is correctly represented by the equation y = 2x + 1, where the number of cuts is "x". The gummy candy will always have at least one piece. Each additional cut adds two pieces, but because the candy is folded in half, there is always one extra piece (the end piece) that must be added to every two pieces made by a cut. For a concrete example, take a string, fold it in half, and then make a cut.
Which of the triangles below is the best example of an isosceles triangle?
This is an isosceles triangle because two sides are of equal length.
The median home cost in the US in 1975 was $40,000. In 1990, an equivalent home cost $140,000. The trend continued into 2005 when the median home cost in the US was approximately $240,000. Assuming that this data's relationship is steady in the future, what is a reasonable estimate of the median home cost in the US in 2025?
$373,333 The data points (1975, 40), (1990, 140), and (2005, 240) all show a rise in home price of $100,000 for every 15 year passage of time. This constant change in dependent variable (y) compared to change in independent variable (x) is the slope characteristic of linear data. Therefore, if the linear trend continues, in 2020 (15 years after 2005), the median home price could be expected to be $340,000 ($100,000 more than $240,000). The year 2025 is 5 years after 2020, which is just 1/3 as much as the previous 15 year increments. Because of the linear trend of this data with its slope of +$100,000/+15 years, it would be reasonable to expect an increase in value of an additional 1/3 of $100,000, or approximately $33,333 past $340,000. Therefore, the final answer is $340,000 + $33,333 = $373,333. It would also be valid to reduce the slope of +$100,000/+15 years to a unit rate of $6,666.67/1 year, multiply that slope by the 20 years that pass between 2005 and 2025 to get $133,333.33, and add that amount of change to the last known data value of $240,000 in 2005 to get $373,333.33 in 2025.
A baseball mitt is on sale for 30% off. If the regular price is $78, what is the sale price, excluding tax?
$54.60 EXPLANATION: There are a couple of different ways to work this problem. You can simply take 30% of $78 (0.3 • 78 = 23.4 or $23.40) and find out how much you will save, then subtract the savings from the original price: $ 78 - 23.40 = $ 54.60. You can also obtain the answer by: 100% - 30% = 70% and 0.70•78 = $ 54.60.
James has saved $35.25. He wants to save his money to buy a bicycle that costs $85.00. His brother's bike cost $92.00. If sales tax is 8%, about how much more must he save to afford purchasing his bike, including tax?
$60 EXPLANATION: . One quick measure to find most sales tax amounts is add a tenth to the original price, so on $85.00 tax would come to about $8.50. $85.00 + $8.50 = $93.50. Notice this is an overestimate so James' target will be a bit more than he actually needs. James needs to save about $93. If he has saved about $35, he will need an additional $58. ($93 - $35 = $58) Therefore, if rounded up this would be the best choice: $60. When dealing with money, generally an overestimate is more reasonable.
The ΔABC in the coordinate plane will be translated 3 units to the right and then 4 units down. Which of the following points correctly expresses the location of vertex C after the translations?
(-2, -2) The original coordinates of point C are (-5, 2). A translation to the right increases the first coordinate, in this case, by 3 units: -5 + 3 = -2. A translation down decreases the second coordinate, in this case, by 4 units: 2 - 4 = -2. The resulting new coordinates for the point C after its translations is (-2, -2).
Trapezoid ABCD is shifted 3 units to the left and 5 units up. What will be the coordinates of Point B after the shift?
(5, 12) Point B is located at (8, 7). To shift left, subtract 3 from the x value. To shift up, add 5 to the y value.
The Huang family is building a circular swimming pool in their backyard with a diameter of 8 meters. They wish to place a decorative rock border along the edge of the pool from point A to point B, as shown by the dotted curve in the diagram. As seen in the diagram, points A and B are directly across from one another. Approximately how many linear meters of rock will be needed to form the decorative border along the edge of the pool from point A to point B?
12.6 m The length of the edge of the pool from point A to point B is a portion of the perimeter of the pool. Because points A and B are directly across from each other and the segment which connects them passes through the center of the circular swimming pool, the segment AB is a diameter, and so the portion of the perimeter of the pool that will have a decorative rock border is exactly half. Because the pool is in the shape of a circle, its perimeter is calculated using the formula for C, the circumference of a circle. C = πd. The circumference of the swimming pool is simply C = π×8 ≈ 3.14×8 ≈ 25.12 meters. Given that the circumference is ~25.12 meters, half that amount is 12.56 meters. Therefore, the best answer option is 12.6.
In Anytown ISD, 13 out of every 20 students ride the bus. Which ratio compares the number of students who ride the bus to those who do not? 13: 20 7: 20 7: 13 13:
13: 7 EXPLANATION: This is the equation that compares riders to non-riders. 13 out of 20 ride the bus. This means that the complement of this relationship, those who do not ride the bus, is 20 - 13 = 7. So the ratio of those who ride the bus TO those who do not ride the bus is: 13:7. When writing ratios, remember that order is important. 13:7 is not the same as 7:13 just like the fractions 13/7 ≠ 7/13 (recall that another way to write the ratio 13:7 is 13/7).
The cement pipe used in a storm drain system is an 8-foot long right circular cylinder with a wall thickness of 3 inches and an outside diameter of 24 inches. Which of the values below best approximates the volume, in cubic feet, of the interior of the pipe? (The formula for the volume, V, of a right circular cylinder with radius r and height h is V = πr2h.)
14.1 If the diameter of the pipe is 24 inches, then the radius of the pipe is 12 inches (because d = 2r, where d = diameter and r = radius). If the wall of the pipe has a thickness of 3 inches, then the interior of the pipe has a radius of 12 - 3 = 9 inches. Because the calculation is to be performed in cubic feet and not in inches, 9 inches must be converted to feet. This can be done using the conversion factor 1 foot/12 inches to cancel inches and bring in feet. 9 inches × 1 foot/12 inches = 9 feet/12, which reduces to ¾ feet, or 0.75 feet. Therefore, the radius to use in the volume calculation is 0.75 feet. The length of the cement pipe was given as 8 feet. Accordingly, the value to use as the height of the right circular cylinder is 8 feet. Finally, substitutions can be made and the appropriate calculations performed, using 3.14 as an approximation for π in the formula V = πr2h so that the volume of the interior of the pipe is discovered to be approximately 14.13 cubic feet. The best answer to select from the options, therefore, is 14.1. V = π(0.75)2(8)= π(0.5625)(8)= π(4.5)≈14.13 cubic feet
What is the volume of the provided right rectangular prism?
1¼ m3 The volume of a right rectangular prism is equal to l \times w \times hl×w×h. Therefore, the volume of this right rectangular prism is 2.5 \times 0.5 \times 1 = 1.252.5×0.5×1=1.25 m3.
What is the area, in square units, of the figure?
45 units2 The area of the figure can be found using the information from the image. One way to address the question is to notice that the pentagon in the diagram is composed of a 6 × 6 unit square and a triangle with a length of 6 units and a height of 3 units. The area of the square and the triangle can be calculated and then added together to get the total area of the pentagon. The area of a square is found with the formula A = s2, where A is the area and s is the length of a side. Or, because a square is a specific type of rectangle, the area of the square can be calculated using A = bh or A = lw, where b is the base, h is the height, l is the length, and w is the width. In this case, the side length is 6 units, which can be seen either through counting boxes in each direction or using subtraction of coordinates on the ends of the square in the x-direction (-5 to +1 is 1 - (-5) = 1 + 5 = 6) and in the y-direction (-2 to 4 is 4 - (-2) = 4 + 2 = 6). Therefore, the area of the square is 62 = 36 square units. The area of a triangle is found with the formula A = 1/2bh, where A is the area, b is the base, and h is the height of the triangle. The base of this triangle can be measured by counting squares along the vertical line x = 1, or by subtracting the highest and lowest y-coordinates along the line x = 1: 4 - (-2) = 4 + 2 = 6 units. The height of the triangle can be measured by counting squares along the horizontal line y = 1, or by subtracting the x-coordinates from the points (1, 1) and (4, 1): 4 - 1 = 3 units. Therefore, the area of the triangle is 1/2(6)(3) = 3 × 3 = 9 square units.The total area of the pentagon is 36 + 9 = 45 square units.
A sixth grade class is asked to estimate the answer to the following question: 75.8 + 326.79 + 488.92 / 11 = _____. Which of the following would be the best answer?
450 EXPLANATION: This question checks for understanding of order of operations. Since this problem involves both addition and division, division MUST be done first. Correct order of operations is Parenthetical expressions, exponents, multiplication/division (left to right whichever comes first), and finally addition/subtraction (left to right whichever comes first). So the 488.92 / 11 is performed first. Since this is an estimation problem, it would be about 500 / 10, which is 50. Then 75.8 + 326.79 is about 400. Adding the 50 and 400 we get about 450
What is the area of the figure provided?
49.5 cm2 The area of the figure is l × w = 5.5 × 9 = 49.5 \text{ cm}^2l×w=5.5×9=49.5 cm2.
The Booster Club at Martin MS is selling spirit buttons for homecoming. The buttons cost $0.75 to make and will be sold for $2 each. How many buttons, b, must be sold to make a profit of $500?
500 = $2 b - $0.75b EXPLANATION: The profit is equal to the selling price minus whatever costs are applicable. So if the profit is to be $500 then enough buttons must be sold to reach that profit. If we are selling the buttons for $2 each but it costs $0.75 to make each one, then there is a profit of $2 - .75 or $1.25 on each button. How many buttons will we have to sell to reach $500 profit: $500 = $1.25b. This option is: $500 = $2b - 0.75 c = 1.25 b. So, the Booster Club will have to sell 400 buttons.
A teacher engages her class in a discussion of the coordinate plane. The students are asked to identify the quadrants, the coordinate axes, and the mathematical notation for various points in the plane. Students are asked to develop a way to quickly identify the quadrant in which various points lie. Which of the following objectives is the teacher most likely trying to address with this lesson? A. developing precise mathematical language when expressing mathematical ideas B. demonstrating how to model and solve real-world problems using mathematics C. augmenting an understanding of estimation and its appropriate uses D. encouraging student use of mathematics manipulatives and technological tools
A. developing precise mathematical language when expressing mathematical ideas The precise use of math language is required when using and describing information in the coordinate plane.
A fifth-grade teacher is beginning a unit on equivalent fractions with her students. If this is an introductory lesson, which of the following activities would be the most effective in helping the students understand the concept of equivalent fractions? A. use pattern blocks to model fractions equivalent to 1/2 of the hexagon B. begin with the concept that 50c is 1/2 of $1; 25c is 1/2 of 50c; 5c is 1/2 of 10c C. find as many fractions as possible equivalent to 1/2 in one minute D. compare pictures showing 1/2 of a variety of different objects
A. use pattern blocks to model fractions equivalent to 1/2 of the hexagon Since this is an introductory activity, concrete, proportional manipulative materials like this should be used for concept development. It is important not to rush past this step and to use a variety of different materials to develop and reinforce understanding of this concept.
What statement is true about the figures?
The area of the 2 figures is congruent, but the perimeter of the second figure is greater. The area is the same because the shapes occupy the same amount of space, but the perimeter is greater for the second figure because it has 2 new external lines where it was cut.
Jana is playing a game called Guess My Rule. Her friends choose a number (input), Jana subtracts 3 and tells her friends the result (output). The game continues until one of her friends can predict what their output number will be and can correctly state Jana's rule. Which graph correctly shows possible values of input/output for Jana's rule?
B EXPLANATION: If we examine Jana's rule, (input a number, subtract 3, give output), then if we input "0", the output would be 0 - 3 = -3. Only option B has the ordered pair (0,-3) as a point. As a side note, the ability to eliminate choices is an excellent test taking and problem solving strategy. If you can eliminate some of your options, and you have to make a guess, then your odds of getting the correct answer is greatly increased. In this particular case, the odds of getting the correct answer to the original question if you had to arrive at your choice by guessing, would be only 25%. By eliminating two choices, your odds of purely guessing, have increased to 50%.
Mrs. Peters teaches a class with native English speakers and English language learners (ELL). She needs to introduce new mathematics terminology for the upcoming instructional unit. Which of the following would be the best strategy for implementing the new terminology? A. Have students practice using the terminology with each other using English in their conversation. B. Mrs. Peters explains each term to the class and then uses the term in a variety of sentences. C. Because math terminology can be difficult, have students create their own spelling for the new terms. D. Have each student write down the new word with the corresponding definition.
B. Mrs. Peters explains each term to the class and then uses the term in a variety of sentences. This is the best strategy because Mrs. Peters will present the terms in context and then provide various examples. This will provide both the native English speakers and ELL students an opportunity to be exposed to the new terms in a variety of contexts.
Mr. Kopko's class is learning about long division. He asks students to use the calculator to check their results. On the following division problem several students are confused by their results. 22/7 = 3R1 = 3.142857 How should Mr. Kopko instruct students to deal with remainders? A. Tell them to multiply their entire result by the numerator and as long as it is identical to the original they have the right answer. B. Tell them to multiply the decimal part of the solution by the denominator to see if it matches the remainder. C. Tell them that if there is any remainder there will be a decimal result in the calculator and not to worry about it. D. Tell them to multiply their entire result by the denominator and as long as it is identical to the original they have the right answer.
B. Tell them to multiply the decimal part of the solution by the denominator to see if it matches the remainder. Multiplying the decimal part of the solution by the denominator should match the remainder and allow the students to check their solutions.
The formula for permutations is: nPr = n!/(n-r)! while the formula for combinations is: nCr = n!/r!(n-r)! As a warm-up, Ms. Caulkins asks her students to compare these two formulas and identify the differences and similarities before she has taught what the formulas are used for. What is her goal in this exercise? A. practice writing fractions with factorials B. activating prior knowledge of reducing fractions C. following classroom procedures that include a daily warm-up D. allowing students to understand the lesson before teaching it
B. activating prior knowledge of reducing fractions Allows them to build on prior concepts with new material.
Mrs. Taylor has 3 ½ yards of material to make spirit flags for the high school football game. Each flag requires ¼ yard strip of material. Which picture above is an accurate picture of this situation and its solution?
C EXPLANATION: This illustrates each yard of the 3 yards divided into 4 parts (1/4 yd) and the ½ yd separated into ¼ yd pieces. There a total of 14 pieces, each equivalent to ¼ yd, so 14 flags can be made.
Maria baked 6 dozen cookies for her classmates. There are 28 students in her class; each child received 2 cookies and Maria gave 6 cookies each to her teacher and her principal. Which equation could be used to find C, the number of cookies she had left over? C = 6 * 12 - 2 * 28 + 2 * 6 C = 6 * 12 - (2 * 28 - 2 * 6) C = 6 * 12 - (2 * 28 + 2 * 6) C = 6 * 12 - 2 * 28 - 6
C = 6 * 12 - (2 * 28 + 2 * 6) EXPLANATION: 6 * 12 gives us the total number of cookies in 6 dozen: 72 cookies. Then if each child receives 2 and there are 28 children, 2 * 28 = 56; if the teacher and the principal each receive 6, that is another 12. So we have 72 - 56 - 12 = 4. Maria will have 4 cookies left.
A math teacher plans her instructional delivery method on resolving the difficulty students have distinguishing between mode and median. She plans to have students first work alone calculating the mode and median of sets of performance results from the school track team. Next, her students will work in groups of 2 or 3 to discuss and interpret their results, and record a summary of the significance of the results on whiteboards. Finally, the groups will present their summaries to the class, along with a teacher-led discussion of the findings. By planning such an activity, the teacher demonstrates that she understands: A. how to use a variety of questioning strategies that encourage mathematical discourse and help students analyze and evaluate their mathematical thinking. B. how technological tools and manipulatives can be used to assist students in developing mathematical thinking. C. how to apply a variety of instructional delivery methods that can help students develop their mathematical thinking. D. how students' prior mathematical knowledge can be used to build conceptual links to new knowledge.
C. how to apply a variety of instructional delivery methods that can help students develop their mathematical thinking. The teacher's plans show she understands how to use individual, small-group, and large-group instruction methods to help students develop their mathematical thinking.
What is the next shape in this pattern?
Circle The pattern is square, 2 circles, triangle, 2 circles, pentagon, 2 circles. After the last square, there was only 1 circle, so there needs to be another circle before moving on to another triangle.
Which of these statements correctly describes the coefficients and degrees of the polynomial shown? 5x7 + 5x5 - 3x - 2
Coefficients: 5, -3, -2 Degrees: 7, 5, 1, 0 The coefficients are the whole numbers preceding a variable. The degrees are the exponents. For example, 5x7 has a coefficient of 5 and a degree of 7. For the degrees, unwritten exponents such as -3x are noted as 1 (because x1 = x) and constants are noted as 0 degrees (because x0 = 1).
Students were asked to name point B. Zoe wrote that B equals -2.4, Julio responded that it equals -2 ¼, Sage said that it equaled -3 ¾ and Kaitlyn said that Zoe and Julio were both correct because the numbers were equivalent. Which student was correct in their response?
Julio Point B is between -2 and -3. It is halfway between -2 and -2 ½ which means it is located at -2 ¼.
Susan 5+2=7 Bob 3+6=9 Jose 9-4=5 Renee 7-2=5 Raul 7+2=9 Susan's second grade class is studying fact families. Each student is given a fact card and asked to find other members of their family. Above are the cards received by 5 students. Which students have cards that belong to the same fact family?
Susan and Renee EXPLANATION: A fact family consists three numbers which are used in two addition problems with the addends reversed (a + b = c and b + a = c) and two subtraction problems (c - b = a and c - a = b). In this case the digits 2, 5, and 7 and only those digits, must be used in each addition and subtraction fact.
Which situation could best be represented by the equation: 12x = 54?
Marty made car payments on her car for 54 months until it was paid off. What is x, the number of years it took Marty to pay off her car? To get the total months, 54, multiply the number of years, x, by the number of months in a year, 12; 12x = 54.
Which situation could best be represented by the equation: 12x = 54? Marty earns $12 for typing a paper. If her rate is $ 54 per hour, what is x, the number of hours it actually took to type the paper? Marty collected 12 dozen eggs every day for 54 days. What is x, the total number of dozens of eggs she collected? Marty had 54 minutes left on his cell phone plan. If he uses 12 minutes, what is x, the number of minutes remaining on his cell phone plan? Marty made car payments on her car for 54 months until it was paid off. What is x, the number of years it took Marty to pay off her car?
Marty made car payments on her car for 54 months until it was paid off. What is x, the number of years it took Marty to pay off her car? EXPLANATION: This is the correct situation for the given equation. 54 months divided by 12 months per year gives us 4 ½ years.
Which of the following activities is the best way for elementary students to learn how to write an equation for a line of best fit on a scatter plot?
Students create a scatter plot of data from a simple experiment where they compare the height to the arm span of various students in class. Students then answer guided questions about the correlation between arm span and height, make predictions, create a line a best fit, and write an equation for the line. This activity shows students the practical application of lines of best fit. Students see the relationship between an experiment, a scatter plot, a line of best fit, and the equation of the line. Students are guided through all aspects of creating the line and using the equation to make predictions.
