Practice Test: Mathematics, Science, and Technology/Engineering (278)
Answer Key, Sample Responses, Evaluation Chart, and Score Calculation Tool
Answer Key
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| Question Number | Your Response | Correct Response |
Related Objectives and Rationale |
|---|---|---|---|
| 1 | B |
Objective 011 Correct Response: B. Students develop their understanding of counting with one-to-one correspondence as they learn to assign a number to every object in a collection. Students can develop their understanding of this concept with counters and a ten frame by picking up a counter and adding it to the ten frame as they count each number from 1 to 10. Incorrect Response: A. This activity does not emphasize the relationship between the counters in the ten frame and the numbers the students count aloud as effectively as Correct Response B. In this activity, the students do not touch the counters directly, and the instructional goal also focuses on counting the total number of counters the teacher placed in the ten frame (i.e., the cardinality of the counters in the ten frame). Incorrect Response: C. This activity supports students' understanding about the concept of hierarchical inclusion because it guides students to realize a collection of 6 objects will always be contained within a collection of 8 objects but that a collection of 10 objects will never be contained within a collection of 8 objects. Incorrect Response: D. This activity supports students' understanding of conservation of number because they observe that the number of objects in a collection does not depend on how the objects are arranged. |
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| 2 | A |
Objective 011 Correct Response: A. The expression 24 times the quantity 10 to the power of 0 plus 10 to the power of negative 2 equals 24 times the quantity (1 plus 1 hundredth) because 10 to the power of 0 equals 1 and 10 to the power of negative 2 equals 1 hundredth) This expression can be evaluated by either multiplying 24 times 1 point 0 1 or by applying the distributive property to obtain 24 times 1 plus 24 times 1 hundredth equals 24 and 24 hundredths . Both strategies yield 24.24. Incorrect Response: B. This is incorrect because 24 hundredths times 1000 is equal to 2 point 4 times 100 which equals 24 times 10 which equals 240 Incorrect Response: C. Every digit in a base-10 number is associated with a unit, such as tens, ones, tenths, or hundredths. For example, the expanded form of 24.24 shows the values its digits represent (i.e., 20 plus 4 plus 0 point 2 plus zero point zero four and the value represented by 2 in the tens place is not 10 times the value represented by 4 in the ones place. Incorrect Response: D. This is true about the first 2 from the left because 20 minus 4 = 16 , but this is not true about the second 2 from the left because 0 point 2 minus 0 point 0 4 = 0 point 0 1 6 |
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| 3 | A |
Objective 011 Correct Response: A. The top number line spans from 4 to 5 in 10 segments of equal length, so each segment has a length of 0.1 unit. The dashed lines that extend on either side of the seventh segment indicate that point A is somewhere between the values of 4.6 and 4.7. The second number line magnifies the distance between these values and again forms 10 segments of equal length, with each segment having a length of 0.01 unit. The following graphic shows how each successive number line helps specify the value of point A. The fourth number line indicates that point A is between 4.6827 and 4.6828. Only the value of 4.68275 is within this range.
The series of number lines from the stem is shown, this time with values labeled at the minimums and maximums. The first number line goes from 4 to 5 in tenths, the second number line goes from 4.6 to 4.7 in hundredths, the third number line goes from 4.68 to 4.69 in thousandths, and the final number line goes from 4.682 to 4.683 in hundred thousandths. Incorrect Response: B. The value of 4.69375 appears to the right of the rightmost tick mark in the third number line and does not fall within the region shown in the fourth number line. Incorrect Response: C. The value of 4.78285 would appear to the right of the rightmost tick mark in the second number line and does not fall within the regions shown in the third and fourth number lines. Incorrect Response: D. The value of 4.79385 would appear to the right of the rightmost tick mark in the second number line and does not fall within the regions shown in the third and fourth number lines. |
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| 4 | A |
Objective 011 Correct Response: A. A factor of a number is a number that divides a given number exactly, leaving no remainder. The factors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105. Since there are 3 prime factors (i.e., 3, 5, and 7), the ratio of factors that are prime numbers to all factors is 3 to 8. Incorrect Response: B. This response may be the result of forming the ratio of prime factors to factors that are not prime. Incorrect Response: C. This response may be the result of considering 1 to be a prime factor. 1 is neither prime nor composite. Incorrect Response: D. This response may be the result of considering 1 to be a prime factor and simplifying a ratio of 4 prime factors and 4 composite factors. |
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| 5 | B |
Objective 011 Correct Response: B. The decimal numbers can be sorted easily by comparing place values: 0.07, 0.077, 0.7, 0.707, 0.71. The fractions can be sorted by converting each to its decimal form: 7 elevenths equals 0 point 6 3 repeating which equals 0 point 6 3 6 3 6 3 and 7 eighths equals 0 point 8 7 5 Note that determining the tenths place value of each fraction is sufficient to determine where they should appear in the sorted set. Incorrect Response: A. This response can be determined to be incorrect by comparing the place values of the numbers or by applying other straightforward number sense strategies. For example, 0.7 is not less than 0.07, and 7 eighths is not less than 7 elevenths . Incorrect Response: C. This response can be determined to be incorrect by comparing the place values of the numbers or by applying other straightforward number sense strategies. For example, 0.7 is not less than 0.07. Incorrect Response: D. This response can be determined to be incorrect by comparing the place values of the numbers or by applying other straightforward number sense strategies. For example, 7 elevenths is not more than 7 eighths . |
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| 6 | D |
Objective 011 Correct Response: D. The total amount of money can be equally shared among 5 people by giving each person 16 dollars divided by 5 equals 3 dollars and 20 cents This division operation can be represented on a number line by starting at 16 and taking 5 jumps of size 3.2 to the left. The final jump ends at 0 because all of the money has been shared among the people. Incorrect Response: A. This number line representation begins at 16 and shows 3 equal jumps of size 5 to the left. This could be chosen because of a misconception about how the remainder should be treated in this context. Incorrect Response: B. This number line representation begins at 16 and shows 5 equal jumps of size 3 to the left. This could be chosen because of a misconception about how the remainder should be treated in this context. Incorrect Response: C. This number line representation begins at 16 and shows 3 equal jumps of size 5 to the left. This could represent dividing 16 dollars by giving 5 dollars to 3 people and 1 dollars to 1 other person, but this does not match the context. |
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| 7 | C |
Objective 011 Correct Response: C. The exact cost of the omelet is 4 dollars and 25 cents plus 3 eggs times 259 dollars over 360 eggs plus 4 dollars and 90 cents Simplifying the middle term of the expression as 3 eggs times 259 dollars over 360 eggs equals 259 dollars for 120 eggs or approximately 260 dollars for 120 eggs leads to the expression 4 point 2 5 plus the quantity 260 divided by 120 plus 4 point 9 0 This differs from the exact value by 1 1 hundred twentieth , or less than one cent. Incorrect Response: A. The actual cost of the eggs is 3 eggs times 259 dollars for 360 eggs is approximately 2 dollars and 16 cents Estimating this cost as 3 eggs times 259 dollars for 360 eggs is approximately 360 dollars over 360 which equals 1 dollar substantially underestimates the cost of the omelet. Incorrect Response: B. This expression could be the result of approximating the cost 3 eggs times 259 dollars for 360 eggs is approximately 2 dollars and 16 cents as 1 egg times 259 dollars over 360 eggs is approximately 260 dollars over 360 which is approximately 72 cents This estimation introduces a very large error in the calculation. Incorrect Response: D. This expression overestimates the price increase by 75 cents (from rounding 4 dollars and 25 cents up to 5 dollars the ingredient cost by 10 cents (from rounding 4 dollars and 90 cents to 5 dollars and the total cost of the eggs at 3 dollars (overestimating that cost by 84 cents The total error introduced by these overestimations greatly exceeds that of Correct Response C. |
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| 8 | C |
Objective 012 Correct Response: C. A person who walks 3 eighths mile in 9 minutes walks at a rate of 9 divided by 3 eighths minutes per mile. Let the 1 whole in the diagram represent 1 mile. When this is broken into 8 parts, each part represents 1 eighth mile, and a group of 3 circled parts represents 3 eighths mile. The bottom figure in the diagram shows that 1 whole can be decomposed into 2 full groups (i.e., 2 times 3 eighths = 6 eighths of a mile, which takes 2 times 9 minutes = 18 minutes) and 2 thirds of a full group (i.e., 2 thirds times 3 eighths of a mile equals 2 eighths of a mile which takes 2 thirds times 9 minutes = 6 minutes). The total time that this diagram represents, 24 minutes, is consistent with the result predicted by the standard fraction division algorithm: 9 divided by 3 eighths equals 9 times eight thirds which = 24. Incorrect Response: A. The distances in this problem could be represented by the equation X plus x plus 8 ninths x equals 2 and 2 thirds, where x equals 12 thirteenths mile and the shorter of the two distances is 8 ninths times 12 thirteenths equals 32 thirty ninths mile. The diagram shown would not be appropriate to use to illustrate the relationships described by this equation or the distances that can be computed from it. Incorrect Response: B. The equation 1 equals one third divided by 10 times t describes the relationships in this problem because distance = rate times time. The diagram shown would not be appropriate to use to illustrate the relationships described by this equation. Incorrect Response: D. The equation 3 x = 1 and 1 third describes the relationships in this problem. The diagram shown would not be appropriate to use to illustrate the relationships described by this equation. |
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| 9 | D |
Objective 012 Correct Response: D. The diagram represents 5 and 3 fourths as 5 groups of 4 fourths (i.e., 5 groups of 1, where each 1 is shown as 4 fourths ) and 1 group of 3 fourths . This can be used to illustrate the relationship 5 and 3 fourths is equal to the quantity 4 times 5 plus 3 all divided by 4 . Incorrect Response: A. The compound fraction 1 over the fraction 4 twenty thirds is equivalent to 1 divided by the fraction 4 twenty thirds This can be interpreted as the number of groups formed by breaking a whole equal to 1 into smaller groups of 4 twenty thirds . It can also be interpreted as the amount of an unknown whole for which 1 represents an amount shared equally by 4 twenty thirds parts of that whole. Neither relationship is effectively represented by the diagram. Incorrect Response: B. This response is incorrect because the equation 23 fourths divided by 4 twenty thirds = 1 is false: 23 fourths divided by 4 twenty thirds is equal to 23 fourths times 23 fourths which equals 529 sixteenths . Incorrect Response: C. The mixed number 5 and 3 fourths can be expressed equivalently as 5 plus the fraction 3 times 6 over 4 times 6 which equals 5 and 18 twenty fourths but the diagram only shows fractions of fourths. The diagram would need to show each rectangular whole being broken into 24 parts to represent fractions of twenty-fourths. |
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| 10 | A |
Objective 012 Correct Response: A. When the fractions of the expression are expressed in terms of their least common denominator, the expression becomes 3 and 100 1 hundred twentieths plus 2 and 12 1 hundred twentieths plus 4 and 30 1 hundred twentieths minus 5 and 15 1 hundred twentieths equals 4 and 127 1 hundred twentieths which equals 5 and 7 one hundred twentieths. . The student's estimate gives a result of 4 plus 2 plus 4 minus 5 = 5 This estimate is 7 one hundred twentieths less than the actual value of the expression. Incorrect Response: B. This response could be the result of evaluating the expression 20 1 hundred twentieths plus 12 1 hundred twentieths plus 30 1 hundred twentieths minus 15 1 hundred twentieths equals 47 1 hundred twentieths . Incorrect Response: C. This response could be the result of evaluating the expression 20 1 hundred twentieths plus 12 1 hundred twentieths plus 30 1 hundred twentieths plus 15 1 hundred twentieths equals 77 1 hundred twentieths . Incorrect Response: D. This response could be chosen as a result of misinterpreting the difference between the estimated value and the actual value. |
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| 11 | A |
Objective 012 Correct Response: A.The shaded area in the first bar is approximately twice that of the unshaded area, the unshaded area in the second bar is approximately 6 times greater than the shaded area, and the ratio of shaded to unshaded area in the third bar is consistent with the ratio 4 to 5 because it has only slightly less shaded area than unshaded area. Incorrect Response: B. This response can be determined to be incorrect by comparing the shaded to unshaded areas in the first two bars. There is too little shaded area in the first bar to be consistent with a 2 to 1 ratio and too much shaded area in the second bar to be consistent with a 1 to 6 ratio. Incorrect Response: C. This response can be determined to be incorrect by comparing the shaded to unshaded areas in the second and third bars. There is too little shaded area in the second bar to be consistent with a 1 to 6 ratio (i.e., the amount of unshaded area is more than 6 times the amount of shaded area) and too much shaded area in the third bar to be consistent with a 4 to 5 ratio (i.e., the ratio indicates that there should be more unshaded area than shaded area but this is not true). Incorrect Response: D. This response can be determined to be incorrect by comparing the shaded to unshaded areas in the second bar. The bar is nearly completely shaded. However, the 1 to 6 ratio indicates that the bar should be mostly unshaded. |
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| 12 | C |
Objective 012 Correct Response: C. The cost for a box of 40 containers can be found by solving the proportion 3 dollars and 25 cents over 6 servings equals x over 40 servings with proportional reasoning or by a direct computation. For the latter approach, multiply both sides of the equation by 40 to obtain x = 40 times 3 point 2 5 over 6 equals 130 over 6 equals 21 dollars and 67 cents Incorrect Response: A. This response could be chosen as the result of scaling the ratio 3 dollars and 25 cents over 6 servings times the ratio 8 over 8 equals the ratio 26 dollars over 40 servings and applying an incorrect scale of 4 fifths to obtain the result 20 dollars and 80 cents . Incorrect Response: B. This response could be chosen as the result of using a scale factor of 6.5 to form approximately 40 servings: The ratio 3 dollars and 25 cents over 6 servings times the ratio 6.5 over 6.5 equals the ratio 21 dollars and 13 cents over 39 point 5 servings. . Incorrect Response: D. This response could be chosen as the result of using an incorrect unit cost per serving and then scaling the unit cost to 40 servings: The ratio 3 dollars and 25 cents over 6 servings equals the ratio 55 cents over 1 serving. This ratio times the ratio 40 over 40 equals the ratio 22 dollars over 40 servings. . The value of 55 cents per serving slightly overestimates the unit price and this error is then increased by a factor of 40. |
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| 13 | B |
Objective 012 Correct Response: B. The discounted price of 104 dollars is 80% of the original price: 104 = 0.8x. The original price is 104 divided by 0.8 = 130 dollars Incorrect Response: A. This response could be the result of calculating the original price as 0 point 2 0 times 104 plus 104 Incorrect Response: C. This response could be the result of calculating the original price as 0 point 8 0 times 104 plus 104 Incorrect Response: D. This response could be the result of calculating the original price as 104 divided by 1 point 2 plus 104 and rounding the result up to 191 dollars |
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| 14 | A |
Objective 013 Correct Response: A. If m and k are integers, then 2m and 2k are even integers because they are multiples of 2, and 2k + 1 is an odd integer because it is 1 more than an even integer. When 2m multiplies the terms 2k and 1 by the distributive property, it creates the products 4mk and 2m, respectively. Both products must be even because they are multiples of 2 (note that 4mk is a multiple of 2 because any multiple of 4 is also a multiple of 2). Therefore, the equation 2 m times open paren 2 k plus 1 close paren equals 4 m k plus 2 m can be used to justify the claim that the product of an even integer and an odd integer is an even integer. Incorrect Response: B. This response is not correct because the right side of the equation cannot represent an odd integer. If m and k are integers, then 4mk + 2m represents an even integer: the terms 2m and 4mk, being multiples of 2, represent even numbers, and any sum of even numbers is also even. Incorrect Response: C. This equation cannot justify this claim because not every term on the right side of the equation is a multiple of 4. While 4mk would be a multiple of 4 if m and k were whole numbers, this would not necessarily be true for 2m (i.e., , 2 m divided by 4 equals 1 half m which is not necessarily a whole number if m is a whole number). Also, consider the counterexample that the product of the even integer 6 and the odd integer 3, 18, is not a multiple of 4. Incorrect Response: D. This claim cannot be justified by the equation because not every term on the right side of the equation is a multiple of 4. See Incorrect Response C. Also, consider the counterexample that 5 and 6 are both consecutive integers but their product, 30, is not a multiple of 4. |
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| 15 | D |
Objective 013 Correct Response: D. The fitness center is open a total of 16 hours as it operates from 5:00 a.m. to 9:00 p.m. The graph in this response fits the description that a relatively large number of people attend the gym for hours 1 to 3 (5:00 a.m. to 8:00 a.m.) and hours 13 to 14 (5:00 p.m. to 7:00 p.m.) and that fewer people attend the gym for hours 4 to 12 (8:00 a.m to 5:00 p.m.) and the hours 15 to 16 (7:00 p.m. to 9:00 p.m.). The graph also shows a small increase in the number of people around lunch hours (where the graph rises almost to the fourth bar), and this value is a little less than half the number of people during peak hours (where the graph crosses slightly past the eighth bar). Incorrect Response: A. This graph shows the fewest number of people attending the gym when it opens and then generally shows the number of people increasing during the day. This is not consistent with the description of how the number of people at the gym changes during the day. Incorrect Response: B. This graph shows the fewest number of people attending the gym when it opens. This is not consistent with the description of how the number of people at the gym changes during the day. Incorrect Response: C. This graph shows the greatest number of people attending the gym in the middle of the day and relatively few people attending the gym at the beginning of the day and in the evening. This is not consistent with the description of how the number of people at the gym changes during the day. |
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| 16 | A |
Objective 013 Correct Response: A. The first square in the pattern has a shaded area of 1 square unit and each new square has 1 fourth the area of the previous square. That is, the shaded area of the square is 1 fourth raised to the first power when n = 2, 1 fourth raised to the second power when n = 3, and, if the pattern continued, 1 fourth raised to the third power when n = 4. It follows that the shaded area of the square when n = 8 is 1 fourth raised to the seventh power Note that this pattern also holds when n = 1 because 1 fourth raised to the power of zero = 1. Incorrect Response: B. This response could be chosen because of the misconception that the shaded area is equal to 1 fourth divided by n where n is the step of the pattern. However, this rule produces a sequence of values that does not match those shown in the pattern: 1 fourth, 1 eighth, 1 twelfth, 1 sixteenth, and so on Incorrect Response: C. This expression represents the amount of unshaded region for n = 8. Incorrect Response: D. This expression could be the result of thinking that term n in the pattern has an area of 1 fourth raised to the power of n but this is incorrect because it assigns the area of square n + 1 to n (i.e., the expression assigns the area of square 2 to square 1, the area of square 3 to square 2, and so on). |
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| 17 | C |
Objective 013 Correct Response: C. The line 2 x minus 3 y = 9 is written in standard form. To convert to slope-intercept form (y = mx + b), solve the equation for y:
Start with the equation 2 x minus 3 y equals 9 and subtract 2 x from both sides to get negative 3 y equals negative 2 x plus 9. Divide both sides by negative 3 to get y equals negative 2 over negative 3 times x plus 9 over negative 3. This equation simplifies to y equals 2-thirds x minus 3. Incorrect Response: A. The equation in this response is incorrect because the coefficient for x is incorrect. See Correct Response C. Incorrect Response: B. The equation in this response is incorrect because the coefficient for x is incorrect. See Correct Response C. Incorrect Response: D. The equation in this response is incorrect because the coefficient of x is incorrect and there is a sign error with the constant term. See Correct Response C. |
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| 18 | B |
Objective 013 Correct Response: B. Let p equal the initial retail price of the kayak. The expression Open paren 1 minus 0 point 3 0 close paren times p represents the sale price, and the expression Open paren 1 minus 0 point 2 0 close paren times open paren 1 minus 0 point 3 0 close paren times p represents the price of the kayak after the coupon is applied to the sale price. Simplifying this expression yields Open paren .80 close paren times open paren .70 close paren times p = .56 times p This means that the new price is 56% of the original retail price, p. Incorrect Response: A. This response may be the result of thinking that the discounts can be added and then applied at once, instead of multiplied in sequence as shown in the rationale for Correct Response B. Incorrect Response: C. This response may be the result of a misconception related to combining the discounts to calculate the new price. Incorrect Response: D. This response may be the result of a misconception related to combining the discounts to calculate the new price. |
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| 19 | B |
Objective 013 Correct Response: B. The rate of change in the total cost is constant since the value increases by 2,500 dollars for every 1 additional day. Incorrect Response: A. This response is incorrect because there is a constant rate of change in the cost to rent a conference room over time. The unit rate is 2,500 dollars per day. Incorrect Response: C. This response is incorrect because the cost to rent the conference room is 2,500 dollars times the number of days for the conference. No additional fixed fee is added to this cost. If such a fee did exist, then the total cost for a 1-day conference (which is the sum of a daily rate and a fixed fee) would be greater than the cost for the second day of a 2-day conference (which is the daily rate). This is not consistent with the data in the table. Incorrect Response: D. This response may be the result of misapplying the equation for the slope of a line between two points: M equals a ratio with a numerator of y 1 minus y 2 and a denominator of x 1 minus x 2 . In this case, the slope (or constant rate of change) is equal to M equals a ratio with the expression 12,500 minus 2,500 for its numerator and the expression 5 minus 1 for its denominator. = 2,500. |
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| 20 | C |
Objective 014 Correct Response: C. The prism has dimensions 5 times 6 times 3 = 90 cubic units. The number of cubes needed to fill the rectangular prism can be represented as follows: 90 minus open bracket the quantity 5 times 6 plus the quantity 4 times 5 plus the quantity 3 times 4 close bracket Using the distributive property, this expression can be rewritten as 90 minus open bracket 5 times the sum of 6 and 4 plus the quantity 3 times 4 close bracket equals 90 minus 5 times the sum of 6 and 4 minus the quantity 3 times 4. which represents 28 more blocks. Incorrect Response: A. This response may be the result of misinterpreting the volume of the prism. The prism has dimensions 5 times 6 times 3 = 90 cubic units. It currently holds 62 blocks and requires 28 more blocks to be filled. This expression is incorrect because it evaluates to 22 more blocks. Incorrect Response: B. This response may be the result of misinterpreting the volume of the prism. The prism has dimensions 5 times 6 times 3 = 90 cubic units. It currently holds 62 blocks and requires 28 more blocks to be filled. This expression is incorrect because it evaluates to 24 more blocks. Incorrect Response: D. This response is most likely the result of a sign error combined with a misconception using the distributive property when rewriting the expression 90 minus open bracket the quantity 5 times 6 plus the quantity 4 times 5 plus the quantity 3 times 4 close bracket This expression is equal to 50, which is not equal to the number of unit cubes needed to completely fill the prism (28). |
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| 21 | A |
Objective 014 Correct Response: A. The net for a cylinder is composed of two congruent circles and a rectangle, as shown.
The net of a cylinder is shown as a rectangle that is attached to a circle on its top edge and an identical circle on its bottom edge. The calculation for the area of the two circular bases is intuitive ( 2 times pie times r squared but the formula for the rectangle (which is the lateral surface area of the prism), circumference of the circle times height of the cylinder = 2 pie r times h is not obvious to most students until they have the opportunity to compare the shape of a cylinder to its net. This realization is essential for students' understanding of surface area for a cylinder. Incorrect Response: B. A cylinder is not a prism since the bases are not polygons, but a cylinder is consistent with the general volume formula (V = area of base times height of the prism) and the general surface area formula (area of bases + lateral surface area) for prisms. This claim is not helpful for students trying to make sense of the relationship between the characteristics of a cylinder and its surface area, namely the relationship between the lateral surface area and the circumference of the circular bases. Incorrect Response: C. This claim is incorrect since a surface area is a sum of areas and not of lengths. Incorrect Response: D. This claim is incorrect since the surface area of the cylinder is the sum of the area of the two circular bases plus the lateral surface area (which is a rectangle, as shown in Correct Response A): 2 pie r times h |
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| 22 | C |
Objective 014 Correct Response: C. A circle with a diameter of d has a circumference of C = pie times d . The circumference of the Earth is 7,918 miles, and its circumference is C is approximately equal to 3 point 1 4 times 7,918, which is approximately 25,000 miles This product may be approximated in a variety of ways, such as by multiplying 3 times 8,000 = 24,000. Incorrect Response: A. This response may be the result of using a value of 2 times the diameter in the calculation for the circumference: C equals 2 times 3 point 1 4 times 16 thousand is approximately 6 point 2 5 times 16 thousand, which is 100 thousand Incorrect Response: B. This response may be the result of using the diameter (7,918) instead of the radius in the calculation C = 2 pie r : C equals 2 times 3 point 1 4 times 7,918 is approximately 6 point 2 5 times 8 thousand, which is 50 thousand Incorrect Response: D. This response may be the result of using a value of one-fourth the diameter 0.2 5 times 7,918 is approximately equal to 2,000 for the radius, r, in the calculation: C equals 2 times 3 point 1 4 times 2 thousand is approximately 6 times 2 thousand, which is 12 thousand |
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| 23 | A |
Objective 014 Correct Response: A. The slopes for each side of the quadrilateral can be determined using the equation M equals a ratio with a numerator of y 1 minus y 2 and a denominator of x 1 minus x 2 : The slope line segment A B equals the difference of 0 and negative 3 divided by the difference of 2 and negative 2. This equals 3-fourths. , The slope line segment B C equals the difference of negative 3 and negative 6 divided by the difference of negative 2 and 2. This equals 3 divided by negative 3, which is negative 1. , The slope line segment C D equals the difference of negative 6 and negative 3 divided by the difference of 1 and 5. This equals negative 3 divided by negative 4, which is 3-fourths. , and The slope line segment D A equals the difference of negative 3 and 0 divided by the difference of 5 and 2. This equals negative 3 divided by 3, which is negative 1. Since opposite sides of the quadrilateral Line segment A B and line segment B C Line segment B C and line segment D A have the same slopes, the quadrilateral must be a parallelogram. We can rule out the choices of square and rectangle by observing that consecutive sides Line segment A B and line segment B C are not perpendicular since the slopes are not opposite reciprocals. Incorrect Response: B. Quadrilateral A B C D cannot be a square since consecutive sides Line segment A B and line segment B C are not perpendicular. (The slopes are not opposite reciprocals.) Incorrect Response: C. Quadrilateral A B C D cannot be a rectangle since consecutive sides ) Line segment A B and line segment B C are not perpendicular. (The slopes are not opposite reciprocals.) Incorrect Response: D. Quadrilateral A B C D cannot be a kite since opposite sides are parallel, as shown in the rationale for Correct Response A. |
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| 24 | D |
Objective 014 Correct Response: D. This response is the result of a vertical translation of the image. The position of each letter changes in a uniform way, and the orientation of the image is preserved. Incorrect Response: A. This response shows the image after a reflection across the y-axis. Incorrect Response: B. This response is the result of either a 180-degree rotation or a series of reflections. A translation cannot produce the image shown since a translation alters a shape's location, not its orientation. Incorrect Response: C. This response is the result of translating the individual letters M, T, and H in the image using different translations. A translation of the entire image would affect all letters uniformly. |
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| 25 | C |
Objective 014 Correct Response: C. Since the shape is a regular pentagon, we can use the fact that the sum of the interior angles is equal to The product of n minus 2 and 180. degrees to find the measure of each interior angle: The ratio with a numerator of the product of n minus 2 and 180 and a denominator of n becomes the ratio with a numerator of the product of 5 minus 2 and 180 and a denominator of 5. This equals the ratio with a numerator of 3 times 180 and a denominator of 5, which simplifies to 540 over 5, which equals 108 degrees. This means the measure of angle 1 is 108 degrees. Also, the measure of angle 3 must be 90 degrees since it is supplementary to a 90-degree angle. Since the lower right region is a quadrilateral, the sum of the interior angles must be 2 times 180 or 360 degrees. We can subtract the known angles to find the measure of angle 2: 360 minus 90 minus 90 minus 108 equals 72 degrees This means that the measure of angle 4 must be equal to 108 minus 72 = 36 degrees, since the measures of angle 2 and 4 must sum to 108 degrees. Since the sum of the interior angles in a triangle is 180 degrees, the measure of angle 5 must be 180 minus 90 minus 36 equals 54 degrees. It follows that the measure of angle 6 is 108 minus 54 equals 54 degrees. Since angles 5, 6, and 7 form a straight line, the measure of angle 7 must be 180 minus 54 minus 54 equals 72 degrees. The diagram below shows the measures for all indicated angles in the figure.
The pentagon described in the stimulus is shown with its angles labeled. The quadrilateral that contains angles 1, 2, 3, and a right angle is shown with its angles labeled 108 degrees, 72 degrees, 90 degrees, and a right angle, respectively; the triangle that contains angles 4, 5, and a right angle is shown with its angles labeled 36 degrees, 54 degrees, and a right angle, respectively; angle 6, which is located between the top left edge of the pentagon and its vertical interior line segment, is labeled 54 degrees; and angle 7, which is the external angle between the overhanging edge and the pentagon, is labeled 72 degrees. Incorrect Response: A. As shown in the rationale for Correct Response C, angle 6 and angle 7 are not congruent. Incorrect Response: B. As shown in the rationale for Correct Response C, The measure of angle 3 minus the measure of angle 5 equals 90 degrees minus 54 degrees, which equals 36 degrees Incorrect Response: D. As shown in the rationale for Correct Response C, The measure of angle 1 plus the measure of angle 6 equals 108 degrees plus 54 degrees, which sums to 162 degrees |
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| 26 | B |
Objective 015 Correct Response: B. There are 16 ounces in 1 pound, so a reasonable estimate is 1,800 times 16 = 28,800. Incorrect Response: A. This response may be the result of the misconception that there are 10 ounces in 1 pound. Incorrect Response: C. This response may be the result of the misconception that there are 64 ounces in 1 pound. Incorrect Response: D. This response may be the result of the misconception that there are 100 ounces in 1 pound. |
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| 27 | D |
Objective 015 Correct Response: D. The total number of people in the survey is equal to the sum of the heights for each interval: 40 + 10 + 20 + 35 + 50 + 30 + 15 + 10 + 5 + 1 + 1 = 217. The number of people who subscribe to fewer than 5 of the services is equal to the sum of the heights for the intervals 0 to 4: 40 + 10 + 20 + 35 = 105. Since this is approximately 50% of the total number of people surveyed ( 105 over 217 is approximately 48 percent this claim is valid. Incorrect Response: A. There are 217 people total in the survey, and 10 + 5 + 1 + 1 = 17 of them subscribe to 7 or more services. This accounts for only 17 over 217 is approximately 8 percent of the total people in the survey. Incorrect Response: B. This response could be the result of misinterpreting a frequency as a percent. It is incorrect because 40 represents a much smaller portion of the people surveyed than 40%. There are 217 people total in the survey, and 40 of them do not subscribe to any services. This accounts for only 40 over 217 is approximately 18 percent of the total people in the survey. Incorrect Response: C. There are 217 people total in the survey, and 50 of them subscribe to exactly 4 services. This accounts for approximately 50 over 217 is approximately 23 percent of the total people in the survey. |
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| 28 | A |
Objective 015 Correct Response: A. The net change in the number of people in the park at the end of each hour is equal to the number of people who entered minus the number of people who exited. The number of people in the park at the end of the fifth hour can be represented with the following expression: An expression showing the sum of the hourly differences between the number of people who entered the park and the number of people who exited the park is shown. The expression is 150 minus 10 plus 175 minus 25 plus 350 minus 75 plus 500 minus 200 plus 350 minus 400. Evaluating the differences leads to the expression 140 plus 150 plus 275 plus 300 minus 50, which equals 815 Incorrect Response: B. This response may be the result of misinterpreting the net change in the fifth hour to be an increase of 50 people instead of a decrease of 50 people. Incorrect Response: C. This response represents the total number of people who entered the park in 5 hours, not the number of customers currently in the park at the end of the fifth hour. Incorrect Response: D. This response is the result of adding all the values in the table instead of subtracting the number of people who exit the park. |
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| 29 | B |
Objective 015 Correct Response: B. A dot plot displays the frequency of each value in a data set using a number line and vertical columns of dots or exes . The mode of a data set is easily identifiable as the value with the tallest column of dots or x's (which represents the value with the greatest number of occurrences in the data set). Incorrect Response: A. Venn diagrams use circles to show relationships among sets that are not mutually exclusive (i.e., students who are enrolled in art class, music class, or both). This representation does not show measures of central tendency for univariate data. Incorrect Response: C. A line graph is particularly effective for showing how a quantity varies with respect to time (e.g., the cost of a commodity over time, the profits of a company by quarter). The measures of central tendency are not easily observable in this context. Incorrect Response: D. A box plot is effective for showing the distribution of univariate data (e.g., the distribution of student scores on a summative assessment). The box shows the quartiles of the data set, and the endpoints of the graph represent the minimum and maximum values of the data set, but the mode cannot be determined from a box plot without additional information. |
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| 30 | B |
Objective 015 Correct Response: B. Given the trend in the data, it is reasonable to assume that the number of customers on Day 7 must be between 98 and 70. Visualizing a smooth curve such as the one shown is also an effective way to arrive at a prediction of 78.
A graph titled Customers Per Day is shown with a horizontal axis labeled days open paren since opening close paren and a vertical axis labeled Number of Customers. 6 data points are shown in the graph at coordinates 1 comma 168, 2 comma 147, 3 comma 119, 4 comma 102, 5 comma 99, and 10 comma 70. A smooth curve closely passes through the collection of data points. It has a vertical value that is slightly less than 80 on day 7. Incorrect Response: A. This value is approximately equivalent to the number of customers on Day 4, which is not consistent with the trend. (The number of customers per day is decreasing at a decreasing rate). Incorrect Response: C. Given the trend of the data, this estimate corresponds to approximately half of the number of expected customers. This estimate is also less than 70 (Day 10), which is not consistent with the trend. Incorrect Response: D. Given the trend of the data, this estimate corresponds to less than half of the number of expected customers on Day 7. This estimate is also less than 70 (Day 10), which is not consistent with the trend. |
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| 31 | D |
Objective 015 Correct Response: D. The total number of service calls that are represented in the graph is equal to the sum of the heights of each interval: 3 + 1 + 2 + 6 + 4 = 16. The intervals 6.0 to 8.9, 9.0 to 11.9, and 12.0 to 14.9 have frequencies of 2, 6, and 4 respectively. Since 12 out of the 16 calls in the graph were 6 minutes or longer, the experimental probability that the next call will be 6 minutes or longer is 12 sixteenths equals 3 fourths equals 75 percent Incorrect Response: A. This response represents the experimental probability that the next service call will last between 6.0 and 8.9 minutes: 2 sixteenths equals 1 sixth is approximately 16 point 6 percent Incorrect Response: B. This response is the experimental probability that the next service call will be shorter than 6 minutes: 4 sixteenths equals 1 fourth equals 25 percent Incorrect Response: C. This response represents the experimental probability that the next call will be longer than 8.9 minutes (using only the two rightmost intervals: 10 sixteenths equals 5 eighths equals 62 point 5 percent |
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| 32 | B |
Objective 016 Correct Response: B. The Earth's tides are the result of the gravitational forces between the Earth, Moon, and Sun. When the Moon is facing toward the Earth, the Earth's oceans are pulled in the direction of the Moon. This results in a bulge (high tide) on the side facing the Moon. At the same time, the centrifugal force generated by the Earth's rotation produces a similar bulge (high tide) on the Earth's opposite side. Incorrect Response: A, C, and D. Other factors, such as ocean currents, surface winds, and severe weather events, can impact tidal height but are not the underlying cause of the cyclical rise and fall of the tides. |
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| 33 | D |
Objective 016 Correct Response: D. Erosion is a geological process by which rock or other geologic materials are transported away from their original place. Incorrect Response: A. Rather than physical erosion, chemical reactions occur at the molecular level during the chemical weathering process to break down a substance. Incorrect Response: B. Deposition occurs following the weathering and erosion of a rock and refers to the settling of the material in a new location. Incorrect Response: C. Flooding can be one of the ways in which erosion can occur, but it is not a process that happens directly to the rock structure. |
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| 34 | B |
Objective 016 Correct Response: B. Removing sediment and debris from along a riverbank can help mitigate the damage caused by flooding. Deepening the river channel increases the amount of water the river can transport at one time, and removing debris can prevent large material, like logs, from damaging structures or causing blockages. Incorrect Response: A. Removing sediment and debris helps keep flowing water within the river channel. During a blizzard, there is a significant amount of precipitation, but the majority of the water falling as snow remains on the land and will not immediately enter the waterway. Incorrect Response: C. Removing sediment and debris along a riverbank would not have an immediate impact on the hazards present in a drought because the river is still able to flow, if it is present at all. Incorrect Response: D. Tsunamis share some hazards with flooding, but mitigation only along a riverbank is unlikely to have a significant impact on a tsunami's effects. |
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| 35 | A |
Objective 016 Correct Response: A. The strategies mentioned are most often used to mitigate extreme heat conditions. Watering high-traffic pavement can help prevent damage to the roads and reduce burn risks. Surfaces can be prevented from reaching dangerous temperatures by creating shade. Green roofs reduce the temperature of the surface by dissipating heat through evapotranspiration. Incorrect Response: B. The strategies mentioned are unlikely to have a significant impact on windstorms because these strategies would not redirect wind or prevent damage to structures. Incorrect Response: C. The strategies mentioned are unlikely to have a significant impact on wildfires because, although wetting pavement would slow a wildfire, it would not prevent or disrupt it significantly. Creating artificial shade or installing green roofs will not prevent or disrupt the spread of fire. Incorrect Response: D. Using water in this way would increase the severity of a drought. While artificial shade would help to reduce evaporation and green roofs can help retain resources, these strategies would most likely not impact the drought in a significant way. |
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| 36 | B |
Objective 016 Correct Response: B. An effective way for students to apply mathematical concepts during the project would be for them to calculate the water that their system can filter in an hour. Students could achieve this by measuring the amount of water filtered in a shorter time and using that result to determine the volume their prototype could filter in an hour or in a day. Incorrect Response: A. Averaging the amount of water used at home does engage students in applying mathematical concepts, but the task is not directly relevant to constructing a filtration system. Incorrect Response: C. Determining the cost to build the current prototype is not an aspect of the project, so estimating the cost to build a larger version would not be an appropriate task for students. Incorrect Response: D. Counting the number of large particles the filter removes is not a task that requires application of an age-appropriate mathematical concept. |
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| 37 | A |
Objective 016 Correct Response: A. A diagram is an example of a model. Diagramming the water cycle allows students to develop a model that they can use to make predictions about the movement of water within the Earth system. Incorrect Response: B. Diagrams can be used as supporting evidence, but creating a diagram does not directly engage students in argument from evidence. Incorrect Response: C. Creating a diagram does not necessarily involve planning or carrying out an investigation. Students may perform an investigation to gather data for the diagram, but that would be an indirect experience. Incorrect Response: D. Students may analyze the information contained in the diagram, but this is not the science practice involved in creating the diagram. |
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| 38 | D |
Objective 016 Correct Response: D. Evaporation is the process by which water molecules absorb enough energy to escape liquid water and enter a gaseous state. As the water evaporates, the volume remaining decreases and students can compare the levels of water in each of the cups. Incorrect Response: A. Adding an indicator dye could potentially improve students' ability to observe the water level, but it is less likely to be a useful method of observing the amount of water lost through evaporation. Incorrect Response: B. Increasing water temperature does increase the rate of evaporation, but it is not a direct measure of the amount of water lost to the atmosphere. Incorrect Response: C. Moisture on the sides of the cups is likely due to condensation, which is the opposite process of evaporation. |
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| 39 | C |
Objective 017 Correct Response: C. A parasitic relationship exists between two organisms when one benefits from the partnership while the other is harmed, generally in a non-fatal manner. The relationship between fleas and dogs is parasitic because the fleas benefit from the blood meal while the dog loses that blood and is often exposed to diseases. Incorrect Response: A. The spider consuming the insect is a better example of a predator-prey relationship than of parasitism because the interaction between the two species does lead to the death of the insect. Incorrect Response: B. In the case of bacteria and the cow, both partners benefit in their mutualistic relationship because the bacteria have a safe place to grow while the cow gains nutrients. Incorrect Response: D. Clownfish and anemones both benefit from their mutualistic relationship. The clownfish gains safety within the anemone's tentacles, and the anemone gains valuable nutrients from the clownfish's waste. |
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| 40 | D |
Objective 017 Correct Response: D. Decomposers, like earthworms, feed on dead and decaying material within the soil. Their digestive processes break down the organic matter and release the nutrients it contained back into the soil through their waste. Incorrect Response: A. Owls are consumers and generally feed on living organisms. Incorrect Response: B. Snakes are consumers that do not typically feed on decomposing matter. Incorrect Response: C. Many species of catfish are omnivorous and eat a wide variety of living organisms. |
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| 41 | C |
Objective 017 Correct Response: C. Plants rely on their roots to absorb water from their surroundings through the process of osmosis. Incorrect Response: A. Structures within the stem are necessary for the transportation of water within a plant but not its absorption from the plant's surroundings. Incorrect Response: B. Flowers are the reproductive structure of a plant and are not responsible for water absorption. Incorrect Response: D. Some specialized leaves do have the ability to absorb water, but it is not the primary way for plants to obtain the water that they need. |
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| 42 | D |
Objective 017 Correct Response: D. Flash floods, volcanic eruptions, and rapid burials are conditions that can lead to the creation of fossils since the organisms are quickly sealed away from processes that would typically cause the remains to decay, such as the presence of oxygen, microorganisms, and scavengers. Incorrect Response: A. Catastrophic changes on a large scale can lead to the evolution of new species, but none of the events presented will typically cause a drastic enough environmental change that would lead to speciation. Incorrect Response: B. Flash floods and volcanic eruptions can cause dramatic changes to landscapes, but the rapid burial of marine organisms does not necessarily happen on a large enough scale to reshape landscapes. Incorrect Response: C. Flash floods and volcanic eruptions can bring additional nutrients to an area that could lead to increased plant growth, but that would not be the case with the rapid burial of marine organisms. |
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| 43 | C |
Objective 017 Correct Response: C. Bacteria represent one of the main types of organisms responsible for the decomposition of a fallen log. Incorrect Response: A. Many species of beetles are decomposers, but they are not responsible for the majority of decomposition occurring in a forest ecosystem. Incorrect Response: B. Several large mammals will opportunistically consume already dead organisms, but they are not generally responsible for decomposition of plant matter into soil. Incorrect Response: D. Small rodents do consume plant matter but are not considered to be decomposers. |
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| 44 | A |
Objective 017 Correct Response: A. The type of symbiosis that occurs when one species benefits and the other is unaffected is known as commensalism. Incorrect Response: B. Parasitism occurs when one species in the relationship benefits mostly at the non-fatal expense of the other. Incorrect Response: C. Mutualism is a type of symbiosis where both partners benefit from the relationship. Incorrect Response: D. Predation occurs when one organism consumes the other for energy. |
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| 45 | C |
Objective 017 Correct Response: C. A graphical representation of the growth of each plant requires students to have an understanding of how to gather and graph quantitative data. Incorrect Response: A. Describing the factors that help plants achieve certain heights would be an application of the science and engineering practice of constructing explanations. Incorrect Response: B. Explaining numerically the genetic characteristics of the fully grown plants is the practice of engaging in argument from evidence. Incorrect Response: D. Drawing a model of what the plant's height will be in the future is an application of the practice of developing and using models. |
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| 46 | D |
Objective 018 Correct Response: D. A chemical reaction involves the formation of a new substance and is typically accompanied by the production of a gas, the formation of a precipitate, a change in color, or a change in temperature. The apple changes color after being cut and exposed to the air due to a chemical reaction called enzymatic browning. Incorrect Response: A. The molecules of the sugar cube dissolve in the tea, forming a solution. This is an example of a physical change. Incorrect Response: B. A liquid expanding in the freezer is undergoing a physical change. The state of matter changes, but the identity of the liquid remains the same. Incorrect Response: C. The water undergoes a physical change as it moves from a liquid to a gas state, which leaves behind any solutes that were previously dissolved in the water. The white residue left behind is not a precipitate because it is not a newly formed substance; the salt was already present in the water. |
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| 47 | C |
Objective 018 Correct Response: C. In a chemical reaction, the number of atoms remains the same, but one or more new substances are formed by rearranging the atoms. Incorrect Response: A. Matter cannot be created or destroyed under the law of conservation of matter. Incorrect Response: B. Atoms of one element can become another element through changes to the nucleus with an atom, but chemical reactions occur though changes to the bonds that exist between atoms. Incorrect Response: D. Atoms can fuse with each other under intense heat and pressure, but these are nuclear reactions rather than chemical reactions. |
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| 48 | A |
Objective 018 Correct Response: A. Sodium chloride is a compound of two different elements, sodium and chlorine, held together by a chemical bond. Incorrect Response: B. Sodium chloride is two elements joined together. An element is a substance that has the same number of protons in its nucleus throughout itself. Incorrect Response: C. A mixture is a combination of two or more pure substances that can be physically separated. Sodium and chlorine form an ionic compound that cannot be separated physically. Incorrect Response: D. The smallest unit of sodium chloride is composed of two atoms. |
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| 49 | C |
Objective 018 Correct Response: C. Potential energy is energy related to the position of an object relative to other objects, such as the floor. In this example, the potential energy of the egg is greatest when the egg is at the point farthest from the floor. Incorrect Response: A. During free fall, the amount of potential energy is decreasing while the amount of kinetic energy is increasing. Incorrect Response: B. At the moment of impact, the last of the potential energy is converted into other forms of energy. Incorrect Response: D. After the collision with the floor, there is no remaining gravitational potential energy. |
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| 50 | B |
Objective 018 Correct Response: B. Objects that have mass exert a large gravitational pull on other objects, with larger objects exerting more pull than smaller objects. In this example, the Sun has a greater mass than Earth, leading to Earth being pulled into the Sun's orbit. Likewise, Earth is close enough and massive enough that it can hold the Moon in its orbit. Incorrect Response: A. The size and shape of a celestial object does not necessarily predict its mass and therefore its gravitational pull. Incorrect Response: C. Inertia is a force that keeps Earth and the Moon moving in a forward direction; coupled with gravity, this forward motion takes the form of a circular revolution. Incorrect Response: D. Acceleration of Earth or the Moon is proportional to their density but is not a direct cause for their revolution around the Sun and Earth. |
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| 51 | D |
Objective 018 Correct Response: D. The law of conservation of mass states that mass is neither created nor destroyed in a chemical reaction. In this example, the total mass of the reactants is 5 + 4 = 9 grams, so the total mass of the products must also be 9 grams. Incorrect Response: A, B, and C. These responses violate the law of conservation of mass in that the mass of the product is less than the mass of the reactants. |
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| 52 | D |
Objective 011 Correct Response: D. Plucking the string when it is depressed at fret position 1 increases its frequency by 8 percent of 440 hertz, which is 0.08 times 440 = 35.2 hertz. Depressing the string in each successive fret position increases its frequency by an additional 35.2 hertz. The frequency produced by plucking the string if it is depressed at fret position 3 is equal to 440 plus 3 times the quantity 35.2 = 545.6 hertz. Incorrect Response: A. This response may be the result of the misconception that the 8% increase means an addition of 8 hertz applied once to the original frequency, such that the result of the calculation 440 + 8 is 448. Incorrect Response: B. This response may be the result of interpreting the 8% increase from each fret as 440 plus the product of 3 and 8 = 464. Incorrect Response: C. This response may be the result of applying the 8% increase to the original frequency of 440 exactly once: 440 plus 0.08 times the quantity 440 = 475.2. |
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| 53 | A |
Objective 018 Correct Response: A. Plucking the guitar string hard produces large vibrations, while plucking it gently produces smaller vibrations. This is because more energy is transferred to the string when it is plucked hard. The greater the amount of energy that is present, the greater the amplitude or the displacement of the string from its resting position. Incorrect Response: B, C, and D. The length of the string being plucked determines the wavelength, period, and frequency of the resulting sound wave, but these characteristics are not affected by the strength of the plucking action. |
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| 54 | B |
Objective 019 Correct Response: B. The property that describes the degree to which a substance can bend without breaking is known as its ductility. Incorrect Response: A. Length is a property that describes a one-dimensional measurement of an object. Incorrect Response: C. Substances with high elasticity are able to return to their original size and shape following the removal of a deforming force. Incorrect Response: D. Hardness is the ability of a material to resist scratching or cutting. |
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| 55 | B |
Objective 019 Correct Response: B. To prevent the phone from slipping out of the case, the students need to incorporate a material that can grip the phone without causing damage. The best material for this task would be rubber. Incorrect Response: A. Glass is brittle and has a smooth, low-friction surface. It would not help the phone stay within the case and would likely break upon impact. Incorrect Response: C. Metal would also typically be a smooth surface that would not provide any assistance in preventing the phone from slipping. Incorrect Response: D. Sandpaper does have the ability to grip the phone, but its gritty texture is likely to cause damage to the phone's surface. |
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| 56 | C |
Objective 019 Correct Response: C. Engineering design is best described as an ongoing, iterative process. If the model is not effective, the most appropriate step to take would be to make changes to the model to better fit the parameters of the problem. Incorrect Response: A. Researching similar studies may help confirm their results, but it will not help the students create a better model. Incorrect Response: B. Starting over completely is usually not necessary if the design works but is not as effective as anticipated. Incorrect Response: D. Unless there is an obvious reason to suspect that there were errors in the testing process, repeating the investigation would not provide helpful information that would allow for improvements to the model. |
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| 57 | B |
Objective 019 Correct Response: B. Cabinet doors, whether constructed of wood, a manufactured wood product, or metal, have considerable mass and thus any magnet used to secure the cabinet door must be strong. Incorrect Response: A. Permission to use magnets is likely not a constraint as there are few safety concerns associated with the use of magnets. Incorrect Response: C. Magnets can easily be manipulated so that the north and south poles of the magnet can be arranged in positions in which they attract each other. Incorrect Response: D. The type of magnets most likely used in this prototype would be bar magnets, not electromagnets, so learning the fundamentals of electromagnetism would not be a constraint. |
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| 58 | D |
Objective 019 Correct Response: D. Constraints refer to limitations on the design and may include factors such as cost, availability of resources, and time. The size and scope of a new water purification system inside a school would be an expensive project, in terms of both material and labor costs. Incorrect Response: A. Ethical considerations are typically addressed during multiple phases of the engineering design process. They are not necessarily associated with a single stage, such as the completion of the prototype. Incorrect Response: B. Defining the problem is typically one of the earliest steps in the engineering design process. It would typically take place well before prototype development. Incorrect Response: C. Acquiring background knowledge will typically take place earlier in the design process than prototype development. |
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| 59 | C |
Objective 019 Correct Response: C. Heat conduction is a type of energy transfer that occurs across a temperature gradient from the hottest material to the coldest. In this example, conduction would occur if a spoon were placed in a water-filled cooking pot that is heated on a hot plate: heat would be conducted from the hot plate burner to the water via the cooking pot and then to the spoon. Incorrect Response: A. A cooler by itself does not create a temperature difference that is needed to demonstrate conduction with the other materials. Incorrect Response: B. A candle is capable of transferring its energy to the metal spoon through the water and pot, but the effect would take a long time and likely be too subtle to be appropriate as a classroom demonstration. Incorrect Response: D. A refrigerator would cool the water to create a temperature difference, but it would also produce a subtler effect. Additionally, it may not be feasible to bring a refrigerator into the classroom. |
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| 60 | D |
Objective 019 Correct Response: D. The engineering design process is an iterative process. After receiving peer suggestions, it is advisable for the students to use the feedback to make improvements to their design. Incorrect Response: A. Replicating the other group's design work is not generally advised due to ethical concerns. Incorrect Response: B. Creating the final design without making any improvements based on feedback skips a vital step in the engineering design process. Incorrect Response: C. It is typically not necessary to completely start over on the engineering design process, since the information gathered in the earlier stages will still be relevant and valuable. |
Open Responses, Sample Responses, and Analyses
| Question Number |
Your Response Read about how your responses are scored and how to evaluate your practice responses |
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|---|---|---|---|
| 61 |
Open Response Item You may type your written response on the assigned topic in the box provided. |
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| First Sample Weak Response |
First Sample Weak Response to Open-Response Item Assignment To find the average heart rate before running, add up all ten numbers and divide by 10 to get 65.5 beats per minute. For example, I added 68 + 72 + 60 + 68 + 70 + 66 + 61 + 56 + 62 + 72 to get 655 and then divided 655 by 10 to get 65.5. Then I did the same thing for the other column of numbers in the table to get 134 beats per minute for the average heart rate after running. That shows exercise made the average heart rate go up. The circulatory and respiratory systems are related. Your heart rate goes up when you run because your heart needs to beat faster to carry more oxygen to your lungs, [which you then have to breathe out]. So basically, all the heart rates and breathing rates in the table go up because your body needs more oxygen. |
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| First Weak Response Analysis |
Analysis of First Weak Response to Open-Response Item Assignment This is an example of a weak response because it is characterized by the following: Purpose: The purpose is only partially achieved. For part one, the candidate calculates the mean (average) heart rates before and after exercise, but the prompt asks for the median. In part two, the response lacks detailed explanation and erroneously claims that oxygen travels from the heart to the lungs; in fact, the bloodstream transports oxygen from the lungs to the heart. Subject Matter Knowledge: The response demonstrates a limited application of subject matter knowledge. The candidate has correctly calculated the mean value for two sets of numbers. However, the candidate has confused mean and median, and the prompt calls for the latter. In part two, the candidate accurately states that heart rates and respiratory rates increase during exercise to provide the body with more oxygen, but they inaccurately describe the role of the heart and lungs in that process. Support: The candidate provides limited support for their answer. While they show the steps used to calculate the average heart rate before exercise, they do not show their work when calculating the rate after exercise, and in both cases the candidate calculates a value outside the scope of the assignment (mean instead of median). Additionally, part two is vague. The candidate discusses the body's need for oxygen but offers few accurate details about how the circulatory and respiratory systems work. Rationale: The response demonstrates a limited understanding of the topic. For example, the candidate does not understand the difference between median and mean values. The candidate's discussion of the circulatory and respiratory systems is also poorly reasoned. They mention the lungs, the heart, and the body's need for oxygen, but they offer an inaccurate explanation of how the body takes in oxygen during physical activity. |
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| Second Sample Weak Response |
Second Sample Weak Response to Open-Response Item Assignment The median is the number in the middle. When looking at the heart rates before exercise, there are 10 numbers total, so the middle is halfway down the column. In this case the median is between 70 and 66. Then you can average those numbers (add them together and divide by 2) to get a median of 68 breaths per minute. Then you do the same thing for the heart rates after exercise. The numbers in the middle of that column are 132 and 123, and if I average them, I get a median of 127.5 beats per minute. If you look at the table, everyone's heart rate and respiratory rate increases after exercise. That's because there's a relationship between the circulatory and respiratory systems that results in these changes. The student's heart beats faster and they breathe harder because their muscles are working harder and need more oxygen. |
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| Second Weak Response Analysis |
Analysis of Second Weak Response to Open-Response Item Assignment This is an example of a weak response because it is characterized by the following: Purpose: The purpose is partially achieved because the response inaccurately calculates the median heart rates in part one. In addition, the candidate only partially explains the relationship between the circulatory and respiratory systems in part two. Subject Matter Knowledge: The response demonstrates a limited application of subject matter knowledge. The candidate knows that the median is the middle number in a set, but they do not realize they must first list those values in order from lowest to highest. The candidate does correctly calculate the mean (average) of the middle numbers they identify in each set, but the final answers are incorrect because they initially misunderstood how to find the median. The response to part two is partial, only showing that the candidate has read the chart and knows that increases in heart rate and respiratory rate are related to the body's need for oxygen. Support: The support is limited. The candidate only details part of the process of finding the median and arrives at the incorrect answers as a result of missing a crucial step (placing numbers in order from lowest to highest). The candidate does calculate averages correctly, but they do not show their work. Finally, the discussion of circulatory and respiratory systems lacks specificity and offers few details about how either system functions. Rationale: The response is poorly reasoned and shows limited understanding of the necessary mathematical and scientific concepts. The candidate does not understand that to find the median, values must be listed in order, and they skip this step in both calculations. In part two, the explanation is vague and repeats the prompt rather than demonstrating the candidate's understanding; the candidate mentions the body's need for oxygen but offers little explanation of how/why heart and respiratory rates increase to meet that need. |
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| First Sample Strong Response |
First Sample Strong Response to Open-Response Item Assignment 1. To calculate the median, we list all numbers in the data set from lowest to highest, then find the value that falls in the middle. There are 10 values in each set provided in the Exhibit, so the median falls between the fifth and sixth numbers.
The data for heart rate before physical activity:
The data for heart rate after physical activity: We can therefore see that the median heart rate almost doubles after physical activity. 2. To see the relationship between circulatory and respiratory systems, we can find the median respiratory rate for comparison.
Respiratory rate before physical activity:
Respiratory rate after physical activity: Cells require oxygen to perform cellular respiration and as a result, give off carbon dioxide as waste. The more one's muscle cells work during exercise, the more oxygen they need and the more carbon dioxide they give off. This means the heart needs to pump faster to deliver more oxygenated blood to muscle cells, which accounts for the increased heart rate after physical activity. In turn, one's breathing rate also increases to bring more oxygen into the lungs, where the bloodstream picks up oxygen. The carbon dioxide that muscle cells produce as waste also enters the bloodstream, returns to the heart, and is pumped to the lungs for exhalation. |
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| First Strong Response Analysis |
Analysis of First Strong Response to Open-Response Item Assignment This is an example of a strong response because it is characterized by the following: Purpose: The response answers both parts of the prompt in full. For part one, the candidate explains how to calculate the median value and then accurately calculates it for the necessary data sets. In addition, for part two, the candidate fully explains the relationship between the circulatory and respiratory systems as evidenced by the data provided in the Exhibit. Subject Matter Knowledge: The response shows substantial and accurate application of subject matter knowledge. The candidate understands how to find the median for a set of numbers and the average between two numbers. The response further shows substantial knowledge about connections between the circulatory and respiratory systems, outlining the role of the heart, lungs, and bloodstream in respiration. Support: The candidate provides specific, high-quality supporting evidence. For example, the candidate lists each data set in full when calculating median heart rates. In addition, the candidate includes detailed calculations of the median respiratory rates, showing their work at each step, and these data support their discussion of circulatory and respiratory systems in part two. Part two also includes detailed, accurate descriptions of how those systems and their organs function. Rationale: The response is ably reasoned and shows a comprehensive understanding of both the mathematical and scientific concepts the assignment addresses. The candidate explains how to calculate the median before showing their work, and they include additional calculations for the median respiratory rate for comparison. These data strengthen their reasoning in part two, where the candidate offers a comprehensive, step-by-step discussion of the relationship between heart rate and respiratory rate. |
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| Second Sample Strong Response |
Second Sample Strong Response to Open-Response Item Assignment To find the median heart rates before and after exercise, list the rates in order from lowest to highest, then find the middle.
56, 60, 61, 62, 66, [the median falls in the middle here - between 66 and 68] 68, 68, 70, 72, 72
115, 120, 123, 128, 130, [the median is between 130 and 132] 132, 140, 142, 150, 160 The median heart rate increased from 67 beats per minute at rest to 131 beats per minute after running in place, demonstrating that exercise causes one's heart to beat faster. The table shows that this holds true for every student measured, and that each student's respiratory rate increased after exercise as well. For example, the student measured in Group 1 saw their respiratory rate increase from 28 to 37 breaths per minute and their heart rate increase from 68 to 128 beats per minute after running. In Group 2, the student's respiratory rate increased from 15 to 21, and their heart rate increased from 72 to 160. This pattern holds for every student measured because the respiratory and circulatory systems work together to provide cells with oxygen. When the student runs, their muscles work harder, their cells use more oxygen, and they breathe faster to keep up with that demand. When they breathe in, red blood cells pick up oxygen from the lungs. Oxygen-rich blood then flows to the heart, which pumps it to the rest of the body. The students' respiratory and heart rates therefore increase to transport oxygen to the muscle cells that need it during exercise. The bloodstream also removes carbon dioxide that cells produce as waste and sends it back to the heart, which then pumps it back to the lungs to be exhaled. |
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| Second Strong Response Analysis |
Analysis of Second Strong Response to Open-Response Item Assignment This is an example of a strong response because it is characterized by the following: Purpose: The purpose is fully achieved. The candidate provides thorough answers to both parts of the assignment. In part one, the candidate accurately calculates the median heart rate and shows their work. In part two, the candidate draws on data from the Exhibit and provides a detailed explanation of how those data reflect the relationship between respiratory and circulatory systems. Subject Matter Knowledge: The response demonstrates an accurate and appropriate application of subject matter knowledge. The candidate knows how to find the median value for a set of numbers and provides accurate calculations for the median heart rates before and after physical activity. Part two demonstrates substantial knowledge of the relationship between the circulatory and respiratory systems, and the candidate accurately applies that knowledge to interpret the data provided in the Exhibit. Support: The supporting evidence is detailed and relevant. In part one, the candidate provides step-by-step calculations to show how they found the median heart rates. In part two, the candidate cites specific data from the Exhibit to support their discussion of circulatory and respiratory systems. That discussion is also detailed, clearly outlining how the heart, lungs, bloodstream, and muscle cells function during physical activity. Rationale: The response demonstrates a comprehensive understanding of the topic, including full explanations of how to calculate the median. In addition, part two is ably reasoned, as the candidate both recognizes patterns in the data provided in the Exhibit and draws on their understanding of circulatory and respiratory systems to explain those patterns. |
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| Review the Performance Characteristics and Score Scale for Written Performance Assignments. | |||
Multiple Choice Question
Practice Test Evaluation Chart
In the evaluation chart that follows, the multiple-choice questions are arranged in numerical order and by test objective. Check your responses against the correct responses provided to determine how many questions within each objective you answered correctly.
Subarea: Mathematics
Objective 011: Apply number theory, structures of numeration systems and operations, and arithmetic properties to the real number system using the Standards for Mathematical Practice.
| Question Number | Your Response | Correct Response |
|---|---|---|
| 1 | B | |
| 2 | A | |
| 3 | A | |
| 4 | A | |
| 5 | B | |
| 6 | D | |
| 7 | C | |
| 52 | D |
out of 8
Objective 012: Apply properties and applications of fractions, ratios, rates, and proportions using the Standards for Mathematical Practice.
| Question Number | Your Response | Correct Response |
|---|---|---|
| 8 | C | |
| 9 | D | |
| 10 | A | |
| 11 | A | |
| 12 | C | |
| 13 | B |
out of 6
Objective 013: Apply knowledge of patterns, algebraic relationships, and functions using the Standards for Mathematical Practice.
| Question Number | Your Response | Correct Response |
|---|---|---|
| 14 | A | |
| 15 | D | |
| 16 | A | |
| 17 | C | |
| 18 | B | |
| 19 | B |
out of 6
Objective 014: Apply concepts of geometry using the Standards for Mathematical Practice.
| Question Number | Your Response | Correct Response |
|---|---|---|
| 20 | C | |
| 21 | A | |
| 22 | C | |
| 23 | A | |
| 24 | D | |
| 25 | C |
out of 6
Objective 015: Apply principles, concepts, and procedures related to measurement and data using the Standards for Mathematical Practice.
| Question Number | Your Response | Correct Response |
|---|---|---|
| 26 | B | |
| 27 | D | |
| 28 | A | |
| 29 | B | |
| 30 | B | |
| 31 | D |
out of 6
Subarea: Mathematics (Objectives 011–015) Total out of 32
Subarea: Science and Technology/Engineering
Objective 016: Apply knowledge of Earth and space science principles and the Science and Engineering Practices to interpret and analyze phenomena.
| Question Number | Your Response | Correct Response |
|---|---|---|
| 32 | B | |
| 33 | D | |
| 34 | B | |
| 35 | A | |
| 36 | B | |
| 37 | A | |
| 38 | D |
out of 7
Objective 017: Apply knowledge of life science principles and the Science and Engineering Practices to interpret and analyze phenomena.
| Question Number | Your Response | Correct Response |
|---|---|---|
| 39 | C | |
| 40 | D | |
| 41 | C | |
| 42 | D | |
| 43 | C | |
| 44 | A | |
| 45 | C |
out of 7
Objective 018: Apply knowledge of physical sciences principles and the Science and Engineering Practices to interpret and analyze phenomena.
| Question Number | Your Response | Correct Response |
|---|---|---|
| 46 | D | |
| 47 | C | |
| 48 | A | |
| 49 | C | |
| 50 | B | |
| 51 | D | |
| 53 | A |
out of 7
Objective 019: Apply knowledge of engineering principles and practices to interpret and analyze design problems.
| Question Number | Your Response | Correct Response |
|---|---|---|
| 54 | B | |
| 55 | B | |
| 56 | C | |
| 57 | B | |
| 58 | D | |
| 59 | C | |
| 60 | D |
out of 7
Subarea: Science and Technology/Engineering (Objectives 016–019) Total out of 28
Practice Test Score Calculation
The practice test score calculation is provided so that you may better gauge your performance and degree of readiness to take an MTEL test at an operational administration. Although the results of this practice test may be used as one indicator of potential strengths and weaknesses in your knowledge of the content on the official test, it is not possible to predict precisely how you might score on an official MTEL test.
The Sample Responses and Analyses for the open-response item may help you determine whether your response is more similar to the strong or weak samples. The Scoring Rubric can also assist in estimating a score for your open response. You may also wish to ask a mentor or teacher to help evaluate your response to the open-response question prior to calculating your total estimated score.
How to Calculate Your Practice Test Score
Review the directions in the sample below and then use the blank practice test score calculation worksheet to calculate your estimated score.
Multiple-Choice Section
| Enter the total number of multiple-choice questions you answered correctly: | 31 | ||
| Use Table 1 below to convert that number to the score and write your score in Box A: | A: | 221 |
Open-Response Section
| Enter the number of points (2 to 4) for your open-response question: | 2 | ||
| Use Table 2 below to convert that number to the score and write your score in Box B: | B: | 24 |
Total Practice Test Score (Estimated MTEL Score)
| Add the numbers in Boxes A and B for an estimate of your MTEL score: | A + B = | 245 |
Practice Test Score Calculation Worksheet: Elementary Mathematics (68)
Table 1:
| Number of Multiple-Choice Questions Correct | Estimated MTEL Score |
|---|---|
| 0 to 25 | 203 |
| 26 to 30 | 212 |
| 31 to 35 | 221 |
| 36 to 40 | 230 |
| 41 to 45 | 239 |
| 46 to 50 | 248 |
| 51 to 55 | 257 |
| 56 to 60 | 266 |
Table 2:
| Number of Open-Response Question Points | Estimated MTEL Score |
|---|---|
| 2 | 24 |
| 3 | 27 |
| 4 | 30 |
Use the form below to calculate your estimated practice test score.
Multiple-Choice Section
| Enter the total number of multiple-choice questions you answered correctly: | |||
| Use Table 1 above to convert that number to the score and write your score in Box A: | A: |
Open-Response Section
| Enter the number of points (2 to 4) for your open-response question: | |||
| Use Table 2 above to convert that number to the score and write your score in Box B: | B: |
Total Practice Test Score (Estimated MTEL Score)
| Add the numbers in Boxes A and B for an estimate of your MTEL score: | A + B = |