Practice Test

Why it is hard

A scale drawing multiplies every length by one factor, and the perimeter shares that factor, so the work is short once the two perimeters are paired. The common loss is reasoning with differences: the perimeter grows by 5, so a side seems to grow by 5 as well. Converting to miles per inch first, then checking that the three real sides add back to 25 miles, guards against that error and against scaling the wrong side.

1. Turning words into math minor Understanding the question

   The 25 miles is the park's perimeter, not a side, so it must be matched with the drawing's perimeter, 5 + 7 + 8 = 20 inches, before any side can be scaled. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

2. Many steps minor Working it out

   Three operations in a row, adding the sides, dividing 25 by 20 for 1.25 miles per inch, and multiplying the 5-inch side, each produce a number the next step needs. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Add the three drawn sides to get the drawing's perimeter, divide the real perimeter by it to find the miles per inch, then multiply the shortest drawn side by that scale.

Traps

• Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
• Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".

The wrong answers, classified

• B Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)

  Reveals a slip Expected pull: weak

  Finds the scale of 1.25 miles per inch but applies it to the 7-inch side, the middle one, giving 8.75 miles instead of the shortest side.

• C Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)

  Reveals a misconception Expected pull: strong

  Keeps a constant difference instead of a constant ratio: the perimeter grows by 5, from 20 to 25, so it adds 5 to the 5-inch side.

• D Misreads the question (Hegarty, Mayer & Monk, 1995)

  Reveals a slip Expected pull: weak

  Pairs the 25 miles with the longest side, 8 inches, instead of the 20-inch perimeter, so each inch seems to be 3.125 miles and 5 inches 15.625.

How the questions are analyzed, and the research behind it

Why it is hard

Scale questions usually convert one length; this one needs a round trip, from the drawing to the real room and back. The real width of 24 feet appears partway through and is a choice, so students who lose track of units stop there. The last conversion, 24/20, then has to be written as a mixed number, and its fraction part is where a hurried student slips.

1. Combines ideas moderate Working it out

   The 960 square feet is a real-world area, so the scale cannot be applied to it directly; the length must go to feet, the width come from the area, and then return to inches. (Embretson, 1998; Daniel & Embretson, 2010)

2. Much to keep track of minor Working it out

   Four lengths in two units are in play, 2 inches, 40 feet, 24 feet and the width on the drawing, and each must stay labeled with its unit. (Sweller, 1988; Carpenter, Just & Shell, 1990)

Strategy: Convert the 2-inch length to real feet, divide the area by it to get the real width, then use the scale to change that width back to inches.

Traps

• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.

The wrong answers, classified

• B An arithmetic slip (Junker & Sijtsma, 2001)

  Reveals a slip Expected pull: weak

  Gets the real width of 24 feet and divides by 20 correctly, but writes the remainder 4/20 as 1/4 instead of 1/5.

• C Units or scale (King et al., 2004)

  Reveals partial knowledge Expected pull: strong

  Finds the room's real width, 960 ÷ 40 = 24 feet, and gives it as the drawing's width without converting back at 1 inch per 20 feet.

• D Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)

  Reveals partial knowledge Expected pull: weak

  Converts the 2-inch length to the real 40 feet and gives that number, answering for the length instead of the width and in the wrong units.

How the questions are analyzed, and the research behind it

Why it is hard

The figure hides the work: it labels four of the six sides, and the missing two are short pieces of the notch that are easy to overlook. Once all six are found, the scale step is a single multiplication, but it must go the right way, since the real garden is larger than the drawing. Tracing the outline and counting six sides guards against the most common loss.

1. Reading a figure, table or graph moderate Understanding the question

   Two sides of the notch carry no labels, so students must find them from the opposite sides: 13 − 8 = 5 cm across and 6 − 3 = 3 cm down. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. Combines ideas minor Working it out

   The perimeter of an L-shaped drawing has to be joined to a scale of 2.5 meters per centimeter, two separate skills in one answer. (Embretson, 1998; Daniel & Embretson, 2010)

3. An answer in an unexpected form minor Choosing the answer

   With no choices, nothing flags a perimeter that is missing two sides or still in centimeters; the student must check the six-sided total alone. (Enright, Morley & Sheehan, 2002)

Strategy: Use the opposite sides to work out the two unlabeled lengths, add all six sides of the drawing, then multiply the total in centimeters by 2.5 meters per centimeter.

Traps

• Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
• Stops one step early: A choice is a number from partway through the solution.
• Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).

The wrong answers, classified

• 75 Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  Reveals partial knowledge Expected pull: strong

  Adds only the four labeled sides, 13 + 3 + 8 + 6 = 30 cm, and scales that to 75 m, missing the two unlabeled sides of the notch.

• 38 Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals partial knowledge Expected pull: moderate

  Finds the drawing's perimeter, 38 cm, correctly and grids it without applying the scale of 2.5 meters per centimeter.

• 15.2 Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)

  Reveals a misconception Expected pull: weak

  Divides the 38 cm perimeter by 2.5 instead of multiplying, shrinking the garden though each centimeter of the drawing stands for 2.5 meters.

How the questions are analyzed, and the research behind it

Why it is hard

The question chains a scale factor to a perimeter, and both links have a trap. Some students scale by adding, since the base grew by 4, and others find the 15 cm leg correctly but then add only some of the sides. Writing all three sides of the large triangle, 15, 15 and 10, before adding is what the question rewards.

1. Reading a figure, table or graph moderate Understanding the question

   The figure carries every number: tick marks show MN = NP, the base MP is 10 cm, and triangle XYZ has 9 cm legs on a 6 cm base. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. Something you must already know minor Working it out

   Students must know that corresponding sides of similar triangles share one scale factor, here 10/6 = 5/3, though the stem never says so. (Anderson & Krathwohl, 2001)

3. Error-prone arithmetic minor Working it out

   The scale factor 5/3 is a fraction, and 9 × 5/3 = 15 is easier to get right by dividing 9 by 3 first. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Compare the two bases to get the scale factor, multiply a leg of the small triangle by it, then use the equal sides to add up the perimeter.

Traps

• Stops one step early: A choice is a number from partway through the solution.
• Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.

The wrong answers, classified

• A Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals partial knowledge Expected pull: moderate

  Finds a scaled leg of 15 cm but adds only it and the 10 cm base, 15 + 10 = 25, leaving out the second 15 cm side.

• B Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals a slip Expected pull: weak

  Finds both legs, 15 cm each, but adds only 15 + 15 and leaves out the 10 cm base; calling all three sides 10 cm also gives 30.

• C Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)

  Reveals a misconception Expected pull: moderate

  Adds 4 cm to each leg because the base grew from 6 to 10, making the legs 13 cm: 13 + 13 + 10 = 36.

How the questions are analyzed, and the research behind it

Why it is hard

Each step is familiar, but the three together, perimeter to width, width to scale factor, and sides to area, demand a plan. The strongest pull is scaling by subtraction, shrinking each side by 1 foot because the width drops from 3 to 2, which gives a neat whole-number area. Others stop at the outer rectangle or add the sides instead of multiplying. Checking that 8/3 to 2 matches 4 to 3 confirms the shape is similar.

1. Combines ideas moderate Working it out

   Perimeter, similarity and area are chained: 14 feet gives the outer width of 3 feet, the 2-foot side sets a scale factor of 2/3, and only then can the area be found. (Embretson, 1998; Daniel & Embretson, 2010)

2. Reading a figure, table or graph minor Understanding the question

   The figure marks the shaded rectangle's 2-foot side running up and down and the outer 4-foot length along the bottom, so the 2 feet must be paired with the outer 3-foot side, which is not labeled. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

3. Error-prone arithmetic minor Working it out

   The scale factor 2/3 makes the shaded length 8/3 feet, and the area 2 × 8/3 = 16/3 has to be written as the mixed number 5 1/3. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Use the perimeter to find the outer rectangle's width, match the shaded width of 2 feet to it, scale the 4-foot length by the same factor, then multiply length by width.

Traps

• Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
• Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.

The wrong answers, classified

• B Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)

  Reveals a misconception Expected pull: strong

  Shrinks each side by the same amount instead of the same factor: the width drops by 1 foot, from 3 to 2, so the length drops from 4 to 3, giving 6.

• C Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: weak

  Finds the shaded sides, 2 and 8/3 feet, correctly but computes the perimeter, 2 × (2 + 8/3) = 28/3 = 9 1/3, instead of the area.

• D Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)

  Reveals partial knowledge Expected pull: moderate

  Finds the outer width of 3 feet and gives the outer rectangle's area, 4 × 3 = 12, instead of the smaller shaded one.

How the questions are analyzed, and the research behind it

Why it is hard

The drawing gives two lengths and only the width matters, so the first decision is which number to scale. The second is direction: the real garden must measure more in yards than the drawing’s 9 in inches, which a student can check before entering anything. Errors come from scaling the length, turning the scale over, or treating 2 to 3 as an increase of one.

1. Turning words into math minor Understanding the question

   “2 inches represents 3 yards” must become 1.5 yards for every inch of drawing, so the 9-inch width is multiplied by 3/2, not by 2/3. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

2. An answer in an unexpected form minor Choosing the answer

   The answer is a decimal entered with no choices to compare, so a result from the 15-inch length or from an upside-down scale looks just as finished. (Enright, Morley & Sheehan, 2002)

Strategy: Find how many yards one inch of the drawing stands for, then multiply the drawing's width, not its length, by that number.

Traps

• Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
• Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).
• Information you don't need: The question gives numbers or facts that play no part in the answer.

The wrong answers, classified

• 22.5 Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)

  Reveals a slip Expected pull: moderate

  Applies the scale to the 15-inch length instead of the 9-inch width: 15 × 3 ÷ 2 = 22.5 yards.

• 6 Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)

  Reveals a misconception Expected pull: moderate

  Turns the scale upside down, multiplying the 9 inches by 2/3 instead of 3/2, so the real garden comes out smaller than the drawing’s number.

• 10 Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)

  Reveals a misconception Expected pull: weak

  Reads 2 inches to 3 yards as adding 1, keeping a constant difference instead of a constant ratio, so 9 inches becomes 9 + 1 = 10 yards.

How the questions are analyzed, and the research behind it

Why it is hard

This is a one-proportion map question, and the only judgments are the direction of the scale and when to round. The phrase to the nearest kilometer tempts some students to round the scale itself before multiplying, which lands on a listed answer two kilometers too far. Rounding only the final 6.75 keeps the result right.

1. Turning words into math minor Understanding the question

   “2 centimeters represents 3 kilometers” gives 1.5 kilometers per centimeter; set up the other way round, the scale shrinks the 4.5 centimeters instead of enlarging them. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

2. Error-prone arithmetic minor Working it out

   The map distance is a decimal, 4.5 × 3 ÷ 2 = 6.75, and the result must then be rounded to the nearest kilometer, two places where a quick estimate drifts. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Work out how many kilometers 1 centimeter on the map stands for, multiply by 4.5, and round only at the very end.

Traps

• Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).
• Rounds too early: Rounds a value partway through, or rounds the wrong way, so the final answer drifts from the exact one.
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.

The wrong answers, classified

• A Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)

  Reveals a misconception Expected pull: moderate

  Turns the scale upside down, multiplying 4.5 by 2/3 instead of 3/2, so the real distance comes out smaller than the map distance.

• B Units or scale (King et al., 2004)

  Reveals a surface strategy Expected pull: weak

  Takes the 4.5 centimeters on the map as kilometers and rounds to 5, never applying the scale that turns each centimeter into 1.5 kilometers.

• D Rounds too early (King et al., 2004)

  Reveals partial knowledge Expected pull: moderate

  Rounds the scale of 1.5 kilometers per centimeter up to 2 before multiplying, so 4.5 × 2 = 9 instead of 6.75.

How the questions are analyzed, and the research behind it

Why it is hard

Unlike most map questions, this one asks for an exact distance, so the decimal 5.4 has to be carried all the way through. The two tempting wrong answers come from opposite places: an upside-down scale shrinks the distance, and a rounded map length gives a result close to the true one. Checking that the answer is one and a half times 5.4 catches both.

1. Error-prone arithmetic minor Working it out

   The map distance is the decimal 5.4, and 5.4 × 3 = 16.2 then halved is a decimal multiplication where rounding 5.4 to make it easier changes the answer. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

2. A very close wrong answer minor Choosing the answer

   7.5 and 8.1 miles are close enough that a rough estimate, 5 × 1.5, lands on the wrong one; only the exact product separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Set up miles over inches equal to 3 over 2, then multiply 5.4 by 3 and divide by 2, keeping the decimal exact throughout.

Traps

• Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).
• Rounds too early: Rounds a value partway through, or rounds the wrong way, so the final answer drifts from the exact one.

The wrong answers, classified

• A Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)

  Reveals a surface strategy Expected pull: weak

  No common working gives 2.6, which sits just below half of 5.4 and matches no step with the scale, so it mostly draws guesses.

• B Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)

  Reveals a misconception Expected pull: moderate

  Inverts the scale, multiplying 5.4 by 2/3 instead of 3/2, as if each mile took up 1.5 inches on the map.

• C Rounds too early (King et al., 2004)

  Reveals partial knowledge Expected pull: moderate

  Rounds 5.4 inches down to 5 before scaling, so 5 × 3 ÷ 2 = 7.5 miles instead of the exact distance.

How the questions are analyzed, and the research behind it

Why it is hard

The figure gives four of the six sides and the scale gives feet per half inch, so two conversions hide behind a simple-looking perimeter question. Students who add only the labeled sides, or who read the scale as 25 feet per inch, reach totals that look reasonable for a pool and are listed. Tracing the outline to count six sides, and doubling 25 before multiplying, avoids both.

1. Reading a figure, table or graph moderate Understanding the question

   Two sides of the notch carry no labels, and the 5 in. arrow spans the full right side, so the missing lengths come from 5 − 2 = 3 and 6 − 3 = 3. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. Combines ideas moderate Working it out

   The perimeter of an L-shaped drawing has to be joined to a scale in which half an inch, not a whole inch, stands for 25 feet. (Embretson, 1998; Daniel & Embretson, 2010)

3. Many steps minor Working it out

   Four stages follow in order, the two missing sides, the 22-inch perimeter, the feet per inch and the product, and a slip at most of them reaches a listed choice. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Use opposite sides to fill in the two missing lengths, add all six sides of the drawing, then double the 25 feet to get the feet one inch stands for, and multiply.

Traps

• Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.

The wrong answers, classified

• A Units or scale (King et al., 2004)

  Reveals partial knowledge Expected pull: strong

  Finds the drawing's 22-inch perimeter but multiplies by 25, using the scale as if 1 inch, not 1/2 inch, stood for 25 feet.

• B Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  Reveals partial knowledge Expected pull: moderate

  Adds only the four labeled sides, 6 + 2 + 3 + 5 = 16 inches, and scales that to 800 feet, missing the two unlabeled sides of the notch.

• D Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: weak

  Multiplies the 6-inch and 5-inch sides to get 30, the outer rectangle's area, and scales that by 50 feet per inch as if it were a length.

How the questions are analyzed, and the research behind it

Why it is hard

A single division settles this question, and the choices are built to punish a misplaced decimal point, since three of them are each a power of ten apart. Students who see that 0.125 is one-thousandth of 125 can answer at once; students who divide by hand often lose a zero. Multiplying the chosen answer back by 125 is a quick check.

1. Error-prone arithmetic moderate Working it out

   The one step is 0.125 ÷ 125, a decimal divided by a large whole number, where the place of the decimal point decides between 0.001, 0.01 and 0.1. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

2. A very close wrong answer moderate Choosing the answer

   0.001 and 0.01 look almost the same, differing by one zero after the decimal point; only exact placement of the point in 0.125 ÷ 125 separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Divide the inches by the feet to get inches per foot, then check by multiplying your result by 125 to see whether you get back 0.125.

Traps

• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.

The wrong answers, classified

• B An arithmetic slip (Junker & Sijtsma, 2001)

  Reveals a slip Expected pull: strong

  Divides 0.125 by 125 but loses one zero in placing the decimal point, giving 0.01, ten times too large.

• C A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)

  Reveals a misconception Expected pull: moderate

  Sees that 125 ÷ 125 = 1 and puts the 1 in the tenths place, where 0.125 begins, instead of the thousandths place.

• D Units or scale (King et al., 2004)

  Reveals a surface strategy Expected pull: weak

  Treats the 0.125 inch as if it already stood for 1 foot, never dividing by the 125 feet it represents, and picks the choice closest to it.

How the questions are analyzed, and the research behind it

Why it is hard

A scale drawing multiplies every length by the same factor, here 3 feet per centimeter, and the question checks whether students use that factor or a difference. The 16 cm side is 4 cm longer than the others, and adding 4 to 36 feels natural but treats the scale as an addition. Students who find the factor from the matching sides first have one multiplication left.

1. Turning words into math minor Understanding the question

   "These two sides each measure 36 feet" refers back to the 12 cm sides, so the scale pairs 12 cm with 36 ft, centimeters on the drawing against feet on the banner. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Find the scale from the matching sides, feet per centimeter, then multiply the 16 cm side by that same factor.

Traps

• Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.

The wrong answers, classified

• A Units or scale (King et al., 2004)

  Reveals a surface strategy Expected pull: weak

  Copies the 16 cm from the drawing as 16 feet, without applying the scale that turns each centimeter into 3 feet.

• B Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)

  Reveals a misconception Expected pull: weak

  Reasons with differences instead of a scale factor and also subtracts, 36 − 4 = 32, though the 16 cm side is the longest.

• C Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)

  Reveals a misconception Expected pull: strong

  Adds the 4 cm difference between 16 and 12 to the 36 feet, keeping a constant difference instead of a constant ratio.

How the questions are analyzed, and the research behind it