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Praxis Elem CKT Math study materials
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Praxis Elem CKT Math practice questions

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Question 1Mathematics content and pedagogical content knowledge for elementary teaching

A student writes 4,507 as 4,000 + 500 + 7. Which statement best describes this decomposition?

  1. It is incorrect because the zero in the tens place must be written as 0.
  2. It is correct only if the number is written in expanded form with exponents.
  3. It is correct because each nonzero digit is matched to its place value. Correct answer.
  4. It is incorrect because 4,507 should decompose as 45 + 7.

Explanation

Expanded form represents each digit multiplied by its place value, so 4,507 equals 4,000 + 500 + 0 + 7. Omitting the zero term is acceptable because 0 × 10 = 0 and adds nothing to the sum. The decomposition faithfully captures the thousands, hundreds, and ones digits.

Question 2Mathematics content and pedagogical content knowledge for elementary teaching

Which comparison is true?

  1. 0.7 < 0.65
  2. 0.4 = 0.40 Correct answer.
  3. 0.09 > 0.1
  4. 3.2 > 3.25

Explanation

Appending a zero to the right of the last decimal digit does not change the value, so 0.4 and 0.40 name the same quantity. Comparing decimals requires aligning place values rather than counting digits.

Question 3Mathematics content and pedagogical content knowledge for elementary teaching

A student claims that 3/4 is larger than 5/6 because 4 and 6 are both larger than the numerators. Which reasoning best diagnoses the error?

  1. The student treated the denominator as a count of pieces rather than the size of each piece. Correct answer.
  2. The student should have converted both fractions to decimals.
  3. The student compared numerators before comparing denominators.
  4. The student confused proper fractions with mixed numbers.

Explanation

A larger denominator means each piece is smaller, so comparing denominators directly reverses the size relationship. Finding a common denominator (twelfths) shows 3/4 = 9/12 and 5/6 = 10/12, so 5/6 is larger. The error is conceptual, not computational.

Question 4Mathematics content and pedagogical content knowledge for elementary teaching

Which expression has the same value as 6 × (4 + 3)?

  1. (6 + 4) × 3
  2. 6 + 4 × 3
  3. (6 × 4) + (6 × 3) Correct answer.
  4. 6 × 4 + 3

Explanation

The distributive property states that a × (b + c) = (a × b) + (a × c). Here 6 × 7 = 42 and (6 × 4) + (6 × 3) = 24 + 18 = 42.

Question 5Mathematics content and pedagogical content knowledge for elementary teaching

A teacher asks students to find 25 × 16 mentally. A student says, "I did 25 × 4 = 100, then 100 × 4 = 400." Which property justifies this strategy?

  1. The commutative property of addition
  2. The identity property of multiplication
  3. The associative property of multiplication Correct answer.
  4. The distributive property over subtraction

Explanation

Regrouping 25 × 16 as 25 × (4 × 4) and then as (25 × 4) × 4 uses the associative property, which allows factors to be grouped differently without changing the product.

Question 6Mathematics content and pedagogical content knowledge for elementary teaching

Which number is a multiple of both 6 and 8?

  1. 14
  2. 24 Correct answer.
  3. 18
  4. 36

Explanation

A common multiple must be divisible by both numbers. 24 = 6 × 4 and 24 = 8 × 3, so 24 is the least common multiple of 6 and 8.

Question 7Mathematics content and pedagogical content knowledge for elementary teaching

A rectangular garden is 12 meters long and 7 meters wide. A student says the perimeter is 84 meters. Which error did the student most likely make?

  1. The student used the wrong unit of measurement.
  2. The student computed area rather than perimeter. Correct answer.
  3. The student added length and width instead of multiplying.
  4. The student doubled only the length.

Explanation

Multiplying 12 by 7 gives 84, which is the area in square meters. The perimeter is 2(12 + 7) = 38 meters, a linear measure.

Question 8Mathematics content and pedagogical content knowledge for elementary teaching

Which unit is most appropriate for measuring the mass of a pencil?

  1. Grams Correct answer.
  2. Meters
  3. Kilograms
  4. Liters

Explanation

A pencil has a small mass, so grams are the reasonable metric unit. Kilograms describe much heavier objects.

Question 9Mathematics content and pedagogical content knowledge for elementary teaching

A bag contains 3 red, 5 blue, and 2 green marbles. If one marble is drawn at random, what is the probability it is blue?

  1. 3/10
  2. 5/8
  3. 1/5
  4. 1/2 Correct answer.

Explanation

There are 10 marbles total and 5 are blue, so the probability is 5/10 = 1/2. Probability is favorable outcomes divided by total outcomes.

Question 10Mathematics content and pedagogical content knowledge for elementary teaching

A stem-and-leaf plot shows the data 12, 15, 15, 18, 21, 23. What is the median?

  1. 17.3
  2. 15
  3. 16.5 Correct answer.
  4. 18

Explanation

With six ordered values, the median is the mean of the two middle values, 15 and 18. That gives (15 + 18) / 2 = 16.5.

Question 11Mathematics content and pedagogical content knowledge for elementary teaching

Which expression represents "five less than three times a number n"?

  1. 5 − 3n
  2. 3(n − 5)
  3. 5n − 3
  4. 3n − 5 Correct answer.

Explanation

"Three times a number" is 3n, and "five less than" that quantity means subtract 5 from it, giving 3n − 5.

Question 12Mathematics content and pedagogical content knowledge for elementary teaching

A student solves 2x + 6 = 14 by writing 2x = 20. Which statement best explains the error?

  1. The student subtracted 14 from both sides.
  2. The student should have multiplied both sides by 2.
  3. The student divided both sides by 2 too early.
  4. The student added 6 to both sides instead of subtracting 6. Correct answer.

Explanation

To isolate 2x, subtract 6 from both sides, giving 2x = 8 and x = 4. The student's result of 20 suggests 6 was added rather than subtracted.

Question 13Mathematics content and pedagogical content knowledge for elementary teaching

Which shape has exactly four lines of symmetry?

  1. A square Correct answer.
  2. A regular pentagon
  3. A rectangle that is not a square
  4. An equilateral triangle

Explanation

A square has four lines of symmetry: two through midpoints of opposite sides and two diagonals.

Question 14Mathematics content and pedagogical content knowledge for elementary teaching

A triangle has angle measures of 50° and 60°. What is the measure of the third angle?

  1. 60°
  2. 80°
  3. 110°
  4. 70° Correct answer.

Explanation

The angles of a triangle sum to 180°. Subtracting 50° and 60° from 180° leaves 70°.

Question 15Mathematics content and pedagogical content knowledge for elementary teaching

A student is asked to round 4,682 to the nearest hundred. Which response shows correct reasoning?

  1. 4,700, because the tens digit 8 means round up. Correct answer.
  2. 4,680, because rounding keeps the tens place.
  3. 4,600, because the hundreds digit is 6.
  4. 5,000, because 4,682 is closer to 5,000 than 4,000.

Explanation

Rounding to the nearest hundred examines the tens digit. Since 8 is 5 or greater, the hundreds digit increases from 6 to 7, giving 4,700.

Question 16Mathematics content and pedagogical content knowledge for elementary teaching

A teacher wants students to understand why the standard algorithm for multi-digit multiplication works. Which instructional approach best supports conceptual understanding?

  1. Having students memorize the steps in order without explanation.
  2. Using area models that partition factors by place value and sum partial products. Correct answer.
  3. Assigning timed drills of multiplication facts.
  4. Showing a video of the algorithm without discussion.

Explanation

Area models connect each partial product to a region whose dimensions are place-value parts of the factors, making the algorithm's structure visible. This supports understanding rather than rote memorization.

Question 17Mathematics content and pedagogical content knowledge for elementary teaching

A student says, "When I add two odd numbers, I always get an odd number." Which response best addresses the misconception?

  1. Agree, because odd plus odd often looks odd.
  2. Provide counterexamples such as 3 + 5 = 8 and explain parity. Correct answer.
  3. Tell the student to memorize the addition table.
  4. Suggest the student only add even numbers.

Explanation

Odd numbers can be written as 2n + 1, and the sum of two such numbers is 2(n + m + 1), which is even. Counterexamples such as 3 + 5 = 8 make the pattern concrete before formalizing it.

Question 18Mathematics content and pedagogical content knowledge for elementary teaching

Which fraction is equivalent to 0.375?

  1. 3/8 Correct answer.
  2. 1/3
  3. 5/8
  4. 3/4

Explanation

0.375 equals 375/1000, which simplifies by dividing numerator and denominator by 125 to 3/8. Checking, 3 ÷ 8 = 0.375.

Question 19Mathematics content and pedagogical content knowledge for elementary teaching

A student writes 2/3 + 1/4 = 3/7. Which statement best describes the error?

  1. The student added numerators and denominators separately instead of finding a common denominator. Correct answer.
  2. The student used the wrong operation.
  3. The student simplified the sum incorrectly.
  4. The student should have multiplied the fractions.

Explanation

Fractions with unlike denominators must be rewritten with a common denominator before adding. The correct sum is 8/12 + 3/12 = 11/12, not 3/7.

Question 20Mathematics content and pedagogical content knowledge for elementary teaching

Which number is prime?

  1. 33
  2. 27
  3. 29 Correct answer.
  4. 21

Explanation

A prime number has exactly two factors, 1 and itself. The number 29 is not divisible by 2, 3, or 5, and its square root is less than 6, so 29 is prime.

Question 21Mathematics content and pedagogical content knowledge for elementary teaching

Which situation is best modeled by the expression 15 ÷ 3?

  1. Finding the area of a 15 by 3 rectangle
  2. Combining 15 red and 3 blue counters
  3. Sharing 15 stickers equally among 3 children Correct answer.
  4. Adding 3 to a number 15 times

Explanation

Partitive division distributes a total equally among a number of groups, giving 5 stickers per child. This matches 15 ÷ 3.

Question 22Mathematics content and pedagogical content knowledge for elementary teaching

A student uses a number line to add 7 + (−4). Which result and reasoning are correct?

  1. −3, because the result takes the sign of the larger absolute value.
  2. 11, because you move 4 units right from 7.
  3. −11, because both numbers are combined as negatives.
  4. 3, because you move 4 units left from 7. Correct answer.

Explanation

Adding a negative number corresponds to moving left on the number line. Starting at 7 and moving 4 units left lands on 3.

Question 23Mathematics content and pedagogical content knowledge for elementary teaching

A teacher presents the pattern 3, 6, 12, 24, ... and asks students to find the next term. Which reasoning is most mathematically sound?

  1. The next term is 36 because the differences are 3, 6, 12.
  2. The next term is 27 because the numbers alternate.
  3. The next term is 48 because each term doubles the previous one. Correct answer.
  4. The next term is 30 because each term increases by 6.

Explanation

Each term is twice the previous term, so 24 × 2 = 48. Identifying the multiplicative rule is more consistent than guessing from a single difference.

Question 24Mathematics content and pedagogical content knowledge for elementary teaching

A rectangular prism has dimensions 2 cm, 3 cm, and 4 cm. What is its volume?

  1. 24 cm³ Correct answer.
  2. 9 cm³
  3. 26 cm³
  4. 12 cm³

Explanation

The volume of a rectangular prism is length × width × height, so 2 × 3 × 4 = 24 cubic centimeters.

Question 25Mathematics content and pedagogical content knowledge for elementary teaching

A student is asked to convert 3 kilometers to meters. Which reasoning is correct?

  1. Add 1,000 because conversion always adds a place value, giving 1,003 m.
  2. Multiply by 1,000 because a kilometer is 1,000 meters, giving 3,000 m. Correct answer.
  3. Divide by 1,000 because meters are larger, giving 0.003 m.
  4. Multiply by 100 because there are 100 meters in a kilometer, giving 300 m.

Explanation

Since 1 kilometer equals 1,000 meters, converting from a larger unit to a smaller unit requires multiplication. Thus 3 km = 3 × 1,000 = 3,000 m.

Question 26Mathematics content and pedagogical content knowledge for elementary teaching

A student explains that 4/5 is larger than 5/8 because 4 is closer to 5 than 5 is to 8. Which reasoning best diagnoses and corrects this misconception about comparing fractions?

  1. The comparison should use a common denominator or benchmark: 4/5 = 32/40 and 5/8 = 25/40, so 4/5 is larger. Correct answer.
  2. The comparison should use cross multiplication of numerators only, giving 4 × 5 versus 5 × 8.
  3. The student is correct because numerators closer in value to denominators always indicate larger fractions.
  4. The fractions cannot be compared unless both have a numerator of 1, so the question is invalid.

Explanation

Comparing fractions requires a common whole, so rewriting with a common denominator (40) shows 32/40 versus 25/40, making 4/5 larger. The student's reasoning confuses the difference between numerator and denominator with the size of the fraction, which is not a valid comparison strategy.

Question 27Mathematics content and pedagogical content knowledge for elementary teaching

A teacher wants students to see that multiplication can be interpreted as repeated groups and as an array. Which pair of representations best supports both interpretations for 3 × 4?

  1. A circle divided into 3 equal parts and a list of the multiples of 4.
  2. A place-value chart with 3 in the tens column and 4 in the ones column.
  3. A fraction strip split into fourths and a set of 3 tally marks.
  4. A number line showing jumps of 4 three times and a 3-by-4 rectangular array of dots. Correct answer.

Explanation

Repeated groups appear as three jumps of 4 on a number line, and the array displays 3 rows of 4 objects, connecting both meanings to the same product. These representations build the equal-groups and area/array interpretations of multiplication.

Question 28Mathematics content and pedagogical content knowledge for elementary teaching

A student claims that 0.6 is less than 0.58 because 6 is less than 58. Which explanation best addresses the error using place value?

  1. 0.58 is larger because the digit 8 is greater than the digit 6.
  2. 0.6 equals 0.60, and 60 hundredths is greater than 58 hundredths, so 0.6 is larger. Correct answer.
  3. 0.6 is less because it has fewer digits after the decimal point than 0.58.
  4. 0.6 and 0.58 are equal because trailing zeros do not change value.

Explanation

Writing 0.6 as 0.60 aligns the place values so both numbers are expressed in hundredths. Comparing 60 hundredths with 58 hundredths shows 0.6 is greater, correcting the mistaken whole-number comparison of 6 and 58.

Question 29Mathematics content and pedagogical content knowledge for elementary teaching

A teacher asks students to estimate 497 × 21 mentally. Which estimate and justification are most appropriate for elementary students?

  1. About 1,000, because 500 ÷ 20 = 25 and then multiplied by 40.
  2. About 100,000, because 500 × 200 = 100,000 by adding a zero to each factor.
  3. About 10,000, because 500 × 20 = 10,000 and both factors were rounded to friendly numbers. Correct answer.
  4. About 5,000, because 500 × 10 = 5,000 using only the tens digit of 21.

Explanation

Rounding 497 to 500 and 21 to 20 gives 500 × 20 = 10,000, a reasonable estimate close to the exact product. Estimation uses compatible numbers whose product is easy to compute mentally.

Question 30Mathematics content and pedagogical content knowledge for elementary teaching

A student is asked to find the area of a triangle with base 10 cm and height 6 cm. The student multiplies 10 by 6 and stops. Which statement best explains the needed correction?

  1. The base and height must be added rather than multiplied, giving 16 square centimeters.
  2. The product is correct because area always equals base times height for any polygon.
  3. The height must be doubled before multiplying, giving 120 square centimeters.
  4. The product 10 × 6 gives the area of a rectangle, so it must be halved to 30 square centimeters. Correct answer.

Explanation

A triangle is half of a parallelogram with the same base and height, so its area is (1/2) × 10 × 6 = 30 square centimeters. The student computed the area of the related rectangle and omitted the factor of one-half.

Question 31Mathematics content and pedagogical content knowledge for elementary teaching

Which statement correctly describes the relationship between the mean and median of the data set 2, 3, 4, 4, 22?

  1. The mean and median are both 7 because the data are evenly spaced.
  2. The mean is 7 and the median is 4, so the mean is pulled upward by the outlier 22. Correct answer.
  3. The mean is 5 and the median is 4 because the outlier is ignored in the mean.
  4. The mean is 4 and the median is 7, so the median is pulled upward by the outlier 22.

Explanation

The sum is 35 and dividing by 5 gives a mean of 7, while the middle ordered value is 4. The large value 22 pulls the mean above the median, illustrating how outliers affect the mean more than the median.

Question 32Mathematics content and pedagogical content knowledge for elementary teaching

A teacher wants students to understand why 3/4 ÷ 1/8 can be interpreted as asking how many eighths are in three-fourths. Which reasoning best supports the answer?

  1. The quotient is 2 because 4 minus 8 leaves a difference of 4 and 3 minus 1 is 2.
  2. Three-fourths is less than one, so the quotient must be less than 1.
  3. Three-fourths equals six-eighths, and there are six groups of one-eighth in six-eighths, so the quotient is 6. Correct answer.
  4. Dividing by one-eighth is the same as multiplying by eight, so the quotient is 24.

Explanation

Measurement division asks how many groups of the divisor fit in the dividend. Since 3/4 = 6/8, there are 6 groups of 1/8, so 3/4 ÷ 1/8 = 6.

Question 33Mathematics content and pedagogical content knowledge for elementary teaching

Which expression correctly uses the distributive property to rewrite 6 × 47 as a sum of two products?

  1. 6 × 40 + 7
  2. 6 × 40 + 6 × 7 Correct answer.
  3. 6 + 40 × 7
  4. 6 × 50 − 6 × 7

Explanation

Decomposing 47 as 40 + 7 and applying the distributive property gives 6 × 40 + 6 × 7. Each part of the decomposed number is multiplied by 6, preserving the total value.

Question 34Mathematics content and pedagogical content knowledge for elementary teaching

A student says, "The number 1 is prime because you can divide it by 1 and by itself." Which response best corrects this misconception?

  1. A prime has exactly two distinct factors, and 1 has only one factor, so 1 is neither prime nor composite. Correct answer.
  2. The number 1 is prime only when it is paired with another prime number.
  3. The number 1 is composite because every number is either prime or composite.
  4. The student is correct because 1 is divisible by 1 and by itself, giving two factors.

Explanation

Primality requires exactly two distinct positive divisors, 1 and the number itself. For 1, those divisors are not distinct, so 1 has only one factor and is classified as neither prime nor composite.

Question 35Mathematics content and pedagogical content knowledge for elementary teaching

A teacher gives students a rule: start at 5 and add 3 each time. Which list correctly shows the first four terms and identifies the pattern type?

  1. 5, 8, 11, 14; arithmetic with a common difference of 3 Correct answer.
  2. 5, 15, 45, 135; geometric with a common ratio of 3
  3. 5, 8, 13, 21; each term is the sum of the two before it
  4. 5, 6, 7, 8; arithmetic with a common difference of 1

Explanation

Adding 3 repeatedly produces 5, 8, 11, and 14, which is an arithmetic sequence with common difference 3. Each term after the first is found by adding the same constant.

Question 36Mathematics content and pedagogical content knowledge for elementary teaching

A student measures the length of a desk as 120 centimeters and wants to express it in meters. Which reasoning is correct?

  1. Since 1,000 centimeters make 1 meter, 120 centimeters equals 0.12 meters.
  2. Since 100 centimeters make 1 meter, 120 centimeters equals 12,000 meters.
  3. Since 10 centimeters make 1 meter, 120 centimeters equals 12 meters.
  4. Since 100 centimeters make 1 meter, 120 centimeters equals 1.2 meters. Correct answer.

Explanation

The metric relationship is 100 centimeters = 1 meter, so converting centimeters to meters means dividing by 100. Dividing 120 by 100 yields 1.2 meters.

Question 37Mathematics content and pedagogical content knowledge for elementary teaching

A teacher wants to build understanding of the standard algorithm for adding 358 + 276. Which classroom representation best makes the regrouping visible?

  1. A number line from 0 to 600 with a single jump of 634
  2. Base-ten blocks showing 8 ones and 6 ones regrouped into 1 ten and 4 ones Correct answer.
  3. A hundreds chart with the numbers 358 and 276 circled
  4. A multiplication array of 358 rows and 276 columns

Explanation

Base-ten blocks let students physically trade 10 ones for 1 ten and 10 tens for 1 hundred, which is exactly the regrouping the algorithm records. Making the exchange concrete connects the procedure to place-value reasoning.

Question 38Mathematics content and pedagogical content knowledge for elementary teaching

A student writes the coordinates of a point as (4, 7) but plots it by moving 4 units up and 7 units right. Which statement best explains the correction?

  1. The first coordinate gives horizontal movement and the second gives vertical movement, so the point is 4 right and 7 up. Correct answer.
  2. The coordinates are interchangeable because addition is commutative, so the plotted point is correct.
  3. The point cannot be plotted because both coordinates are positive.
  4. The point should be plotted 4 units up and 7 units right because the vertical coordinate is listed first.

Explanation

In an ordered pair (x, y), the first value is the horizontal position and the second is the vertical position. The point (4, 7) is therefore 4 units right and 7 units up from the origin.

Question 39Mathematics content and pedagogical content knowledge for elementary teaching

Which statement correctly describes why a square is also a rectangle in the hierarchy of quadrilaterals?

  1. A square is not a rectangle because rectangles must have unequal adjacent sides.
  2. A square is a rectangle because both shapes have exactly one pair of parallel sides.
  3. A square is a rectangle only because both shapes have four equal sides.
  4. A square has four right angles and opposite sides equal, which satisfies the definition of a rectangle. Correct answer.

Explanation

A rectangle is defined as a quadrilateral with four right angles, and a square meets that definition. The square adds the extra condition of four equal sides, so it is a special rectangle.

Question 40Mathematics content and pedagogical content knowledge for elementary teaching

A student says the probability of rolling an even number on a fair six-sided die is 1/2 because there are three even outcomes. Which statement best evaluates this reasoning?

  1. It is incorrect because the sample space has only three outcomes.
  2. It is correct only if the die is rolled an even number of times.
  3. It is incorrect because even numbers are more likely than odd numbers on a die.
  4. It is correct because the three even outcomes are equally likely and the sample space has six outcomes. Correct answer.

Explanation

The sample space {1, 2, 3, 4, 5, 6} has six equally likely outcomes, and {2, 4, 6} contains three of them. The ratio 3/6 simplifies to 1/2, so the student's reasoning is sound.

Question 41Mathematics content and pedagogical content knowledge for elementary teaching

A teacher asks students to solve 5 × 23 using partial products. Which set of partial products is correct?

  1. 20 and 3
  2. 100 and 15 Correct answer.
  3. 50 and 23
  4. 115 and 5

Explanation

Decompose 23 into 20 + 3, then multiply each part by 5: 5 × 20 = 100 and 5 × 3 = 15. Adding the partial products gives 115.

Question 42Mathematics content and pedagogical content knowledge for elementary teaching

A student is asked to round 3.467 to the nearest tenth. Which response shows correct reasoning?

  1. 3.47, because rounding to the nearest tenth keeps two decimal places
  2. 3.467, because the number is already between 3 and 4
  3. 3.4, because the tenths digit stays the same when rounding
  4. 3.5, because the hundredths digit 6 rounds the tenths digit 4 up to 5 Correct answer.

Explanation

Rounding to the nearest tenth examines the hundredths digit, which is 6. Since 6 is 5 or greater, the tenths digit 4 increases to 5, giving 3.5.

Question 43Mathematics content and pedagogical content knowledge for elementary teaching

A teacher wants students to recognize that the same quantity can be named by equivalent fractions. Which task best targets that understanding?

  1. Writing the decimal form of 1/2 as 0.5
  2. Shading 1/2 of a rectangle and then showing the same shaded region as 2/4 and 4/8 Correct answer.
  3. Ordering 1/2, 1/3, and 1/4 from least to greatest
  4. Finding the sum of 1/2 and 1/3 using common denominators

Explanation

Showing one shaded region labeled as 1/2, 2/4, and 4/8 makes visible that different fraction names can describe the same amount. This is the core idea of equivalent fractions.

Question 44Mathematics content and pedagogical content knowledge for elementary teaching

A student computes 8 − 3 + 2 and gets 3 by subtracting 3 + 2 first. Which statement best explains the correct evaluation?

  1. The expression cannot be evaluated without parentheses.
  2. The student is correct because subtraction should always be done after addition.
  3. The expression equals 13 because 8 + 3 + 2 adds all the numbers.
  4. Addition and subtraction are performed left to right, so 8 − 3 = 5 and 5 + 2 = 7. Correct answer.

Explanation

When an expression contains only addition and subtraction, operations are applied from left to right. Thus 8 − 3 = 5, and 5 + 2 = 7.

Question 45Mathematics content and pedagogical content knowledge for elementary teaching

A rectangular garden is 9 meters long and 6 meters wide. A student says the area is 30 square meters. Which error did the student most likely make?

  1. The student squared the width instead of multiplying length by width.
  2. The student multiplied the length by 2 instead of by the width.
  3. The student added the length and width and doubled the sum instead of multiplying the dimensions. Correct answer.
  4. The student used the perimeter formula and divided by 2.

Explanation

Area of a rectangle is length times width, so 9 × 6 = 54 square meters. The student's 30 comes from 2(9 + 6), the perimeter formula, suggesting the student doubled the sum of the dimensions instead of finding their product.

Question 46Mathematics content and pedagogical content knowledge for elementary teaching

A student is asked to find the greatest common factor of 24 and 36. Which response is correct and shows a valid method?

  1. 6, found by choosing the smallest shared factor greater than 1
  2. 18, found by subtracting 36 and 24
  3. 12, found by listing factors of both numbers and choosing the largest shared one Correct answer.
  4. 72, found by multiplying the two numbers together

Explanation

Listing factors of 24 and 36 reveals shared factors including 1, 2, 3, 4, 6, and 12. The greatest of these shared factors is 12.

Question 47Mathematics content and pedagogical content knowledge for elementary teaching

A student solves 3/4 × 2/5 and writes 6/20, then says the answer is 6/20 in simplest form. Which statement best evaluates the work?

  1. The product should be 8/15 because the fractions are cross-multiplied.
  2. The product 6/20 is correct and is already in simplest form.
  3. The product 6/20 is correct but should be simplified to 3/10. Correct answer.
  4. The product should be 5/9 because numerators and denominators are added.

Explanation

Multiplying numerators and denominators gives 6/20, which is a correct product. Dividing numerator and denominator by the common factor 2 yields 3/10, the simplest form.

Question 48Mathematics content and pedagogical content knowledge for elementary teaching

A teacher displays the data 4, 4, 5, 6, 9 and asks which measure of center best represents a typical value when one value is much larger than the rest. Which response is most appropriate?

  1. The mean, because it always equals one of the data values.
  2. The median, because it is less affected by extreme values than the mean. Correct answer.
  3. The mode, because every data set has exactly one mode.
  4. The range, because it describes the middle of the data.

Explanation

The median depends on position in the ordered list, so a single very large value shifts it less than the mean. For skewed data, the median often better represents a typical value.

Question 49Mathematics content and pedagogical content knowledge for elementary teaching

A student claims that a shape with four right angles must be a square. Which counterexample best refutes the claim?

  1. A triangle with one right angle and two acute angles
  2. A rectangle that is 6 cm by 3 cm, which has four right angles but unequal adjacent sides Correct answer.
  3. A rhombus with four equal sides but no right angles
  4. A trapezoid with exactly one pair of parallel sides

Explanation

A 6 cm by 3 cm rectangle has four right angles, satisfying the given condition, yet its adjacent sides are unequal so it is not a square. This single example shows the claim is false.

Question 50Mathematics content and pedagogical content knowledge for elementary teaching

A teacher wants students to understand the meaning of the remainder when 17 is divided by 5. Which interpretation is most appropriate?

  1. The quotient is 3.4, so the remainder is 4.
  2. The remainder is 12 because 17 minus 5 equals 12.
  3. Seventeen objects shared among 5 groups give 3 in each group with 2 left over. Correct answer.
  4. The remainder is 5 because the divisor is 5.

Explanation

Dividing 17 into 5 equal groups gives 3 per group and leaves 2 objects that cannot form another full group. The remainder 2 is always less than the divisor 5.