PRAXIS Elementary Education: Math Content Knowledge 1 — Questions and Answers
Question 1: A student is asked to place 3/4 and 7/8 on a number line. Which fraction is greater, and how do you know?
- 3/4, because 3 is closer to 4 than 7 is to 8
- 7/8, because it has a larger numerator
- 7/8, because it is closer to 1 whole than 3/4 (Correct answer)
- They are equal because both are close to 1
Correct answer: 7/8, because it is closer to 1 whole than 3/4
7/8 is greater because it is closer to 1 whole: 3/4 = 6/8, and 6/8 < 7/8. Using benchmark fractions confirms 7/8 > 3/4.
To compare 3/4 and 7/8, convert to a common denominator: 3/4 = 6/8. Then compare 6/8 and 7/8 — since 7 > 6, 7/8 > 6/8 = 3/4. Alternatively, use benchmark reasoning: 3/4 is 1/4 away from 1 (= 2/8 away), while 7/8 is only 1/8 away from 1. So 7/8 is closer to 1 and thus greater. The answer 'larger numerator means larger fraction' only works when denominators are the same. Understanding fraction comparison strategies is essential for elementary math instruction.
Question 2: What is the value of the digit 6 in the number 3,645?
- 6
- 60
- 600 (Correct answer)
- 6,000
Correct answer: 600
In 3,645, the digit 6 is in the hundreds place. Its value is 6 × 100 = 600.
Place value understanding is foundational in elementary mathematics. In the number 3,645: 5 is in the ones place (value = 5), 4 is in the tens place (value = 40), 6 is in the hundreds place (value = 600), 3 is in the thousands place (value = 3,000). The value of a digit is the digit multiplied by its place value. The digit 6 in the hundreds place has a value of 6 × 100 = 600. This understanding supports multi-digit operations and number sense development.
Question 3: A rectangle has a length of 8 units and a width of 5 units. What is its area?
- 13 square units
- 26 square units
- 40 square units (Correct answer)
- 80 square units
Correct answer: 40 square units
Area of a rectangle = length × width = 8 × 5 = 40 square units.
The area of a rectangle is calculated using the formula A = l × w (length times width). With l = 8 and w = 5: A = 8 × 5 = 40 square units. Area is measured in square units because it represents a two-dimensional space. Perimeter (the distance around) would be 2(8 + 5) = 26 units. Students often confuse area and perimeter, so understanding the conceptual distinction — area covers a surface, perimeter borders it — is important for teaching.
Question 4: Which property is demonstrated by: 4 × (3 + 2) = (4 × 3) + (4 × 2)?
- Commutative property of multiplication
- Associative property of multiplication
- Distributive property (Correct answer)
- Identity property of multiplication
Correct answer: Distributive property
The distributive property states that multiplication distributes over addition: a(b + c) = ab + ac.
The distributive property is one of the fundamental properties of operations and states: a × (b + c) = (a × b) + (a × c). In the example, 4 × (3 + 2) = (4 × 3) + (4 × 2) = 12 + 8 = 20. The commutative property states that order doesn't matter (a × b = b × a). The associative property states that grouping doesn't matter for three factors: (a × b) × c = a × (b × c). The identity property states that any number times 1 equals itself. The distributive property is foundational for mental math, multi-digit multiplication, and algebra.
Question 5: A student says that 0.3 is greater than 0.25 because '3 is less than 25.' What misconception does this reveal?
- The student does not understand place value in decimals.
- The student confuses multiplication and division.
- The student has a whole number bias when comparing decimals.
- Both A and C — place value misunderstanding and whole number bias (Correct answer)
Correct answer: Both A and C — place value misunderstanding and whole number bias
The student treats decimals like whole numbers, ignoring place value (0.3 = 0.30, which is greater than 0.25). This is the 'whole number bias' and a place value misconception.
This common decimal misconception is called 'whole number thinking' or 'whole number bias' — students compare decimal digits as if they were whole numbers, reasoning '25 > 3, so 0.25 > 0.3.' To correct this, teachers can: (1) have students append a zero to make place values explicit (0.30 vs. 0.25), showing 30 hundredths > 25 hundredths; (2) use number lines or base-ten blocks; (3) explicitly discuss that tenths are larger units than hundredths. Both a place value misunderstanding and whole number bias are present, making D correct.
Question 6: What is the greatest common factor (GCF) of 24 and 36?
- 4
- 6
- 12 (Correct answer)
- 72
Correct answer: 12
Factors of 24: 1,2,3,4,6,8,12,24. Factors of 36: 1,2,3,4,6,9,12,18,36. Greatest common factor = 12.
The GCF is the largest number that divides evenly into both numbers. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Common factors: 1, 2, 3, 4, 6, 12. Greatest common factor = 12. Alternative method using prime factorization: 24 = 2³ × 3; 36 = 2² × 3². GCF = 2² × 3 = 4 × 3 = 12. The GCF is used to simplify fractions (e.g., 24/36 = 2/3 when divided by GCF 12) and is foundational for work with fractions.
Question 7: A class of 24 students is divided equally into groups. Which number of groups is NOT possible?
- 3 groups
- 4 groups
- 5 groups (Correct answer)
- 6 groups
Correct answer: 5 groups
24 ÷ 5 = 4.8, which is not a whole number. 24 is not divisible by 5, so 5 equal groups of whole students is impossible.
For 24 students to be divided into equal groups, 24 must be divisible by the number of groups. Divisibility checks: 24 ÷ 3 = 8 ✓; 24 ÷ 4 = 6 ✓; 24 ÷ 5 = 4.8 ✗ (not a whole number); 24 ÷ 6 = 4 ✓. Since 5 is not a factor of 24, 24 students cannot be divided into 5 equal groups of whole students. Understanding divisibility and factors is fundamental to division instruction and fraction concepts at the elementary level.
Question 8: A student solves 43 − 18 and gets 35. What error did the student likely make?
- The student added instead of subtracting.
- The student forgot to regroup (borrow) and subtracted 3 − 8 as 8 − 3 = 5. (Correct answer)
- The student made an arithmetic error on the tens digit only.
- The student rounded to the nearest ten.
Correct answer: The student forgot to regroup (borrow) and subtracted 3 − 8 as 8 − 3 = 5.
The typical error is subtracting the smaller digit from the larger within a column (8 − 3 = 5 in ones column), rather than regrouping. 43 − 18: if student does 8 − 3 = 5 (ones) and 4 − 1 = 3 (tens), they get 35.
This is a classic regrouping error in subtraction. The correct process: 43 − 18. Since 3 < 8, regroup from the tens: borrow 10 from 40, making the ones 13 − 8 = 5, and tens 3 − 1 = 2, giving 25. The student instead computed 8 − 3 = 5 (reversing the subtraction in the ones place) and 4 − 1 = 3 (tens), getting 35. This 'smaller from larger' error is one of the most common subtraction mistakes and indicates the student does not understand regrouping. Remediation should use base-ten blocks to make regrouping concrete.
Question 9: What is 2/3 + 1/4?
- 3/7
- 3/12
- 11/12 (Correct answer)
- 8/12
Correct answer: 11/12
To add fractions with unlike denominators, find the LCD (12). 2/3 = 8/12; 1/4 = 3/12. 8/12 + 3/12 = 11/12.
Adding fractions with unlike denominators requires finding a common denominator. Least Common Denominator of 3 and 4 = 12. Convert: 2/3 = 8/12 (multiply numerator and denominator by 4); 1/4 = 3/12 (multiply by 3). Add: 8/12 + 3/12 = 11/12. Adding numerators without finding common denominators (3/7) is a common student error. 11/12 cannot be simplified further. Understanding fraction addition with unlike denominators is a key 4th-5th grade standard.
Question 10: Which of the following numbers is prime?
- 21
- 27
- 29 (Correct answer)
- 33
Correct answer: 29
29 is prime — its only factors are 1 and 29. 21 = 3×7, 27 = 3×9, 33 = 3×11.
A prime number has exactly two factors: 1 and itself. Testing: 21 = 3 × 7 (composite); 27 = 3 × 9 = 3³ (composite); 29 — not divisible by 2, 3, 5 (the primes up to √29 ≈ 5.4), so 29 is prime; 33 = 3 × 11 (composite). 29 is the only prime in the list. Teaching students to test divisibility by primes up to the square root of a number is an efficient primality test. Understanding prime and composite numbers supports factoring, GCF/LCM calculations, and fraction work.
Question 11: A bag contains 4 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly picking a blue marble?
- 1/3
- 3/7
- 3/9 (Correct answer)
- 1/4
Correct answer: 3/9
Total marbles = 4+3+2 = 9. Probability of blue = 3/9 = 1/3. Both 3/9 and 1/3 are equivalent; 3/9 is the un-simplified form.
Probability = (number of favorable outcomes) / (total number of equally likely outcomes). Total marbles: 4 + 3 + 2 = 9. Blue marbles: 3. P(blue) = 3/9. This simplifies to 1/3, but the answer 3/9 is also correct (they are equivalent). Option A (1/3) equals 3/9 in simplified form, and both are correct. Option B (3/7) incorrectly uses only the non-red marbles as the total. Understanding probability concepts, including identifying the sample space (all possible outcomes), is part of elementary statistics and data standards.
Question 12: A student is learning to multiply 24 × 3. Which strategy shows decomposition (breaking apart numbers)?
- Using repeated addition: 24 + 24 + 24
- Breaking 24 into 20 + 4: (20 × 3) + (4 × 3) = 60 + 12 = 72 (Correct answer)
- Using a times table chart
- Counting by 24s three times
Correct answer: Breaking 24 into 20 + 4: (20 × 3) + (4 × 3) = 60 + 12 = 72
Decomposing 24 into 20 + 4 and distributing the multiplication uses the distributive property and builds number sense.
Decomposition strategies in multiplication involve breaking one or both factors into easier-to-work-with components, then applying the distributive property. Breaking 24 into 20 + 4: (20 × 3) + (4 × 3) = 60 + 12 = 72. This strategy builds conceptual understanding, connects to the distributive property, and helps students compute mentally. Repeated addition (3 × 24 = 24+24+24) is a valid strategy but reflects a repeated-addition conception of multiplication rather than decomposition. Counting by 24s is similar to repeated addition.
Question 13: What is the perimeter of a square with a side length of 7 cm?
- 14 cm
- 21 cm
- 28 cm (Correct answer)
- 49 cm
Correct answer: 28 cm
Perimeter of a square = 4 × side = 4 × 7 = 28 cm.
The perimeter of any polygon is the total distance around its boundary, calculated by summing all side lengths. A square has four equal sides, so: P = s + s + s + s = 4s = 4 × 7 = 28 cm. The area would be s² = 7² = 49 cm² — which is option D. Students commonly confuse area and perimeter; conceptually, perimeter is a one-dimensional measure (length), while area is two-dimensional. Units for perimeter are linear (cm), while units for area are square (cm²).
Question 14: What type of angle is formed when two perpendicular lines intersect?
- Acute angle
- Obtuse angle
- Right angle (Correct answer)
- Straight angle
Correct answer: Right angle
Perpendicular lines intersect at exactly 90°, forming four right angles at the intersection point.
Perpendicular lines are lines that intersect at right angles (90°). When two perpendicular lines cross, they form four right angles. Angle types: acute = less than 90°; right = exactly 90°; obtuse = greater than 90° but less than 180°; straight = exactly 180°. Understanding perpendicularity is foundational for geometry — it applies to rectangles (all corners are right angles), coordinate grids (x and y axes are perpendicular), and construction geometry. The symbol ⊥ denotes perpendicular lines.
Question 15: Which of the following correctly shows the order of operations for: 3 + 4 × 2 − 1?
- 14 − 1 = 13 (left to right)
- 3 + 8 − 1 = 10 (multiply first) (Correct answer)
- 7 × 2 − 1 = 13 (left to right)
- 3 + 4 = 7, then 7 × 1 = 7
Correct answer: 3 + 8 − 1 = 10 (multiply first)
Order of operations (PEMDAS): multiplication before addition/subtraction. 4 × 2 = 8 first, then 3 + 8 − 1 = 10.
The order of operations (PEMDAS/BODMAS) specifies: 1. Parentheses/Brackets; 2. Exponents/Orders; 3. Multiplication and Division (left to right); 4. Addition and Subtraction (left to right). In 3 + 4 × 2 − 1: first perform multiplication: 4 × 2 = 8, giving 3 + 8 − 1. Then left to right: 3 + 8 = 11, then 11 − 1 = 10. This is a critical concept because without a standard order, the same expression could yield different values. The common mistake is computing left to right without prioritizing multiplication/division.
Question 16: A pattern shows: 2, 5, 8, 11, 14... What is the rule and the next term?
- Add 2 each time; next term is 16
- Add 3 each time; next term is 17 (Correct answer)
- Multiply by 2; next term is 28
- Add 4 then subtract 1; next term is 17
Correct answer: Add 3 each time; next term is 17
The pattern increases by 3 each time (arithmetic sequence with common difference 3). 14 + 3 = 17.
To identify a number pattern, find the difference between consecutive terms: 5−2=3, 8−5=3, 11−8=3, 14−11=3. The pattern is an arithmetic sequence with a common difference of +3. The next term: 14 + 3 = 17. Identifying and extending patterns is foundational algebraic thinking and is included in elementary math standards. Students learn to express rules (add 3 each time), predict future terms, and eventually represent patterns with variables.
Question 17: Which fraction is equivalent to 2/3?
- 3/4
- 4/6 (Correct answer)
- 6/10
- 3/5
Correct answer: 4/6
4/6 is equivalent to 2/3 because 2/3 × 2/2 = 4/6. Both represent the same point on the number line.
Equivalent fractions represent the same value using different numerators and denominators. To find a fraction equivalent to 2/3, multiply or divide both numerator and denominator by the same non-zero number: 2/3 × 2/2 = 4/6. Verification: 4÷6 = 2÷3 = 0.666... Cross-multiplication test: 2×6 = 12 = 3×4 = 12 ✓. Understanding equivalent fractions is essential for comparing fractions, adding/subtracting fractions with unlike denominators, and simplifying fractions. 6/10 simplifies to 3/5, not 2/3.
Question 18: A teacher wants to introduce multiplication using equal groups. Which problem best illustrates this concept?
- What is 6 more than 4?
- If there are 4 bags with 6 apples each, how many apples total? (Correct answer)
- If apples cost $6 each, how much change from $10?
- Share 24 apples equally among 6 friends.
Correct answer: If there are 4 bags with 6 apples each, how many apples total?
The equal groups model of multiplication: 4 bags × 6 apples each = 4 groups of 6 = 24. This directly models multiplication as repeated equal groups.
Multiplication can be understood through several models: equal groups, arrays, number lines, and area models. The equal groups model is typically introduced first: a certain number of equal-sized groups. Option B (4 bags with 6 apples each) directly models 4 × 6 as 4 equal groups of 6. Option A is addition. Option C involves money and change (subtraction/addition). Option D is division (sharing 24 into 6 equal groups). Starting with equal groups builds intuitive meaning for multiplication before introducing abstract notation.
Question 19: What is the least common multiple (LCM) of 4 and 6?
- 2
- 12 (Correct answer)
- 24
- 4
Correct answer: 12
Multiples of 4: 4,8,12,16... Multiples of 6: 6,12,18... Least common multiple = 12.
The Least Common Multiple (LCM) is the smallest positive integer divisible by both numbers. Method 1 (list multiples): Multiples of 4: 4, 8, 12, 16, 20... Multiples of 6: 6, 12, 18, 24... First common multiple = 12. Method 2 (prime factorization): 4 = 2²; 6 = 2 × 3. LCM = 2² × 3 = 12. The LCM is used primarily for finding common denominators when adding or subtracting fractions with unlike denominators. For 1/4 + 1/6, the LCD = 12, giving 3/12 + 2/12 = 5/12.
Question 20: A container holds 2.5 liters. If it is 3/4 full, how much liquid is inside?
- 1.5 liters
- 1.75 liters
- 1.875 liters (Correct answer)
- 2.0 liters
Correct answer: 1.875 liters
3/4 of 2.5 = 0.75 × 2.5 = 1.875 liters.
Finding a fraction of a quantity: 3/4 of 2.5 = (3 ÷ 4) × 2.5 = 0.75 × 2.5. To compute: 0.75 × 2.5 = (0.75 × 2) + (0.75 × 0.5) = 1.5 + 0.375 = 1.875 liters. Alternatively: 3/4 × 2.5 = 3 × 2.5 / 4 = 7.5 / 4 = 1.875. This problem integrates fractions and decimals in a real-world context — an important connection for upper elementary students and for teachers who must be able to identify appropriate problem-solving approaches.
Question 21: Which of the following best describes a proportional relationship between two quantities?
- One quantity increases as the other decreases.
- The ratio between the two quantities is constant. (Correct answer)
- The quantities are always equal.
- One quantity is always double the other.
Correct answer: The ratio between the two quantities is constant.
In a proportional relationship, the ratio y/x is constant (y = kx for some constant k). The ratio never changes even as the values do.
A proportional relationship exists when two quantities are related by a constant ratio (y = kx, where k is the constant of proportionality). For example, if one apple costs $0.50, then 2 apples cost $1.00, 3 cost $1.50, etc. — the ratio (price/apples) is always $0.50. Understanding proportional reasoning is a major goal of upper elementary mathematics and bridges arithmetic and algebra. It underpins percentages, unit rates, scale drawings, and probability. All of options B's special cases (equal, always double) describe proportional relationships, but B is the general definition.
Question 22: A student is asked to estimate 48 × 52. Which estimation strategy gives the most reasonable result?
- Round both to nearest ten: 50 × 50 = 2,500 (Correct answer)
- Round down: 40 × 50 = 2,000
- Round up: 50 × 60 = 3,000
- Multiply exactly: 48 × 52 = 2,496
Correct answer: Round both to nearest ten: 50 × 50 = 2,500
Rounding both to the nearest ten gives 50 × 50 = 2,500, which is the closest estimate to the actual answer of 2,496.
Estimation involves producing a reasonable approximation quickly. Rounding 48 and 52 to the nearest ten gives 50 × 50 = 2,500, which is extremely close to the exact answer of 2,496 (only 4 off). Rounding down to 40 × 50 = 2,000 underestimates by about 20%. Rounding up to 50 × 60 = 3,000 overestimates by about 20%. The multiply-exactly option gives the precise answer but is not an estimation. Teaching estimation builds number sense and helps students check whether computed answers are reasonable.
Question 23: What is the definition of a variable in elementary mathematics?
- A number that changes depending on the operation
- A symbol (often a letter) representing an unknown or changing quantity (Correct answer)
- A type of geometric shape
- A rule for ordering numbers
Correct answer: A symbol (often a letter) representing an unknown or changing quantity
A variable is a symbol, often a letter, that represents an unknown quantity or a quantity that can take different values.
Variables are introduced in elementary mathematics as part of early algebraic thinking. A variable is a symbol (typically a letter like n, x, or □) that represents an unknown quantity or a quantity that varies. In the equation n + 3 = 7, n is an unknown with a specific value (4). In a rule like y = 2x, x can take many values and y changes accordingly. Understanding variables prepares students for formal algebra and supports the generalization of arithmetic patterns. Elementary students often encounter variables as boxes or blanks before transitioning to letters.
Question 24: A number line from 0 to 1 is divided into 8 equal parts. A point is marked at the 5th tick mark. What fraction does this represent?
- 5/10
- 5/8 (Correct answer)
- 4/8
- 1/5
Correct answer: 5/8
With 8 equal parts, each part is 1/8. The 5th tick mark represents 5/8.
When a number line from 0 to 1 is divided into n equal parts, each part has a length of 1/n and the kth tick mark represents k/n. With 8 equal parts, each part = 1/8, and the 5th tick mark = 5/8. This represents 5 of the 8 equal portions between 0 and 1. Number lines are a critical representation for fractions because they make the distance/measurement interpretation visible and help students understand that fractions are numbers with specific positions, not just parts of a whole.
Question 25: Which of the following is the correct way to interpret the expression 3⁴?
- 3 × 4 = 12
- 4 × 4 × 4 = 64
- 3 × 3 × 3 × 3 = 81 (Correct answer)
- 3 + 3 + 3 + 3 = 12
Correct answer: 3 × 3 × 3 × 3 = 81
3⁴ means 3 is the base and 4 is the exponent: multiply 3 by itself 4 times: 3 × 3 × 3 × 3 = 81.
Exponential notation: in the expression bⁿ, b is the base and n is the exponent (or power). The exponent indicates how many times the base is used as a factor. 3⁴ = 3 × 3 × 3 × 3 = 9 × 3 × 3 = 27 × 3 = 81. Common misconceptions: multiplying base by exponent (3 × 4 = 12) or using the exponent as the base (4³ = 64). Understanding exponents is introduced in upper elementary grades and is foundational for scientific notation, area/volume formulas, and algebraic concepts.
Question 26: A data set shows the following temperatures (°F) for one week: 72, 68, 75, 80, 72, 65, 72. What is the mode?
- 65
- 68
- 72 (Correct answer)
- 75
Correct answer: 72
The mode is the value that appears most frequently. 72 appears three times, more than any other value.
Measures of central tendency: Mean (average) = sum ÷ count; Median = middle value when ordered; Mode = most frequently occurring value. Data set ordered: 65, 68, 72, 72, 72, 75, 80. Mode = 72 (appears 3 times). Mean = (72+68+75+80+72+65+72) ÷ 7 = 504 ÷ 7 = 72 (coincidentally also 72). Median = 72 (4th value in ordered list). While all three measures are 72 in this case, students must know the mode is determined by frequency, not calculation.
Question 27: What is the standard algorithm for multi-digit multiplication, and what conceptual understanding should accompany it?
- Repeated addition; understanding multiplication as adding the same number many times
- The partial products method followed by regrouping; understanding place value and the distributive property (Correct answer)
- Estimation and rounding; understanding that precise answers aren't needed
- Using a calculator; understanding that tools support computation
Correct answer: The partial products method followed by regrouping; understanding place value and the distributive property
The standard multiplication algorithm uses partial products and regrouping, grounded in place value and the distributive property.
The standard multi-digit multiplication algorithm works by multiplying each digit of one factor by each digit of the other, accounting for place value (essentially applying the distributive property repeatedly). For example, 23 × 45 = 23 × (40 + 5) = (23 × 40) + (23 × 5) = 920 + 115 = 1,035. The algorithm packages these partial products efficiently with regrouping. Conceptual understanding — knowing why the algorithm works through place value and the distributive property — is essential before or alongside procedural fluency, as recommended by standards frameworks.
Question 28: What does 'unitizing' mean in the context of place value?
- Measuring objects with standard units
- Understanding that a group of ones can be treated as a single unit at a higher place value (Correct answer)
- Ordering numbers from smallest to largest
- Converting fractions to decimals
Correct answer: Understanding that a group of ones can be treated as a single unit at a higher place value
Unitizing is the understanding that 10 ones can be treated as a single ten, 10 tens as a single hundred, etc. — foundational to place value.
Unitizing is a critical conceptual development in place value understanding. It involves the ability to simultaneously see a quantity in two ways: as individual units and as a group. For example, 10 individual ones can be unitized as 1 ten. This conceptual shift is essential for understanding regrouping in addition and subtraction, for understanding that the '2' in '28' represents 2 tens (20), not 2 ones, and for later work with fractions and measurement. Without unitizing, students may not truly understand place value and may rely on rote procedures without meaning.
Question 29: A teacher uses base-ten blocks to teach subtraction with regrouping. What is the purpose of this manipulative?
- To speed up computation by having students count blocks
- To provide a concrete, visual model that makes the concept of exchanging units tangible (Correct answer)
- To replace the need for standard algorithms
- To test whether students understand addition before subtraction
Correct answer: To provide a concrete, visual model that makes the concept of exchanging units tangible
Base-ten blocks make the abstract concept of regrouping (exchanging 1 ten for 10 ones) concrete and visible, supporting conceptual understanding.
Base-ten blocks (units, rods/longs, flats, cubes) provide a concrete representation of the base-ten number system. When teaching subtraction with regrouping, students can physically exchange one ten-rod for ten unit blocks to model 'borrowing,' making the abstract process tangible. This concrete experience precedes pictorial representations (drawings) and abstract notation (the algorithm). The CPA (Concrete-Pictorial-Abstract) progression, advocated in mathematics education research, emphasizes that concrete manipulative experiences build the conceptual foundation for procedural fluency.
Question 30: What is the mean (average) of the following data set: 12, 15, 18, 21, 24?
- 17
- 18 (Correct answer)
- 19
- 21
Correct answer: 18
Mean = sum ÷ count = (12+15+18+21+24) ÷ 5 = 90 ÷ 5 = 18.
The mean (arithmetic average) is calculated by: (1) summing all values, and (2) dividing by the count. Sum = 12 + 15 + 18 + 21 + 24 = 90. Count = 5. Mean = 90 ÷ 5 = 18. Note that the median (middle value in ordered set) is also 18 for this symmetric data set. The mean represents the balance point of the data — if you redistributed the total equally among all data points, each would equal the mean. Understanding the mean as a balance point builds conceptual understanding beyond the calculation.
Question 31: Which of the following correctly represents the number 4,307 in expanded form?
- 4,000 + 300 + 70
- 4,000 + 300 + 7 (Correct answer)
- 4,000 + 30 + 7
- 400 + 300 + 7
Correct answer: 4,000 + 300 + 7
4,307 = 4,000 + 300 + 0 + 7. The tens digit is 0, so there is no tens term. Expanded form: 4,000 + 300 + 7.
Expanded form represents each digit multiplied by its place value: 4 × 1,000 = 4,000; 3 × 100 = 300; 0 × 10 = 0 (no tens); 7 × 1 = 7. Full expanded form: 4,000 + 300 + 0 + 7 = 4,000 + 300 + 7. The key feature is the zero in the tens place — students must recognize that it contributes nothing to the value. Expanded form develops place value understanding and prepares students for multi-digit computation by making the value of each digit explicit.
Question 32: A student walks 1/2 mile to school and then 3/4 mile to the library. How far does the student walk in total?
- 4/6 mile
- 1 1/4 miles (Correct answer)
- 5/6 mile
- 1 1/2 miles
Correct answer: 1 1/4 miles
1/2 + 3/4 = 2/4 + 3/4 = 5/4 = 1 1/4 miles.
Adding fractions with unlike denominators: LCD of 2 and 4 = 4. Convert 1/2 = 2/4. Then: 2/4 + 3/4 = 5/4. Convert to mixed number: 5/4 = 1 1/4. Real-world fraction problems require students to apply fraction addition in context, recognize when to convert to a common denominator, and interpret results as mixed numbers when the sum exceeds 1. This is a typical 4th-grade standard problem connecting fraction computation to measurement contexts.
Question 33: Which of the following best illustrates the concept of 'conservation of number' in early childhood mathematics?
- A child counts 5 objects scattered randomly and then rearranged in a line and gets 5 both times. (Correct answer)
- A child understands that 5 is between 4 and 6.
- A child can count to 100 by memory.
- A child knows that 5 + 0 = 5.
Correct answer: A child counts 5 objects scattered randomly and then rearranged in a line and gets 5 both times.
Conservation of number (Piaget) is the understanding that the count/quantity of objects remains the same even when they are rearranged.
Conservation of number, a concept from Piaget's cognitive development theory, refers to the understanding that the quantity of a set of objects remains constant regardless of how they are arranged, spread out, or moved. A child who has achieved conservation of number understands that 5 blocks spread out in a line is still 5, not more, even though the arrangement looks different. Children who lack conservation may believe that spreading objects out makes 'more' of them. This concept develops in early childhood (typically around ages 5-7) and is foundational for understanding counting, addition, and number sense.
PRAXIS Elementary Education: Math Content Knowledge 1
A student is asked to place 3/4 and 7/8 on a number line.
Which fraction is greater, and how do you know?