Practice Set · Pre-Algebra · Young Fermats · Class 1
Pre-Algebra Class 1: Integers on the Number Line & Absolute Value
The 7 lesson slides from Class 1 of the SOMATH Young Fermats — Pre-Algebra course, followed by 25 practice questions with hidden step-by-step answer explanations. Free to use at home, in class, or as a warm-up before enrolling.
What this lesson covers. Class 1 of our Young Fermats Pre-Algebra course anchors the whole year. Students learn the number-line home of every kind of number they will meet — natural numbers, whole numbers, integers — plus the two ideas they will lean on all semester: divisibility rules and absolute value as distance from zero. Below are the 7 slides used in class, then 25 practice questions with step-by-step explanations that stay hidden until your child clicks to reveal.
25 Practice Questions
From the list -6, 0, 3, -1, 5, 0.5, -0.25, list every number that belongs to Z (the integers).
Show answer & explanation
-6, 0, 3, -1, 5.
Why. Integers are the whole numbers together with their negatives, with no fractional part. So -6, 0, 3, -1, and 5 are integers. The decimals 0.5 and -0.25 are NOT integers because they have fractional parts.
Which of these numbers are in W (the whole numbers) but NOT in N (the natural numbers)? List: 0, 1, 5, -2, 12, -7.
Show answer & explanation
Only 0.
Why. N = {1, 2, 3, ...} starts at 1. W = {0, 1, 2, 3, ...} adds 0. So the only whole number that is NOT a natural number is 0. The negatives -2 and -7 are not in W at all.
On a number line, point A is at -6 and point B is at 10. What integer is at the MIDPOINT of AB?
Show answer & explanation
M = 2.
Why. Midpoint = (a + b) / 2 = (-6 + 10) / 2 = 4 / 2 = 2. Check: distance from -6 to 2 is 8, and distance from 2 to 10 is 8. Same both ways — that's what midpoint means.
Order from LEAST to GREATEST: -|-7|, -(-5), 0, |-3|, -(4 - 9).
Show answer & explanation
-7, 0, 3, 5, 5
Why. Simplify each: -|-7| = -7. -(-5) = 5. 0 = 0. |-3| = 3. -(4 - 9) = -(-5) = 5. Order: -7 < 0 < 3 < 5 = 5.
Fill each blank with < , > , or = : (a) -(-8) ___ |-8| (b) -|-6| ___ -(-6) (c) -|4| ___ 0
Show answer & explanation
(a) = (b) < (c) <
Why. (a) -(-8) = 8 and |-8| = 8, so equal. (b) -|-6| = -6 and -(-6) = 6, so -6 < 6. (c) -|4| = -4, and -4 < 0.
At 6 a.m. the temperature was -12 °F. It rose 3 °F per hour for 5 hours. What was the temperature at 11 a.m.?
Show answer & explanation
3 °F
Why. Total rise = 3 × 5 = 15 °F. New temperature = -12 + 15 = 3 °F.
A submarine at -180 feet rises 25 feet per minute for 4 minutes. What is its new depth?
Show answer & explanation
-80 feet.
Why. Total rise = 25 × 4 = 100 feet. Since up is positive: -180 + 100 = -80 feet. The submarine is still below sea level, just closer to the surface.
Maya's checking account started at -$35 (overdrawn). She deposited $80, then paid a $22 bill. Write a signed-number expression for her ending balance and evaluate it.
Show answer & explanation
-35 + 80 - 22 = $23
Why. Start -35, add deposit +80, subtract bill 22. -35 + 80 = 45. 45 - 22 = 23. Ending balance = $23.
Evaluate: |-9| + |4 - 11| - |-2| × |5|.
Show answer & explanation
6
Why. |-9| = 9. |4 - 11| = |-7| = 7. |-2| × |5| = 2 × 5 = 10. Now use order of operations (multiplication first): 9 + 7 - 10 = 16 - 10 = 6.
Solve for x: (a) |x| = 12 (b) |x - 4| = 7. List every solution.
Show answer & explanation
(a) x = 12 or -12. (b) x = 11 or -3.
Why. (a) |x| = 12 means x is 12 units from 0, so x = 12 or x = -12. (b) |x - 4| = 7 means x - 4 = 7 (→ x = 11) OR x - 4 = -7 (→ x = -3). Absolute-value equations usually have TWO solutions.
True or false, with a one-sentence reason: (a) |a × b| = |a| × |b| for every pair of integers. (b) |a| is always positive or zero, never negative.
Show answer & explanation
(a) True. (b) True.
Why. (a) Check with a = -4, b = 3: |-4 × 3| = |-12| = 12, and |-4| × |3| = 4 × 3 = 12. Absolute value drops the sign either way, so it's always true. (b) By definition, |a| is a distance from 0, and distances are never negative.
List every integer x that satisfies |x| < 4.
Show answer & explanation
-3, -2, -1, 0, 1, 2, 3 (seven integers).
Why. |x| < 4 means x is LESS than 4 units from 0, so -4 < x < 4 (strict inequality, endpoints NOT included). The integers strictly between -4 and 4 are -3, -2, -1, 0, 1, 2, 3.
A drone starts at 0 on a number line and moves: +6, -10, +4, -3. (a) Where does it end up? (b) What is the TOTAL DISTANCE it traveled?
Show answer & explanation
(a) -3. (b) 23.
Why. (a) Final position = signed sum: 6 - 10 + 4 - 3 = -3. (b) Total distance = sum of absolute values: |6| + |-10| + |4| + |-3| = 6 + 10 + 4 + 3 = 23. Position tells you WHERE the drone ended; distance tells you HOW FAR it flew.
Points P and Q are on the number line with |P| = 7 and Q = -2. Find every possible distance from P to Q.
Show answer & explanation
Distances are 5 or 9.
Why. |P| = 7 means P = 7 or P = -7. Distance from P to Q = |P - Q|. Case 1: P = 7, Q = -2 → |7 - (-2)| = |9| = 9. Case 2: P = -7, Q = -2 → |-7 - (-2)| = |-5| = 5. So the distance is either 5 or 9.
Test each number for divisibility by 3 AND by 4: (a) 132 (b) 246 (c) 528.
Show answer & explanation
(a) 132 is divisible by both. (b) 246 is divisible by 3 only. (c) 528 is divisible by both.
Why. Rule for 3: digit sum is a multiple of 3. Rule for 4: last two digits form a multiple of 4. (a) 132: digits 1+3+2 = 6 ✓ by 3. Last two digits 32 = 4 × 8 ✓ by 4. Both. (b) 246: 2+4+6 = 12 ✓ by 3. Last two digits 46, and 46 ÷ 4 = 11 R 2, NOT by 4. Only 3. (c) 528: 5+2+8 = 15 ✓ by 3. Last two 28 = 4 × 7 ✓ by 4. Both.
Find the SMALLEST positive integer that is divisible by both 6 and 8.
Show answer & explanation
24.
Why. This is the LCM of 6 and 8. Multiples of 8: 8, 16, 24, 32, ... The first one that's also a multiple of 6 is 24 (since 24 = 6 × 4 = 8 × 3). So LCM(6, 8) = 24.
Which of these numbers are divisible by 9? Numbers: 234, 405, 512, 729, 1234.
Show answer & explanation
234, 405, and 729 are divisible by 9. 512 and 1234 are not.
Why. Rule for 9: digit sum is a multiple of 9. 234: 2+3+4 = 9 ✓. 405: 4+0+5 = 9 ✓. 512: 5+1+2 = 8 ✗. 729: 7+2+9 = 18 ✓. 1234: 1+2+3+4 = 10 ✗.
List every multiple of 6 between 30 and 70 (inclusive).
Show answer & explanation
30, 36, 42, 48, 54, 60, 66.
Why. Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ... Those between 30 and 70 inclusive: 30, 36, 42, 48, 54, 60, 66. That's seven numbers.
How many multiples of 3 are there from 1 to 60?
Show answer & explanation
20.
Why. The multiples of 3 from 1 to 60 are 3, 6, 9, ..., 60. Count them: 60 ÷ 3 = 20. So there are 20 multiples of 3 in that range.
For each number, list every set from {N, W, Z} it belongs to: (a) 7 (b) 0 (c) -4.
Show answer & explanation
(a) N, W, Z. (b) W, Z (not N). (c) Z only.
Why. N = {1, 2, 3, ...}; W = N ∪ {0}; Z = W ∪ {negatives}. So (a) 7 is a positive counting number — it's in all three. (b) 0 is in W and Z, but N starts at 1, so 0 is NOT in N. (c) -4 is negative, so only in Z.
True or false: (a) Every natural number is an integer. (b) Every integer is a whole number. (c) 0 is a natural number.
Show answer & explanation
(a) True. (b) False. (c) False.
Why. (a) N ⊂ Z, so every natural number IS an integer. True. (b) Not every integer is whole — negatives like -3 are integers but not whole numbers. False. (c) N starts at 1, not 0. So 0 is a whole number but not a natural number. False.
Give ONE number in each category, or write 'impossible': (a) A whole number that is not a natural number. (b) An integer that is not a whole number. (c) A natural number that is not an integer.
Show answer & explanation
(a) 0. (b) -1 (or any negative). (c) Impossible.
Why. (a) 0 is in W but not N — the only such number. (b) Any negative integer works, like -1: it's in Z but not in W. (c) Every natural number IS an integer since N ⊂ Z, so this is impossible.
A stock opened Monday at $50. It changed each day: Mon +4, Tue -6, Wed +2, Thu -3, Fri +5. What was the closing price on Friday?
Show answer & explanation
$52.
Why. Track running total from $50: Mon 50 + 4 = 54. Tue 54 - 6 = 48. Wed 48 + 2 = 50. Thu 50 - 3 = 47. Fri 47 + 5 = 52. Friday close = $52. (Shortcut: sum the changes: +4 - 6 + 2 - 3 + 5 = +2, so 50 + 2 = 52.)
In a card game, positive cards add to your score and negative cards subtract. Amara drew: +5, -3, +8, -4, and a mystery card X. Her final score was 10. What is X?
Show answer & explanation
X = 4.
Why. Sum without X: 5 - 3 + 8 - 4 = 6. With X: 6 + X = 10, so X = 4.
An elevator starts at floor 0 and makes these moves: +4, -7, +3, -2, +6. (a) What floor does it end on? (b) What is the total DISTANCE (in floors) it traveled?
Show answer & explanation
(a) Floor 4. (b) 22 floors.
Why. (a) Signed sum: 4 - 7 + 3 - 2 + 6 = 4. (b) Total distance = sum of absolute values: |4| + |-7| + |3| + |-2| + |6| = 4 + 7 + 3 + 2 + 6 = 22 floors. Same distance-vs-position idea as the drone problem in Q13.
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About the Young Fermats Pre-Algebra course
Class 1 is one of 48 rolling classes in our Young Fermats Pre-Algebra program for grade 5–6 students (ages 10–12). The full arc runs from integers on the number line through systems of linear equations and probability — the direct on-ramp to Algebra 1 in 7th or 8th grade. Classes are small-group (max 6 students), 120 minutes per week, and students can start any Monday because the syllabus is rolling.
Every new family starts with a free 30-minute evaluation in our 226 W 79th Street classroom plus a written diagnostic delivered within 48 hours. The diagnostic maps your child's current level against the readiness signals for Pre-Algebra and Algebra 1 — and gives you a clear next step whether you enroll or not.
Ready to see the whole 48-class arc?
Class 1 is a taste — Young Fermats runs 48 rolling classes, students can join any Monday, and the syllabus takes them from integers through systems of equations. Small-group (max 6), 120 minutes a week, at 226 W 79th Street. First class is free. Cancel any time with 15 days' notice.
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