Heschel School · Rising 8th Grade Summer Math Packet · Solved & Explained
Heschel School Rising 8th Grade Summer Math Packet — Solved & Explained
A complete walkthrough of The Heschel School's Rising Grade 8 Summer Review Packet — the 10-question Integers + Order of Operations + Fraction set that bridges 7th-grade pre-algebra and 8th-grade Algebra I. Every problem solved step by step, with a SOMATH video walkthrough. Written by Upper West Side math tutors three blocks from Heschel.
Watch the full walkthrough
A SOMATH math teacher walks through every problem on this page. Use the video for full reasoning and use the written solutions below as a quick checking aid.
About this packet — and about Heschel math
The Abraham Joshua Heschel School at 30 West End Avenue on the Upper West Side runs a leveled middle-school math program. Per their published curriculum, most Heschel students finish Algebra I in 8th grade; the Enriched track finishes Algebra I in 7th grade and Geometry in 8th. (See the Heschel Middle School Math page.)
The Rising 8th Grade Summer Review Packet is the bridge document Heschel uses to keep students sharp between 7th and 8th grade. The first page — the page we're solving here — targets three foundations that everything in Algebra I depends on:
- Integers and substitution. Plugging signed numbers into algebraic expressions without losing a negative sign.
- Order of operations (PEMDAS). Especially the rules for exponents on negatives and what absolute value bars include or exclude.
- Fractions and mixed numbers. Including the rules for multiplying and dividing mixed numbers and signed fractions.
These are exactly the skills that decide whether a Heschel student lands in the Standard, Accelerated, or Enriched 8th-grade track in the fall — and they're the same skills that show up on the SHSAT, ISEE Upper Level, and SSAT Upper Level in 8th grade.
The given values used throughout problems 1–4 of the packet:
a = 4, b = −6, c = −12, d = 2
Jump to a problem
Step-by-step solutions
Integers & substitution
Problem 1
Find the value of |c − b| ÷ a when a = 4, b = −6, c = −12.
Final answer: 3/2 (or 1.5)
|c − b| ÷ a
= |(−12) − (−6)| ÷ 4
= |−12 + 6| ÷ 4
= |−6| ÷ 4
= 6 ÷ 4
= 3/2
Why this matters: The classic 7th-grade error is dropping the inner negative — students write c − b = −12 − 6 = −18 instead of −12 + 6 = −6. Subtracting a negative adds. The absolute value bars then strip the sign at the end, not in the middle.
Problem 2
Find the value of −b² + 4 · d when b = −6, d = 2.
Final answer: −28
−b² + 4·d
= −(−6)² + 4·(2)
= −(36) + 8
= −28
Why this matters: −b² means "take b, square it, then negate." The exponent applies only to b — not to the leading minus sign. So even though b is already negative, b² = 36 and −b² = −36. This is the single most-missed convention on the 8th-grade Algebra I diagnostic at every NYC private school.
Problem 3
Find the value of b − b/d when b = −6, d = 2.
Final answer: −3
b − b/d
= (−6) − (−6)/(2)
= (−6) − (−3)
= −6 + 3
= −3
Why this matters: Division comes before subtraction (PEMDAS), so we do b/d first, then subtract. Then the same "subtract a negative = add" rule shows up again. Two sign rules, one problem.
Problem 4
Find the value of (a + d) / (b − c) when a = 4, b = −6, c = −12, d = 2.
Final answer: 1
(a + d) / (b − c)
= (4 + 2) / ((−6) − (−12))
= 6 / (−6 + 12)
= 6 / 6
= 1
Why this matters: The fraction bar is a grouping symbol — finish the numerator and the denominator completely before dividing. Without that habit, students try to "cancel" pieces and lose the whole problem.
Order of operations & absolute value
Problem 5
Simplify: (6 − (−4)² − 15) / (−5)
Final answer: 5
(6 − (−4)² − 15) / (−5)
= (6 − 16 − 15) / (−5) ← (−4)² = +16
= (−25) / (−5)
= 5
Why this matters: Contrast with Problem 2 carefully. Here the negative is inside the parentheses: (−4)² = (−4)·(−4) = +16. In Problem 2 the negative was outside: −(−6)² = −36. Parentheses change the rule.
Problem 6
Simplify: 1/3 − (−5/6) · 3/5
Final answer: 5/6
1/3 − (−5/6) · 3/5
= 1/3 − (−15/30) ← multiply first
= 1/3 − (−1/2)
= 1/3 + 1/2
= 2/6 + 3/6
= 5/6
Why this matters: Multiplication comes before subtraction. The shortcut students miss: (−5/6)·(3/5) — the 5's cancel and you get −3/6 = −1/2 in one step. Either way, subtracting a negative becomes addition, then you need a common denominator (6).
Problem 7
Simplify: 2.4 − |−3.61| − (−8.3)
Final answer: 7.09
2.4 − |−3.61| − (−8.3)
= 2.4 − 3.61 − (−8.3) ← |−3.61| = 3.61
= 2.4 − 3.61 + 8.3
= (−1.21) + 8.3
= 7.09
Why this matters: Decimals follow the same sign rules as integers. The absolute value bars only neutralize the sign of what's inside them — they don't change the minus sign sitting in front of them.
Problem 8
Simplify: −|62 + (−77)|
Final answer: −15
−|62 + (−77)|
= −|62 − 77|
= −|−15|
= −(15)
= −15
Why this matters: Two sign moves in one expression. Inside the bars, 62 + (−77) = −15. The bars strip that sign, giving 15. Then the leading minus sign (which lives outside the bars) finally takes its turn, making the answer negative.
Problem 9
Simplify: (−9 + 4)² − |12(−3) + 80 ÷ 20|
Final answer: −7
(−9 + 4)² − |12(−3) + 80 ÷ 20|
= (−5)² − |−36 + 4|
= 25 − |−32|
= 25 − 32
= −7
Why this matters: The full PEMDAS workout. Inside each grouping symbol first, then exponents and absolute value, then subtraction last. Inside the bars, 80 ÷ 20 = 4 before adding to −36 — division and multiplication beat addition.
Mixed numbers & signed fractions
Problem 10
Simplify: 2½ · (−3⅕) ÷ 8/15
Final answer: −15
2½ · (−3⅕) ÷ 8/15
= (5/2) · (−16/5) ÷ (8/15) ← convert to improper fractions
= (5/2) · (−16/5) · (15/8) ← divide = multiply by reciprocal
= (5 · −16 · 15) / (2 · 5 · 8)
= (−1200) / (80)
= −15
Why this matters: Three big habits in one problem: (1) convert mixed numbers to improper fractions before doing anything else, (2) replace ÷ with × reciprocal as soon as you see it, (3) track the sign separately. The fast version cancels first: the 5's cancel, the 8 cancels with the 16 leaving 2, and you're left with (−2)(15)/2 = −15.
If your child can do these in 30 minutes, they're ready
Heschel's leveling decision — Standard, Accelerated, or Enriched — isn't just about can a student solve these. It's about fluency: can they do all ten in roughly half an hour, without a calculator, with no sign errors? That's the bar for the Accelerated track. The Enriched track expects this kind of work to be near-automatic in 7th grade.
From our classroom three blocks away, the three patterns that decide the level are:
- Whether the student notices when a negative is inside or outside the exponent / absolute value — Problems 2, 5, 7, 8, 9.
- Whether they cancel fractions before multiplying instead of multiplying huge numerators and denominators first — Problems 6 and 10.
- Whether the fraction bar is treated as grouping — Problems 1, 4, 5.
Keep going — Part 2 & other UWS posts
Next up: Heschel Rising 8th Grade Summer Packet — Part 2: Compare & Order Rational Numbers (Solved) — problems 11, 12, and 13 walked through step by step.
If you're a Heschel parent, you may also know families at neighboring UWS schools. We've solved the rising-grade summer packets that show up on the UWS most often:
- 1st Grade Math Practice — Shapes & Measurement (20 questions)
- 1st Grade Math Practice — Addition & Subtraction Within 20 (Harder)
- NY Regents Phase-Out 2027 — UWS Parent Guide
Heschel parent? We're three blocks away.
School of Math (SOMATH) is at 226 W 79th St, 1st Floor — about a 6-minute walk from Heschel at 30 West End Avenue. We tutor rising 7th, 8th, and 9th graders through the Standard, Accelerated, and Enriched tracks of Algebra I and Geometry, and we know this packet well.
Free 30-minute evaluation with a written diagnostic within 48 hours — even if you don't enroll. We tell you which Heschel math track your child is most likely to land in, and exactly what to work on this summer to land in the next track up.
School of Math (SOMATH) is an independent math tutoring program. We are not affiliated with The Heschel School. The "Rising Grade 8 Summer Packet" is published by Heschel for incoming 8th graders; this post solves and explains the problems for educational use by Heschel families and any 7th-to-8th-grade math student. The official packet and answer key live on the Heschel Middle School Summer Reading page.
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