Heschel School · Rising 8th Grade Summer Packet · Part 2 · Compare & Order Rational Numbers
Heschel Rising 8th Grade Summer Packet — Part 2: Compare & Order Rational Numbers (Solved)
Part 2 of The Heschel School's Rising Grade 8 Summer Review Packet — three problems that decide whether a student really understands what a number line means. Comparing and ordering signed fractions, decimals, repeating decimals, exponents, and absolute values, 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 the full reasoning and the written solutions below as a quick checking aid.
About this page of the packet
If Part 1 tested whether a student can evaluate an expression with signed numbers, Part 2 tests something deeper: whether they can compare two expressions without computing one full numerical value. That skill — fluency with the size and sign of a number — is what separates the Standard track from the Accelerated and Enriched tracks at The Abraham Joshua Heschel School (30 West End Avenue).
These three problems combine every concept from page 1 — absolute value, exponents on negatives, signed fractions, decimals — and add one new idea: the repeating decimal. Eighth-grade Algebra I (8.NS.A.1) opens with proving why every rational number is either a terminating or repeating decimal, so the notation matters.
How to read the bar notation in Problem 12:
1.45 means 1.454545454545… (the 45 repeats forever).
A bar above one or more digits is the standard middle-school notation for a repeating decimal. Without the bar, "1.45" would just mean one-and-forty-five-hundredths.
Jump to a problem
Step-by-step solutions
Compare using <, >, or =
Problem 11
Compare using <, >, or =: −|−13⁶| ___ (−13)⁷
Final answer: −|−13⁶| > (−13)⁷
Left side: −|−13⁶|
= −|13⁶| ← 13⁶ is already positive (even exponent), so −13⁶ inside the bars is just the negative of a positive number; the bars strip the sign either way
= −(13⁶)
= −4,826,809 (negative, around −4.8 million)
Right side: (−13)⁷
= −(13⁷) ← odd exponent keeps the negative sign
= −62,748,517 (negative, around −62.7 million)
Compare: Both numbers are negative. On the number line, the one closer to zero is larger.
−4,826,809 is closer to zero than −62,748,517, so:
−|−13⁶| > (−13)⁷
Why this matters: You don't actually need to multiply out 13⁶ or 13⁷ to answer this. The whole question is about sign and size. Both sides are negative — that part is decided by the leading minus signs and the parity of the exponents. The right side has a higher power of 13, so it's further from zero, which makes it smaller (more negative). This sign-and-magnitude reasoning is the single most-tested idea on the 8th-grade Algebra I diagnostic.
Problem 12
Compare using <, >, or =: 16/11 ___ 1.45
Final answer: 16/11 = 1.45
Convert 16/11 to a decimal by long division:
16 ÷ 11 = 1 remainder 5
50 ÷ 11 = 4 remainder 6
60 ÷ 11 = 5 remainder 5 ← we've seen remainder 5 before — the cycle starts repeating
50 ÷ 11 = 4 remainder 6 ← same as step 2
60 ÷ 11 = 5 remainder 5 ← same as step 3
So 16/11 = 1.4545454545… = 1.45
That is exactly the number on the right.
16/11 = 1.45
Why this matters: The most common student error here is to write < because 16/11 ≈ 1.4545… and they round it to 1.45, then compare 1.45 to "1.45" without noticing the bar. The bar over the 45 means the digits repeat forever — they're literally the same number. Recognizing repeating-decimal notation is a 7th-grade Common Core standard (7.NS.A.2.d) and an 8th-grade Algebra I prereq (8.NS.A.1).
Order from least to greatest
Problem 13
Order the numbers below from least to greatest. (Your final answer should be written in the form of the original numbers.)
−0.09, 9/10, −|−8/9|, −0.95, −|−4/5|, −10/9
Final answer: −10/9, −0.95, −|−8/9|, −|−4/5|, −0.09, 9/10
Step 1 — strip the absolute-value bars, keeping the outside sign:
−|−8/9| = −(8/9) = −8/9
−|−4/5| = −(4/5) = −4/5
Step 2 — convert each number to a decimal (just enough to compare):
−0.09 → −0.09
9/10 → +0.90
−|−8/9| = −8/9 → −0.888…
−0.95 → −0.95
−|−4/5| = −4/5 → −0.80
−10/9 → −1.111…
Step 3 — order the decimals from least (most negative) to greatest:
−1.111… < −0.95 < −0.888… < −0.80 < −0.09 < +0.90
Step 4 — write the answer back in the original form (as the problem asks):
−10/9, −0.95, −|−8/9|, −|−4/5|, −0.09, 9/10
Why this matters: This is the synthesis problem of the whole packet. Three different traps are baked into the same list: (1) the absolute-value bars on the two middle numbers (it's easy to forget the outside negative and put them on the positive side of the line), (2) the two improper fractions −10/9 and 9/10 that look almost identical but live on opposite sides of zero, and (3) the temptation to compare "−0.09" with "−0.95" as if smaller-looking digits mean smaller numbers — on the negative side, the rule flips. The student who writes them in original form (as the problem requires) shows the grader they tracked both the value and the notation.
What this page tells us about the Heschel placement
Page 1 of the packet is about executing sign rules. Page 2 is about seeing past the notation — recognizing that −|−13⁶| and (−13)⁷ are just two negative numbers in different costumes, and that 1.45 is literally the same number as 16/11.
From our classroom three blocks away, the students who land in Accelerated or Enriched 8th-grade math at Heschel are the ones who:
- Estimate before computing. They look at Problem 11 and say "both negative, the bigger exponent is further from zero" before reaching for a calculator.
- Read notation carefully. They notice the bar in 1.45, the bars around |−8/9|, and the negative sign sitting outside the bars in −|−4/5|.
- Use a number line in their head. They picture −10/9, −0.95, and −0.09 on a line and can see at a glance which is most negative.
Other Heschel & UWS posts we've published
- Heschel Rising 8th Grade Summer Packet — Part 1 (Integers, Order of Ops, Fractions)
- 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.
Want your child in a Young Fermats — Algebra Ignite class at SOMATH?
Real Algebra I course for Grades 7–8 (ages 12–14). Linear equations and inequalities, systems, exponents, factoring, quadratics and functions — the same depth honors students see, at a small-group pace.
See the Young Fermats — Algebra Ignite course → Book free evaluation