75-Archetype SyllabusMathematics 1 • ARCH-M1-06 • Spec M3

Full 6-Card Lesson 15-Question Timed Drill

UAT-UK SPEC CODE: M3 (Ratio and proportion)ASSUMED MATHS 1 PREREQUISITE: M3.10FREQUENCY: MEDIUM

2D/3D Geometric Similarity Scaling (k, k^2, k^3) & Frustums

Deconstructed from 9 past UAT-UK questions (4% of all historical Mathematics 1 papers). Students using standard A-Level classroom methods spend an average of 145s on this archetype—blowing past the 90s ESAT budget. Using the 2-line Sub-90 heuristic below compresses solve time to 35s (saving 110s per question).

Historical Frequency

9 Past Qs

4% of Mathematics 1

Classroom Trap Time

145s

+55s over 90s budget

Sub-90 Target Time

35s

Banks +110s on the clock

Dominant UI Format

5-Option A–E

Maths 1 Skill: M3.10

Card 1 & Card 2 Preview: The Classroom Time Trap vs. Sub-90 2-Line Rule

Excerpted from our interactive 6-Step Topic Lesson for ARCH-M1-06.

Launch All 6 Active-Recall Cards

CARD 1 OF 6 • THE CLASSROOM TIME TRAP (145s)

Why Students Bleed Time:

For mathematically similar 3D solids, the volume ratio is the cube of the linear ratio:

\frac{V_{\text{large}}}{V_{\text{small}}} = \left(\frac{4}{3}\right)^3 = \frac{64}{27}

. Since

108 \div 27 = 4

V_{\text{large}} = 4 \times 64 = 256\text{ cm}^3

(~95s).

In 9 past UAT-UK questions (4% of Mathematics 1), candidates lose 110s by applying linear (k) or area (k²) scale factor to a 3d volume. Instead, apply M3 with Maths 1 skill M3.10.

CARD 2 OF 6 • THE 2-LINE SUB-90 HEURISTIC (<35s)

Line 1 (Cube the Ratio):

k = \frac{4}{3} \implies k^3 = \frac{64}{27}

**Line 2 (Cancel

27

):**

108 \times \frac{64}{27} = 4 \times 64 = 256\text{ cm}^3

(20s).

Sanity & Limit Check: Verify units/signs against factor_of_two_or_stoichiometry_trap (Scaling volume by k² or k instead of k³ (e.g. 192 * (3/4) = 144 instead of 192 * (3/4)³ = 81)) and subsystem_isolation_trap (Using frustum height ratio h_bottom / h_total instead of full similar cone ratio h_top / h_total).

Examiner Distractor Blueprint for ARCH-M1-06

FOIL #1factor_of_two_or_stoichiometry_trap

Scaling volume by k² or k instead of k³ (e.g. 192 * (3/4) = 144 instead of 192 * (3/4)³ = 81)

Applying linear (k) or area (k²) scale factor to a 3D volume.

Open Factor-of-2 / Stoichiometry Clinic

FOIL #2subsystem_isolation_trap

Using frustum height ratio h_bottom / h_total instead of full similar cone ratio h_top / h_total

Applying similarity scaling to the non-similar frustum slice instead of the whole apex cone.

Open Subsystem Isolation Trap Clinic

FOIL #3intermediate_stop_trap

Outputting the top small cone volume V_small instead of frustum volume V_large - V_small

Forgetting to subtract the removed apex cone from the full cone.

Open Intermediate Stop Trap Clinic

FOIL #4fence_post_or_domain_trap

Taking A₂/A₁ = (V₂/V₁)^(3/2) instead of (V₂/V₁)^(2/3)

Inverting the fractional power when bridging between Volume and Surface Area.

Open Fence-Post / Domain Trap Clinic

Try 1 Live Sample Question Under the 90-Second Stopwatch

Press W to open the 2-line dry-wipe scratchpad, select A–E, and lock in to reveal the Dual-View solution.

Full 6-Card Lesson Launch All 15 Questions (ARCH-M1-06)

ARCH-M1-06: 2D/3D Geometric Similarity Scaling (k, k^2, k^3) & Frustums (Sample)Mathematics 1

00:00.0

SUB-90 ZONE (<35S)

0.0s35s Sub-9090s ESAT Budget150s Classroom

Question 1 of 1ARCH-M1-06 (6-Card Lesson)Format: Standard 5-Option

Right-click any option to cross out • Keys A–E + Enter

Question 1 of 1 • 90s Target

Ready to Start Question 1?

The question stem, diagram, and answer options are hidden behind this pre-start gate. When you click below, the question will reveal and your 90-second split stopwatch will immediately begin ticking.

Mathematics 1•ARCH-M1-06•Standard 5-Option

or press Enter to start