Reasoning · Grade 6-2 Shortest Distance and Surface Area

Problem

Fold a patterned cone net

The four quarter-circle nets are striped: parallel to the bottom radius, close diagonals, nested right angles, wider diagonals the other way. The cones below are banded parallel to the base, uneven across the seam, close chevrons, wide chevrons. Match each cone to its net.
1 2 3 4
Your answer
How to solve
Strategy Create a Physical Representation — The honest way to do this is to cut out four paper quarter-circles, draw the four patterns on them and roll each one up — the answer is then in your hand. But rolling also teaches the rule that lets you do it by eye: because paper bends without stretching, a mark's distance from the sector's centre is exactly its distance from the apex measured down the slant. So the only thing that matters about a stripe is how its distance from the centre changes as it travels round, and how it arrives at the two straight radii, since those two radii become the one seam. I read those two facts off each net in turn and keep a short list of which cone each net must give, so no cone gets used twice.
1STEP 1

What rolling the sector does to every point

Distance from the centre becomes distance from the apex.

distance from the sector's centre = distance from the apex down the slant
2STEP 2

Turn that into two things to look for

So watch only the shape and count of the bands.

3STEP 3

Net ③ folds into plain horizontal bands — that is cone 1

Net 3 gives plain bands: cone 1.

4STEP 4

Net ① folds into the lopsided cone — that is cone 2

Net 1 comes out lopsided: cone 2.

5STEP 5

Nets ② and ④ both give bent bands, so look at which way they bend

Nets 2 and 4 both give bent bands.

6 stripes → 6 bands
6STEP 6

Net ②'s stripes join up in pairs, so it makes only about half as many bands

Net 2's stripes join, so it makes fewer bands.

9 stripes, joined in pairs → about 4 bands
7STEP 7

Match the two chevron cones by which way their bands bend

By band count, 4 is cone 3 and 2 is cone 4.

8STEP 8

Write the four labels and check nothing is used twice

All four labels are used exactly once.

Answer
3, 1, 4, 2
Every label is used exactly once, as a four-to-four matching requires. The band counts agree: net ③'s seven nested right angles give the seven or so hoops on cone 1; net ④'s six chord-parallel stripes each close up into one full band, matching the many narrow bands of cone 3; net ②'s nine stripes pair off across the back into about four broad bands, matching the small number of wide bands on cone 4. The one net whose pattern meets one radius at a right angle and the other one parallel — net ① — is matched to the one cone that is not left-right symmetric, cone 2, and no other net could produce that. As a second, independent check on the last pair, look at what happens at the seam: net ④'s stripes are furthest from the sector's centre at the two radii, so its bands sit at their lowest at the seam and climb away from it, while net ②'s stripes are closest to the centre at a radius, so its bands peak at the seam and drop away. Rolling four paper cones reproduces all four matches.
Takeaway

The centre of the sector becomes the tip and the two straight edges become one seam — so a stripe's distance from that centre is its height on the cone, and counting stripes tells you how many bands to expect.

  • What rolling the sector does to every point
  • Turn that into two things to look for
  • Net ③ folds into plain horizontal bands — that is cone 1
  • Net ① folds into the lopsided cone — that is cone 2
  • Nets ② and ④ both give bent bands, so look at which way they bend
  • Net ②'s stripes join up in pairs, so it makes only about half as many bands
  • Match the two chevron cones by which way their bands bend
  • Write the four labels and check nothing is used twice