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Geometry and Measurement 6-8 / Week 06 / Tuesday
2/6
Week 06 · Triangles with Rules, and Slices

Tuesday

One, many or none
// When three straws make a roof, and what a cut face shows
⏱ about 20 min

Tuesday: One, Many or None

"Three straws, one roof," Rocket says. "Six, eight, ten. Done. Can I build a different one with the same straws?"

He tries. Every time, the triangle comes out the same shape, just turned around. "No. It is stuck."

"Now try three corners," Raven says, handing him the protractor. "Fifty, sixty and seventy degrees. What do you notice?"

Rocket draws a tiny one and a huge one. Both fit. "Same shape, any size. There are lots."

Nova hovers over the two drawings, her light measuring each corner. "The corners always added to 180 on our straw frames," she says.

"And ninety, ninety and twenty?" Raven asks.

Rocket tries. The two ninety-degree rays run side by side forever. "None. They never meet."

Nova hums. "One, many or none. Every set of three measures is one of those."

Three measures, three outcomes

What you knowHow many trianglesWhy
Three sides that closeexactly oneThe straws lock into one shape.
Three sides that do not closenoneThe short straws cannot reach.
Three corners adding to 180°manySame shape, any size.
Three corners not adding to 180°noneThe rays never meet as a triangle.
Two sides and the corner between themexactly oneThe third side has one place to go.
Two corners and the side between themexactly oneThe two rays cross in one spot.

The crew measured the three corners of every straw triangle they built. Each time the corners added to 180 degrees.

For now, treat that as the crew's rule from the Blueprint Table. In week 8 you will see why it is always true.

Three corners that add to 180 fit a triangle, but they do not fix its size. A tiny one and a huge one both work.

A triangle with corners of 50° and 60° and the third corner marked ?

The third straw

If you know two straws, the third is not free. It must be shorter than the two added together.

Straws 5 and 7: the third must be shorter than 12 cm. It must also be longer than 2 cm, or the short pair cannot reach.

Between those two numbers, any length closes. Each choice gives a different triangle.

HOW MANY TRIANGLES?
  • Read the question.
  • Tap your answer.
Three straws: 6, 8 and 10 cm. How many different triangles can they make?
Three corners: 50°, 60° and 70°. How many different triangles have these corners?
Three corners: 90°, 90° and 20°. How many different triangles have these corners?
Three straws: 4, 5 and 12 cm. How many different triangles can they make?
TWO SIDES, TWO CORNERS
  • Read the question.
  • Tap your answer.
Straws 6 cm and 8 cm with a 40° corner between them. How many different triangles?
Corners 70° and 80° at the two ends of a 10 cm straw. How many different triangles?
Corners 100° and 90° at the two ends of a 10 cm straw. How many different triangles?
Corners 45°, 45° and 90°. Do they add to 180, so a triangle can have them?
THE THIRD STRAW
  • Read the question.
  • Tap your answer.
Two straws are 5 and 7 cm. The third straw must be shorter than how many cm?
Two straws are 5 and 7 cm. The third straw must be longer than how many cm?
Two straws are 9 and 4 cm. The third straw must be shorter than how many cm?
Two straws are 3 and 10 cm. The third straw must be longer than how many cm?
TEST THREE MEASURES
  • ?Ask: are they three sides, three corners, or a mix?
  • ?Three sides: add the two shorter ones and compare with the longest.
  • ?Three corners: add them and compare with 180.
  • ?Decide: exactly one, many or none. Then build or draw it to check.
WHY THIS EXERCISESides use the straw rule. Corners use the 180 rule. Each test tells you one, many or none.
Draw two triangles with corners 50°, 60° and 70°: one small and one large. Label the corners.

One, many or none. Tomorrow is Build Lab: straw frames, then a clay block under the string.

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