Where do these live?

• CCSS-M · 2.MD.B.6
  Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

Still 2.MD.B.6, but stretched. By dropping 30 and 70 rather than 50, the task forces children to measure from a landmark that is not the middle, which is the more general skill the standard is building toward.

A good second or third session, once children are comfortable estimating to 100. The odd landmarks are deliberately harder.

1. Start: 0 at 6.9%
2. Drop 1: 100 at 93.1%
3. Drop 2: 30 at 32.8%
4. Drop 3: 70 at 67.2%

When to dropAfter two cards, drop 100 to set the range. Then drop 30 and 70, the odd landmarks that are the point of the task. Watch whether the class can still measure from them: if they place 45 halfway between 30 and 70, they have reasoned well, even though 50 is never shown.

Drop unlocks after 2 cards placed.

• 20

  at 24.1%
  In the first third, a little left of the 30 drop.
• 45

  at 45.7%
  Between 30 and 70, and left of the true centre. Tests reasoning between two odd landmarks.
• 65

  at 62.9%
  Just left of 70. Close to a landmark but not on it.
• 85

  at 80.2%
  Between 70 and 100, in the last stretch.

• Is 45 closer to 30 or to 70? Roughly how far between them?
• 65 is near a landmark. Which one, and which side of it?
• Which card is the furthest from any landmark we have?

• Looks like
  45 placed at the visual centre of the whole line.
  Means
  Defaulting to the middle of the line rather than reasoning from the landmarks given.
  Respond with
  Cover 0 and 100 with your hands and ask them to place 45 using only 30 and 70. Then uncover and check.

Ask whether 30 and 70 were harder to use than 50 would have been, and why.

Landmarks do not have to be halves and quarters. You can reason from any known point, and 30 and 70 are just as useful as 50 once you commit to them.