**Challenge Narrative**

Students make a usable yardstick by labeling a blank yardstick with smaller units (ultimately 1-inch or ½ inch) using lengths created by folding 8 ½ inch by 11 inch pieces of paper. Students may start by marking 6 inch lengths on their yardstick. If they think they are done at that point, ask them to include smaller units. Ask them to check the precision of their units. Is the inch from 1-2 the same as the inch from 30 to 31? If you use the start or the end of the stick to measure, do you get the same length?

**Launch**

Students work in same 3-person groups.

Supplies needed:

- Blank yardstick or meterstick
- Pencil
- Markers
- 8 ½ in by 11 in paper

**Challenge Statement**

I have always been wondering how much stuff would fit into our classroom. For example, if we wanted to fill it with cubes that are 1 ft by 1ft by 1 ft, i.e. 1 ft^3, how many cubes could we fit into the room? I got these yardsticks for us to use to find the answer, but unfortunately the yardsticks don’t have any smaller units marked and I am not even sure they are exactly one yard long. Each group, take one of the blank yardsticks and fill in smaller units, so that we can use it to figure out how many cubes we could fit into the room. You can only use a standard sheet of paper to mark the yardstick.

**Task Solution**

There are many different solutions. Here is one:

- Use 6 inch paper length from warm-up to mark multiples of 6. (6, 12, 18, 24, 30, 36)
- Use half of 6 in paper to mark other multiples of 3 (3, 9, 15, 21, 27, 33)
- Use 11 in side of whole sheet of paper to mark multiples of 11 (11, 22) and everything we already have +11 and everything we already have -11 (14, 17, 20, 23, 25, 26, 28, 29, 30, 31, 32, 34, 35) and (16, 19, 13, 10, 8, 5, 4, 2, 1).

Other solutions can involve measuring a 1 inch distance (e.g. 12-11) and then marking all missing inches.

**Synthesis**

Maker ideas:

- Group work: listening to and offering ideas
- Scouting: learning by observing others
- Hidden assumptions: units we are using are more precise than we thought.

**Mathematical ideas:**

- Conversion within the same measurement system.
- Writing and using numerical expressions.
- Decomposition of numbers.
- Attend to precision
- Using appropriate tools

Groups come together with their yardsticks and compare how close they are to each other. They share their methods and discuss the advantages and disadvantages of each methods - which methods give more precise measurements? If we use an imprecise unit, for example if our 1-inch unit is 1/16 in too long, then if we iterate it, after 12 inches, the tick mark is off by 12/16 = ¾ of an inch.