Maker Challenge: Make a Yardstick

In the warm-up, students individually explore how to measure 6 inches with a letter sized piece of paper (8 ½ in by 11 in). They share their strategies and teach each other their methods.

In the main challenge, groups of 3 students label a blank yardstick with as many tick marks as they can. 

Activities
: 

Warm Up

15 minutes

[Could take place in prior class; could give additional time to allow for more methods to share and additional time where student teach each other their methods.]

Narrative

Students use pieces of 8 ½ in by 11 in paper to find a length of exactly 6 inches. There are many ways to do this by folding the paper in different ways and subtracting and adding different lengths created by the folds. If students are stumped or don’t know how to start, encourage them to fold one side in half or across a diagonal and to find the lengths this creates. Alternatively, if many students are stuck but others have ideas that work, have one of the students share their strategy, without showing the entire solution, then continue with more individual work time. There are many different ways to find 6 inch lengths. It is not critical that students find the most efficient one, i.e. using the fewest steps. 

Launch

Form teams of 3 (which will stay together throughout cycle). Teams need unlined pieces of 8 ½  inch by 11 inch paper.

Task Statement

Here are some pieces of 8 ½  inch by 11 inch paper.

  • Individually, explore how you can use one piece of paper to find a length that is exactly 6 inches.

  • In your group, share your methods with each other and choose one to share with the whole group. Be prepared to explain the method to the class.

  • Whole class discussion: Share 3 or 4 different methods. (pre-chosen by teacher). Ask groups that didn’t share if their method was like one of the ones shown. If applicable, invite groups that have different methods to show theirs.

Task Solution

Solutions vary. Here is one using 5 steps. It can be done in as few as two. 

Synthesis

As students share their solutions, encourage them to write down their steps using numerical expressions and/or equations, e.g.: -- Step 1. Fold the long edge in half (11”÷2 = 11/2” = 5 ½”.)  Step 2. Fold the short edge onto the long edge. The part that doesn’t share an edge is (8 ½” - 5 ½ “= 3”).  Step 3. Mark 3” twice (3” x 2 = 6”) Some solutions may be using estimates while others use more exact folding methods. Discuss the advantages of both in terms of precision: Folding methods are more precise than eyeballing and fewer folds give less room for compounding errors. 

Task 1 - Make a Yardstick

30 minutes

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.

Materials for this Lesson
: 

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

Timeline
: 

Warmup: 15 minutes

  • Launch: 2 minutes
  • Individual: 3 minutes
  • Groups: 5 minutes
  • Share: 5 minutes

Task 1: 30 minutes

  • Launch: 3 minutes
  • Build: 20 minutes
  • Synthesis: 7 minutes