Following the lesson(s) on pacing, cut-outs of feet are created and glued to machine tape. The use of cut-outs of feet raises opportunities to consider which attributes of a foot are important for the measure of length.

Students use their foot-strip ruler to measure lengths of different objects, many of which are chosen to (a) pose the problem of what to do about non-whole number units and (b) what to do when the number of units is not sufficient to measure the distance.

The first (a) places students in the position of seeing the need for parts of units and motivates splitting a unit. Unit-splits are gateways to measurement approaches to fractions.

The second, (b), places students in the position of learning more about ideas of iteration—the repeated translation of identical units and their acculumation that results in a measure.

Because the same length will have different measures, students can informally see that different definitions of length-1 are possible, and in this activity, they correspond to differences in the length of each student's foot. Each student gets to name his or her foot. We aim to help students see that for a fixed distance, different measures are possible. This raises the need for standardization (agreeing about length-1).

• Attributes/ Qualities
  
• Guided re-invention of tools
  
• Iteration of units
  
• Partitions of units

Each student first traces an outline of his or her foot . That outline is cut-out and labeled with the student's name.

Prompts a conversation about the important attributes of the cut-out of the foot, by asking students if we can substitute other things for feet:

• What could we substitute for your foot? Could we use a (rectangular) strip of paper (show a cut-out with construction paper that is as long as one of the foot cut-outs) that is just as long as your foot? Why? Does it matter how fat or skinny the strip is? Why? How about:
  • Two-by fours?
  • String?
  • Thread?
  • Rubber bands? (stretch a rubber band if there is no objection)
    
• Teacher note. It is important to prompt consideration of which attributes of a foot are being repesented by the paper strip or string. For example, feet have curvature-along the outline and also arches. Does the paper strip represent the length from heel to big toe, from heel to little toe? Width? The big idea is that any object can be considered as a bundle of any number of potential attributes.
  

Students' ways of thinking:
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Students pointed out that the length from heel to big toe and the length from heel to little toe are different. The teacher used the students' thinking to let students think about a solution for the problem. Students came to the agreement that they would use the length from heel to big toe as substitution for their feet.

• If students agree that rectangular construction paper strips can be used in place of their feet, they each construct a ruler by gluing paper strips to machine tape. Students are asked to mesaure objects in the room that will produce (a) approximately whole number units; (b) the need to use parts-of-units, and (c) the need to re-use the units, because the length is greater than that of the ruler.

Let students try to solve these problems, then have a whole-class conversation about:

(1) How did you place your units? Why did you do that?

(2) How did you measure? What was involved? Why did you do it that way? Try to promote an image of iteration as copying a unit n times. Be sure that you hold the unit apart from its copies. Enact at least two iterations of whole numbers so that everyone can see what you are doing.

(3) What did you do if the length being measured was longer than your ruler? Why did you do that?

(4) What did you do if the length being measured included a part of your foot-unit but not the whole unit? [See Parts of Units]

(5) How did you mark your ruler? Why did you mark it in that way?

(6) Where did you begin to measure? What did you label that, and why?

(7) Let's pick one of these (choose an object). What were some measures of this length? Why were they different? (be sure to pick someone with a smaller foot and someone with a larger foot to make the comparison clearer. If big feet are a problem, introduce a giant foot (or your foot), and then use that as the comparison.

Assessing student thinking during instruction forms a solid foundation for changing, elaborating or simply continuing a lesson, as needed. Here we note some of the tasks and situations developed by teachers to both assess student reasoning and to support its development.  

•  Mal-ruler. Create a mal-foot-strip ruler with units that are not identical and with uneven spaces between units. Measure several objects with the ruler and report the results to the class. Ask students to comment on what they see as potential difficulties with the mal-ruler. Ask students to propose remedies and rationales for these remedies. Be careful not to simply accept convention as a rationale—it is important that students understand function, not simply form.

•  Where do the marks go? Create a foot-strip ruler with labels for each unit in the middle of the unit and at the end of the unit. Ask students to compare the placement of the labels and ask them to speculate why rulers are conventionally labeled at the end of the unit (to signify the distance or length of each unit). The consequences of different choices of marks can be made explicit by having students move their finger from the start to the end of the unit. Have students compare that motion to the motion made by starting at the unit and then traveling until they see the mark somewhere within the unit.

•  Zero-point . Have students use their ruler to measure something beginning at one or two. Do they compensate for the shift in origin or do they simply read off whatever number that they see? (zero-point) If they do, what happens when they have a fractional unit in the measure (e.g., 4 ½)? What if they start at a fractional unit (e.g., 1 ½) and end at another fractional unit?

•  Zero-point. Give students a footstrip ruler with ½ unit markings. Have students use their ruler to measure something that ends at the ½ unit marking. What do they call the measure? Some students who are thinking about units as labels and not lengths may treat 3 ½ as 4 and ½ because they think: 1, 2, 3, --4 because it is the fourth count, and ½ because it is not a whole number.