Students are each provided a thin rectangular strip of paper, named after them. Each personal unit is at least one foot long to help create the need to partition the unit to measure objects. Alternatively, each student is given a foot-long length and informed that it is one foot. This choice may allow students to capitalize on what they know about foot measure (Or any other standard measure). The choice is up to the teacher: Will the convention foreclose student thinking or support it? The assessments conducted in the previous lesson might be the best guide. If students are firmly convinced that only conventional measures are possible, it is likely better that unconventional measures be introduced. In part A of the lesson, various splits of units are explored and symbolized. The lesson employs paper folding as a model of partitioning. In part B of the lesson, the splits are put to use measuring objects and deciding when measures are equivalent.

If students say that rounding is good enough, introduce a four foot long strip, called a “giant foot,” and ask students to measure something about ¼ as long as the strip. In this case, the visual perception of the distance makes the need for a part-unit clear.

In this activity, we will use the language, “of,” and model fractions as “a measured in b.”

For example, if we have a length measured in feet, then “1/2 of 1 foot” means split or divide the foot into 2 congruent parts. ½ is 1 copy of the 2 parts. What has been done is that the new unit of measure is 2, each of which is half as long as the original measure, the foot. 2/2 means 2 copies of the new unit, 3/2 means 3 copies of the new unit, or “3 halves (giving the new unit a name that relates it to the old unit, the foot). It is also possible to give the ½ unit a new name, so that if the length of the personal unit is “Bob,” then ½ Bob could be called something else too, such as “click.”

Whole Group Demonstration/Discussion, Individual or Pair Problem Solving

   Teacher Demonstration

Let's try to create parts of our units by folding our strip paper. Try to create 2 parts. Demonstrate splitting the unit by folding into 2 congruent pieces. Ask: What do I call the length from here (start) to here (midpoint), moving your fingers along the paper-strip? [Students will likely say “one half”]

Emphasize that each partition is congruent (by folding), that there are 2 partitions, and that the new measure of the length is 2: 2 halves. ½ means 1 copy of the new unit-the half. [It could be called anything but we usually call it a half to remind ourselves of its relation to the original unit – ½ foot or ½ <personal unit name>. Be sure to emphasize that the part-of-the unit traveled is half as long as the unit, and that the unit is twice as long as the part. Be sure that students replicate your motion and not just point to the part of the unit.

1 foot is 2 times as long as ½ foot.

½ foot is ½ times as long as 1 foot.

Ask: What might 2/2 mean? How many copies? Of what? What about 3/2?

Teacher note: The intention of Fraction Strips I is to provide students with experience repeatedly splitting a unit and then observing the resulting partitions of that unit. For example, ½ means that the unit is split into 2 congruent partitions and we move from the beginning to the end of the first partition (visible as a fold line on the paper). We use an informal language “of” to signify copies of splits of a unit, as in ½ of 1 unit-name. We explore splits of 2, 3, and 5, and combinations of 2 and 3. We suggest an implicit model of addition when we ask about continuing a motion from, for example, 1/8 of a strip to the next fold (another 1/8), and symbolize that distance traveled as 2/8.

There are many problems in Fraction Strips I. We suggest letting students solve at most 2 problems at a time, then sharing their solution strategies, then continuing with the lesson in this manner. Alternatively, you might want all students to solve selected problems and then assign different problems to different students or groups of students.

Mal-ruler . A teacher, Deb Lucas, constructed a mal footprint ruler that was intended to probe students' thinking about iteration and identical units. The mal-ruler was constructed by mixing different length foot-prints, by leaving gaps between units, and by overlapping units. She measured a few objects with the ruler, reported results, and asked students what they thought (and why) of the resulting measurement. Was it accurate? What did a simple count of the number of feet tell us?

Summarizing, Representing Units in Measure Walks. Deb Lucas asked students to summarize what they knew about rulers. During the course of one such summary-discussion, a controversy arose about “space,” with some students saying that the marks denoting the “end of units” and others denoting more densely notated marks as meaning, “less space.” To follow up on this emerging idea of “space,” Rich asked students to represent, with drawings and with a drawing of a ruler, three different types of measure walks. One walk traversed a distance on the floor heel-to-toe, step-by-step. The second mixed the units, heel-to-toe, and an exaggerated stride. For Rich, this corresponded to a situation in which the units were mixed. The third walk included a jump, along with heel-to-toe. Rich intended this as corresponding to gaps between units. Student solutions were quite variable and overall, a good way of provoking representational discussion—what about the rulers corresponded to what about the walks? Students did some surprising things: what Rich took as gaps, students saw as another way of traversing the length. But during the course of discussion, Rich asked what would happen if we measured again using each method? Most students suggested that the heel-to-toe was likely most repeatable. Some students suggested that for longer distances, use of strides was better, as long as they could be calibrated to feet. A few students suggested the jumps. They acknowledged that they were least replicable (Rich jumped several times from the same starting point), but the jumps had the advantage of quick measure.