Chapter 12Notes for InstructorsContentMeasurement is the focus of this chapter. I was surprised to discover that many studentshad difficulty with this chapter. In retrospect, I think I really should have emphasizedThe Measurement Process highlighted in the green box in Section 12.1. Students shouldunderstand area as a “covering” problem and volume as a “filling” problem.Manipulatives and Other ResourcesI used Polydron and Power Solids (Relational Geosolids are similar to Power Solids) inthis chapter, both of which can be found in the Mathskellar. The polydron were particularlyhelpful when calculating Surface area. Students really needed to see the 3-D polyhedron laidout as a 2-D net to calculate the surface area. The power solids were helpful in comparingvolumes. There is rice in the Mathskellar that can be used to fill the Power Solids, but Ithink it is best to use water if at all possible. It is difficult to level the rice, making the visualcomparison of the volumes of the cone, cylinder, and sphere a little less than accurate.Tangrams are discussed in Section 12.1. After I taught this lesson, I discovered that thereis a set of overhead tangrams in the manipulative kits. There should be at least one of thesekits in the Mathskellar. It is in a black ETA bag.There is a copy of the Connected Mathematics Sequence in the Mathskellar. It a juniorhigh curriculum sequence. The books Covering and Surrounding and Filling and Wrappingare particularly relevant to this chapter. It might be good to do some exercises from thissequence in class, or at least show the curriculum to your class. It is likely that some of yourstudents will be preparing to be middle school teachers.Notes and Suggestions:Notes on Section 12.1: The Measurement Process• I think it is important to use non-standard units of measurement in this section toreally illustrate the measurement process. The pen unit discussed in problem number4 of Section 12.1 can lead to a nice discussion about tilings.• Some of the students in my class were familiar with Unit analysis because of Chemistryclass or another science class, but others had difficulty with this concept. I would saythe split in my class was about 50/50. When covering Unit Analysis, I think it isimportant to remind students that one is the multiplicative identity. Since we aremultiplying by one, we actually do not change our original quantity, only its units.• There are blocks and multi-link cubes that could be used for problem 5 in Section 12.1.Notes on Section 12.2: Area and Perimeter• When I ask students to approximate the area of a figure on a grid, the lower boundis determined by the squares completely contained within the figure and the upper1bound is given by all the squares that intersect the figure. As the grid is refined, thelower bound and upper bound should approach each other. Note that the unit does notchange as the grid is refined. For example, on question 5 of Measurement Worksheet I,students will need to use the fact that 4 small squares equals 1 unit to determine thelower and upper bounds. On question 6, they will need to use the fact that 16 smallsquares equals 1 unit.• I think students should understand the area formula for a rectangle. (They may needto be reminded of the area model for multiplication.) From this, they should be ableto derive the area formulas for parallelograms, triangles, and trapezoids.• You should define π as the ratio of the Circumference of a circle to its diameter. Withthis definition in hand, there is some motivation for the area formula for a circle. (Seethe diagram at the bottom of page 773 in Section 12.2 in which sectors of a circle arerearranged to be a near-parallelogram.)• There is another motivation for the area formula of a circle given in problem 27 ofSection 12.2. I am including a diagram with a similar diagram with the documentationfor this chapter.• Students may have a bit of difficulty with the unit analysis in this section becausethey do not understand that units2= units × units. They will need to use linearconversion factors twice to to change units of area.• I like problems 11 and 12 in Section 12.2.Notes on Section 12.3:The Pythagorean Theorem• I am including a diagram that represents the Pythagorean Theorem for the 3-4-5 righttriangle with this documentation.• I really like problems 25 and 26 in Section 12.3. We did these exercises as a class. Todo this, I magnified the shapes shown right above problem 25 in Section 12.3 on thephotocopier. I magnified them several times and even used the large 11×17 paper.I had each student cut out the set of shapes. (I brought the Mathskellar scissors toclass.) Then we did problems 25 and 26 together as a class.Notes on Section 12.4:Surface Area and Volume• I think it is a good idea to use nets to motivate the surface area formula for a rightprism. Many of my students did not understand why perimeter came into play in thisformula.• The Power Solids (or Relational Geosolids) were helpful in understanding the volumeformulas for spheres and cones.2• I think it would have been a good idea to have a deck of playing cards to illustratethe volume formula for an oblique prism. (see the discussion at the top of page 800 inSection 12.4.)• Carl Lee has a nice Wingeom file that illustrates figure 12.26. You might ask himabout it.WorksheetsI have included three worksheets with this