# MATH 5379 Measurement Concepts in K-8 Mathematics

Summer I 2010

For our course syllabus, click Section 001 or Section 002.

NOTE: Technical information such as prerequisites, text materials, course format and assignments, and other details can be found in the syllabus, a copy of which is provided in a link at the top of this page.

### Class list of key ideas

A summary of the key ideas of measurement, taken from the first journal entries:
• Measurement involves comparing an attribute of an object to a unit with the same attribute.
• Estimation and personal benchmarks
• All measurement involves approximation and error.
• Understand conceptually how measurement tools work
• Students need to develop formulas conceptually, not be told them.
• Concrete measuring experiences are crucial to understanding measurement.
• Language and vocabulary skills
• Students need to communicate their understanding to others, orally and on paper, in order to deepen it.
• Standard units
• Conversions within and between systems of units (Equivalence can be addressed by allowing students to measure the same quantities in different units individually, and discussing their equivalence together.)
• Scale of units may affect students' ability to understand and select them.
• Measurement should be integrated throughout the year as a means of obtaining meaningful data with which to work.
• Understand measurement as an important daily life activity
• The distinction between perimeter and area needs special attention.

### "College-level" problems

Problems appropriate for writing up in the two-problem paper.
• Session 1: The River Problem --- solution and deconstruction
• Session 6: Area of a Triangle --- a rigorous treatment including the equivalence of all three orientations (sides as base) and obtuse triangles
• Session 7: Formula 1 --- a rigorous response to the question, for which shapes is it true that Area = base x height?
• Session 7: Area of a Trapezoid --- a rigorous treatment justifying the area formula for all possible trapezoids
• Session 7: Hungry Cow Deconstruction --- a comparative deconstruction of the two area problems
• Session 8: Triangular inches --- use the particular problems posed to illustrate your response to the overall question of the properties and usability of the triangular inch as a unit of area
• Session 10: Formula 2 --- a rigorous response to the question, for which shapes is it true that Volume = (Area of base) x height?
• Session 10 supplemental: Tetrahedral inches
• Session 11: Algebra of units
• Session 12: Design a Can --- solution and deconstruction

### Links for specific class meetings

• Session 1 The River Problem:
(In all solutions below, the tree or other landmark on the opposite side of the river is at point A, while the point directly across the river from it (on the near side) is called point B. Solutions almost always begin by having someone make a right angle at point B by walking up or downstream from B beside the river.)
• Solution 1: scale map
Walk along the river an arbitrary distance to point C. Measure length BC and angle ACB (angle ABC should measure 90 degrees). Then make a scale map on paper, duplicating the angle measurements. Now measure the desired length AB on the map, and use the scale (proportion) to calculate the length of the real AB.
• Solution 2: similar right triangles on the ground
Walk along the river an arbitrary distance to point C (and mark it). Then turn 90 degrees and walk away from the river an arbitrary distance, to point D. Mark point E at the intersection of lines BC and AD (this will probably require people stationed at B, C and D, and a fourth person adjusting location until (s)he is on both lines. Now triangles ABE and DCE are similar. Measure lengths BE, EC, and CD, and find AB by proportion.
Source: Wayne State University's Problem of the Week 8, from fall 1999
• Solution 3: congruent right triangles on the ground
Very similar to Solution 2, but a little extra care makes the triangles congruent and avoids the need to calculate. Walk along the river an arbitrary distance to point E (and mark it). Now continue walking along line BE (from E) and duplicate length BE, to a new point C with BE=EC. Then turn 90 degrees and walk away from the river until you can see point E line up with point A. This new point D forms a triangle DCE which is congruent (not just similar) to ABE. Measure DC, which equals AB.
Source: Math Forum
• Solution 4: congruent right triangles through a mirror
Stand immediately across the river (point B) from the tree (point A). Your friend walks some distance along the shore perpendicular to the line from you to the tree (i.e., perpendicular to line AB) and (from this new point C) holds up a mirror parallel to this line, facing you (standing at point B, you can see yourself in it. Wave hello to yourself!). You now walk away from the tree and river along line AB until you can see the tree (point A) reflected in the mirror (which is still at point C). At this point (point D) you are exactly as far from point B as the tree is! One variation of this solution used in class (without a mirror) was to have the person at point C measure angle ACB and then tell a person walking along line AB away from the river when to stop (at point D) so that angle BCD has the same measure as angle ACB. An interesting point is that this solution doesn't actually require the angle to be measured in units, merely copied.
Source: Kribs-Zaleta
• Solution 5: special right triangles
There are two "special" right triangles known by geometry students, the 45-45-90 and 30-60-90 triangles (so called because those are the vertex angle measurements in degrees). The 45-45-90 triangle is isosceles, and you can recreate one by walking along the river from point B, perpendicular to line AB, until the angle between the lines from where you are to points A and B is exactly 45 degrees. Then you are at a point C which is exactly as far from point B as point A is!
The 30-60-90 triangle is half an equilateral triangle, and is known to have sides in proportion 1, square root of 3, and 2 (the hypotenuse). To recreate this, walk along the river from point B, perpendicular to line AB, until the angle between the lines from where you are to points A and B is exactly 60 degrees (this will involve less walking than getting to 45 degrees). Then you are at a point C which forms a 30-60-90 triangle ACB, with the desired length AB exactly the square root of 3 times length BC.
Source: Kribs-Zaleta
• Solution 6: right triangle trigonometry (tangent)
Walk along the river an arbitrary distance to point C. Measure length BC and angle ACB (angle ABC should measure 90 degrees). Then write the equation tanACB=AB/BC, substitute, and solve for AB.
Source 1: learner.org's measurement course, Session 5C
Source 2: The Math Page
Source 3: a story from the Peace Corps in Togo! --- if you don't read any others, read this one!
• Solution 7: Laws of Sines and Cosines
Pick two arbitrary points B and C on the near side, two more points D and E such that ABD and ACE are lines, measure all the lengths on the near side of the river, and blast away with the Laws of Sines and Cosines. Requires absolutely no angle measurements, but a lot of trig. The solution (see link) also discusses precision.
Source: River Width assessment from the Illinois Learning Standards performance assessments for Grades 11 and 12 mathematics
• a related problem: Napoleon's River
the classic solution, and why it's impractical, from UC Davis
• Session 2
• Interactive activity from learner.org on determining a circle's area by decomposing and reassembling it
• Session 4
• Session 6
• Session 7 Area posters (from 2006)
• parallelogram area 1 and 2
• trapezoid area 1, 2a and 2b
• obtuse triangles 1, 2a and 2b
• Session 8
• The area of Texas is 268,601 square miles, or 620,307 triangular miles, or 691,200 sq.km., or 1,596,258 tr.km. (Actually, reports of Texas's area on the Internet vary, say between 678,000 and 696,000 sq.km.; the figures given here in triangular units are obtained simply by converting from the corresponding square units.)
• Session 9 Battista article in two parts: one, two
• Session 10
• learner.org's approach to volumes of prisms and cylinders, pyramids and cones, includes an interactive activity comparing spheres and cones to cylinders
• Here's one approach to showing how 3 pyramids fit in a cube (see also the animation at right), but you can google 3 pyramids cube and you'll get half a million hits.
• Session 12
• Session 14