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Can You Solve This Pier Puzzle? | Physics Girl

  Classroom Resource Information  

Title:

 Can You Solve This Pier Puzzle? | Physics Girl

URL:

 https://aptv.pbslearningmedia.org/resource/pier-puzzle-physics-girl/pier-puzzle-physics-girl/

Content Source:

 PBS
Type: Audio/Video

Overview:

This math brainteaser challenges you to find a simple, elegant solution to a seemingly complex problem! Can you figure it out?

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
31. Justify whether conjectures are true or false in order to prove theorems and then apply those theorems in solving problems, communicating proofs in a variety of ways, including flow chart, two-column, and paragraph formats.

a. Investigate, prove, and apply theorems about lines and angles, including but not limited to: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; the points on the perpendicular bisector of a line segment are those equidistant from the segment's endpoints.

b. Investigate, prove, and apply theorems about triangles, including but not limited to: the sum of the measures of the interior angles of a triangle is 180?; the base angles of isosceles triangles are congruent; the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length; a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity.

c. Investigate, prove, and apply theorems about parallelograms and other quadrilaterals, including but not limited to both necessary and sufficient conditions for parallelograms and other quadrilaterals, as well as relationships among kinds of quadrilaterals.

Example: Prove that rectangles are parallelograms with congruent diagonals.

Unpacked Content
Evidence Of Student Attainment:
Students:
  • Make, explain, and justify (or refute) conjectures about geometric relationships with and without technology.
  • Explain the requirements of a mathematical proof.
    1. Present a complete mathematical proof of geometry theorems including the following: vertical angles are congruent. when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent. points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
      Critique proposed proofs made by others.
    2. Present a complete mathematical proof of geometry theorems about triangles, including the following: measures of interior angles of a triangle sum to 180o. base angles of isosceles triangles are congruent. the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. the medians of a triangle meet at a point.
      Critique proposed proofs made by others.
    3. Present a complete mathematical proof of geometry theorems about parallelograms, including the following: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
      Critique proposed proofs made by others.
Teacher Vocabulary:
  • Same side interior angle
  • Consecutive interior angle
  • Vertical angles
  • Linear pair
  • Adjacent angles
  • Complementary angles
  • Supplementary angles
  • Perpendicular bisector
  • Equidistant
  • Theorem Proof
  • Prove
  • Transversal
  • Alternate interior angles
  • Corresponding angles
  • Interior angles of a triangle
  • Isosceles triangles
  • Equilateral triangles
  • Base angles
  • Median
  • Exterior angles
  • Remote interior angles
  • Centroid
  • Parallelograms
  • Diagonals
  • Bisect
Knowledge:
Students know:
  • Requirements for a mathematical proof.
  • Techniques for presenting a proof of geometric theorems.
Skills:
Students are able to:
  • Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.
  • Generate a conjecture about geometric relationships that calls for proof.
Understanding:
Students understand that:
  • Proof is necessary to establish that a conjecture about a relationship in mathematics is always true, and may also provide insight into the mathematics being addressed.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.31.1: Define vertical angle, transversal, parallel lines, alternate interior angles, corresponding angles, perpendicular bisector, line segment, equidistant, endpoints, interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, median, intersection, opposite sides, opposite angles, diagonals, parallelogram, bisector, and converse.
GEO.31.2: Develop a process that demonstrates the logical order of properties to form a proof.
GEO.31.3: Arrange statements to form a logical order.
GEO.31.4: Identify measures of vertical angles, alternate interior angles, corresponding angles, measures of interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, and median.
GEO.31.5: Illustrate vertical angle, transversal, parallel lines, alternate interior angles, corresponding angles, perpendicular bisector, line segment, equidistant, endpoints, interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, median, intersection, opposite sides, opposite angles, diagonals, parallelograms, bisectors, and their properties.
GEO.31.6: Find the measure of the third interior angle of a triangle when given the measure of the other two interior angles.

Prior Knowledge Skills:
  • Define a right angle, Pythagorean Theorem, converse, and proof.
  • Define exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, corresponding angles, and transversal.
  • Identify attributes of triangles.
  • Identify exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, and corresponding angles.
  • Identify a transversal.
  • Apply properties to find missing angle measures.
  • Discover the Angle Sum Theorem (sum of the interior angles of a triangle equal 180 degrees).
  • Identify parallel lines.
  • Define supplementary angles, complementary angles, vertical angles, adjacent angles, parallel lines, perpendicular lines, and intersecting lines.
  • Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
  • Recognize and demonstrate that two right triangles make a rectangle.
  • Recognize polygons.
Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.

Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
32. Use coordinates to prove simple geometric theorems algebraically.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given coordinates and geometric theorems and statements defined on a coordinate system, use the coordinate system and logical reasoning to justify (or deny) the statement or theorem, and to critique arguments presented by others.
Teacher Vocabulary:
  • Simple geometric theorems
  • Simple geometric figures
Knowledge:
Students know:
  • Relationships (e.g. distance, slope of line) between sets of points.
  • Properties of geometric shapes.
  • Coordinate graphing rules and techniques.
  • Techniques for presenting a proof of geometric theorems.
Skills:
Students are able to:
  • Accurately determine what information is needed to prove or disprove a statement or theorem.
  • Accurately find the needed information and explain and justify conclusions.
  • Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.
Understanding:
Students understand that:
  • Modeling geometric figures or relationships on a coordinate graph assists in determining truth of a statement or theorem.
  • Geometric theorems may be proven or disproven by examining the properties of the geometric shapes in the theorem through the use of appropriate algebraic techniques.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.32.1: Apply formulas, and properties of polygons, angles, and lines to draw conclusions from the given information.
GEO.32.2: Identify properties of perpendicular and parallel lines, properties of polygons.
GEO.32.3: Illustrate polygons created by given coordinates on a coordinate plane.
GEO.32.4: Identify distance formula, circle formula, Pythagorean Theorem, midpoint.

Prior Knowledge Skills:
  • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
  • Demonstrate an understanding of an extended coordinate plane.
  • Draw and label a 4 quadrant coordinate plane.
  • Draw and extend vertical and horizontal number lines.
  • Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
  • Recall how to graph points in all four quadrants of the coordinate plane.
  • Define ordered pairs.
  • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
  • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
  • Identify which signs indicate the location of a point in a coordinate plane.
  • Recall how to plot ordered pairs on a coordinate plane.
  • Identify the length between vertices on a coordinate plane.
  • Calculate the perimeter and area using the distance between the vertices.
  • Define a right angle, Pythagorean Theorem, converse, and proof.
  • Recognize examples of right triangles.
  • Demonstrate how to find square roots.
  • Solve problems with exponents.
Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.

Tags: angles, geometry, real life, reflection
License Type: Public Domain
For full descriptions of license types and a guide to usage, visit :
https://creativecommons.org/licenses
AccessibilityVideo resources: includes closed captioning or subtitles
Comments
  This resource provided by:  
Author: Kristy Lacks
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