Example of an Understanding By Design Curriculum

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Unit: Unit w/No Topics - Ratio and Proportion using Similarity
Subject: Mathematics
Minutes for Assessments: 40

Brief Summary of Unit

(Describe the context for this unit within the curriculum, and the curricular aims of the unit.)

Ratio and proportion are key concepts in mathematics and are essential to understanding scale drawings and the relationships between many geometric figures. Similarity is the first step in determining lengths, perimeter, and area when doing a comparison between shapes. Triangle similarity is essential to the understanding of congruence and vice versa.

Students will not struggle so much with the mathematics of ratio and proportion but more so the implications and applications of it. For instance determining when to use indirect measurement and getting the proportions set up correctly is very difficult for most students. After the proportion is set up, the math is simple. The uncoverage refers to the application of ratio, proportion, and similarity. Understanding why a proportions works will help students understand how to use one more effectively.

Math of the lesson in math lack student engagement and therefore students never really develop an enduring understanding of the content. Similarity is an area of mathematics where student engagement can be limitless. Students can learn indirect measurement by using mirrors to determine the height of immeasurable objects. Cartoons or pictures can be used to scale up or down according to a certain ratio. Students can determine the consistency of a “Hot Wheels” car to that of a real version of the car using measurement and proportion. Similarity can be used to “guess” the number of Skittles in a jar. This content has a wealth of opportunities to make the material meaningful and engaging to students.

Stage One - Desired Results

Mathematical Problem solving and Communication:

A. Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to solving non-routine and multi-step problems.

B. Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results.

C. Present mathematical procedures and results clearly, systematically, succinctly and correctly.

Numbers and Operations:
Identify and/or use proportional relationships in problem solving settings.
Geometry:
Use properties of congruence, correspondence and similarity in problem-solving settings involving two- and three- dimensional figures.

What will students understand (about what big ideas) as a result of the unit? “Students will understand that…”

Big Ideas: ratio, proportion, similarity, and triangle similarity

Understandings:

  1. A ratio is the relationship of two or more quantities or measurements.
  2. A proportion can be used when two objects are similar
  3. Indirect distances or lengths can be found using ratios because of triangle similarity
  4. The relationship of variables to unknowns

What arguable, recurring, and thought-provoking questions will guide inquiry and point toward the big ideas of the unit?

  1. How are ratio and scale related mathematically?
  2. What does it mean to have a ratio of a:b? Is that different than b:a? Why?
  3. How do you determine if figures are similar? What does it mean when they are similar?
  4. How do you use similarity to solve problems?
  5. What is the difference between similarity in two- and three-dimensional figures?

What key knowledge and skills are needed to develop the desired understandings and meet the goals of the unit? What knowledge and skill relate to the content standards on which the unit is focused?

Students will know:

  • Key terms – ratio, proportion, scale, perimeter, area, hypotenuse, leg, altitude, right triangle, geometric mean, similar triangles, similar figures,  indirect measurement, scale drawing, similarity ratio
  • Similarity theorems – AA Similarity, SAS Similarity, SSS Similarity, Side-splitter Theorem, Altitude to Hypotenuse Similarity Theorem, Triangle – Angle Bisector, Perimeters & Areas of Similar Figures
  • Mathematical calculations with the cross product property
  • Mathematical rules for multiplying monomials and binomial

Students will be able to:

  • Ratio and proportion – write ratios and solve proportions
  • Similar polygons –  identify and apply similar polygons
  • Proving triangles similar –  use and apply AA, SAS, and SSS similarity statements
  • Proportions in triangles – find and use relationships in similar right triangles
  • Perimeter and areas of similar figures –  use the side – splitter theorem and the triangle angle bisector theorem