Example of what a Understanding By Design Curriculum would look like.

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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 binomials
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
Stage Two - Assessment
Logo Design - Design a logo for an advertising company that uses similarity throughout the design. Explain where similarity exists and how you can demonstrate it.

Volume Similarity – Complete a lab experiment on volume of similar shape. Explain what has been discovered and show the formula for determining the volume of similar items
 

Lima Bean Grow – Each student will be given a lima bean and be required to inscribe it in a rectangle.  They will measure the dimensions of the rectangle and determine the ratio of growth as the bean is immersed in water over a given number of hours.  They will also do predictions based upon the ratio of growth. Students will be required to problem solve and write about this experience.

Scale Drawing – Students will be given a small piece (a square) of a larger picture and a scale factor.  They will then use the factor to redraw the picture piece in the scaled up or down ratio. 

Challenge Similarity – Students will be given one challenge problem in which they will use the skills learned to figure out a ratio, measurement, or scale factor.  They will be required to write all steps out and accompany them with explanations as to why they were performed.
Quizzes –  There will be 3 quizzes given, one on ratio and proportion, the second on  similarity in two- and three-dimensional shapes, and the third on applications of similarity.

Test – A test will be given on the entire unit of study for similarity.  It will include, calculation problems, applications, algebraic and numeric proportions, and a written explanation of a concept.

Vocabulary – Students will be required to do the following for all of the identified key terms:  define in their own works, draw an example of it, write a word problem using it, and solve the word problem

Writing – Students will be given an application problem and they must determine all areas, and reasons why, peers could have difficulty with it.

Proof – Students will be required to prove two triangles are similar, given a choice of three situations
Stage Three - Learning Plan
1.  Present students with a jar of skittles and a huge rectangular prism that is empty.  State that the ratio of skittles in the jar to the rectangular solid is 3:16.  How would you figure out the number of skittles in the rectangular prism?  (the prism is too large to actually fill with skittles) – H 

2.  Students will learn about ratio and proportion as it applies to area and volume using a discovery activity where students measure and calculate area and volume to determine a pattern – T, E1,

3.  Discuss the vocabulary that they will need and have them start with an activity called Vocab Twist, then use a foldable activity to reinforce the learning – E1

4.  The ticket out the door (first week) will be writing short word problems using the vocabulary correctly – E1, R, E2

5.  Introduce the topics they will be learning about (essential questions) and look at the performance tasks along with the rubrics for each one - W

6.   Students will write about the struggles and triumphs of their group each week - R

7.  Going Fishing – Students will determine the number of fish in a pond using a simulation of tagging fish.  They will use their knowledge of ratio and proportion to figure out the number of fish.  Students will be required to problem solve and write about this experience – T, R

8.   Students will complete a scale drawing of their room with measurements and calculations of paint and carpet (both real and scale) and explain the process, reflecting on challenges and reasons – H, T, E1

9.  Identify similarity in photos and justify the similarity both mathematically and in written form – E1, R, T

10.  Students will perform peer reviews on many of the performance tasks – R, E2, T

11.  Students will work on a discovery activity that had objects at each station and they will be required to determine and justify similarity -– E1, R, T

12.  Students will use the rubric for the Design of the logo to peer review and justify the score for each category (they will score their own work as well)  – R, E2, T

13.  Students will explain what and why for all of their errors on all tests and quizzes - R, E2

14.  Students will apply similarity to immeasurable objects outside to determine lengths - E1, T

15.  Students will use a workmat when solving any problems -  E1, E2, T

16.  Similarity Scavenger Hunt -  Students will be given a scavenger hunt in which they will apply the skills they have learned to new situations and in doing so complete a scavenger hunt around the building – H, T

17.  Students will utilize the double side journal and note taking guide along with foldables for all of the notes from direct instruction – R, E2, T

18.  Students will practice their skills of ratio, proportion, and similarity by determining if a hotwheels car holds a ratio to the real car by measuring and justifying reasons and calculations.  Students will peer review others work. – W, E1, E2, T
 
Contact:
OnHand Schools
412-325-8000 x103 or x104
tdemarco@onhandschools.com
giacono@onhandschools.com
http://www.onhandschools.com