Applying Standards Based Constructivism:
A Two-Step Guide for Motivating Students
Proportion Problems

Popular Name: Proportion Problems
Grade Level: 7th/8th Grade
Discipline: Mathematics 
Standards and Performance Indicators Context
 
MST Standard 3: Mathematics
Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, applying mathematics in real-world settings, and solving problems through the integrated study of numbers systems, geometry, algebra, data analysis, probability, and trigonometry.  Students will:
  • Understand and apply ratio, proportions, and percents through a wide variety of hands-on explorations.
  • Apply concepts of ratio, and proportions to solve problems.
  • Represent numerical relationships in one-and-two dimensional graphs.
  • Apply the concept of similarity in relevant situations.
  • Explore relationships involving points, lines, angles, and planes.
Core Curriculum Outline Connection
Learning Objectives (which will become the dimensions of the assessment’s rubric.)
Students will be able to demonstrate that similar polygons have corresponding angles with the same measure and corresponding sides forming the same ratio.
EXPLORATORY PHASE
 (estimated time: 2 – 45 minute classes)
  • Teacher divides the students into groups of four.
  • Each student is given a reading selection that involves a real-world application of proportion.  (Selections of such readings are available on several websites regarding similarities, ratios, and proportions).
  • After each student completes the reading, he/she writes an answer tone of those questions:
    • A question that came to my mind while reading this selection was?
    • One thing that struck me as important based on the reading was?
  • Students share their responses within their four-student group.
  • Student groups report-out to the whole class, two of their responses.
  • Following the reporting-out, the groups are asked to brainstorm a definition of explanation of the term “proportion.”
  • Groups report-out their definitions and explanations and the teacher records their responses on the board or overhead projector, noting where there are similarities.
  • Whole class debrief regarding any preliminary conclusions or other thoughts about proportion.
Point projection activity: Creating proportional triangles (30 minutes)
  • Students are provided rulers, protractors, colored pencils and unlined paper.
  • Each student is instructed to follow the steps contained on the directions sheet.
  • Following the lesson is a diagram that represents what the directions are asking students to do.  It is prepared as a guide for the teacher and not as an aid to the students.  The exploration is intended to stimulate student interest through their engagement. 
Directions Sheet:
  1. Read all the directions before you begin.
  2. Using a ruler, draw a small triangle on the left side of the paper a little more than 1 _ inches from the left edge of the paper.
  3. Label the triangle ABC, putting a letter at each vertex.
  4. Make a pencil point about one inch to the left of the triangle.
  5. Label the pencil point P.
  6. Using a ruler, draw lines that go from the pencil point P to the three vertices of the triangle and extend to the right edge of the paper.
  7. Measure the distance along each line running from the pencil point P to the three vertices (These may be different distances.)
  8. Now measure the same distances from the vertices of the triangle down the lines stretching to the right edge of the paper and put a mark at these new distances.
  9. Draw a new triangle by connecting these three marks.
  10. Label this triangle A’B’C’.
  11. Repeat this proves a third time to make a third triangle.
  12. Label the third triangle A”B”C” .
Spontaneous/Planned Interventions
Many students will experience confusing attempting to follow the above directions.  The teacher will have to work-the-room and assist with suggestions and questioning techniques and, the case of the use of a protractor, perhaps direct instruction.  If time becomes a factor, the teacher may wish to have a student who understands the directions assist a student who doesn’t.

Point projection activity: creating a rule (30 minutes)
Students follow these directions:
  1. In pairs discuss how to create a table to compare the sides and the angles of three triangles.
  2. Individually record your data in the table.
  3. In pairs, compare you individual data and account for any discrepancies.
  4. In pairs, develop a rule that applies to both your and your partner’s triangles.
  5. Prepare to share your rule with the other students.
  • The pairs rejoin their group-of-four to share their findings with each other and to devise a way to solve problems regarding the proportion.  For example, “How would you figure out how large a triangle 5 times bigger than your original one would be?  Or one 10 times?”  These methods should include cross products or equivalent fractions (See concluding page for additional information/explanation).

Proportion Problems Jigsaw (Solving problems 15 minutes, jigsaw/sharing 20 minutes – 5 minutes per presentation in the jigsaw group).
  • Students count off by fours.
  • Each group goes to one corner of the room
  • The teacher assigns each group to a problem.
    • #1 group: Studies shadow problems (how tall is a tree that casts a shadow of x length when a rod of y length casts a shadow of z length.)  They develop diagrams  and methods for solving such problems using proportion.
    •  #2 group: Studies recipes.  How do double, triple, and cut a recipe in half using proportions.
    • #3 group: Studies models.  How to figure out the actual size of cars, airplanes, etc. from scale models.
    • #4 group: Studies maps.  How do maps make use of proportion regarding distances? 
  • Through a jigsaw process the four groups share what they have learned with each other. (Audience beyond the teacher.)
Whole Class Debrief/Reaction (10 minutes)
  • The teacher conducts a whole class debrief using the class’s initial definitions, explanations and preliminary conclusions about proportion as a focus for this reflection.
DISCOVERY PHASE
(estimated time: one 45 minute class)

Performance Task (including planned interventions and audience beyond the teacher.)
Students will demonstrate their understanding of similarity by creating word problems involving proportion.
Task Specifications for Developing the Student-Generated Product/Process
  • Each student will develop one proportion-based word problem.
  • The problem is to be expressed in a paragraph or less.
  • The problem should be consistent with the dimensions and criteria expressed in the performance task rubric.

Assessment of Performance Task
Dimensions of Student-Created Word Problems Involving Proportion
 Criteria for a score of
4
 Criteria for a score of
3
 Criteria for a score of
2
 Criteria for a score of
1
Multi-level

30%
 Requires 3 or more levels of computation &/or reasoning

30 points
 Requires 2 or more levels of computation &/or reasoning

20 points
 Requires only one level of computation &/or reasoning

10 points
 Is too simplistic



5 points
Realistic

20%
 Is unique real-world application of proportion

20 points
 Is a real-world application of proportion

15 points
 Application to the real-world that is a bit far fetched

10 points
 The size of the graph mIs not a real-world application of proportion

5 points
Able to be solved mathematically

20%
 Can be solved using cross products of equivalent fractions

20 points
  
  
 Can’t be solved using cross products of equivalent fractions

0 points
Correct use of terminology

30%
 All terms are correctly used (e.g. corresponding angles, corresponding sides, ratio, proportion vertices)

20 points
 Most terms are correctly used (e.g. corresponding angles, corresponding sides, ratio, proportion vertices)

15 points
 Some terms are correctly used (e.g. corresponding angles, corresponding sides, ratio, proportion vertices)

10 points
 No terms are correctly used (e.g. corresponding angles, corresponding sides, ratio, proportion vertices)

5 points
Resources To Be Made Available To Students
  •  For the Exploratory Phase: Rulers with inches or metric units, color pencils, protractors, unlined paper maps, recipes, scale models.
  • For the Discovery Phase:  Rulers with inches or metric units, color pencils, protractors
Suggestions for the Teacher
  • To provide an audience beyond the teacher have the student-created problems solved by other class members.
  • Consider having the problems graded by class members using the rubric.
  • To determine the impact of this lesson on a student’s understanding of proportion, administer a traditional assessment of the topic.
Background Information for the Teacher
  • Cross Products and Equivalent Fractions
The following way is offered as a shorthand way to distinguish a cross- products from an equivalent fractions approach.
The problem:  A large bouquet of flowers is made up of daisies, roses, and tulips in the ratio of 8:3:4.  If there is a total of 75 flowers in the bouquet, how many are tulips?
The Cross Products Approach
Tulips  4 = X
Total   15  75

15X = 300
    X = 20
Equivalent Fractions Approach
daisies : roses : tulips : total
   8        :    3     :   4      :    15

Once the student sees that the multiplier factor is 5,
it's logical to them that all of the flower's numbers
should also be multiplied by 5
  • Point Projection Exemplar for Teacher Reference
proportion