Formative Assessment 112

Assessment Problem: 

Given is a sketch of the Bet-Shean Temple (BST) restoration (Stern, 1992).
The sketch is drawn on a scale of 1:200.
a) What are the dimensions of the main hall of the BST in reality?
b) What is the ratio between the perimeter of the main hall in the sketch and the perimeter of the main hall in reality? What is the connection between this ratio and the given scale?
c) what is the ratio between the perimeter of the main hall in reality and the perimeter of your classroom?
d) What is the ratio between the area of the main hall in the sketch and its area in reality?
e) If the sketch of the BST is delineated on a scale of 1:400, how would the dimensions of the main hall in reality be affected compared to the previous sketch? Explain.
f) If the sketch of the BST is delineated on a scale of 1:100, what would be the ratio between the areas of the main hall in the new sketch and the area of the main hall in the original sketch? Explain.
g) The sketch is enlarged in a photocopier by 50%, what are the new dimensions of the main hall in the enlarged  sketch?
h) The given sketch is enlarged in a photocopier, so that the area of the main hall after the enlargement is 243cm square. Find out what the scale of the new sketch is. Explain your answer.
BSK-task Second Level:
a) A model of the BST was constructed on a scale of 1:50 in one of the museums. In the backyard of the BST there was a rectangular pool used to store water. Its dimensions in the model are: 4cm by 6cm by 8cm.
1) How many cubic meters of water could be stored in the pool?
2) What is the ratio between the volume of the pool in the model and the volume of the pool in reality? How is this ratio connected to the given scale? Explain
b) In the backyard of the BST model there was an additional round water well whose diameter is 5cm and depth 6cm.
1) How many cubic meters of water could be stored in this well?
2) What is the ratio between the volumes of the well in the model and the well in reality? How is this ratio connected with the given scale? Explain.

Common Responses: 

The pre-service teachers who worked on this problem felt that the problem-solving approach taken in the proportional reasoning course caused them to “construct their understanding of the proportional reasoning concepts through an enjoyable and efficient process,” (p. 339).  In addition, they successfully incorporated the proportional reasoning tasks into their first years of teaching.

Mathematical Issues: 

Proportional reasoning is often challenging for students and pre- or in-service teachers. The BST problem positively influenced the content and pedagogical knowledge of pre-service teachers. Freudenthal (1983) articulated three categories: 1. Comparing two parts of a single whole, as in the ‘‘ratio of girls to boys in a class is 15 to 10’’, or ‘‘a segment is divided into the golden ratio’’. 2. Comparing magnitudes of different quantities with an interesting connection, as in ‘‘miles per gallon’’, or ‘‘people per square kilometre’’, or ‘‘unit price’’. These comparisons are not generally called ratios, but rates, or densities. 3. Comparing magnitudes of two quantities that are conceptually related, but not naturally thought of as parts of a common whole, as in ‘‘the ratio of sides of two triangles is 2 to 1’’. These comparisons are sometimes referred to as scaling. Teachers may wish to consider an appropriate order for types of proportional reasoning tasks. The following is a viable

Domain: 
Patterns & Functions
Theme: 
Interpreting Functions
Common Core Standards: 
7.RP: Ratios and Proportional Relationships
Research References: 
Ben-Chaim, D., Keret, Y., & Ilany, B.-S. (2007). Designing and implementing authentic investigative proportional reasoning tasks: The impact on pre-service mathematics teachers’ content and pedagogical knowledge and attitudes. Journal of Mathematics Teacher Education, 10(4), 333-340.