



Solutions for Session 10, Part C
See solutions for Problems: C1  C2  C3  C4  C5  C6 C7  C8  C9

Problem C1  
a.  The mathematical content is patterns and proportional reasoning. 
b.  This prepares students for reasoning with proportions, predicting how many of a particular item will be in a strip of a given length, and using drawings to help solve problems. 
c.  This content requires the use of proportional reasoning and all the skills related to patterns: finding, describing, explaining, and using patterns to predict. 
d.  Students could approach these problems in several ways:
•  Draw two strips and count the number of each shape in two strips 
•  Double the number for two strips to get the number for four strips 
•  Count by ones, twos, or threes to get the number for four or six strips 
•  Draw two, four, or six strips and count the number of each shape drawn 

e.  To extend students' understanding, ask questions such as the following:
•  If there are three circles, how many strips are there? 
•  For the first strip: How many more plus signs than circles in four strips? 
•  For the second strip: How many more Xs than circles in six strips? 
•  For the second strip: Could there be 10 Xs in some number of strips? 

f.  Very few textbooks contain problems of this type. 
<< back to Problem C1




Problem C2  
a.  The mathematical content is balance, the notion that the objects on the lower pan on a pan balance weight more than the objects on the higher pan, and that one object that balances with two objects is the heaviest of the three objects. 
b.  This prepares students for understanding equality as balance. 
c.  This content introduces equality as balance to set the stage for solving equations by manipulating equations while maintaining balance. 
d.  Students might use the following reasoning to solve this problem: Since Scale B has both a cube and a sphere on one side, and Scale A has only a sphere on one side, Scale B must have heavier blocks.

e.  To extend students' understanding, ask students to describe other ways to solve the problem or to make up weights for each block that would preserve the balance, and ask questions such as the following:
•  Are there any blocks that are on both Scale A and Scale B? 
•  What will happen to the balance if you remove one sphere from both scales? 

f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem C2





Problem C3  
a.  The mathematical content is function as a machine that takes one input, does something to that input, and returns exactly one output. 
b.  This prepares students for understanding function. 
c.  This content gives students an intuitive understanding of function. By using the machine metaphor and choosing shapes rather than numbers, students realize that you must get exactly one output for every input, thus gaining an intuitive understanding of function. 
d.  Students approach these problems by examining and comparing the given inputs and outputs and describing what is the same and what is different in each case. 
e.  To extend students' understanding, ask them to make their own shape machines. 
f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem C3





Problem C4  
a.  The mathematical content is function as a machine that takes one input, does something to that input, and returns exactly one output. 
b.  This prepares students for understanding function. 
c.  This content gives students an intuitive understanding of function. By using the machine metaphor and choosing shapes rather than numbers, students realize that you must get exactly one output for every input, thus gaining an intuitive understanding of function. 
d.  Students approach these problems by examining and comparing the given inputs and outputs and describing what is the same and what is different in each case. 
e.  To extend students' understanding, ask them to make their own shape machines. 
f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem C4





Problem C5  
a.  The mathematical content is function as a machine that takes one input, does something to that input, and returns exactly one output. 
b.  This prepares students for understanding function. 
c.  This content gives students an intuitive understanding of function. These machines extend students understanding of function by using numbers as inputs and outputs. 
d.  Students approach these problems by examining and comparing the given inputs and outputs and describing what is the same and what is different in each case. 
e.  To extend students' understanding, ask them to make their own number machines. 
f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem C5





Problem C6  
a.  The mathematical content is function as a machine that takes one input, does something to that input, and returns exactly one output. 
b.  This prepares students for understanding function. 
c.  This content gives students an intuitive understanding of function. These machines extend students understanding of function by using numbers as inputs and outputs. 
d.  Students approach these problems by examining and comparing the given inputs and outputs, and describing what is the same and what is different in each case. 
e.  To extend students' understanding, ask them to make their own number machines. 
f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem C6





Problem C7  
a.  The mathematical content is function as a machine that takes one input, does something to that input, and returns exactly one output. 
b.  This prepares students for understanding function. 
c.  This content gives students an intuitive understanding of function. These machines extend students understanding of function by using numbers as inputs and outputs. 
d.  Students approach these problems by examining and comparing the given inputs and outputs and describing what is the same and what is different in each case. 
e.  To extend students' understanding, ask them to make their own number machines. 
f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem C7





Problem C8  
a.  The mathematical content is twostep functions depicted as a flow chart showing one input that goes through two operations and gives one output. 
b.  This prepares students for understanding twostep functions. 
c.  This content gives students an intuitive understanding of multistep functions. These machines extend students understanding of function by using more than one operation. 
d.  Students approach these problems by following the flow chart. Start with an In number, take it through the first operation, put that number through the second operation, and then write the Out number. 
e.  To extend students' understanding, ask them to make their own twosteppers. 
f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem C8





Problem C9  
a.  The mathematical content is twostep functions depicted as a flow chart showing one input that goes through two operations and gives one output. 
b.  This prepares students for understanding twostep functions. 
c.  This content gives students an intuitive understanding of multistep functions. These machines extend students understanding of function by using more than one operation. 
d.  Students approach these problems by following the flow chart. Start with an In number, take it through the first operation, put that number through the second operation, and then write the Out number. 
e.  To extend students' understanding, ask them to make their own twosteppers. 
f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem C9


