



Solutions for Session 10, Grades 68, Part D
See solutions for Problems: D1  D2  D3  D4  D5

Problem D1  
For the first set of scales the answer is 1 cube on the left pan. For the second set of scales the answer is 1 cylinder on the right pan
a.  The mathematical content is balance, the notion that the objects on the lower pan on a pan balance weigh more than the objects on the higher pan, and that one object that balances with two or more objects is the heaviest.
This prepares students for understanding equality as balance and the notion that adding or removing the same items from both sides of a balance retains balance.

b.  This content introduces equality as balance, setting the stage for solving equations by manipulating them while maintaining balance. 
c.  This content illustrates methods for solving linear equations. 
d.  Students approach these problems in several ways:
•  From Scale B, 2 cubes balance 1 cylinder. 
•  Replace the 2 cylinders on Scale A with 4 cubes. 
•  Then 2 spheres balance 6 cubes. 
•  That means 3 cubes balance 1 sphere. 
•  Replace the sphere on Scale C with 3 cubes. That gives 5 cubes on the left. 
•  The 3 cylinders on the right need 6 cubes. To balance the scale, add 1 more cube. 

e.  To extend students' understanding, ask questions such as the following:
•  Look at Scale A. Which block weighs the most? How do you know? 
•  Look at Scale B. Which weighs more, a cylinder or a cube? 
•  How many cubes balance 1 cylinder? How do you know? 

f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem D1




Problem D2  
For the first set of scales the answers are: cylinder = 3, cube = 4, sphere = 6, cone = 1.
For the second set of scales the answers are: cylinder = 5, cube = 1, sphere = 2, cone = 4.
Possible responses to the questions to consider:
a.  The mathematical content is the concept of variable and solving four equations in four variables. 
b.  This content prepares students for understanding that they can replace a variable with its value. 
c.  This content illustrates methods for solving systems of linear equations. 
d.  Students approach these problems in several ways. One way is shown for each problem.
•  For the first set of scales: Compare Scales A and D. Replace the cylinder, cone, and sphere on Scale D with 10 pounds. That means that the cube weighs 4 pounds (14  10). Replace the 2 cubes on Scale C with 8 pounds (2 * 4). That means the sphere weighs 6 pounds (14  8). Replace the sphere on Scale B with 6 pounds. Then the 2 cylinders weigh 6 pounds (12  6), and 1 cylinder weighs 3 pounds (6 ÷ 2). Then on Scale A, replace the cylinder and the cube with 9 pounds (6 + 3), to show the cone weighs 1 pound (10  9). 
•  For the second set of scales: From Scale G, if 2 spheres and 2 cylinders weigh 14 pounds, then 1 sphere and 1 cylinder weigh 7 pounds (14 ÷ 2). Replace 1 cylinder and 1 sphere on Scale H with 7 pounds to show that 1 cone weighs or 4 pounds (11  7). Replace 1 cylinder and 1 sphere on Scale F with 7 pounds to show that 1 cube weighs 1 pound (8  7). Replace 2 cones on Scale E with 8 pounds (4 + 4) to show that 1 cylinder is 5 pounds (13  8). Since we know that 1 cylinder and 1 sphere weigh 7 pounds, then 1 sphere is 2 pounds (7  5). 

e.  To extend students understanding, ask them to describe other ways to solve the problems and ask questions, such as the following about the first set of scales:
•  If each cylinder on Scale B weighs 5 pounds, how much does the sphere weigh? How do you know? 
•  If the sphere on Scale C weighs 4 pounds, how much does each cube weigh? How do you know? 
•  Compare Scales A and D. How much does the cube weigh? How do you know? 

f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem D2





Problem D3  
Answer: helmet = $29, bell = $10, lock = $24
Possible responses to the questions to consider:
a.  The mathematical content is the concept of variable and solving three equations in three variables. 
b.  This content prepares students for understanding that they can replace a variable with its value. 
c.  This content illustrates methods for solving systems of linear equations. 
d.  Students approach these problems in several ways. Two ways are shown.
•  Because the first two pictures each show a bell, and the 1st price is $5 more than the 2nd, we know that the helmet costs $5 more than the lock. In the 3rd picture, the helmet and lock together cost $53. If the helmet was $5 cheaper, the two would cost the same, and the combined price would be$48 (53  5). Then each item would cost $24 (48 ÷ 2). Thus, the lock costs $24 and the helmet costs $29 (24 + 5). Because the helmet and bell together cost $39, the bell is $10 (39  29). 
•  Adding all the pictures together shows that 2 helmets and 2 locks and 2 bells would cost 39 + 34 + 53, or $126. Because 2 of each item costs $126, then 1 of each item would cost $63 (126 ÷ 2). The 1st picture shows that the helmet and bell cost $39, so the lock must cost $24 (63  39). Similarly, the helmet costs $29 (63  34), and the bell costs $10 (63  53). 

e.  To extend students' understanding, ask questions such as the following:
•  How are the first two pictures the same? 
•  How are they different? 
•  What causes the difference? 
•  How does this difference help you solve the 3rd picture? 

f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem D3





Problem D4  
Answers: 1, 3, 3; 2, 6, 9; 3, 9, 18; 4, 12, 30, 5, 15, 45
The rule for the number of blocks for stair N is N = 3S, so for Stair 10 it takes 3 * 10, or 30 blocks.
Possible responses to the questions to consider:
a.  The mathematical content is function. 
b.  This content prepares students for making tables of values to represent data and for writing functions from those tables. 
c.  This content illustrates the representation of a function as a scenario, a table of values, and an equation. 
d.  Students approach these problems in several ways. One way is shown here. Each stair requires 3 more blocks than the prior stair. The 1st stair requires 3 blocks, the 2nd stair requires 2 * 3 blocks, the 3rd stair requires 3 * 3 blocks, so the Sth stair will require 3 * S blocks. 
e.  To extend students' understanding, ask them to find the rule for the total number of blocks (T) needed to build S stairs. (That rule is T = (3 / 2)S(S + 1).) 
f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem D4





Problem D5  
For the first set of shapes the rule is B = 2 + 3(N  1) = 2N + (N  1) = 3N  1
For the second set of shapes the rule is B = 1 + N^{3}
Possible responses to the questions to consider:
a.  The mathematical content is inductive reasoning and writing functions. 
b.  This content prepares students to identify, continue, and generalize patterns. 
c.  This content illustrates the difference between linear and nonlinear functions. 
d.  Students approach these problems in several ways. One way is shown here:
•  For the first set of shapes: The 1st shape uses 2 blocks. The 2nd shape uses 2 blocks + 1 set of 3 blocks. The 3rd shape uses 2 blocks + 2 sets of 3 blocks. The Nth shape will use 2 blocks + (N  1) sets of 3 blocks. 
•  For the second set of shapes: The 1st shape uses a 1 * 1 * 1 cube of 1 block with 1 block on top. The 2nd shape uses a 2 * 2 * 2 cube of 8 blocks with 1 on top. The 3rd shape uses a 3 * 3 * 3 cube of 27 blocks with 1 on top. The Nth shape uses a N * N * N cube of N^{3} blocks with 1 on top. 

e.  To extend students' understanding, ask questions such as the following:
•  How would you build the 9th or 10th or 12th shape? 
•  How does telling how to build the shape help you write the function rule? 
•  Describe how to build each shape in problem the first set of shapes so that you produce each of the different function rules shown in the answers. 

f.  Answers will vary. Very few textbooks contain problems of this type. 
<< back to Problem D5


