


From Arithmetic to Algebra 




In the following four problems, note how the
arithmetic solutions use logic and involve a probing of the conditions of
the problem—procedures which often easily transfer to the more abstract
process of algebraic solutions, where problems ultimately are reduced to
equations. 




In a class with a total of 85 students, 380
oranges were distributed with each girl getting 5 oranges and each boy 4.
How of the students are girls, and how many are boys? 




380 oranges 

85 students 

5 oranges per girl 

4 oranges per boy 

Give each student 4 oranges: 4 x 85 = 340 

380 – 340 = 40 

There are 40 girls and 85 – 40 = 45 boys 






380 oranges 

85 students 

5 oranges per girl 

4 oranges per boy 

Let x be the number of boys 

Let y be the number of girls 

4x + 5y = 380 and x + y = 85 

Solve the above system to find x and y 




A bottle and a cork together cost $1.10, but the
bottle costs a dollar more than the cork. What is the cost of each? 




Systematic GuessandCheck Approach 






Let b = cost of bottle 

Let c = cost of cork 

b + c = 1.10 

b = c + 1.00 

Solve the above system 




One raja says to another, “If you give me one
camel, then we will have an equal number of camels” The other replied, “No,
if you give me one camel, then I will have double the number you have.” How
many camels does each raja have? 




Teacher: By how much do the two numbers differ? 

Student: 

Teacher: Why? 

Student: 

(continued) 




Teacher: Can one of the numbers be odd and the
other, even? 

Student: No 

Teacher: Why? 

Student: 

(continued) 




Teacher: Can both numbers be even? 

Student: Yes 

Teacher: Are you sure? What do we get, then,
when 1 is added to the larger number? Is it even or odd? Could an odd
number be the double of the lesser number? 

(continued) 




Student: 

Teacher: So what do we know about the larger
number? Is it even or odd? 

Student: It’s odd. 

Teacher: Why, then, must the smaller number be
odd as well? 

(continued) 




Student: 

Teacher: Can the numbers be large? 

Student: No, when you add 1 to the larger
number, it becomes double the smaller number. 

Teacher: Let’s try some values. 

(continued) 





Let x be the number of camels owned by the first
raja. 

Let y be the number owned by the second. 

x + 1 = y – 1 

y + 1 = 2(x – 1) 

Solve the above system. 




Ninetynine girls and one boy are in a
mathematics class. How many girls must leave the room so the the percent of
girls becomes 98 percent? 








Use a guessandcheck strategy, focusing on the
percent of boys in the class when the total number of students in the class
is varied. 




Let x = the number of girls who must leave the
class 

99  x
= 98 

100 –
x 100 

Solve the above proportion. 

