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Patterns, Functions, and Algebra
Session 9 Part A Part B Part C Part D Part E Homework
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Session 9 Materials:

Session 9, Part C:
Algebraic Structures (50 minutes)

In This Part: Units Digit | A New Algebraic Structure | Properties | More Properties

Mathematicians have always been interested in solving equations. Over the past 150 years they have studied techniques for solving equations, properties of operations that allow one to develop strategies for solving equations, and, eventually, entire systems in which one can calculate, and hence solve, equations. Note 4

These algebraic structures have become the primary focus of modern algebra. An algebraic structure is a collection of objects and operations that can be used to calculate and solve equations. The objects can be numbers, polynomials, geometric figures, points in space, card shuffles, or just about any mathematical object you can think of. The operations are usually binary operations, operations that combine two objects and form another of the same type.

Examples of systems include the system of integers and the system of rational (whole and fractional) numbers. Here, the operations are the usual operations of arithmetic -- addition, multiplication, etc. The structural approach to algebra has enormously widened the kinds of systems in which algebraists work, and hence has changed the face of what's considered "algebra."

Algebraic structures come up naturally in mathematical investigations. In Part C, we will investigate units digit arithmetic. Our goal here is to look at the underlying structure of this arithmetic, not just the calculations involved in it. Suppose, for example, that you are looking at the last digit, or units digit, of whole numbers. (Note: In Parts C and D of this session, an "*" is used to represent multiplication.)

Find the units digit of:
(22 * 43 + 59 * 27) * (47 + 1,432 * 268 * 21,343)

One way to do this is to carry out the entire calculation and then to look at the units digit. But there's no need for that much work; you can predict what the units digit will be without making the explicit calculations. For example, the units digit of 22 * 43 will be 2 * 3 = 6. And the units digit of 59 * 27 will be the units digit of 9 * 7 (that is, 3), so (22 * 43 + 59 * 27) will have the same units digit as 6 + 3. In other words, you can replace the numbers in the calculation by their units digits, turning the very large problem into a more manageable one:

(2 * 3 + 9 * 7) * (7 + 2 * 8 * 3)

Then, you can simplify as you go, so that, for example, (2 * 3 + 9 * 7) becomes 6 + 3, which becomes 9. These calculations depend upon order of operations. Look at the tip in Problem C1 below if you are unfamiliar with this concept.

Problem C1


Using this line of reasoning, find the units digit of 2,314 * 426 + 573 * 234.
Note 5

Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
The order of operations -- which operations you do first, second, and so on -- are: inner parentheses, exponents, multiplication or division, addition or subtraction.   Close Tip


Problem C2


True or false: The units digit of 2,314 * 426 + 573 * 234 is the same as that of 2312 x 422+ 576 x 232.

Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Try to answer this question by doing as little calculation as possible.   Close Tip


Problem C3


Find the units digit of
(312 * 423 + 57 * 57) * (28 + 1,045 * 68 * 68 * 68)

video thumbnail

Video Segment
In this video segment, Prof. Cossey asks participants to find the units digit of the product in Problem C3 above. Try Problem C3 yourself, then compare your work to that of the onscreen participants.

What shortcuts did you use while working on Problem C3? Can these shortcuts always be used in this algebraic structure?

You can find this segment on the session video, approximately 5 minutes and 21 seconds after the Annenberg Media logo.



Problem C4

write and reflect  

Some people say that "the units digit of the sum is the units digit of the sum of the units digits" and "the units digit of the product is the units digit of the product of the units digits." Explain what they mean. Is what they say correct? Why or why not?


The kind of reasoning you are doing in Problem C4 is one which begins to address the structure of units digit arithmetic.

Take it Further

Problem C5


Explain why "taking the units digit" is the same as "divide by 10 and take the remainder."

The fact that these are the same explains why "taking the units digit" is a modular arithmetic.


Next > Part C (Continued): A New Algebraic Structure

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