Lesson 6: Algebraic Expressions The Distributive Property
Students use the structure of an expression to identify ways to rewrite it.
Students use the distributive property to prove equivalency of expressions.
The previous five lessons introduced the graphs of the functions students will study in this algebra course. These next lessons change the focus from graphs to expressions and their structure. In Grades 5-8, the term "expression" was described but not formally defined, and many subtleties may have been overlooked. For example, the associative property may not have been made explicit; for instance, is really ( ) . The lessons that follow will formally define algebraic expression and the equivalency of algebraic expressions, and simultaneously introduce students to the notion of a recursive definition, which later becomes a major aspect of Grade 9 (recursive sequences). Lesson 6 begins with an expression-building competition. As this is a strange, abrupt change of direction for students from the previous five lessons, it may be worthwhile to mention this change.
You have learned in middle school to find equations for straight line graphs such as the ones that appear in Lesson 1, but as we saw in Lessons 2 and 3, not all graphs are linear. It would be nice to develop the machinery for developing equations for those too, if at all possible. Note that Lesson 4 indicates that graphs still might be very complicated, and finding a single equation for them might not be possible. We may, however, be able to find equations that produce graphs that approximate the graphs, or sections thereof. Also, Lesson 5 points out the value of finding the point of intersection of two graphs. Our goal is to develop the algebraic tools to do this.
Exercise 1 (13 minutes)
The following is known as the "4-number game." It challenges students to write each positive integer as a combination of the digits and ; each used at most once, combined via the operations of addition and multiplication only, as well as grouping symbols. For example, can be expressed as ( )( ). Students may use parentheses or not, at their own discretion (as long as their expressions evaluate to the given number, following the order of operations). Digits may not be juxtaposed to represent larger whole numbers; so you cannot use the numerals 1 and 2 to create the number 12 for instance.
Play the 4-numer game as a competition within pairs. Give the students 3 minutes to express the longest list of numbers they can, each written in terms of the digits , and . Students may want to use small dry erase boards to play this game or pencil and paper. Optionally, consider splitting up the tasks (e.g., 1-8, 9-20, 21-30; 31-36) and assign them to different groups.
Below are some sample expressions the students may build, and a suggested structure for displaying possible expressions on the board as students call out what they have created. When reviewing the game, it is likely that students will have different expressions for the same number (see answers to row 7 and 8 given below). Share alternative expressions on the board and discuss as a class the validity of the expressions.
Challenge the students to come up with more than one way to create the number .
After students share their results, ask these questions:
What seems to be the first counting number that cannot be created using only the numbers 1, 2, 3, and 4 and the operations of multiplication and addition?
What seems to be the largest number that can be made in the 4-number game?
(1+2 )( 3)( 4) =36
We can now launch into an interesting investigation to find structure in the game.
Suppose we were playing the 2-number game. What seems to be the largest number you could create using the numbers 1 and 2 (each at most once) and the operations of multiplication and addition?
( 1+2) = 3
Suppose we were playing the 3-number game. What seems to be the largest number you could create using the numbers 1, 2, and 3 (each at most once) and the operations of multiplication and addition?
( 1+2)( 3) = 9
Encourage students to generalize the pattern for the 5-number game, and the -number game, and think about why this pattern for an expression gives the largest attainable number.
Add 1 and 2, and then multiply the remaining numbers. For the 5# game, the largest number would be (1+2)(3)(4)(5) = 180. For the n# game, the largest number would be (1+2)(3)(4)(5)… )