A basic skill of algebra is being able to express words in algebraic symbols. Students had some practice with this in seventh grade, but now some of the expressions are more complex, such as “the quantity of three less than some number divided by 3”. Also, students need to develop the nomenclature and vocabulary of algebraic expressions; i.e., (x + y)^3 is read as “x + y the quantity to the third power”.
Also important is for students to start seeing that the fraction bar represents division. They have seen this in seventh grade, but it must continue to be repeated at the start of the algebra course to develop the habit of seeing division represented this way. Students need to see that x/3 is the same as 1/3(x), so problems and examples will use both.
Warm-Ups.
1. Evaluate. (x-5)^3 when x = 3. Answer: (3-5)^3=(-2)^3=-8
2. How many terms are in the expression 2xy + y/(x+1)? Answer: Two terms.
3. The sum of 1 + 2 + 3 + …. + n is given by the formula n(n+1)/2. Use the formula to find the sum of 1 + 2 + 3 + … + 20. Answer (20∙21)/2 = 210
4. What is the value of 100 – 10/0.1? Answer: 100-100 = 0
5. Write as a power (i.e., with an exponent). x∙x∙x Answer: x^3
Problem 1 requires students to recall what exponents represent, and that a negative number raised to the third power (or any odd-numbered power) will be a negative number. Problem 2 requires students to know the definition of “term” presented in the previous lesson. The plus sign between the two expressions serves as an indicator that the two expressions so separated are terms. Prompts may include referring to the plus sign, as well as asking whether 2xy represents a product, and whether y/(x+1) represents division. Problem 3 is simple substitution; prompts may include “What do you plug in, and where?” In calculating the value of the expression, attention should be drawn to the fact that 20 can be divided by the 2 in the denominator resulting in a simpler calculation: 10 × 21 = 210.
Stating expressions correctly. I start out by writing y + 3 on the board. “Can someone please read the expression I put on the board?” Students may be slow to respond, since it seems so obvious they may suspect a trick. “I promise it isn’t a trick, the answer is as easy as it looks,” I might say if there are no responses. Someone will eventually respond: “y plus three”.
“Correct.” I now write 7(y + 3) on the board. “Someone read this expression.”
There will likely be a variety of responses, some of which may be “seven y plus 3”, or “seven times y plus 3”.
“When we have a number or variable next to an expression in parentheses, what operation are we doing?” I’m waiting to hear “multiplying”.
“We’re multiplying; you know this from seeing the distributive rule. What are we multiplying here? Seven times what? I heard someone say ‘Seven times y plus 3’ If I were to write this, that would be 7y + 3. Is that what we have here?”
General consensus is “No.” My point having been made, and not wanting to play “read my mind” with the students, I explain: “We’re multiplying seven by a sum: the sum of y plus 3, and we can say it that way: Seven times the sum of y plus 3. Or instead of ‘the sum of’ we can say ‘the quantity y plus 3’. Anything in parentheses can be referred to as ‘the quantity’. Let’s try some.”
8(x – y) Answer: Eight times the quantity x minus y
9y – x Answer: Nine times y minus x, or nine y minus x.
“How about x^3?” They have seen this before and will say x to the third power.
“What about (x – 3)^3?” If students are slow raising hands, I’ll say “Remember what I said about how we refer to anything in parentheses.” If someone says “x – 3 to the third power, I will say that they have described x – 3^3.”
“What about a/4?” I will usually hear. “a over 4”.
“That’s correct. So is ‘a divided by 4’. And one fourth of a is also correct.”
More: 1/3(n) – 3 Answer: One third of n minus 3.
1/3(n-3) Answer: One third of (or times) the quantity n minus 3.
2/5(x –y)^5 Answer: Two fifths of (or times) the quantity x minus y to the fifth power.
From Words to Algebra. Students have had practice translating from words to algebra in seventh grade. The lesson on this is found in Post 5 and the techniques discussed there can be repeated. In fact, much of this lesson will be a repetition of what they have learned in seventh grade.
“Much of this is familiar to you, but some of it you may have forgotten. How would I write five more than some number?”
They should remember it is 5 + x or x + 5, though I may need to remind them that “some number” –since it isn’t named—is going to be a variable.
“How about 5 more than two times a number?” Answer: 5 + 2x
“Try 3 times the sum of 3 and y” Answer: 3(3 + y) or 3(y+3)
“Now I’m willing to bet $100 that even though you’ve had this before, the first answer I hear will be wrong.” I do this because it’s a dare, and students tend to respond out of curiosity and/or to prove me wrong. “How would you write 4 less than some number?”
Generally, when I preface the question with a dare, the first answer I hear is usually correct, and in this case will be x – 4. “A common mistake is to write it 4 – x”.
More Examples.
Three more than four times some number. Answer: 4x + 3
Five less than 6 times some number. Answer: 6x – 5
A certain number added to 3 times the number. (This might require some prompts.)“How would we write a ‘certain number’?” x
“How would we write 3 times that number?” 3x.
“How do we write the sum of the two?” x + 3x.
“Can that be simplified?” Yes; 4x.
5 divided by the sum of 4 and some number. Answer:
Some students will use the divide sign. I make a point to say that now we use the fraction bar for algebraic expressions.
I write the following on the board:
“One tree is half as tall as another. Let h be the height of the taller tree. Represent the height of the smaller tree. Answer: h/2 or ½(h).
“Joy is x years old. Tom is 1 year younger. Write an algebraic expression for Tom’s age. Answer: x – 1
I introduce the next problem by referring to the previous one. “We were told that Joy is x years old, so obviously x represents Joy’s age. The next one is similar: (This is projected on the board):
“Ken has 4 fewer than the number of stamps that Len has. Write an algebraic expression for the number of Ken’s stamps.”
“The number of Ken’s stamps is described in terms of the number of Len’s stamps. So what are we going to let x represent?” Students will generally go with x representing the number of Len’s stamps, so the answer will be: x – 4.
I provide a variation on this. Ken has 4 fewer than one-third as many stamps as Len has.”
“What does the expression look like now for the number of Len’s stamps?” Answer: 1/3(x) – 4
Homework. The homework problems should be both straight translation, and word type problems such as the latter examples.