Accelerated Seventh Grade Math, and Eighth Grade Math
So far, discussion of seventh grade math has been focused on regular Math 7 classes, with some blending of accelerated Math 7 topics. Those topics may or may not be do-able, and are dependent on the ability of the students in said classes.
At this juncture, we now depart from the regular Math 7 curriculum and address the accelerated Math 7 program. Because those topics are the same as those addressed in Math 8, this chapter covers both grades. I do not address the geometry topics as mentioned in the introduction.
The topics to be covered are as follows:
Functions
Graphing Linear Functions
Slope and Constant Rate of Change
Graphing Proportional Equations and Direct Variation
Slope-Intercept Form of Linear Equations, and Graphing
Writing Equations in Slope-Intercept Form
Linear Equations and Functions
Functions are ultimately an abstract concept which at the novice level must be taught in its most basic form. This would be the linear equation for starters. As students progress through subsequent math courses, the definition essentially stays the same, except there are more functions introduced: cubics, trigonometric, exponential, logarithmic and so on.
It is most easily described and defined in terms of values that are dependent on one another. The cost of oranges that are $3 per pound is dependent on how many pounds are bought, and so forth. Functions are introduced as rules that assign a number to another one; these rules are in the form of equations.
Typically, textbooks introduce relations first, as ordered pairs of numbers and then functions are introduced as a special type of relation in which for each “input value” there is one and only one “output value”. Students are then tasked in the same lesson as finding examples of relations that aren’t functions, and those that are, by virtue of whether each input has one output. This is confusing and tends to cause cognitive overload.
I introduce the difference between relations and functions in the subsequent lesson and then turn to graphing. At that point, having digested the initial concept of what a function is, students are better prepared to build on that foundation.
Warm-Ups. The Warm-Ups include some percentage problems and some equations that relate directly to this particular lesson.
1. Solve the equation. 3y-12 = y-2 Answer: 2y = 10; y = 5
2. In the equation y = -4x + 2 what is the value of y if x = 3?
Answer: y = -4(3)+2 = -12+2 = -10
3. A rental company charges $50 for a moving truck plus an additional fee of $0.55 per mile that the truck is driven. If the truck drives 100 miles, what is the total cost for the rental? Answer: 50 + (0.55)(100) = 50 + 55 = $105
4. An egg tray with 12 eggs weighs 440 grams. The empty tray weighs 20 grams. What is the average weight of an egg? Answer: Weight of 12 eggs: 440-20 – 420; average weight = 420/12 = 35 g
5. A bowling alley charges a $5 fee for renting shoes, and $2 per game. If Jerry pays $25 for bowling, how many games did he bowl? Answer: Let x = number of games bowled, so 2x = cost for x games; 25 = 5 + 2x; 2x = 20; x = 10 games.
With the exception of Problems 1 and 4, the Warm-Ups focus on how the value of one thing affects another; that is, independent and dependent variables. This section addresses such dependencies in terms of what a function is, and how they work.
The Wind-up: What We Know So Far. The discussion of Warm-Up answers leads to the topic of functions. Taking Problem 1, I point out that we have been solving equations that have one variable. But looking at Problem 2 we see there are two variables. The problem tells us what the value of x is, so we are able to find out what y is.
“What about Problem 5? Does that have two variables?” The general answer will be “no”. “We saw that the problem was solved by writing the equation 25 = 5 + 2x, because we were given the total cost of Jerry’s visit to the bowling alley. So far we have been dealing with equations that have one variable. Now we’ll look at equations that have two variables.”
I write the following equation on the board:
A typical question and answer would then be:
“What does the 5 represent?” The cost of shoe rental.
“What does the x represent?” The number of games.
“What does the 2 in 2x represent?” The cost of each game; $2.
“So y would be what?” The total cost of x games.
“Problem 5 is actually a special case, where the total cost is $25, but there can be many different cases if we use two variables. The value of y changes depending on what x is; that is, the number of games. If the number of games increases, what happens to y?”
I hope to hear “y increases” but I’ll accept “It costs more,” and then paraphrase.
“If we have an equation y = 2x, the value of y varies, depending on the values of x. If x is the number of pounds of oranges, and oranges cost $2 per pound, then y represents the cost of the oranges. If we bought 2 pounds one week and 3 pounds the next, the cost y would change from what to what?”
Answer: From $4 and $6.
“The cost depends upon the price, because the values of y change as the values of x change. If x decreases what happens to y? Answer: Decreases.
The Pitch: What is a Function. “Instead of saying ‘the cost depends upon the price’, we can also say that ‘the cost is a function of the price or: y is a function of x.
The equation y = 2x is a rule that says for each input value for x, say 1, 2, 3, 4, the output value, y, is twice that amount: 2, 4, 6, 8.
“If you travel at 40 mph, and the time of travel is t hours, then the distance d depends on the value of t. And we can express this relationship as d = 40t.
I have them write in their notebooks:
A function is a rule which assigns to each input value one and only one output value.
I’ll get into the “one and only one output value” in the next lesson.
The “input values” are called the “domain”, and the “output values” are called the “range.”
“We saw that when we have an equation like d = 40t, as t changes, so does d. The d value is an output value; t is an input value. In this equation d is dependent on t. So which value, d or t is the dependent value?” Answer: d.
“So the independent value is t; it’s free as a bird.”
Examples. The examples we focus on express functions in various forms.
“Functions can be in the form of an equation, a list of ordered pairs, a table, or a graph.”
1. For y=2x the y values are dependent on the x values. Whatever x equals affects what y equals. If the x values are 1, 2, 3, 4, 5 what are the associated y values?”
Answer: 2, 4, 6, 8, 10
“List these as ordered pairs. I’ll do the first one; you do the rest. (1,2). What would we get if x is zero?” (0,0)
“Now do 2, 3, 4 and 5” (2,4), (3, 6), (4, 8), (5,10)
“Now let’s write this as a table. We’ll put the x values in the top row and y values in the second row. Write in your notebook and fill in the y values.”
I circulate around to see that they have done this correctly:
2. For the equation y = 2x + 3, I have them write the ordered pairs, and then a table using 0 through 4 for the x values. Answer: (0,3)(1, 5)(2, 7)(3,9)(4,11)
“What is the domain?” (If they’ve forgotten, I remind them they’re the input values) 0, 1, 2, 3, 4 “What is the range?” 3, 5, 7, 9, 11
“What’s dependent on what? Which is the dependent value and which is the independent value?” Answer: y is dependent, x is independent.
“So the domain values are independent; range values are dependent.” I have them write that down in their notebook.
3. A boat travels at 5.5 mph. The distance it travels is a function of the time traveled. If it travels for 2 hours at 5.5 mph, how far has it travelled? 11 miles. 4 hours? 22 miles. Let d = distance and t = time. Write an equation for this relationship.
I announce this as a “challenge problem” and circulate around the room offering guidance while they work it. Answer: d = 5.5t
What are the independent and dependent variables? Independent: t, Dependent: d
Homework. The homework should be problems like the examples; write ordered pairs and a table for various functions; identify functions as in Example 3, identify the domain and range, dependent and independent variables.