Writing “double, then add one”
Explain variables and function-value notation, then substitute consistently.
Begin with the activity ↓Writing “double, then add one”
Take the number on a card, double it, then add one. Input 3 becomes 6 and then 7. Naming the input x and output y lets us shorten the rule to y = 2 × x + 1.
Take the number on a card, double it, then add one. Input 3 becomes 6 and then 7. Naming the input x and output y lets us shorten the rule to y = 2 × x + 1.
The letter x is a place for a chosen allowed input, not one permanently fixed number. At x = 0, y = 1; at x = 2, y = 5. Within one calculation, each occurrence of x has the same chosen value.
The product 2 × x is often shortened to 2x. Thus y = 2x + 1 means the same rule: multiply x by two, then add one. It is not digit concatenation or the same as x + 2.
Naming this function f gives f(x) = 2x + 1. The notation f(x) means the output of function f at input x, not f multiplied by x. Here f(3) names the output seven obtained from input three.
Here the domain is the integers zero through four and the codomain is the integers. The expression can be calculated for other numbers, but that alone does not enlarge this activity’s domain. Distinguish calculability from the stated allowed inputs.
Change the input below and follow the doubled intermediate value to the final output. Explain why the words, expression and function value describe the same relationship. Next we turn the pairs into table rows and points.
Listen one paragraph at a time
Hold on to this distinction
In f(3), three is the input and the whole expression f(3) is the output. Expand f(3) = 2 × 3 + 1 = 7 to distinguish their roles.
Change it and check
Predict the result for a new input. Select it to compare the words, intermediate calculation and function value.
f(2) = 2 × 2 + 1 = 5
f(input) denotes the output at that input.
The domain is the integers 0,1,2,3,4. Here y and f(x) are two names for the same output; f(x) is not multiplication.
Try with paper and objects
Put the same four into two boxes and write “2 × box + 1” beside “f(box).” Calculate the first to explain why the function value is nine.
Check and explain another example
Check your idea 1
Check your idea 2
Apply it in a new setting
Write an answer yourself
Return to the explanationCompare with a solution
9 · output value
f(4) = 2 × 4 + 1 = 9.
Does your explanation include these ideas?
Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.
Continue, or pause here
Turn each table row into a coordinate point and check whether intermediate inputs are allowed.