Lines - Adding
What does it mean to Add Functions (Lines)?
To be helpful with the transition beyond elementary school mathematics we need to support students in developing more powerful and general ways of making sense of core mathematical ideas. These more powerful ideas can build on concepts introduced in elementary curricula. To illustrate how this can work, the Common Core State Standards emphasize “Operations and Algebraic Thinking” as beginning in elementary grades. Functions are then introduced as an 8th grade standard. By building on and extending ideas related to comparisons (=, >, <) and operations (+, -, * , ÷) with numbers in elementary grades to comparisons (=, >, <) and operations (+, -, * , ÷) with functions in subsequent grades, the function-based approach to algebra supports very general approaches to understanding key mathematical ideas that will support student advancement through high school and beyond.
For this activity we can ask in what ways is adding a linear function to another linear function analogous to, say, adding 3 + 1? Activities extending this introduction are included.
- What do you notice? What do the sliders do? What seems to be happening? How does the Purple line depend on the Green line and the Blue line?
- Does the Purple line always stay horizontal ("flat")? If not, what combinations of the Green and the Blue lines make THE PURPLE LINE HORIZONTAL?
- See if you can find at least two ways of making the Purple line horizontal. Record the combinations of m's and xi's for each way that works.
- Write two lines in slope-intercept form that when added the combined function will be flat. (You can use the environment below or a graphing calculator to try out your combination).
- Can you come up with other combinations?
- If you had to explain how to make more combinations to someone else, what would you say?
- Now turn off the Purple line by de-selecting "add". Click the red "subtract box". What is the same and what is different about the lines?
- Similar to above, can you find at least two ways of making the RED line horizontal? Record the combinations of m's and xi's for each way that works.
- Can you come up with a pair of linear functions in slope-intercept form that when one is subtracted from the other the resulting function is flat? To try your lines you can change h(x) from f(x) + g(x) to f(x) - g(x) in the environment below (double-click on h(x) to make the change). Then enter your lines in f(x) and g(x).
- Can you come up with other combinations?
- If you had to explain how to make more combinations to someone else, what would you say?
- What do you think we might get if we multiplied two lines?