11 Variation · g10-11-01 · 5–10 min
Which power controls the multiplier?
Does doubling x always double y?
Predict: For y=kxⁿ, predict y(2x)/y(x) for n=1,2,3.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Constant k = 2; Positive input x = 1 | |
Case 2
Constant k = 2; Positive input x = 1 | |
Case 3
Constant k = 0; Positive input x = 2 | |
Revise: My claim needs the condition… because…
Exit: y=3x². If x is tripled, what happens to y?
11 Variation · g10-11-02 · 5–10 min
A fixed product instead of a fixed ratio
What remains fixed in inverse variation?
Predict: Predict the new output when x is multiplied by t in y=k/x.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Constant k = 4; Input x = 1; Input multiplier t = 2 | |
Case 2
Constant k = 4; Input x = -2; Input multiplier t = 2 | |
Case 3
Constant k = 0; Input x = 0; Input multiplier t = 2 | |
Revise: My claim needs the condition… because…
Exit: y=12/x. Find y at x=3 and x=6.
11 Variation · g10-11-03 · 5–10 min
Change one factor or both?
In joint variation, do two doublings still give a doubling?
Predict: For y=kxz, compare doubling x alone with doubling both x and z.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Constant k = 2; Input x = 1; Input z = 2 | |
Case 2
Constant k = 2; Input x = -1; Input z = 2 | |
Case 3
Constant k = 2; Input x = 0; Input z = 2 | |
Revise: My claim needs the condition… because…
Exit: y=2xz; initially x=3,z=4. What if x doubles and z halves?
11 Variation · g10-11-04 · 5–10 min
The fixed part changes the ratio
Does a straight graph always mean direct proportion?
Predict: Compare y=a+bx with y=bx while keeping the same rate b.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Fixed part a = 3; Rate b = 2; Input x = 2 | |
Case 2
Fixed part a = 0; Rate b = 2; Input x = 2 | |
Case 3
Fixed part a = 3; Rate b = 2; Input x = 0 | |
Revise: My claim needs the condition… because…
Exit: A charge is 20+3x. Compare x=2 and x=4.
