From Numbers to Functions
How can we describe change itself?
Featured character: The Relationship
The story unfolds
The mathematics beneath the story
Grade 8: unit rate as slope, linear relationships, functions, and y=mx+b. Algebra I: solving and comparing linear equations/functions, rational coefficients, and later rational expressions where appropriate.
Big idea: Slope is a ratio of changes. Proportional linear functions are the special case with intercept zero; rational coefficients express precise relationships.
Teacher investigation
Compare y=(3/2)x and y=(3/2)x+4. Interpret 3/2 as a change of 3 vertical units per 2 horizontal units. Explain the effect of +4.
Work with a partner before reading the interpretation. Draw or build a representation; label the whole and the unit.
Reveal the mathematical interpretation
Slope is a ratio of changes. Proportional linear functions are the special case with intercept zero; rational coefficients express precise relationships.
Same slope, different starting values
On the shifted line, (0,4) and (2,7) give (7 − 4)/(2 − 0) = 3/2. At (2,7), the coordinate ratio 7/2 is not the slope. The triangles measure changes along each line.
The unit becomes a rate of change
Fractions as two unrelated numbers, equivalence as a trick, and procedures before meaning can resurface in algebra. Keep the fraction as one coefficient, keep units in slope, and move between graph, equation, and context rather than remaining in one model.
Study a completed solution
For y = (3/2)x + 4, points (0,4) and (2,7) give slope (7 − 4)/(2 − 0) = 3/2. The intercept is 4. To solve (3/2)x + 4 = 10, subtract 4: (3/2)x = 6, then multiply both sides by 2/3: x = 4. Substitution gives 6 + 4 = 10.
Compare approaches. Solve the same equation by multiplying every term by 2: 3x + 8 = 20, so 3x = 12 and x = 4. Compare clearing denominators with isolating the coefficient: each uses equivalent equations and applies the same operation to both sides.
Analyze an incorrect response
“At (2,7), slope is 7/2.”
Where does the reasoning first break? Why might it seem reasonable? Repair it and justify the repair before opening the interpretation.
Compare your error analysis
That is a ratio of coordinates from the origin, not the ratio of changes on this line. Use two points: Δy/Δx = 3/2. Also critique “3x + 4 = 20” after doubling the original equation: the constant 4 must become 8.
An error is evidence to investigate, not a label for the learner.
Complete the reasoning
For y = (2/3)x + 1, moving from x = 3 to x = 9 changes y from __ to __. The slope is (__ − __)/(9 − 3) = __.
Check the completed reasoning
3 to 7; (7 − 3)/6 = 2/3. The intercept does not change the rate.
Try a new situation independently
A tank starts with 5 liters and fills at 3/4 liter per minute. Write a function, find when it holds 11 liters, and explain whether volume is proportional to elapsed time. Verify using a second method.
Compare after trying it
V(t) = 5 + (3/4)t for t ≥ 0 while filling. V = 11 gives t = 8 minutes. Check: 5 + (3/4) × 8 = 11. A table increasing by 3 liters every 4 minutes agrees. The initial 5 liters mean the relationship is not proportional.
Make the teaching decision
- Listen for
- The learner uses changes between two points for slope and preserves equality when operating on an equation.
- Reduce support when
- The learner translates a new context to a linear equation, interprets both coefficients, and checks a solution by substitution or another method.
- Return to a model when
- If y/x is used for a line with an intercept, draw a slope triangle between two points on that line. If clearing denominators loses a term, mark every term on both sides.
A short sequence to compare
Compare y = (3/2)x and y = (3/2)x + 4 at x = 0 and 2. Their changes match even though their starting values differ.
Check again in 1–2 weeks
A line passes through (2,5) and (6,8). Find its slope and equation, then find x when y = 11. Explain why 5/2 is not its slope.
Teacher interpretation for the delayed check
m = (8-5)/(6-2) = 3/4. Using (2,5), b = 7/2, so y = (3/4)x + 7/2. For y = 11, x = 10. Check: 15/2 + 7/2 = 11. The ratio 5/2 compares coordinates from the origin, which is not on this line.
Record accuracy, explanation, and independent transfer separately. These are instructional cues, not fixed mastery cutoffs. Revisit a model when evidence calls for it.
A misconception to anticipate
Students mistake slope for a single coordinate or confuse y-intercept with rate of change.
Instructional response
Connect a ratio table, slope triangle, and equation using consistent units.
Pause and reflect
What part of y=(3/2)x+4 still tells the same story as equivalent ratios?
What might a student at your grade already know? What representation would make the reasoning visible? What evidence would convince you that understanding is secure?
The next chapter begins
Return to the first loaf: units, equivalence, and multiplicative relationships have been present throughout.
Bring it to your classroom
Grade 8–Algebra I · Three-page student practice: the original two pages plus an optional “Think Like a Mathematician” extension with nonroutine problems and justification. Separate teacher key includes solutions, developmental guidance, and specific cues for reducing or restoring model support.