Lesson 19c · Explicit Remediation

Read the Axes, Tell the Story

First name what the graph measures. Then choose the direction. Only then calculate and explain.

Step 1: read the two axis labels

Every point $(x,y)$ contains two facts. Read the horizontal fact first and the vertical fact second.

Horizontal x-axis
time in hoursThe input: when are we measuring?
Vertical y-axis
tank fullness in %The output: what is changing?
Point $(2,80)$ means: after $2$ hours, the tank is $80\%$ full. The $80$ comes from the $y$-axis, so it is a percentage reading.

Step 2: choose increase or decrease before calculating

Compare the ending $y$-value with the starting $y$-value. Say the direction out loud.

ending $y$ is larger
→ INCREASES → positive slope
ending $y$ is smaller
→ DECREASES → negative slope

Language check: “increases” means goes up. “Decreases” means goes down. A negative result describes a decrease.

Step 3: calculate, attach units, and finish the sentence

If a graph's $y$-axis is labelled in percent, subtracting two readings gives a change in percentage points.

Worked example · Water tank

The tank changes from $80\%$ full at hour $2$ to $56\%$ full at hour $4$.

  1. Axes: $x=$ time in hours; $y=$ tank fullness in percent.
  2. Direction: $56$ is smaller than $80$, so fullness decreases.
  3. Changes: $\Delta y=56-80=-24$ percentage points; $\Delta x=4-2=2$ hours.
  4. Slope: $m=\frac{-24}{2}=-12$.
  5. Meaning: The tank fullness decreases by 12 percentage points per hour.
Use this sentence frame:
The __________ increases / decreases by __________ y-units for every $1$ x-unit.
Practice · follow the named step in each part
  1. Problem 1 · Read only, no calculation
    Graph labels: $x=$ days; $y=$ pages drawn in a sketchbook. One point is $(3,18)$.
    1. Name what the $x$-axis measures, including its unit.
    2. Name what the $y$-axis measures, including its unit.
    3. Write one complete sentence explaining the point $(3,18)$.
    Sentence frame for (c): After ___ days, ___ pages have been drawn.
  2. Problem 2 · Choose the direction first
    Water trough: $x=$ time in hours; $y=$ water volume in litres. It changes from $90$ L at hour $0$ to $66$ L at hour $3$.
    1. Circle one word: the water volume INCREASES / DECREASES.
    2. Choose one sign: the slope is POSITIVE / NEGATIVE.
    3. Calculate the slope with units.
    4. Complete the explanation sentence.
    The water volume __________ by __________ litres per hour.
  3. Problem 3 · Percentage points
    Storage meter: $x=$ days; $y=$ storage used in percent. It changes from $35\%$ on day $1$ to $55\%$ on day $5$.
    1. State what the $y$-axis measures.
    2. Circle one word: storage used INCREASES / DECREASES.
    3. Calculate the slope. Use the unit “percentage points per day”.
    4. Complete the explanation sentence.
    The storage used __________ by __________ percentage points per day.
  4. Problem 4 · Different axis units
    Drawing challenge: $x=$ days; $y=$ completed drawings. The graph passes through $(2,7)$ and $(5,19)$.
    1. Calculate $\Delta y$ and $\Delta x$ separately.
    2. Calculate the slope and write its units.
    3. Answer yes or no: should you turn this slope into a percentage? Then give the reason.
    Reason frame for (c): No, because the axes measure different things: ___ and ___.
  5. Problem 5 · Physical percentage gradient
    Loading ramp: rise $=3$ m; horizontal run $=12$ m. Both measurements are lengths.
    1. Calculate the slope $m=\frac{\text{rise}}{\text{run}}$.
    2. Convert the slope into a percentage gradient.
    3. Complete the interpretation sentence.
    The ramp rises ___ m for every $100$ m horizontally, so its gradient is ___%.
  6. Problem 6 · Put all four steps together
    Outdoor temperature: $x=$ hours after noon; $y=$ temperature in °C. The line passes through $(2,14)$ and $(6,6)$.
    1. Name both axes and their units.
    2. Circle one word: temperature INCREASES / DECREASES.
    3. Calculate the slope with units.
    4. Write a complete interpretation sentence without using a frame.
Show answers
Problem 1a
The $x$-axis measures time in days.
Problem 1b
The $y$-axis measures the number of pages drawn.
Problem 1c
After $3$ days, $18$ pages have been drawn.
Problem 2a
DECREASES, because $66$ L is less than $90$ L.
Problem 2b
NEGATIVE, because a decrease gives a negative change in $y$.
Problem 2c
$m=\frac{66-90}{3-0}=\frac{-24}{3}=-8$ L/hour.
Problem 2d
The water volume decreases by $8$ litres per hour.
Problem 3a
The $y$-axis measures storage used in percent.
Problem 3b
INCREASES, because $55\%$ is greater than $35\%$.
Problem 3c
$m=\frac{55-35}{5-1}=\frac{20}{4}=5$ percentage points per day.
Problem 3d
The storage used increases by $5$ percentage points per day.
Problem 4a
$\Delta y=19-7=12$ drawings; $\Delta x=5-2=3$ days.
Problem 4b
$m=\frac{12}{3}=4$ drawings per day.
Problem 4c
No. The axes measure different things: drawings and days. The rate stays $4$ drawings/day.
Problem 5a
$m=\frac{3}{12}=0.25$.
Problem 5b
Percentage gradient $=0.25\times100\%=25\%$.
Problem 5c
The ramp rises $25$ m for every $100$ m horizontally, so its gradient is $25\%$.
Problem 6a
The $x$-axis measures time in hours after noon. The $y$-axis measures temperature in °C.
Problem 6b
DECREASES, because $6$ °C is less than $14$ °C.
Problem 6c
$m=\frac{6-14}{6-2}=\frac{-8}{4}=-2$ °C/hour.
Problem 6d
The temperature decreases by $2$ °C per hour.
Tutor notes

Require the spoken axis sentence before any subtraction: “$x$ measures ___; $y$ measures ___.” If the direction word and sign disagree, stop there and compare the two $y$-values rather than continuing the calculation.

For explanations, accept the sentence frame first. The goal is accurate mathematical language, not elegant prose. Release the frame only in Problem 6d.

After 19cContinue to Lesson 20 only when she can name both axes, choose increase/decrease, attach rate units, and explain one result in a complete sentence.
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