Lesson 26 · Rate of change

Constant Rate in Tables

Equal steps in x should cause equal steps in y.

A riding log may show days in one column and distance in another. A constant rate means that whenever the input changes by the same amount, the output also changes by the same amount.

$x: 0\to2$
$\Delta x=2$
→
$y: 5\to11$
$\Delta y=6$
→
$\frac{\Delta y}{\Delta x}=\frac62=3$

Important: compare changes, not the raw table values. If the $x$-steps are different sizes, divide each $y$-change by its matching $x$-change.

Worked example
hours $x$024
distance $y$ (km)1713

Each $\Delta x=2$ hours and each $\Delta y=6$ km, so the rate is $6\div2=3$ km/hour. The starting value is $1$ km, so $y=3x+1$.

Read the changes
  1. Problem 1 · Constant increase
    $x$0246
    $y$5111723
    1. Calculate each $\Delta x$.
    2. Calculate each $\Delta y$.
    3. Calculate the rate $\frac{\Delta y}{\Delta x}$.
    4. Choose CONSTANT or NOT CONSTANT.
    x-changes → y-changes → divide → decide →
  2. Problem 2 · Battery decrease
    hours $x$0123
    battery $y$ (%)90827466
    1. Choose INCREASES or DECREASES.
    2. Calculate the rate with units.
    3. Interpret the rate using “percentage points per hour”.
    direction → changes → rate → sentence →
  3. Problem 3 · Spot a non-constant pattern
    $x$0123
    $y$25914
    1. List the three changes in $y$.
    2. Choose CONSTANT or NOT CONSTANT.
    3. Explain your choice in one sentence.
    differences → compare → explain →
  4. Problem 4 · Fill a missing table value
    $x$1357
    $y$8?2026
    1. Calculate the constant rate using two complete columns.
    2. Calculate the missing value.
    3. Check the rate on both sides of the missing value.
    known pair → rate → missing value → check →
  5. Problem 5 · Table to equation
    $x$024
    $y$405468
    1. Calculate the constant rate $m$.
    2. State the starting value $b$.
    3. Write the equation $y=mx+b$.
    rate → x=0 value → equation →
Contexts and independent checks
  1. Problem 6 · Feed cost
    days $x$0369
    cost $y$ (€)0122436
    1. Calculate the rate with units.
    2. Interpret the rate in a sentence.
    3. Write an equation for cost after $x$ days.
    changes → units → meaning → equation →
  2. Problem 7 · Compare two tables

    Plan A: $(0,10),(2,18)$. Plan B: $(0,4),(3,19)$.

    1. Calculate Plan A's rate.
    2. Calculate Plan B's rate.
    3. Choose the plan with the greater rate and state the difference.
    rate A → rate B → compare →
  3. Problem 8 · Uneven x-steps
    $x$1410
    $y$71943
    1. Calculate the rate from the first pair of columns.
    2. Calculate the rate from the second pair.
    3. Choose CONSTANT or NOT CONSTANT.
    divide matching changes → compare rates →
  4. Problem 9 · Missing-coordinate review

    A line with slope $3$ passes through $(2,5)$ and $(8,y)$.

    1. Calculate $\Delta x$.
    2. Calculate $\Delta y$.
    3. Calculate $y$.
    Δx → mΔx → ending y →
  5. Problem 10 · Inequality retention

    A water container starts with $50$ L and loses $6$ L per hour. Let $h=$ hours.

    1. Write an inequality for when at most $14$ L remain.
    2. Solve and interpret the inequality.
    model → negative coefficient → reverse sign →
  6. Problem 11 · Decreasing arithmetic retention

    A training plan uses $72,65,58,51,\ldots$ jumps. The values decrease.

    1. Calculate the signed common difference using next term minus previous term.
    2. Write an explicit formula in the form $a_n=a_1+(n-1)d$, keeping negative $d$ in parentheses.
    3. Calculate $a_{10}$.
    next − previous → signed formula → substitute →
Show answers
Problem 1a
$\Delta x=2,2,2$.
Problem 1b
$\Delta y=6,6,6$.
Problem 1c
Rate $=6\div2=3$.
Problem 1d
CONSTANT.
Problem 2a
DECREASES.
Problem 2b
Rate $=-8$ percentage points/hour.
Problem 2c
Battery level decreases by $8$ percentage points per hour.
Problem 3a
The changes are $+3,+4,+5$.
Problem 3b
NOT CONSTANT.
Problem 3c
Equal $x$-steps produce different $y$-changes.
Problem 4a
Using $(1,8)$ and $(5,20)$: $m=12\div4=3$.
Problem 4b
From $x=1$ to $x=3$, $y$ rises by $3\times2=6$, so the missing value is $14$.
Problem 4c
$(14-8)\div(3-1)=3$ and $(20-14)\div(5-3)=3$.
Problem 5a
$m=(54-40)\div(2-0)=7$.
Problem 5b
$b=40$.
Problem 5c
$y=7x+40$.
Problem 6a
$m=12\div3=€4$/day.
Problem 6b
Feed cost increases by €4 per day.
Problem 6c
$y=4x$.
Problem 7a
Plan A: $(18-10)\div2=4$.
Problem 7b
Plan B: $(19-4)\div3=5$.
Problem 7c
Plan B is greater by $1$ y-unit per x-unit.
Problem 8a
$(19-7)\div(4-1)=12\div3=4$.
Problem 8b
$(43-19)\div(10-4)=24\div6=4$.
Problem 8c
CONSTANT.
Problem 9a
$\Delta x=8-2=6$.
Problem 9b
$\Delta y=3\times6=18$.
Problem 9c
$y=5+18=23$.
Problem 10a
$50-6h\leq14$.
Problem 10b
$$50-6h\leq14\quad\mid-50$$$$-6h\leq-36\quad\mid\div(-6)$$$$h\geq6$$At most $14$ L remain after at least $6$ hours.
Problem 11a
$d=65-72=-7$.
Problem 11b
$a_n=72+(n-1)(-7)$.
Problem 11c
$a_{10}=72+9(-7)=9$.
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