Riemann Sum Slider: A 40-Minute Classroom Lesson

Objective and materials

By the end of the period, students can state and justify which of the left, right, and midpoint sums overestimates or underestimates a definite integral on an increasing function, and can explain, without being told, why more rectangles closes the gap faster for the midpoint sum than for the left or right sum.

• A shared screen or projector open to /interactives/riemann-sum-slider, with the $x^2$ preset selected (it loads first, on the interval from 0 to 2)
• One notecard or half sheet per student for written predictions
• A whiteboard, or one shared document, where the class prediction gets recorded before every reveal
• Optional: one device per pair so pairs can drive the slider themselves in an optional extension after the exit ticket

Nothing needs to be built ahead of time. The slider already opens on $f(x) = x^2$ with the bounds set to 0 and 2, $n$ set to 8, and the left method selected, so the only setup is loading the page and leaving the slider untouched until Move 1, when setting $n$ to 4 becomes the reveal.

The hook, 4 minutes

Draw a curve on the board that clearly bends upward, no axes needed, and ask the class for the area underneath it between two points. Let a few students offer methods. Someone will suggest rectangles. Ask that student to come up and sketch four of them under the curve using only a straightedge.

Whatever they draw, ask the room a single question: if two different people each drew four honest rectangles under this same curve, could they get two different areas? Take a show of hands and move straight into Move 1 without resolving it. The slider resolves it in the next six minutes.

Guided exploration: four teacher moves, 26 minutes

Each move follows the same shape: state exactly what is about to change, collect a written prediction from every student before the change happens, then click and compare. Do not click ahead of the prediction. The prediction is the lesson; the click is only the check.

1. Move 1, left sum baseline, 0:04 to 0:10. The slider is still sitting at its default of $n = 8$ from setup; nothing has changed yet. Before revealing the rectangles, have every student write down a single number: their guess for the left sum, closer to 0, to 3, or to 6. Then click nothing yet, just point at the readout showing the exact integral equals $2.6667$, and ask whether their guess for the left sum should land above or below that number, and why, using the shape of the curve rather than arithmetic. Now reveal: set $n$ to 4 with the left method already selected. The left sum readout shows $1.7500$. Ask which students guessed low, and whether the four rectangles they can now see explain why.
2. Move 2, right sum and the over and under rule, 0:10 to 0:16. Before clicking the Right button, ask the class to predict whether the right sum will be bigger or smaller than the left sum's $1.7500$, and to predict a specific number. Click Right: the readout shows $3.7500$, and the caption above the error meter now reads that the right sum is an overestimate here. State the general rule out loud and have students write it in their own words: on a curve that is increasing across the whole interval, a left edge always sits at the lowest point of its own strip and a right edge always sits at the highest point of its own strip, so left always undershoots and right always overshoots, no matter how many rectangles you use.
3. Move 3, increasing $n$, 0:16 to 0:24. With the right method still selected, ask students to predict what the printed error number will do, not the error meter bar, when $n$ moves from 4 to 8: will it drop to about half its value, to a quarter, or barely change? Drag $n$ to 8 and read the new error out loud: about $0.5208$, compared with $1.0833$ at $n = 4$, which is close to half. Repeat the prediction for the midpoint method: switch to Midpoint at $n = 4$ first, note the error is $0.0417$, then ask whether doubling $n$ to 8 will again cut that number in half or by something more. Drag to $n = 8$: the midpoint error drops to about $0.0104$, one quarter of what it was. Name the pattern: left and right sums roughly halve their error each time $n$ doubles, while the midpoint sum roughly quarters its error, so midpoint gets accurate far faster. Teacher note: the error meter bar itself is log-scaled, so it will barely move on screen through this whole move even though the printed number is doing exactly this halving and quartering; tell students to watch the number, not the bar.
4. Move 4, midpoint versus trapezoid, 0:24 to 0:30. Drag $n$ back to 4. Ask students to predict, before clicking, whether the midpoint sum at $n = 4$ is the exact average of the left sum and the right sum they already recorded, which is $\frac{1.7500 + 3.7500}{2} = 2.7500$. Most will predict yes. Click Trapezoid at $n = 4$: the readout shows $2.7500$, matching that average exactly. Then click back to Midpoint: the readout shows $2.6250$, a different number. Ask the class to reconcile the two readouts, and land on the correction: the trapezoid sum is built from the two edge heights on every strip, so it is always exactly the average of the left and right sums; the midpoint sum instead samples the height at the center of each strip, which is a different number whenever the curve is not a straight line.

That table is the answer key for Moves 1 through 4 for $f(x) = x^2$ on the interval from 0 to 2, where the exact integral is $\frac{8}{3} \approx 2.6667$. Every value in it is exactly what the slider displays, so it doubles as a projector-free reference if a laptop fails partway through.

Check for understanding, 4 minutes

Move away from the screen for this part; it tests whether the rule transfers, not whether students can read a readout. Pose the question without the slider: $f(x) = x^3$ is increasing across the interval from 0 to 1. Without computing anything, will a left sum with $n = 10$ land above or below the exact integral, and will a right sum with the same $n$ land above or below it?

Give students thirty seconds of silent thinking, thirty seconds to compare with a neighbor, then take a thumbs up or down on both parts at once. Cold-call one thumbs-down student to hear the reasoning of a neighbor who answered correctly before moving on.

Exit ticket, 6 minutes

1. On an increasing function, which of the left, right, and midpoint sums is guaranteed to underestimate the exact integral, and which one has no guarantee either way without knowing whether the curve bends up or down?
2. If you double the number of rectangles, which shrinks faster: the error in a left sum or the error in a midpoint sum?
3. True or false, and correct it if false: the midpoint sum is always the average of the left sum and the right sum.
4. For f(x) = x^2 on the interval from 0 to 2 with n equal to 4, the left sum is 1.7500 and the right sum is 3.7500. Without using the slider, state the trapezoid sum.

No-tech variant with graph paper

Run the identical four moves with nothing but paper when a screen is not available. Print or draw one set of axes per student showing $y = x^2$ from $x = 0$ to $x = 2$, with gridlines every 0.25 units on both axes, and mark the curve at $x = 0, 0.25, 0.5$, up through 2 so students can connect the dots with a smooth curve rather than draw the parabola freehand.

1. Move 1, left sum by hand. Students split the interval from 0 to 2 into four strips of width $0.5$ using their straightedge, then draw a rectangle over each strip whose height matches the curve at that strip's left edge. Before they draw, have them predict and write down whether the total area of their four rectangles will land above or below the true area under the curve. Then they compute the area of each rectangle by hand, using height times $0.5$, and add the four, arriving at $1.75$.
2. Move 2, right sum by hand. Repeat with rectangles drawn to the height of each strip's right edge instead, predicting first, then computing to get $3.75$. Have students shade the left-sum rectangles in one color and the right-sum rectangles in a second color on the same graph, so the true area's location between the two totals is visible as the sliver each shading fails to cover or overcovers.
3. Move 3, increasing $n$ by hand. Give the second half of the class eight narrower strips of width $0.25$ instead of four, using the same axes, and have them repeat the left-sum computation only. Compare the class's two left-sum totals, $1.75$ at four strips against $2.1875$ at eight strips, against the true value of about $2.6667$, and confirm on the board that doubling the strip count roughly halved the gap.
4. Move 4, midpoint by hand. Students find the middle $x$-value of each of their four original strips, $0.25$, $0.75$, $1.25$, and $1.75$, read the curve's height at each from their graph, and compute the midpoint sum by hand to get $2.625$. Ask them to also average their left and right totals, $\frac{1.75 + 3.75}{2}$, and to notice by hand what Move 4 shows on screen: that average is $2.75$, not $2.625$.

The check for understanding and the exit ticket run exactly as written above; neither one needs a screen. Collect the graph paper itself as the artifact of the lesson: a left-sum shading, a right-sum shading, and the four numbers a student computed by hand are a complete record of whether the objective was met.

Common misconceptions

• Assuming the right sum is always the bigger one, full stop, rather than tying the direction to whether the function is increasing. On a decreasing function the rule flips: the left edge is the highest point on its strip and the right edge is the lowest, so left overestimates and right underestimates. None of the slider's five presets is purely decreasing across the interval it uses, so this case will not show up on screen; put a simple decreasing line such as $f(x) = 4 - x$ on the board and have students check both sums by hand to see the flip.
• Treating the midpoint sum as the average of the left and right sums. It is the trapezoid sum that equals that average exactly, on every function, every time. The midpoint sum uses a genuinely different sample point, and only matches the trapezoid sum when the function is a straight line; on a line both midpoint and trapezoid land exactly on the true integral, while the left and right sums still disagree with each other unless the line is flat.
• Believing more rectangles eventually makes the sum exact rather than merely closer. For the left or right sum, the slider's error meter still reads a small but clearly nonzero number at $n = 200$ for a curved function. For the midpoint or trapezoid sum on that same preset, the error has already dropped below the display's floor by $n = 200$, so the readout rounds to $0.0000$ instead; check this claim with a left or right sum if you want it visibly confirmed on screen, rather than with midpoint or trapezoid. Either way, the sum converges toward the exact integral as $n$ grows without bound, it does not reach it at any finite $n$.
• Expecting the error to shrink at the same rate for every method. Left and right sums are first order: doubling $n$ roughly halves the error. Midpoint and trapezoid are second order: doubling $n$ roughly quarters the error. A student who has only ever used one method will not expect this difference until they watch both error readouts side by side.
• Reading a rectangle's height off the wrong endpoint when sketching by hand. In the no-tech variant, the single most common drawing error is a student who draws a left-sum rectangle but reads the curve's height from the strip's right edge out of habit. Have students label which edge they are using before they draw, not after.