Interactive Math Activities with Simulations

These interactive math activities use ten free simulations to make abstract ideas visible, from adding fractions to reading a box plot. In math, a simulation is not a lab. It is a picture that changes the moment the numbers change. Students drag a coefficient and watch the parabola move, or drop a thousand balls and watch a probability turn into a shape. That instant link between the symbol and the picture is what many students never get from a worksheet. Each activity below gives the grade level, a driving question, the steps and the result students should find, with every number checked in the simulation. Students open them on Simulic from a link or QR code and don't need an account.

For general routines (projector demos, pair work, homework), read how to use interactive simulations in the classroom. Its math example covers the constant c in a quadratic; the quadratic activity here uses a different simulation and a different question.

Number sense: fractions, ratios and integers

1. Why isn't 1/3 + 1/4 equal to 2/7?

Grades 4–6 · Fraction models – comparing and adding

Driving question: A student writes 1/3 + 1/4 = 2/7. How can you show them why that's wrong without just telling them?

1. Set the two fractions to 1/3 and 1/4. Students estimate the sum first: more or less than 1/2?
2. Watch how both bars are cut into twelfths, and read the step-by-step sum.
3. Go back to the defaults, 2/3 and 3/4. Which is bigger, and by how much?
4. Challenge: find two fractions with different denominators whose sum simplifies.

Expected result: 1/3 + 1/4 = 4/12 + 3/12 = 7/12, a little more than 1/2. The answer 2/7 is less than 1/3, so it can't be right. For the defaults, 8/12 < 9/12, and the sum is 17/12 = 1 5/12. One answer to the challenge: 1/6 + 1/3 = 3/6 = 1/2.

2. Will the two paints be the same color?

Grades 6–7 · Proportion playground – comparing ratios

Driving question: Cup A has 4 drops of blue and 6 of yellow. Cup B has 6 blue and 9 yellow. Same color or not?

1. On the Paint screen, students predict before looking at the cups.
2. Check the result and the simplest ratio shown for each cup.
3. Change cup B to 6 blue and 8 yellow and predict again.
4. On the Apples screen, compare 6 apples for $3 with 4 apples for $2.

Expected result: both cups are 2 : 3, so the colors match. Many students say B is "more yellow" because it has more yellow drops. With 6 : 8 (3 : 4) the colors differ. Both apple deals cost $0.50 per apple, the same unit rate. For a whole-class version where students move two hands to match a ratio, try ratio and proportion.

3. Why is subtracting a negative the same as adding?

Grades 6–7 · Adding and subtracting integers on a number line

Driving question: Why does 5 − (−4) give 9?

1. Start with the default, (−3) + 7, and explain the result using the two arrows.
2. Switch to subtraction and set 5 − (−4). Which way does the red arrow point?
3. Press "Flip sign of b" and compare with 5 + 4.
4. Find every b that makes (−3) + b = 0.

Expected result: (−3) + 7 = 4, because the red arrow of length 7 overshoots zero by 4. Subtracting −4 means adding its opposite, so the arrow points right and lands on 9, exactly where 5 + 4 lands. Only b = 3 gives zero: opposites cancel.

Algebra: equations, lines and quadratics

4. What keeps an equation balanced?

Grades 6–8 · Equation balance – solving linear equations

Driving question: You may change an equation however you like, as long as it stays true. Which moves are allowed?

1. On the Basics screen, students find the hidden masses of the shapes by balancing.
2. On the Variables screen, build 3x + 2 on the left and x + 8 on the right. Change the value of x until the scale balances.
3. On the Operations screen, press New equation and solve it with the operation buttons (+1, −x, ÷2 and so on), which act on both sides at once.
4. Finish with Game level 1 or 2.

Expected result: 3x + 2 = x + 8 balances at x = 3 (11 on each side). Subtract x from both sides (2x + 2 = 8), subtract 2 (2x = 6), divide by 2 (x = 3). Students who add or remove tiles on only one side see the scale tip at once. That instant feedback is what makes "do the same to both sides" stick.

5. What does the slope really measure?

Grades 8–9 · Graphing lines – slope and y-intercept

Driving question: Two points are on a line. How can you tell how steep it is, and where it crosses the y-axis?

1. On the Slope screen, read the rise and run for the default points (−2, −1) and (2, 3).
2. Drag P₂ until the line is flat, then until it is vertical.
3. On the Slope-intercept screen, read the default equation and change only c.
4. Open the systems of two linear equations simulation and solve x + y = 3, x − y = 1 by reading the intersection.

Expected result: rise 4, run 4, slope 1. A flat line has slope 0. A vertical line has an undefined slope: with P₂ dragged to (−2, 3), the equation becomes x = −2. The default slope-intercept line is y = (3/2)x − 1, and changing c slides it up or down without changing its steepness. The two lines in the system meet at (2, 1). Press the Parallel example to show a system with no solution.

6. Where does the vertex go when a changes sign?

Grades 9–10 · Quadratic function – vertex, axis of symmetry, variation table

Driving question: For y = x² − 2x − 3, if you change only a from 1 to −1, does the parabola just flip over the x-axis?

1. Read the default: vertex, roots, Δ and the variation table.
2. Students sketch where they think the new parabola will be when a = −1.
3. Set a = −1 and compare.
4. Find the maximum of y = −x² + 4x using the simulation (a = −1, b = 4, c = 0).

Expected result: the default has Δ = 16, roots at −1 and 3 and its vertex at (1, −4). Most students predict a vertex at (1, 4). In fact the axis moves too, because it is x = −b/2a. The new vertex is (−1, −2), Δ = −8 and there are no x-intercepts. The maximum of −x² + 4x is 4, reached at x = 2.

Trigonometry and geometry

7. How tall is the building?

Grades 9–11 · Trig ratios of an acute angle – measuring height

Driving question: You stand 10 m from a building and look up at 35°. How tall is it?

1. In Right triangle mode, change the angle and watch sin, cos and tan. Then keep the angle and change side AB.
2. Switch to "Measure building height" with the defaults: d = 10 m, α = 35°, eye height 1.5 m. Students calculate first.
3. Discuss the error if you forget the eye height.

Expected result: the ratios depend only on the angle, not on the size of the triangle. 10 × tan 35° ≈ 7.00 m, plus 1.5 m of eye height, gives 8.50 m. Forgetting the eye height underestimates by 1.5 m, about 18%. For older students, continue on the unit circle: set m = 0.5 to see the two solution families of sin x = 0.5 (x = π/6 + 2kπ and 5π/6 + 2kπ), then push m to 1.2 to show that sine has no solution while tangent always does.

8. If the sides triple, does the area triple?

Grades 8–10 · Similar triangles – measuring height with shadows

Driving question: A triangle is enlarged with scale factor k. What happens to its perimeter and its area?

1. With the default k = 1.5, read the side ratios, the perimeter ratio and the area ratio.
2. Students predict the area ratio for k = 2 and k = 3, then check.
3. Switch to "Measure height with shadows". Students use the person's height and the two shadow lengths to calculate the tree's height before reading it.

Expected result: every side ratio and the perimeter ratio equal k, but the area ratio is k²: 2.25 for k = 1.5, 4 for k = 2 and 9 for k = 3. In the shadow mode, the sun's rays make the same angle everywhere, so tree height ÷ person's height = tree shadow ÷ person's shadow.

Probability and statistics

9. Where will the balls land?

Grades 7–10 · Galton board – Plinko probability

Driving question: A ball bounces left or right at each of 12 pegs with equal chance. Which bin is most likely, and how likely is the end bin?

1. On the Intro screen, drop 1 ball, then 10. Students sketch the shape they expect after 100.
2. Drop 100 balls several times, with the theoretical curve shown and the bin labels in percent.
3. On the Lab screen, set p = 0.7 and compare the mean and standard deviation with theory.

Expected result: the middle bin (k = 6) is most likely at about 22.6%. Each end bin has 1 chance in 4,096, about 0.02%, so it almost never fills. The theoretical mean is 6 and the standard deviation about 1.73. With p = 0.7 the hump shifts right to a mean of 8.4 and narrows slightly (σ ≈ 1.59).

10. Which average can one outlier wreck?

Grades 6–10 · Summary statistics – dragging data points

Driving question: One student scores far above the rest. Which measures change, and which hardly move?

1. Read the default data set: 5, 7, 7, 8, 9, 10, 12, 14, 15, 25.
2. Drag the 25 to 30, then to 16. Record the mean, median, standard deviation and IQR each time.
3. Students explain why 25 is marked as an outlier.

Expected result: the default has mean 11.2, median 9.5, mode 7, Q₁ = 7, Q₃ = 14 and standard deviation about 5.51. The fence is 14 + 1.5 × 7 = 24.5, so 25 is an outlier. Dragging it to 30 raises the mean to 11.7 and the standard deviation to about 6.81, while the median stays at 9.5. At 16, the mean drops to 10.3 and the outlier disappears.

Matching activities to common errors

Activity	Common error	Question to attach
1. Fractions	Adding tops and bottoms	Number: 1/3 + 1/4 as a decimal (0.583 ± 0.005)
2. Ratios	Comparing differences instead of ratios	Multiple choice: same color or not?
4. Equations	Changing only one side	Number: x in 3x + 2 = x + 8
6. Quadratics	"Changing a only flips the graph"	Prediction: new vertex when a = −1
7. Trig	Forgetting the eye height	Number: building height (8.50 m ± 0.05)
9. Probability	"All bins are equally likely"	Number: probability of the middle bin (0.226 ± 0.01)

How to run these in class

• Use the projector for the hook. Activities 1, 2 and 6 start with a prediction the whole class can argue about. Open the link in present mode (add ?present=1), take a vote, then reveal. Teaching with a projector explains present mode and the Show QR code button for moving onto devices.
• Lock predictions in. Prediction questions on a link keep the simulation locked until each student commits, and can be asked again afterward. This is the core of Predict–Observe–Explain, and it works in math as well as science.
• Change the numbers for each class. On the Share page, create a link per class and pin different starting values, such as a different data set in activity 10 or different coefficients in activity 6. A number question with a tolerance then checks each class's own answer. See using simulations for formative assessment for using the results table.

Keep each activity to 10–20 minutes and finish on paper: one written step, one sketch or one sentence that connects the picture to the algebra.

FAQ

Are these math simulations only for demonstrations?

No. Most of them work best in pairs on phones or laptops, with a target to reach. Use the projector for the prediction, then hand over the QR code.

Which activity suits elementary grades?

Activity 1 (fraction models) works from about grade 4. Activities 2 and 3 suit grades 6–7.

Do the simulations show the working, or just the answer?

Many show both. The fraction, quadratic, systems and statistics simulations print the steps or formulas next to the picture. Ask students to write the step before they read it.

Can students use these at home?

Yes. A plain play link opens full screen on any device. Attach a question set to the homework link and check the results table before the next lesson.