Teaching objectives
From counting factors to graphing curves. The lab, with 23 rooms, starts with the properties of powers, continues with the exponential function and ends with the logarithm as its inverse question. It is split into three blocks: powers (rooms 1-11), the exponential function (12-16) and the logarithm (17-23).
The properties are not presented as rules: they are discovered by writing the power as a literal product of factors. In a quotient, pairs of equal factors are crossed out, and in a power of a power, groups are counted. There are several kinds of room: challenges where the student types a number, multiple-choice questions, speed tests with a queue of questions, sliders that move the base or the exponent on a graph, and tutorials in which the student works out the points of a curve and marks them on the graph. On the graphs the exponential is orange and the logarithm teal; asymptotes are dashed red and the line y = x is grey.
What you'll learn
The lab takes the student from multiplying factors to understanding the exponential and the logarithm as inverse functions, with the graph as support.
- Feel exponential growth through the chessboard and rice legend: 263 ≈ 9.22 × 1018 (room 1).
- Deduce the three properties (product, quotient and power of a power) by counting factors and groups, and chain them to leave a single power (rooms 2 and 3).
- Know when two powers can be combined (common base or common exponent) and that sums and differences are never simplified with those rules (rooms 4 and 5).
- Understand why a0 = 1 and what a negative exponent means (room 6).
- Go from fractional exponents to roots and compute them, with and without a negative sign (rooms 7-10).
- Break composite bases into prime factors so they can be compared (room 11).
- Describe the graph of y = ax according to the base: it passes through (0, 1), is always positive and increases if a > 1 or decreases if 0 < a < 1 (rooms 12 and 13).
- See that the exponential eventually overtakes any power of x and relate it to the cost of a brute-force computation (room 14).
- Sketch an exponential by hand (asymptote, fixed point and two or three points) and then do it without a list of steps (rooms 15 and 16).
- Understand the logarithm as the missing exponent, and as the inverse of the exponential, with graphs symmetric about y = x (rooms 17-19).
- Know the properties of the logarithm and their limit: the logarithm of a product is the sum of the logarithms, but that of a sum does not split (room 20).
- Sketch a logarithm and recognise the four types of graph (exponential and logarithm, increasing and decreasing) (rooms 21-23).
Key mathematical ideas
- Product, quotient and power of a power: am · an = am+n (the factors are joined); am / an = am−n (n pairs are crossed out); (am)n = am·n (n groups of m factors).
- Common base or common exponent: with the same base, exponents are added or subtracted; with the same exponent, the bases are multiplied or divided, as in 34 · 54 = (3 · 5)4. If nothing is shared, they cannot be combined, and a sum or difference such as 34 + 32 is not simplified with these rules.
- Zero and negative exponents: going down one step in the ladder 23, 22, 21… divides by 2. So 20 = 1, 2−1 = 1/2 and a−n = 1/an. The statement a0 = 1 is not a convention: it is an / an.
- Fractional exponent: a1/2 = √a and am/n is the n-th root of am. For example, 84/3 = (∛8)4 = 16 and 43/2 = 8; with a negative sign the result is inverted: 9−1/2 = 1/3.
- Factorising bases: 610 / (45 · 273) = (210 · 310) / (210 · 39) = 3.
- The exponential y = ax: always positive, passes through (0, 1) and has the asymptote y = 0. It is increasing for a > 1 and decreasing for 0 < a < 1. To graph it: asymptote, fixed point (0, 1), f(1) = a, f(−1) = 1/a and one more point.
- The exponential wins: x10 < 2x holds for good from x ≈ 58.77 on (the first integer is 59). With n = 50, n2 is 2500 operations and 250 is more than a thousand trillion.
- The logarithm: loga b = c means ac = b. It exists only for b > 0, has the asymptote x = 0 and passes through (1, 0). To graph it: asymptote, (1, 0), (a, 1), (1/a, −1) and one more point. It is the inverse of the exponential: 2log2 x = x and log2 2x = x.
- Properties of the logarithm: log(x · y) = log x + log y, log(x / y) = log x − log y, log(xn) = n · log x. By contrast, log(x + y) is not log x + log y: log 2 ≈ 0.301 and log 1 + log 1 = 0.
Level: secondary school (ESO), curricular. The lab is long and can be split by blocks: powers (1-11), exponential (12-16) and logarithm (17-23). In class, rooms 1 (the prediction is made before counting), 14 (the challenge of finding from which x the exponential wins) and 18 (the two inverse machines and the mirror y = x) work well.
Room-by-room contents
Room 1 · The rice board
The legend of the inventor of chess, who asked for one grain of rice for the first square, 2 for the second, 4 for the third, and so on, doubling up to square 64. Before seeing anything, the student predicts how much rice the last square was worth by choosing among four options, from a handful to more than all the rice grown in history. Then they go through the board square by square with a “Next square” button, the number of grains doubling and a pile of dots filling up. Square n holds 2^(n−1) grains, and the last one, 2^63 ≈ 9.22 × 10^18. To finish, they write the grains of two squares as a power.
Student tasks
- Predict how much rice square 64 was worth.
- Go through the board doubling the grains square by square up to the 64th.
- Write the exponent of the grains on square 10 (9).
- Write the exponent of the grains on square 20 (19).
Room 2 · The properties, crossing out factors
The properties of powers are discovered by writing each power as a product of factors. In the product of powers with the same base, all the factors are joined. In the quotient 3^5 / 3^3, the student crosses out pairs of equal factors and counts what is left; then they repeat it with 5^7 / 5^4. In a power of a power, (2^2)^3 is counted as 3 groups of 2 factors, with a button to count each group; then they repeat it with (4^2)^4. Each example uses a different base, and moving on to the next one hides the previous one. It ends with three questions asking for the general rule: a^m · a^n, a^m / a^n and (a^m)^n.
Student tasks
- Cross out the pairs of factors in 3^5 / 3^3 and in 5^7 / 5^4.
- Count the groups of factors in (2^2)^3 and in (4^2)^4.
- Choose what a^m · a^n is equal to (a^(m+n)).
- Choose what a^m / a^n is equal to (a^(m−n)).
- Choose what (a^m)^n is equal to (a^m·n).
Room 3 · Challenge: simplify in a chain
Three expressions that mix product, quotient and power of a power. The student applies the properties step by step until a single power of a is left and writes only the final exponent. If they get stuck, the hint says where to start: combine the product in the numerator first, or resolve the power of a power first.
Student tasks
- Simplify (a^3 · a^2) / a^4 and write the exponent (1).
- Simplify (a^2)^3 / a^2 and write the exponent (4).
- Simplify (a^5 · a) / (a^2)^2 and write the exponent (2).
Room 4 · Different bases and exponents
A box with the rule: to combine two powers in a product or quotient, something has to be shared, the base or the exponent. Four numerical examples illustrate it (common base, common exponent, nothing in common, addition or subtraction) and a note says that sums and differences of powers are never combined with these rules. Then six multiple-choice questions in which the student decides what can be done: with the same base the exponents are added, with the same exponent the bases are multiplied, and when nothing is shared or it is a sum or difference, it cannot be combined.
Student tasks
- Calculate 3^4 · 3^2 (3^6, same base).
- Calculate 3^4 · 5^4 ((3 · 5)^4, same exponent).
- Decide what happens with 3^4 · 5^2 (it cannot be combined).
- Decide what happens with 3^4 + 3^2 (a sum is not simplified with these rules).
- Calculate 10^5 / 2^5 ((10 / 2)^5, same exponent).
- Decide what happens with 6^7 − 6^3 (a difference is not simplified this way).
Room 5 · Speed test: simplify and calculate
A timed test of 12 four-option questions on product, quotient, power of a power and power of a product or quotient. Some cannot be simplified (different bases or a sum of powers) and the student has to recognise it. Each question has a time limit: a miss sends the question to the back of the queue and raises the time available, a hit lowers it. If the score at the end is negative, the test is repeated.
Student tasks
- Answer 12 timed questions, such as a^4 · a^3, (a^2)^4 or (a · b)^3.
- Recognise the ones that cannot be combined, such as a^3 · b^2 or a^5 + a^3.
- Repeat the missed questions, which go to the back of the queue.
- Pass the test with a non-negative score.
Room 6 · The ladder: zero and negative exponents
A ladder with 2^3 = 8, 2^2 = 4 and 2^1 = 2, and three gaps below: 2^0, 2^−1 and 2^−2. Each step down divides by 2, and the student fills in the gaps with whole numbers or fractions (“1/2”). Then two multiple-choice questions justify the pattern: a^0 = 1 because a^n / a^n is a^0 and also 1, and a^−n is 1/a^n.
Student tasks
- Calculate 2^0 following the pattern (1).
- Calculate 2^−1 (1/2).
- Calculate 2^−2 (1/4).
- Choose why a^0 = 1 (because a^n / a^n gives a^0 and gives 1).
- Choose how to write a^−n as a fraction (1/a^n).
Room 7 · Fractional exponents
It starts from a^0.5 = √a with three examples: 4^0.5 = 2, 9^0.5 = 3 and 16^0.5 = 4. A graph of 4^x with a slider for the exponent, from −2 to 3, and a readout of 4^x; when the slider lands on a recognisable exponent (1/2, −1/2, 1/3, 2/3, 3/2, 5/2…), its root form appears. Then two calculations with the same pattern, 4^1.5 = 4^1 · 4^0.5 and 4^3.5 = 4^3 · 4^0.5; the first can be checked by moving the slider to 1.5 and the second is outside its range.
Student tasks
- Move the slider of 4^x and read what appears at the fractional exponents.
- Calculate 4^1.5 (8).
- Calculate 4^3.5 (128).
Room 8 · Practice: fractional powers
The rule is not given at the start. First an example, 4^(3/2) = √(4^3) = 8, and a guided reflection: if 8^(4/3) = x, cubing gives x^3 = 8^4 = 4096, and the cube root of 4096 is 16. Then a multiple-choice question about what a^m/n is (the n-th root of a^m). Only after answering it do the rule box and the series of challenges appear: calculating fractional and negative powers and writing others as a root or a fraction.
Student tasks
- Deduce the value of 8^(4/3) by solving x^3 = 8^4 (16).
- Choose what a^m/n is (the n-th root of a^m).
- Calculate fractional and negative powers from the practice series.
- Write a^(2/5) as a root.
- Write 5^−3 as a fraction (1/125).
- Write a^(−3/4) as a root and a fraction.
Room 9 · Speed test: fractional powers
A timed test of 10 questions like those in the previous room, with the same queue and time mechanism as room 5: fractional and negative powers with four options, such as 16^(1/2), 27^(2/3), 8^(4/3) or 9^(−1/2), and exercises in writing a power as a root or a fraction. A miss sends the question to the back of the queue and raises the time available.
Student tasks
- Answer 10 timed questions on fractional and negative powers.
- Calculate values such as 27^(2/3) (9), 8^(4/3) (16) or 9^(−1/2) (1/3).
- Repeat the missed questions.
- Pass the test with a non-negative score.
Room 10 · Factorise and simplify: composite bases
The powers 6^10, 4^5 and 27^3 do not share a base at first sight. A modelled example breaks them into primes: 6^10 = 2^10 · 3^10, 4^5 = 2^10 and 27^3 = 3^9, and the fraction 6^10 / (4^5 · 27^3) becomes (2^10 · 3^10) / (2^10 · 3^9). The student crosses out the factors of 2 and the factors of 3 in two boxes, and the result is 3. Then two new expressions that they must factorise and simplify themselves.
Student tasks
- Cross out the factors of 2 and of 3 in the modelled example.
- Factorise and simplify 9^4 / 3^6 (9).
- Factorise and simplify 8^5 / (2^3 · 4^6) (1).
Room 11 · The exponential function
The graph of y = a^x with a slider for the base a, from 0.2 to 4, the asymptote y = 0, the point (0, 1) and a table of values for x from −2 to 2. The student moves the base and watches how the curve changes: with a > 1 it increases and with 0 < a < 1 it decreases. Three multiple-choice questions fix what was observed.
Student tasks
- Choose which point all the curves pass through, however a is moved ((0, 1)).
- Decide whether a^x can be 0 or negative (no, it is always positive).
- Say how the function behaves if 0 < a < 1 (it decreases).
Room 12 · Challenge: get the exponential
The same graph, with a slider for the base from 0.1 to 5 and a red dot on the graph marking the target. The student adjusts a and presses Check; it is accepted if the curve passes through the point within a small margin. There are three targets in a row: an exponential through (2, 9), a decreasing one through (1, 0.5) and one that satisfies f(3) = 8. If interrupted, the challenge resumes where it was left.
Student tasks
- Get an exponential that passes through (2, 9) (a = 3).
- Get a decreasing exponential that passes through (1, 0.5) (a = 1/2).
- Get an exponential with f(3) = 8 (a = 2).
Room 13 · The exponential always wins
The graphs of y = x^2, y = x^3 and y = 2^x on the same axes, with a zoom slider from ±3 to ±20 and a readout of values (at x = 10: 100, 1000 and 1024). Zooming out shows the exponential overtaking the other two. A text compares an algorithm costing 2^n with one costing n^2: with n = 50, 2500 operations against more than a thousand trillion. The final challenge: find a value of x from which x^10 < 2^x holds forever. The smaller the better; on success the real minimum is shown, x ≈ 58.77, and the first integer, 59.
Student tasks
- Zoom out and see where 2^x overtakes x^2 and x^3.
- Write a value of x from which x^10 < 2^x holds forever.
- If the value does not satisfy the inequality or only does for a while, try a larger one.
Room 14 · Tutorial: graphing an exponential
An algorithm in five steps: horizontal asymptote, fixed point (0, 1), calculation of (1, a) and (−1, 1/a), an extra point and joining up the curve. It is first shown modelled with y = 3^x. Then the student does it with y = (1/2)^x: for each point they first calculate the value and then tap it on the graph; the curve is drawn at the end.
Student tasks
- Calculate f(0) and mark (0, 1).
- Calculate f(1) = (1/2)^1 and mark (1, 0.5).
- Calculate f(−1) = (1/2)^−1 and mark (−1, 2).
- Calculate f(2) = (1/2)^2 and mark (2, 0.25).
Room 15 · Practice: graph it yourself
With no list of steps, the student decides what to calculate to draw two exponentials, first the increasing y = 4^x and then the decreasing y = (1/3)^x. For each one they calculate the fixed point, f(1) and f(−1) and mark them on the graph; the curve appears once the three points are done.
Student tasks
- Calculate and mark the points of y = 4^x: (0, 1), (1, 4) and (−1, 0.25).
- Calculate and mark the points of y = (1/3)^x: (0, 1), (1, 1/3) and (−1, 3).
Room 16 · The inverse question
The logarithm is introduced as the missing exponent. The student finds the exponent in five equations: 2^? = 8, 3^? = 9, 9^? = 3, 2^? = 1/4 and 5^? = 1, and writes it as log_2 8, log_3 9… A fractional exponent appears (a square root is raising to 1/2), a negative one and a zero.
Student tasks
- Calculate log_2 8 (3).
- Calculate log_3 9 (2).
- Calculate log_9 3 (1/2).
- Calculate log_2 (1/4) (−2).
- Calculate log_5 1 (0).
Room 17 · The inverse machines
Two animated machines: the Raiser, which takes a number x and calculates 2^x, and the Logarithm, which calculates log_2 x. The student feeds a 5 into the Raiser and its output into the Logarithm, and checks that 5 comes back out; then a 3 into the Logarithm and its output into the Raiser, and the 3 comes back. The machines are removed and the graph remains with the mirror y = x: the student drags a point along the exponential and sees its reflection in the logarithm. To finish, a round-trip calculation: knowing that log_2 1.919 ≈ 0.94, calculate 2^0.94 without a calculator.
Student tasks
- Feed 5 into the Raiser and its result into the Logarithm (5 comes out).
- Feed 3 into the Logarithm and its result into the Raiser (3 comes out).
- Drag the point along the exponential and watch its reflection in the logarithm.
- Calculate 2^0.94 knowing that log_2 1.919 ≈ 0.94 (1.919).
Room 18 · The graph of the logarithm
The graph of y = log_a x with a slider for the base a, from 0.25 to 4, the vertical asymptote x = 0, the point (1, 0) and a table with x = 0.5, 1, 2, 4 and 8. A text contrasts it with the exponential: there the asymptote was horizontal and here it is vertical, and the domain is bounded on the left. Three multiple-choice questions: the domain, why log(−5) does not exist and what happens if 0 < a < 1.
Student tasks
- Choose the domain of log_a x (the positive numbers).
- Choose why log(−5) does not exist (there is no exponent c with a^c = −5).
- Say what log_a x is like if 0 < a < 1 (decreasing).
Room 19 · The properties of the logarithm
A table sets the properties of the power beside those of the logarithm: multiplying becomes adding, dividing becomes subtracting and the exponent becomes a multiplier. With two sliders, x and y between 1 and 20, the student checks that log(x · y) and log x + log y give the same result. Then three calculations, the last one a counterexample: a sum inside the logarithm does not split.
Student tasks
- Check with the sliders that log(x · y) = log x + log y.
- Calculate log(x · y) knowing that log x = 0.7 and log y = 0.3 (1).
- Calculate log(x / y) knowing that log x = 0.9 and log y = 0.2 (0.7).
- Calculate log(2) with the sliders and compare it with log 1 + log 1 = 0 (0.301).
Room 20 · Tutorial: graphing a logarithm
The equivalent algorithm for the logarithm: vertical asymptote, fixed point (1, 0), calculation of (a, 1) and (1/a, −1), an extra point and joining up the curve. It is shown modelled with base 2 and then the student does it with y = log_1/2 x: they calculate the value of each point and tap it on the graph.
Student tasks
- Write the value of y at the fixed point (1, y) (0).
- Calculate log_1/2 (0.5) (1) and mark the point.
- Calculate log_1/2 2 (−1) and mark the point.
- Calculate log_1/2 (0.25) (2) and mark the point.
Room 21 · Practice: graph it yourself
With no list of steps, the student graphs two logarithms: first y = log_3 x, increasing, and then y = log_1/4 x, decreasing. For each one they calculate three points and mark them on the graph; the curve is drawn at the end.
Student tasks
- Calculate and mark log_3 1, log_3 3 and log_3 (1/3) (0, 1 and −1).
- Calculate and mark log_1/4 1, log_1/4 (1/4) and log_1/4 4 (0, 1 and −1).
Room 22 · Final recognition
Five rounds to tell the four types of graph apart: increasing exponential, decreasing exponential, increasing logarithm and decreasing logarithm. In three rounds a formula is given and the student chooses among four small graphs; in the other two a graph is given and they choose among four formulas. A miss suggests looking at the asymptote and at whether the curve rises or falls.
Student tasks
- Choose the graph of y = 3^x (increasing exponential).
- Choose the graph of y = log_4 x (increasing logarithm).
- Choose the graph of y = (1/3)^x (decreasing exponential).
- Given a decreasing exponential, choose its formula among four (y = (1/2)^x).
- Given a decreasing logarithm, choose its formula among four (y = log_1/2 x).
Rooms to project
The most striking ones to show and discuss in class.