Worked examples, practice problems and past exams: the order that teaches, and where to find them

Practice problems teach in a particular order. First one worked example read line by line, then the same problem redone with the solution hidden, then a sibling with new numbers, then a mixed set where you have to choose the method, and only then a past exam under time. Do them in the other order, past exam first, and you learn what you cannot do and nothing about how. Here is the order, why each step is there, a set built for one calculus idea, and where to find problems that are worth doing.

The order, and why it is an order

Six steps. Each is there because skipping it costs something specific.

1. One worked example, read line by line. For a procedure you are meeting for the first time, studying a worked example teaches the pattern faster than solving cold: Sweller and Cooper (1985) found that algebra students who studied worked examples learned the procedure with less time and fewer errors than students who solved the same problems unaided. The theory is why a worked example teaches and a wall of text does not; the mechanics of reading one so that it sticks are in how to study math. Skipping it means learning the pattern by trial and error, which is slow and leaves wrong lines in.
2. The redo, cold. The same problem, solution hidden, marked line by line. This is retrieval: Roediger and Karpicke (2006) found that producing the material from memory beat rereading it for retention a week later. Skipping it means the example felt understood and was not.
3. A sibling with new numbers. The procedure, not the answer, is what you are learning, and the sibling is the first proof that you have the procedure rather than the memory of one solution.
4. The faded set. A problem or two with the first line given, then with none. Renkl and Atkinson (2003) argue from cognitive load that the move from examples to problems should be gradual, with the learner taking over one step at a time, and their experiments on fading supported it for near transfer. Skipping it is the jump that makes step 5 feel impossible.
5. The mixed set. Six to ten problems of several types in random order, so that you choose the method. Rohrer and Taylor (2007) found that students who practised shuffled problems did worse during practice and better a week later than students who did one type at a time. Kornell and Bjork (2008) found the same for learning to recognise which category an example belongs to: mixing beat blocking, and learners believed the opposite. Skipping it means the test is the first time you ever chose. The full case is why mixing problem types beats blocked practice.
6. A past exam under time, graded with the scheme, then the error log. It trains the one thing none of the first five can: the whole paper, with a clock. The practice-test method has how to take and read one. Skipping it means the first full paper you sit is the one that counts.

The order is the four moves applied to problem sets: answer first at steps 2 to 6, mix at step 5, and the spacing lives in when you come back to the misses. The review by Dunlosky and colleagues (2013) rates practice testing high and interleaved practice moderate, which is steps 2 and 5 in their language.

How many problems?

Fewer than you think at the start and more at the end. One example at step 1, one redo at step 2, one or two siblings at step 3, two at step 4. A mixed set of six to ten at step 5. One past paper at step 6 per unit.

Ten examples skimmed teach less than one read closely; that is the line from the maths guide, and it is the reason the counts start at one. The place to spend more problems is the mixed set, where every problem is a decision, not the blocked set, where the heading made the decision for you.

Worked set: the product and quotient rules, start to finish

MCV4U idea 7, product and quotient rules, with the chain rule (idea 8) joining at step 5. Every derivative below was recomputed before it was written.

Step 1, the example. Differentiate f(x) = x²·eˣ. The product rule: (uv)′ = u′v + uv′. Here u = x², u′ = 2x, v = eˣ, v′ = eˣ. So f′(x) = 2x·eˣ + x²·eˣ. Factor the common x·eˣ: f′(x) = x·eˣ(2 + x). Read it like a mechanic: which factor is u, which is v, what each derivative is, where the two terms came from, why the factoring is optional and the two terms are not.

Step 2, the redo cold. The same f, solution hidden. The student writes 2x·eˣ + x²·eˣ and stops without factoring. Marked line by line against the example: right. Factoring is presentation, not procedure.

Step 3, the sibling. g(x) = x³·sin x. u = x³, u′ = 3x², v = sin x, v′ = cos x. g′(x) = 3x²·sin x + x³·cos x. Right.

Step 4, the fade. h(x) = (2x + 1)/x², with the first line given. The quotient rule: (u/v)′ = (u′v − uv′)/v². With u = 2x + 1, u′ = 2, v = x², v′ = 2x, the given first line is h′(x) = [2·x² − (2x + 1)·2x] / x⁴. The student finishes it: the numerator is 2x² − (4x² + 2x) = 2x² − 4x² − 2x = −2x² − 2x = −2x(x + 1). Over x⁴, one x cancels: h′(x) = −2(x + 1)/x³. The student's first attempt had the numerator the other way round, (2x + 1)·2x − 2·x², and got +2(x + 1)/x³. That is the common wrong line for this rule, and the reason the quotient rule is harder than the product rule: the minus is not symmetric, so the order of the two terms matters, where in the product rule it does not. Into the error log: "quotient rule: derivative of the top first, times the bottom, minus."

Step 5, the mixed set. Six problems, two product, two quotient, two chain, in the order they were drawn. Answers under the set; attempt them all first.

1. y = x·cos x
2. y = (3x² + 1)⁴
3. y = eˣ / x
4. y = (x² + 1)·sin x
5. y = x / (x + 1)
6. y = √(x² + 4)

Answers: (1) product: cos x − x·sin x. (2) chain: 4(3x² + 1)³ · 6x = 24x(3x² + 1)³. (3) quotient: (eˣ·x − eˣ·1)/x² = eˣ(x − 1)/x². (4) product: 2x·sin x + (x² + 1)·cos x. (5) quotient: [1·(x + 1) − x·1]/(x + 1)² = 1/(x + 1)². (6) chain, with y = (x² + 4)^½: ½(x² + 4)^(−½) · 2x = x/√(x² + 4).

The student's score: 4 of 6, with problems 2 and 6 missed. In problem 2 the inner derivative, 6x, was forgotten; in problem 6 the ½ power was differentiated as if it were a square. Both are chain-rule misses, both go on the log, and both become tomorrow's list, which starts again at step 1 with a chain-rule example.

Step 6, the past paper. The teacher's released unit test, or the department's old exam, done in 40 minutes with the book closed, then graded with the scheme and read into the error log by idea, naming the wrong line each time. Chapter 4 of the free playbook is that reading, line by line.

Where to find practice problems

In roughly the order they are worth your time.

• Your textbook's end-of-section sets. The odd-numbered answers at the back are a marking scheme for free. Do the odd ones; the even ones are for a second pass when you want to be unsure.
• Your teacher's review package and released tests. The closest thing to the exam's own style, written by the person writing the exam.
• The department's or the course's past exams, where they are released. The past-exams section below says how to use one.
• Open textbooks published under licences that allow reuse. Many are free, and their exercise sets are real. Check the licence page of any book before you rely on it, and prefer a book whose answers are included.
• The course's own practice bank, which is where the app on this page appears once; the end panel says the rest.

Where not to look

• Answer apps' solution banks. In our own reading of their review banks, the posted answers are frequently wrong, and tonight you have no way to tell which. A wrong worked solution teaches a wrong line as confidently as a right one teaches a right one.
• Worked solutions you read and never redo. Reading a solution is step 1 without steps 2 to 6. It feels like practice and produces recognition, not recall.
• Problem sets from a different course's syllabus. A first-year set used for a Grade 12 course teaches methods the exam will not ask for and skips ones it will.
• Anything you would paste in for a mark. Not because of the rule, though there is one, but because it is not practice. When a step-by-step answer teaches and when it replaces the thinking draws the line.

Past exams: how to use one

Never first. A past exam before the order above tells you that you cannot do the course, which you knew, and tells you nothing about how to.

One per unit, under time, closed book. Graded with the marking scheme, not by feel. Then the error log by idea: for each lost mark, the idea it belonged to and the line that went wrong. Each missed idea goes back to step 1 of the order, with a worked example, on the next evening. A second past paper two weeks later shows the change, and the change is the measurement.

Where to find them: your teacher or department first; many will hand over a released test on request. Universities often release past papers through the course page or the library. A past paper from another school or another year is still useful when the syllabus matches, and the way to check is to hold its questions against the course's idea list; the MCV4U course map is the example of such a list. This page hosts no past exam and copies no question from one.

Science and other subjects

The order is the same in every subject with problems in it; what counts as a problem changes.

• Chemistry: the same six steps on the calculation chapters (moles, gas laws, equilibrium, pH), with the names and definitions as cards alongside, which is how the science-study guide treats the subject.
• Physics: a sketch before step 1 and before every problem after it. The example teaches the equation; the sketch is what tells you which equation.
• Biology: the data-analysis and experimental-design questions are the problems, and the released tests are the past papers. The order holds; the siblings are other data sets.

The mistakes

• Past exam first. It diagnoses and does not teach.
• Reading solutions as practice. Recognition is not recall.
• Twenty of one kind. Blocked practice feels better and tests worse, in study after study.
• Never timing anything. The exam has a clock; meet it before the day.
• No error log. A problem done and not marked is a problem not done, and a problem marked wrong without the line named will be wrong again.

The order on one page

1. One worked example, read line by line.
2. The redo, solution hidden, marked.
3. A sibling with new numbers (one or two).
4. A faded set, first line given, then none.
5. A mixed set of six to ten, methods shuffled, so you choose.
6. A past exam under time, graded with the scheme, read into the error log by idea.

The misses go back to step 1. The counts are small at the start and larger at the end, and the mixed set is where the problems earn their time.