Back to the on-screen lesson ·
Turn a problem set into pairs — an example explained, then a problem like it solved cold — and fade the examples as your own solutions start working.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will lay out a session that pairs each worked example with a problem of the same kind, uses a solutions manual as a source of examples, and fades the support.
Last lesson you explained a worked example to yourself, line by line. That leaves one question: what comes after the example? This lesson answers it with a problem of the same kind, solved with the example covered — and then another pair, and then less and less support until you are solving on your own.
| Term | What it means |
|---|---|
| Worked example | A problem solved step by step for you to study, from a textbook, a teacher, a solutions manual or a coworker. |
| Paired problem | A problem of the same kind, solved straight after the example with the example covered. |
| Faded example | A worked example with its later steps hidden, for you to finish. |
| Fading | Withdrawing the examples bit by bit as your own solutions start working. |
| Problem alone | A problem with no example in sight: the end point of the session. |
When you meet a new kind of problem, trying it cold wastes effort: you search about and may practice a wrong method. Reading a pile of examples wastes it the other way: everything makes sense and nothing is practiced. The middle path is to pair them.
Study one worked example, explaining it to yourself. Turn it over. Solve one problem of the same kind. Check it. Then a second pair. Then a faded example: an example with its last step or two hidden, which you finish. Then problems alone. When a paired problem goes wrong, go back to the matching line of the example — not the whole chapter.
Your textbook already has the parts. Many books put worked solutions or answers to the odd-numbered exercises in the back or in a solutions manual. Use an odd one as the example and the even one after it as the problem.
Another way: picture
Think of learning to ride with training wheels that come off one at a time. First you ride beside someone who shows you; then you ride a short stretch; then one wheel is off; then both.
Another way: steps
Sweller and Cooper (1985) taught high school students algebra. Some solved a set of problems; others studied worked examples paired with problems. On similar problems afterwards, the example group solved them faster and made fewer errors. The advantage was weaker on problems that did not look like the examples — which is one reason the later lessons in this course mix problem kinds.
The IES practice guide (Pashler and colleagues, 2007) turned this into a recommendation: alternate worked examples with problems for the student to solve, and shift from more examples early on to more problems later. Research on fading, much of it by Renkl and Atkinson, found that taking the steps away gradually works better than jumping straight from full examples to problems.
Worked examples help most when a kind of problem is new. Once you can do it, they can get in the way: reading a solution you no longer need costs time and adds nothing. Kalyuga and colleagues (2003) called this the expertise reversal effect.
So watch your own results. Two paired problems right in a row, without peeking, means drop the examples for that kind and move to problems alone. A paired problem you cannot start means the example has not done its work yet: go back to its key line and self-explain it. Neither is a failure; both tell you which way to move.
Anything that shows a solution step by step: a textbook example, a solved exercise in a solutions manual, a teacher's board work, a coworker showing you a task. A final answer alone is not a worked example — the answers in the back tell you whether you were right, not how to get there, so use them to check a paired problem, not to learn from.
There are 5,280 feet in a mile, 3,600 seconds in an hour, and 16 cups in a gallon. Converting between units is a skill a nursing, trade or science course may assume you have, and it is ideal for pairing.
A fictional student, Lena, starts with a worked example: 5 miles to feet is 5 × 5,280 = 26,400 feet. She says why: one factor, miles to feet. She covers it and solves 3 miles to feet: 15,840 ✓. Second pair: 2 hours to seconds is 2 × 60 × 60 = 7,200 — two factors chained. She solves 4 hours: 14,400 ✓. Then a faded example: 3 gallons to cups, with the last line hidden; she writes 3 × 16 = 48 cups. Finally, alone: 2 weeks to minutes. She chains three factors, 2 × 7 × 1,440, and gets 20,160 — right, with no example in sight. Six items, about twenty minutes, and every example was followed by something she did herself.
"Read all the examples first, then do the problems." By the time you reach the problems, the first example has faded and nothing links each problem to its example.
"Real learning is struggling with problems on my own." For a new kind of problem, struggling cold mostly practices searching. Struggle is worth it once you have seen the method, which is what the pairs are for.
"Keep the example open while I do the problem." Then the problem is copying. Cover it, and look back only at the line you need.
"Once examples help, keep using them." As you get good, they stop helping. Fade them out.
Study the first worked example, explaining each line.
3x + 5 = 20 → 3x = 15 (take 5 from both sides) → x = 5 (divide by 3)
The example shows the order of moves: undo the adding before the multiplying.
Cover it and solve a problem of the same kind.
Paired problem: 4x + 7 = 31 → 4x = 24 → x = 6 ✓
Solving straight after the example, with it hidden, uses the method while it is fresh.
Do a second pair with a small change.
Example: 2x − 3 = 9 → 2x = 12 → x = 6. Problem: 5x − 4 = 21 → x = 5 ✓
The subtraction shows what changes and what stays the same.
Finish with problems alone.
Alone: 7x + 2 = 30 → x = 4 ✓; 6x − 1 = 29 → x = 5 ✓
Two right without an example means the examples can go for this kind.
Find which exercises have worked solutions.
Solutions manual: odd numbers 11, 13, 15 worked in full
Each odd exercise can serve as a worked example for the even one after it.
Study 11's solution, then solve 12 with 11 covered.
Exercise 12 attempted: stuck at step 3 ✗
The stuck point tells you exactly which line of the example to revisit.
Go back to the matching line of the example only.
Exercise 11, step 3: convert grams to moles before using the ratio
Returning to one line keeps the session moving; rereading the chapter would not.
Finish 12, then pair 13 with 14 and solve 15 cold.
12 ✓, 14 ✓, 15 solved before looking, then checked ✓
Exercise 15 has a solution, but using it only to check is the fading.
Study a worked example of the pattern.
hablar (to speak) → yo hablé: drop -ar, add -é
The example shows the ending rule for regular -ar verbs.
Cover it and try a verb of the same kind.
cantar (to sing) → yo canté ✓
Same kind, new verb, example hidden: the paired problem.
Take the next verb with no example.
A fictional student has three worked examples and six problems on solving equations like 3x + 5 = 20. Which plan pairs worked examples with problems?
A fictional plan fades a worked example of 4 steps backwards: each round hides one more step, starting from the last, until the whole problem is the student's. Complete the worked count.
Find how many steps the second round still shows.
Round 1: all 4 shown. Round 2: a shown, the last one hers
Backward fading hides the last step first, so she finishes the example before she has to start one.
Find the round where nothing is shown.
Round b: every step hidden → a problem alone
Backward fading ends with the whole problem in the student's hands.
Add up the steps she writes herself across all the rounds.
0 + 1 + 2 + … one more each round = c steps
Each round asks a little more of her, so the jump from example to problem is never all at once.
Lay out a fictional six-slot session on a new kind of problem. Use: worked example, similar problem, faded example, problem alone.
| Item | |
|---|---|
| Slot 1 | |
| Slot 2 | |
| Slot 3 | |
| Slot 4 | |
| Slot 5 | |
| Slot 6 |
A fictional chemistry textbook has worked solutions for the odd-numbered exercises in its solutions manual. A student is assigned exercises 21 to 26. She studies each odd one's solution and solves the even one after it cold — except that the last two are both solved cold. Fill in how she uses each exercise.
| Use it as | |
|---|---|
| Exercise 21 | |
| Exercise 22 | |
| Exercise 23 | |
| Exercise 24 | |
| Exercise 25 | |
| Exercise 26 |
A fictional session: 4 pairs of a worked example and a problem, then 2 problems alone. Explaining an example takes 8 minutes and a problem takes 6. How many problems does she solve herself, and how many minutes does the session take?
Problems she solves: a
Minutes for the session: b
A fictional new hire at a store has to learn the office's sales spreadsheet. A coworker offers to show her some formulas. Lay out a six-slot session that pairs examples with her own attempts and fades the support.
| Item | |
|---|---|
| Slot 1 | |
| Slot 2 | |
| Slot 3 | |
| Slot 4 | |
| Slot 5 | |
| Slot 6 |
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A fictional student's homework is on unit conversions. Build her six-slot session from the items offered, pairing examples with problems and fading the support.
| Item | |
|---|---|
| Slot 1 | |
| Slot 2 | |
| Slot 3 | |
| Slot 4 | |
| Slot 5 | |
| Slot 6 |
You can pair examples with problems. Take your next problem set, mark which exercises have worked solutions, and write the pairs down before you start. Next: pairing pictures with words. You can stop in the middle of any of this. Closing the page is the whole of stopping — your place and anything you had typed are already saved, and the lesson opens again where you left it.
12. Your turn: Spanish past tense, paired, step 3
bailar (to dance) → yo bailé ✓; now fade to a new kind of verb
Two right in a row means move on — perhaps to -er verbs with a new example.