Two speeds of the same method

Here is a question you have certainly met. Four shirts, three pairs of trousers, two hats. How many different outfits? Most people know the answer is 4 x 3 x 2 = 24 and could not say why the multiplication is allowed. And when the question gets slightly harder, the multiplication is the first thing to go wrong. ⚠️ SO START WITH THE OTHER METHOD, THE ONE YOUR SPECIFICATION ACTUALLY NAMES: SYSTEMATIC LISTING. Write out every outfit, but in a strict order. All the outfits with the first shirt, then all with the second, and so on. Within each shirt, all the trousers in order. Within each pair of trousers, both hats. Do that and something becomes visible. ⚠️ THE LIST HAS A SHAPE. Four blocks, each containing three smaller blocks, each containing two lines. ⚠️ THAT SHAPE IS THE MULTIPLICATION. 4 x 3 x 2 IS NOT A TRICK THAT HAPPENS TO WORK. IT IS A DESCRIPTION OF THE LIST YOU WOULD HAVE WRITTEN. Which is why the word SYSTEMATIC is in the name of this topic, and it is not about being tidy. A systematic list can be CHECKED, because you can see whether a branch is missing. A haphazard list cannot, because nothing tells you what should have been there. And a systematic list REVEALS THE STRUCTURE, which is what lets you multiply instead. A haphazard one hides it. ⚠️ SO LISTING AND MULTIPLYING ARE NOT TWO DIFFERENT SKILLS. THEY ARE THE SAME METHOD AT TWO SPEEDS, and the systematic order is what makes them agree. Step 5 covers the part that decides which speed to use, and it is where the marks are.

Five words this topic runs on

Five terms, defined and nothing more. Which situations are which comes next.

Match each situation to what it is

A shirt, a pair of trousers and a hat chosen together, written down as one complete choice
Writing out every possibility in a fixed order, changing the last item fastest, so none can be missed
Working out 4 times 3 times 2 instead of writing the twenty-four lines out
However you choose the shirt, there are still three pairs of trousers available
A digit already used cannot be used again, so the second stage has one fewer option than the first

AN OUTCOME, because it is one complete result rather than one of the choices within it
SYSTEMATIC LISTING, because the fixed order is what guarantees nothing is missed
THE PRODUCT RULE, because the choices at each stage are being multiplied together
INDEPENDENT STAGES, because the number of later options does not depend on the earlier choice
A RESTRICTION, because a rule has changed how many options are available

When multiplying stops working

Two-digit numbers are made from the digits 1, 2 and 3. A student says there are 3 x 3 = 9 if digits may repeat, and also 9 if they may not. What is wrong with the second claim?

With no repeats the second stage has only two options left, whichever digit was chosen first, so the count is 3 x 2 = 6 rather than 9
Nothing is wrong; the restriction does not affect the count
The product rule can never be used once there is a restriction of any kind
The stages should be added rather than multiplied when there is a restriction

Which speed, and how to tell

The whole skill is deciding which of these you are in. The third column is where the marks are at the top of the paper.

True about counting outcomes

Select the TWO statements that are true.

A systematic list reveals the structure that makes the multiplication valid, rather than being a slower alternative to it
The product rule needs each stage to offer the same number of choices whatever was chosen before
Multiplying the stages always gives the right total, whatever restrictions apply
Listing systematically rather than haphazardly is a matter of neat presentation

How to write a count that earns method marks

Your specification's guidance for this topic says two things, and both are about what you write down rather than what you work out. ⚠️ FIRST: USE THE PRODUCT RULE EXPLICITLY ONCE THE TOTAL IS LARGE, RATHER THAN LISTING EVERY OUTCOME BY HAND. Listing twenty-four outfits is defensible. Listing four hundred is a way of running out of time and making a copying error, and it does not show anything the multiplication would not have shown better. ⚠️ SECOND, AND IT IS WORTH MORE: SHOW THE WORKING AT EVERY STEP. A CORRECT FINAL ANSWER WITH NO WORKING CAN LOSE METHOD MARKS IF IT IS WRONG - and on a counting question it is very easy to be wrong by one. So write the count as a sentence with numbers in it, not as a bare product. Not 4 x 3 x 2 = 24, on its own. But: "There are three stages. 4 choices of shirt, then 3 of trousers whatever the shirt, then 2 of hat whatever the rest. 4 x 3 x 2 = 24." ⚠️ NOTICE WHAT THAT SENTENCE DOES. THE PHRASE "WHATEVER THE SHIRT" IS YOU STATING THAT THE STAGES ARE INDEPENDENT, which is the condition the whole method rests on. A marker can see you checked it. In the bare product, nobody can. Two more habits. When you are unsure, LIST A SMALL VERSION FIRST. Work out a cut-down case by hand, multiply it, and see whether the two agree. If they do, the structure is what you thought and you can scale up. If they do not, you have found the restriction before it cost you the question. ⚠️ AND IF THE QUESTION SAYS "LIST", LIST. It is asking for the outcomes themselves, and a total, however correct, is not an answer to it.

Six numbers, written out and then multiplied

Make every two-digit number you can from the digits 1, 2 and 3, with no digit used twice. List it systematically. Fix the first digit, then run through the possible second digits in order. Starting with 1: 12, 13. Not 11, because no repeats. Starting with 2: 21, 23. Starting with 3: 31, 32. Six numbers, and you can see it is six without counting them one at a time: three blocks of two. ⚠️ NOW READ THE STRUCTURE OFF THE LIST. THREE CHOICES FOR THE FIRST DIGIT. FOR EACH ONE, TWO REMAIN. 3 x 2 = 6. The multiplication did not replace the list. It described it. Change one thing and watch the second factor move. ⚠️ IF REPEATS WERE ALLOWED, THE SECOND STAGE WOULD HAVE THREE OPTIONS RATHER THAN TWO, and the count becomes 3 x 3 = 9. The three extra are 11, 22 and 33. The method did not change. One number in it did. Now a case where the plain product genuinely breaks. A cafe offers 3 starters and 4 main courses, so 3 x 4 = 12 meals. ⚠️ BUT ONE PARTICULAR STARTER CANNOT BE HAD WITH ONE PARTICULAR MAIN, because they share an ingredient the kitchen runs out of. Now the number of mains available depends on which starter you chose: 4 for two of the starters, but only 3 for the other one. ⚠️ THE STAGES ARE NO LONGER INDEPENDENT, SO THERE IS NO SINGLE SECOND FACTOR TO MULTIPLY BY. Two honest ways through. Count all 12 and subtract the one pairing that is not offered: 12 - 1 = 11. Or split into cases: the two unrestricted starters give 2 x 4 = 8, the restricted one gives 1 x 3 = 3, and 8 + 3 = 11. ⚠️ BOTH AGREE, AND BOTH BEAT GUESSING WHETHER THE PRODUCT RULE STILL APPLIES.

Count the outfits

A wardrobe holds 4 shirts, 3 pairs of trousers and 2 hats. Every shirt goes with every pair of trousers, and every hat goes with everything. How many different complete outfits are possible?

Order the method for a counting question

A reliable order for answering a counting question you have just been given.

Work out how many stages the selection has
Count how many choices are available at each stage
Check whether those numbers stay the same whatever was chosen earlier
If they do, multiply them together and write down why you were allowed to
If they do not, list systematically or split the problem into separate cases

The counting run

Five questions on listing and the product rule. Three lives.

Complete the counting facts

One complete result of a whole selection, rather than one of the choices within it, is an _____. Writing out every possibility in a fixed order so that none is missed or repeated is _____. Multiplying together the number of choices available at each stage is _____. Stages where the number of choices does not depend on what was chosen earlier are _____.

outcome systematic listing the product rule independent stages a restriction place value a factor a sample space

Spot the true counting facts

Tap the TWO statements that are true.

The multiplication in the product rule describes the structure a systematic list would have had
The plain product rule breaks when the number of later choices depends on what was chosen earlier
Multiplying the stages together always gives the right total, whatever the restrictions
Listing systematically rather than haphazardly is purely a matter of presentation

Write the count so a marker can follow it

Assemble the kind of working step 7 describes, for the wardrobe with 4 shirts, 3 pairs of trousers and 2 hats.

Explain when you are allowed to multiply

A classmate multiplies the stages together on every counting question and gets some of them wrong. They cannot see what distinguishes the ones that work. Write them an explanation.

Explain what a systematic list is, and why the fixed order matters rather than being tidiness
Explain how the structure of such a list turns into the multiplication
State the condition the product rule needs, and give an example where it holds
Give an example where a restriction breaks it, and show one honest way to get the right total anyway
Finish by explaining what their working should say, so that a marker can see they checked the condition rather than guessed

Systematic Listing

Get the method right under pressure

What you'll cover