In and Out Boxes

The short answer

An in and out box is a small table with an In column and an Out column, and one hidden rule that turns every In value into its Out value. The child's job is to work out that rule from the pairs already filled in, then use it on the pairs that are not. You will also see this activity called a function machine or what's my rule, and all three names describe the same thing. This page explains how to teach it; to print a sheet, use Find the Rule.

What the child is actually doing

Reading numbers off a table is not the task. The task is working something out from evidence: looking at a pair of values, forming a guess about what was done to the In value, and then testing that guess against a pair they have not used yet. When one guess explains both filled pairs, the child has the rule, and the rest of the card follows from it.

The rule is the thing, not the next number

A rule maps each In value to its Out value. It is not a promise about what comes next down the column. That is what separates this from a repeating pattern, where a child looks along a row and continues it. Here the child looks across, from In to Out, and the order the rows are printed in does not matter at all.

How to teach it, step by step

Work through one card together before the child works alone.

• Cover the Out column and read the In values aloud, so the child hears that they are not in order and not evenly spaced.
• Uncover one filled pair and ask what could have been done to that In value. Take every reasonable answer for now.
• Uncover the second filled pair and ask which of those answers still works. Usually only one does.
• Say the surviving rule out loud in plain words, such as adds 6, or times 5.
• Apply it to the remaining In values one at a time, writing each Out value as you go.

Start with add and take away

Begin where a rule can only add the same number or take away the same number. There are two things to decide, which of the two operations and which number, and a child can hold both in mind at once. Stay here until the child can say the rule in words before writing anything down.

Add times tables once the first is secure

The next step lets a rule multiply as well, so the child now chooses between three operations rather than two. Keep the times rules small. Sheets from Find the Rule use times 2 through times 5 at this step, which keeps the arithmetic light enough that finding the rule stays the hard part.

A worked example

Take a card from the step that includes times rules. Its In values are 1, 7, 4 and 10, printed in that order, and only the first two Out values are shown: 5 under the 1, and 35 under the 7.

Start with the first pair. What turns 1 into 5? Adding 4 works. Times 5 works too. Both are rules this sheet is allowed to use, so one pair does not settle it.

Now test both against the second pair. Adding 4 to 7 gives 11, not 35. And 7 x 5 = 35, so times 5 survives. One rule explains both filled pairs, which makes it the rule, and the two empty boxes are 20 and 50.

The arithmetic in that check, 7 x 5 = 35, is verified content from the Quick Math Sheets fact library. The rule itself is a different kind of claim. It is not a fact about numbers in general, it is a statement about this one card, and the child settles it by finding a second pair that it also explains.

Common mistakes, and what to do

The child continues the column without using the rule

This is the one to watch for, and it is easy to cause by accident. Write In values that count up by one and the Out values count up by one as well, so the child can fill the whole column by adding one each time without ever thinking about the rule. The sheet then looks finished and correct while the actual task went untouched. Printed sheets from Find the Rule prevent it by construction: the In values on every card are spread apart and printed out of order, so continuing the column gets the child nowhere. If you are writing tables by hand, spread the In values out and shuffle them before you write.

The child finds a rule that fits one pair only

Ask for the second pair every time. A rule that explains one pair is a guess, and a rule that explains two is an answer. Asking the child which second pair they checked turns that into a habit instead of a correction you have to keep making.

The child reverses the rule on the empty boxes

A child who has just worked out takes away 3 will sometimes add 3 when filling the boxes in, especially if the numbers there happen to be larger. Point back at a filled pair and ask which column the rule starts from. In goes in and Out comes out, in that direction, every time.

Keep it steady

Short sessions beat long ones here. One card talked through out loud is worth more than a full page filled in silence. And if the child works out the rule and then wants to invent one for you to solve, let them, because making a rule and finding a rule are the same skill from two sides.

Practice it

Print a sheet at Find the Rule. It gives four cards on one page, each hiding its own rule. On every card two of the four columns show both values and two leave the Out box empty, with room on the card for the child to record the rule before filling those boxes.

The control is called what the rule can do, and it offers two settings. Adds or takes away holds every rule to one of those two. Adds, takes away or times brings in the times rules described above. A new version prints four fresh cards at the same setting, so one setting can carry several sessions. There is also an optional teacher page carrying each card's rule, if you want to check a finished sheet quickly.

You can build the same sheet inside the algebra generator if you are already making other sheets there.

Related

• Algebra worksheets, the hub this sheet sits under.
• Complete the pattern, where the child continues a repeating pattern rather than recovering a rule. Useful first if the idea of a rule is still new.
• Missing numbers, for a child who is still getting comfortable with the order of numbers.