Breaking a problem into smaller steps

Find the question before choosing a calculation

You have 26 stones altogether. Exactly two must stay loose, outside separate full groups of four. How many full groups are there? The total stone count is given; the group count is unknown. Let g count full groups. It must be zero or a positive whole number.

Restate the question: “How many groups of four account for all the stones except the two fixed loose ones?” We are counting groups, not measuring mass or finding the stone total again. Keep the word “exactly”: three loose stones would not meet the conditions.

Separate the parts

Think of the total as grouped stones plus two loose stones, as in T = 4 × g + 2. A sketch can mark the two loose stones separately and leave space for groups. Each stone belongs to one part; none should be omitted or counted twice.

Before asking how many groups fit, ask how many stones belong in groups. Set the two fixed loose stones aside: 26 − 2 = 24. This intermediate answer is 24 grouped stones. It is not 24 groups, and setting stones aside does not remove them from the overall arrangement.

Choose a plan from the structure

Our plan is to set aside the loose stones, organise the rest into full groups of four, and rebuild the total to check. Each step answers a smaller question that prepares the next. We work from a known total back to its grouped part, rather than guessing the group count.

Do not choose an operation from a single word. This problem mentions a “total,” but the total is already known. We subtract two because those stones lie outside the groups. We divide the remaining count by four because each full group must contain four stones.

Carry out one meaningful step at a time

After setting aside two, organise the remaining 24 stones. Six groups of four use them all, so 24 ÷ 4 = 6. The result is six full groups. Check the grouping directly: 6 × 4 = 24, with no grouped stones left over.

Keep labels beside your work: 26 total stones; 24 grouped stones; six full groups. These labels show how the meaning changes between steps. An unexplained list “26, 24, 6” hides that reasoning. A short sentence or a picture can make each step understandable.

Rebuild the story, including every condition

Six full groups contain 24 stones. Return the two loose stones: 6 × 4 + 2 = 26. The total matches, every group contains four, exactly two remain loose, and six is an allowed group count. Answer the question in words: there are six full groups.

Some totals cannot satisfy this story. With one stone, you cannot set aside two. With 27, setting aside two leaves 25: six groups of four plus one extra stone. Calling the extra stone loose would make three loose stones. Neither total permits a solution under the original conditions.

Try a smaller case and return to the problem

If the plan is unclear, try six total stones with the same conditions. Set aside two; four remain; they form one full group. A two-stone total leaves no grouped stones and gives zero groups. These small cases help show what each step means before returning to 26.

If a check fails, revisit your steps: did you forget the loose stones, confuse groups with stones, or ignore a remainder? Correct the method and check again. Keep the given total, group size and loose count unchanged. Sometimes “no solution under these conditions” is the correct conclusion.

Worked examples

Try it yourself

1. There are 18 stones under the same four-per-group and exactly-two-loose conditions. Find the group count and check it.

   Show the answer and reasoning

   Four full groups: 18 − 2 = 16 grouped stones, and 16 ÷ 4 = 4 groups.

   Rebuild the total: 4 × 4 + 2 = 18. Sixteen is an intermediate stone count; four is the requested group count.

2. Keeping the same conditions, what happens with a total of one stone? What happens with two?

   Show the answer and reasoning

   One stone gives no solution. Two stones give zero full groups and exactly two loose stones.

   One cannot supply the required two loose stones. With two, setting both aside leaves none to group; 4 × 0 + 2 = 2.

3. For 27 stones, someone reports six full groups and exactly two loose stones. Does this account for every stone?

   Show the answer and reasoning

   No. Six groups and two loose account for only 26 stones. The original problem has no solution.

   After setting aside two, 25 remain. Six groups use 24 and leave one extra; a partial group or an unreported extra stone violates the conditions. A seventh full group would need four stones, not the one that remains.

What to carry forward

• Name the question, the given quantities and the required conditions.
• Use meaningful intermediate answers to connect smaller steps.
• Check the original story; revise the method without quietly changing the problem.