Numbers on a number line

Activities to try

1. 1

   Build and test a number-line inequality

   Why it works: Inclusion is not an arbitrary drawing convention: whether the circle is filled depends on whether the boundary value itself is a solution. The arrow/segment represents the whole set of satisfying values, and its direction is fixed by what the symbol means.

   1. Choose an inequality symbol (<, ≤, >, ≥) and a boundary value, then draw the matching number line: decide whether the boundary circle is open or closed and which way the arrow points.
   2. Test several sample values by asking of each, 'does this value satisfy the inequality?', and mark those that satisfy it (e.g. in one colour) and those that do not (another colour).
   3. Switch the symbol from ≥ to > (or ≤ to <) on the same boundary and observe what changes: the boundary circle hollows, and a value sitting exactly on the boundary flips from satisfying to not satisfying.
   4. Deliberately draw the arrow the wrong way, then test values along it to see that a whole stretch of the line fails the check — confirming direction matters.
   5. Read a given diagram (e.g. a filled circle at −2 with a right-pointing arrow) back into symbols and words.
2. 2

   Translate worded constraints into symbols and diagrams

   Why it works: Everyday boundary phrases map precisely onto strict or inclusive symbols, and testing whether the boundary value is allowed decides between an open and closed circle.

   1. Translate worded constraints into inequality symbols: 'at least 12' → x ≥ 12, 'at most'/'no more than' → ≤, 'more than' → strict >, 'less than' → strict <.
   2. Build a phrase→symbol reference table and, for each phrase, test the boundary value against the words ('is 12 allowed if you must be at least 12?').
   3. Draw the number-line diagram for a compound statement such as 'x is at least 1 and less than 4' (closed circle at 1, open circle at 4, segment shaded between).
   4. Check your diagram by reading it back into words and comparing with the original statement.
3. 3

   Solve a simple linear inequality and plot its solution set

   Why it works: Solving an inequality mirrors solving an equation, but the answer is a set shown on the number line; special care is needed when multiplying or dividing by a negative because the direction reverses.

   1. Solve a one-step or two-step inequality such as 3x + 1 < 10 using the same inverse operations as an equation (subtract 1, then divide by 3, giving x < 3).
   2. Represent the resulting solution set on a number line with the correct open or closed circle and arrow.
   3. Check the solution by testing a value inside the range and a value outside it.
   4. (Extension) Explore what happens when both sides are multiplied or divided by a negative number, using test values to discover that the inequality direction reverses.