Half a strip, half a group and halfway along a number line can describe the same fraction in different ways. To connect the pictures, first decide what counts as one whole. Then check that its parts represent equal amounts.
This lesson begins with sharing and drawing. Use it when you can count objects and compare equal groups, with an adult reading or helping you fold paper if useful. You will work with halves, quarters and eighths, explain equal shares and locate fractions between zero and one.
Start by naming the whole
A whole might be one complete paper strip. It might also be a group of sixteen counters considered together. Point to the whole before you say what fraction is selected. “One quarter” tells us a relationship to a whole; it does not always mean one object.
When one whole is divided into four equal parts, each part is one quarter. We write one quarter as 1/4. The bottom number, called the denominator, tells us there are four equal parts in the whole. The top number, called the numerator, tells us that one of those parts is selected.
Use large drawn circles or paper cards rather than small loose objects if handling or swallowing risks matter. You can circle groups with a pencil without cutting anything. An adult can draw or move objects while you explain what to do.
Worked case 1: share sixteen counters fairly
Draw sixteen circles and four large spaces. Put one circle into each space in turn. Continue until all sixteen circles have been assigned. Each space receives four counters. Check that no counter appears in two spaces and none is left outside.
The four shares contain 4, 4, 4 and 4 counters. Together they return the whole group: 4 + 4 + 4 + 4 = 16. One share contains one quarter of the group, which is four counters. It does not contain one quarter of a counter.
Now point to two of the four shares. They contain eight counters, which is two quarters of the original group. There are another eight counters in the other two shares, so the selected amount is also half the group. Keep all sixteen counters as the whole while making that comparison.
A useful explanation is: “The whole has sixteen counters. There are four equal shares. I selected one share, so I selected one quarter, or four counters.” Saying both the fraction and the number of counters prevents the two answers from being confused.
Worked case 2: connect a share to an area
Take one rectangular paper strip. Fold it in half so that its ends meet, then fold that half in half again. Unfold it. For a rectangular strip folded accurately in this way, the creases divide the strip into four equal-sized sections.
Mark one section with a cross. The marked area is one quarter of the strip. It is only one piece of paper in this model, whereas the quarter of the sixteen-counter group contained four separate counters. Both are one of four equal shares of their stated whole.
Equal parts must have equal amounts of the whole. For these rectangular strips with the same height, equal widths give equal areas. Simply drawing three dividing lines is not enough: lines placed unevenly can produce four unequal parts.
| Whole strip divided into | Selected equal parts | Fraction selected | Amount covered |
|---|---|---|---|
| 2 equal parts | 1 | 1/2 | Half the strip |
| 4 equal parts | 2 | 2/4 | Half the strip |
| 8 equal parts | 4 | 4/8 | Half the strip |
Lay strips with equal whole lengths underneath each other and align their left ends. The selected sections finish at the same place. This is visible evidence that 1/2 = 2/4 = 4/8. These are equivalent fractions: different fraction names for the same amount of the same whole.
Do not decide that the strip with four selected pieces must show the largest amount. Its pieces are smaller. Compare the covered area, using the same whole, rather than counting selected pieces alone.
Worked case 3: place a fraction on a number line
A number line uses distance to show quantity. In these diagrams, the distance from zero to one is one whole unit. The three whole units are exactly the same length and start at the same position.
- On the halves line, divide the whole distance into two equal intervals. Starting at zero, travel one interval. You reach 1/2.
- On the quarters line, divide the same whole distance into four equal intervals. Travel two intervals from zero. You reach 2/4 at the same point.
- On the eighths line, divide the same whole distance into eight equal intervals. Travel four intervals from zero. You reach 4/8 at that same point.
Count intervals, not tick marks. A quarters line has four spaces but five boundary marks when both zero and one are marked. Zero is the starting position. You have travelled zero intervals when your finger is still on zero.
For 3/4, use the quarters line and travel three of its equal intervals from zero. The point is after 1/2 and before 1. Three equal steps out of four identify the point.
Check an unequal-part claim
A learner points to the bottom strip in the picture and says, “It has four pieces. The marked piece must be one quarter.” Look at the sizes. The right-hand piece is three times as wide as each small piece, so the four pieces are not equal shares.
The claim is incorrect. The number four describes how many pieces were drawn, but it does not guarantee quarters. To repair the picture, redraw four equal-width sections across the same rectangle. Alternatively, fold a fresh equal-sized strip into four matching sections and use it as a guide.
Keep a second check in mind: identical fraction names do not promise identical physical amounts when the wholes differ. Half of a long strip can be longer than half of a short strip. Both are halves of their own wholes. Compare like-sized wholes when using strips to compare fraction sizes.
Four matching and explanation tasks
Task 1: match a group to a strip
A group of sixteen counters is separated into four equal shares. One share is circled. Match it to either one of four equal parts of a strip or one of two equal parts of that strip. Explain your choice.
Answer: match it to one of four equal strip parts. Each model selects one quarter of its stated whole. The counter share contains four counters; the strip model selects one quarter of one strip. The objects differ, but the fraction relationship matches.
Task 2: match two fraction names
Four of eight equal sections of a strip are crossed. Which matches that amount on an equal-sized whole: one of two equal sections, or one of four equal sections?
Answer: one of two equal sections matches. Four eighths and one half cover the same area. One quarter covers only half as much area as one half when the whole strips are equal-sized.
Task 3: correct a number-line count
A line from zero to one is divided into four equal intervals. A learner counts zero as “one”, points to the next tick and calls it 2/4. What is that tick’s correct label?
Answer: the first tick after zero is 1/4. Only one of the four intervals has been travelled. The middle tick is 2/4, which is also 1/2. Count the travelled spaces aloud to make the correction.
Task 4: explain the three-quarter point
A whole strip has four equal sections with three crossed. Draw a zero-to-one line that matches it and name the point. How many quarter-length steps remain to reach one?
Answer: divide the line into four equal intervals and mark the point after three steps from zero. Label it 3/4. One quarter-length step remains to reach 1, because four quarter-length steps make the whole distance.
Try a short explain-and-check routine
Choose one model and tell a partner: “My whole is ___. I divided it into ___ equal parts. I selected ___ parts. The fraction is ___.” Ask your partner to make a different model with the same fraction.
If matching pictures is difficult, stay with halves first. Fold one strip, draw its two equal parts and locate halfway on a line. Introduce quarters when the learner can explain why both halves are equal. An adult can read the fraction names aloud while the learner points to the corresponding parts.
Questions about visual fractions
Can the whole be more than one object?
Yes. A stated group, such as sixteen counters together, can be the whole. Name that group before finding a fraction of it.
Does one quarter always mean one item?
No. One quarter of a sixteen-counter group is four counters. One quarter means one of four equal shares of the stated whole.
Why must the parts be equal?
The denominator names equal-sized shares. Four unequal pieces do not automatically represent quarters, even when there are exactly four pieces.
Why are 1/2 and 2/4 at the same point?
Two quarter-length steps cover the same distance as one half-length step. With the same zero-to-one unit, their endpoints coincide.
Do I count the mark at zero as the first part?
No. Zero is where you start. Count each equal interval you travel from zero, rather than counting the boundary marks.
Can I compare strips of different whole lengths?
They can show fractions of different wholes, but covered length alone will not fairly compare the fractions. Use equal-sized whole strips for that comparison.
Do fraction parts have to be coloured?
No. Crosses, outlines, labels or pointing can identify selected parts. The pictures here use crosses and labels as well as shading.
For an adult supporting the activity
Ask the learner to connect one action, one drawing and one explanation rather than memorise isolated pictures. The Institute of Education Sciences mathematics practice guide recommends carefully chosen representations and number lines. Keep the whole visible while the learner explains each connection.
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