See what the two numbers mean
The full rectangle represents one whole paper ribbon, including both blue and white parts. It is divided into four equal parts. Only the blue parts with dots are shaded.
Denominator: 4
How many equal parts make one whole?
Numerator: 3
How many of those parts are selected?
Build it yourself
Change the pieces. Watch the quantity.
3/4 means three copies of 1/4. One whole is still four fourths.
Try this: choose thirds, then select five pieces. Three pieces fill the first whole. Two more fill part of the next. You have 5/3, not 5/6: one strip is still the chosen whole.
Catch the common mistake
Pause and explain
Draw six equal parts of one whole and shade five. Why is the fraction 5/6 rather than 5/1?
See the reasoning
The unit size is one sixth because six equal parts make the whole. Five of those units are selected, so the numerator is 5 and the denominator remains 6.
More explanation, worked examples, and independent practiceRead the complete lesson, including answers, teaching support, and scoring rubrics.
Full explanation
Learning goal: Explain what the numerator and denominator tell us, build a unit fraction, and combine unit fractions to describe a quantity. This lesson targets Ohio 3.NF.1. Allow two 30-minute sessions with paper strips, crayons, and counters. Students should already be able to share a group equally and count equal groups.
Start with one sheet of paper. That sheet is our whole. Fold it into four parts of equal size. One part is one fourth, written 1/4. A fraction names a quantity in relation to a chosen whole. It is not just a pair of numbers separated by a line.
The bottom number, called the denominator, tells how many equal parts make one whole. The top number, called the numerator, tells how many of those parts we are counting. In 3/4, each part is one fourth and we count three of them. Say “three fourths,” not “three out of anything.” Three fourths can describe three parts of a shape, a length, or a quantity beyond one whole when more fourths are counted.
A unit fraction has 1 as its numerator. One sixth, 1/6, is the quantity formed by one of six equal parts of a whole. Three sixths is three copies of that unit fraction: 1/6 + 1/6 + 1/6. The parts must have equal size. Four sections made by uneven cuts are not automatically fourths. Count alone cannot establish a fraction.
The whole also matters. Half of a large sheet can be larger than half of a small sheet. Both quantities are one half of their own whole. When comparing physical pieces, first check that the wholes have the same size. Within one whole, cutting into more equal parts makes each individual part smaller. A sixth of the same strip is smaller than a third because the strip has been shared into more pieces.
Additional worked examples
Example 1: Make and explain 3/8. Draw a rectangle and divide it into eight equal-size sections. Shade three. Step 1: Identify the entire rectangle as one whole. Step 2: Count eight equal parts, so each has size 1/8. Step 3: Count three shaded parts. The shaded quantity is three copies of 1/8, or 3/8. The denominator remains 8; selecting three parts does not change the size of each part.
Example 2: Critique an unequal partition. A strip has four sections, but one section takes half the strip. A student calls that large section 1/4 because it is one of four sections. The conclusion is incorrect: fourths must have equal size. Fold another same-size strip into four equal pieces. The large section covers two of those pieces, so it is 2/4 of the strip, not 1/4. The repair is to measure or fold equal parts, not just recount sections.
Example 3: Count beyond one whole. Place two identical rectangles side by side. Divide each into three equal parts. Select all three parts of the first and one part of the second. Each piece is 1/3 of one rectangle, and four such pieces make 4/3. Do not rename the quantity 4/6: the chosen whole is one rectangle, not both rectangles together.
Independent practice and scoring
Guided practice: Fold a strip into six equal parts. Point to one sixth, then shade five sixths. Explain why the same denominator appears in 1/6 and 5/6. Expected reasoning: the unit size remains one sixth, but the number of selected units changes from one to five.
Independent work: (1) Draw a whole divided into four equal parts and shade 3/4. (2) Write 5/8 as a sum of unit fractions. (3) Draw two same-size wholes in thirds and show 5/3. (4) Explain why one of three unequal pieces cannot automatically be called 1/3. Students must make the models themselves; choosing a drawing does not demonstrate that skill.
Answer guide: (1) Three of four equal areas shaded. (2) Five copies of 1/8. (3) One complete whole and two thirds of another. (4) Thirds require three equal-size parts of a specified whole. Award 2 points per task: 1 for the correct model or statement and 1 for explaining equal parts and the chosen whole. Use the errors to plan feedback rather than treating the score as a standardized proficiency level.
Support and extension: Let students fold real paper before drawing. Ask them to finish “Each part is __ because the whole has __ equal parts.” Extend by asking whether two different-looking partitions can both show fourths. Equal area is required; identical appearance is not. End with an exit slip explaining 2/3 in words and a labeled drawing.
Ohio learning standard: 3.NF.1 · Framework: math-2017. Visual activities supplement the full lesson and its independent performance tasks.