Math // Fractions

Fractions as parts of shapes

Partition shapes into parts with equal areas and express the area of each part as a unit fraction of the whole

Ages
8 to 9
Type
Concept
Builds on
2 topics
Leads to
0 topics
Standards
1 code

Quick check

If you ask the learner to fold a square piece of paper into 8 equal parts, can they tell you what fraction of the whole square each part represents?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Partition a rectangle into 6 equal-area parts and label each 1/6
  • Show two different ways to partition a square into 4 equal parts
  • Given a pre-partitioned shape, write the unit fraction for one part

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

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Fractions as parts of shapes

Ages 8 to 9 // Concept

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Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Equal parts of a shape means parts of equal AREA — the same amount of the whole — not parts of the same SHAPE. A square can be split into four equal parts as strips, as quadrants, or as four triangles (both diagonals), and each is 1/4 because the areas are equal. The denominator counts how many equal-area regions the whole is cut into, and each region is 1/b of the whole. Learners should partition shapes several ways, check equal area by cutting and rearranging or by counting grid squares, contrast valid partitions with unequal 'four-piece' traps, and read a pre-partitioned shape by counting equal regions rather than cut lines.

Common misconceptions

  • Believes equal parts must be the same SHAPE (rejects a valid differently-shaped 1/4)
  • Counts pieces without checking equal area — calls any shape cut into 4 pieces 'quarters', even unequal ones
  • Writes the unit fraction from the number of cut lines rather than the number of regions
  • Thinks only the 'standard' partition (e.g. four small squares) is correct

Useful representations

  • A square partitioned several ways into 4 (strips, quadrants, both diagonals)
  • Squared/grid paper to measure and compare areas
  • Overlay or cut-and-rearrange to prove two differently-shaped parts are equal
  • Area-model rectangle labelled in unit fractions

Teaching practices

  • Define equal parts as equal AREA, and show several correct partitions of one shape that look different but cover the same amount
  • Contrast valid equal-area partitions with tempting invalid ones (four pieces of different sizes) so 'four pieces' does not automatically mean 'quarters'
  • Use overlay/rearrangement or a squared grid to check that differently-shaped parts really have equal area
  • When reading a pre-partitioned shape, count the equal REGIONS (not the cut lines) and check they are equal before writing 1/b
  • Connect to the unit-fraction meaning: each equal-area part is 1/b of the whole

Activities to try

  1. 1

    Quarter a square two different ways

    Why it works: Parts that look completely different can each be 1/4 as long as their AREAS are equal and together they cover the whole.

    1. Take a square and cut (or draw cut lines) to make 4 equal parts.
    2. Now make 4 equal parts a DIFFERENT way — e.g. four vertical strips, then four quadrants, then four triangles using both diagonals.
    3. For each attempt, check that every part covers the same amount: cut and rearrange the parts to see if they match, or draw the square on grid paper and count the squares (putting half squares together) in each part.
    4. Label each part 1/4 and state in words why it is one quarter.
    5. Compare a set of four triangles with a set of four strips: confirm both read 1/4 even though they look nothing alike.
  2. 2

    Spot the trap: four pieces vs four equal parts

    Why it works: Counting four pieces is not enough — a quarter must be one of four EQUAL-area parts. Unequal pieces are not quarters even though there are four of them.

    1. Look at a square cut into four pieces of clearly different sizes. Decide: is each piece a quarter? Explain.
    2. Check by comparing the areas — on squared/grid paper, or by cutting and rearranging the pieces to see if they match.
    3. State whether 'four pieces' is the same as 'four equal parts', and why.
    4. Sort several partitioned shapes into 'shows quarters' and 'does not': include an equal-area-but-different-shape example (YES) and a four-unequal-pieces example (NO).
  3. 3

    Read a pre-partitioned shape: count regions, not lines

    Why it works: The denominator is the number of equal regions the whole is cut into, so you count the parts, not the cuts; and the regions must be equal in area before you name the fraction.

    1. Given a shape already cut into equal parts, first check the regions really are equal in area.
    2. Count the number of equal REGIONS (the parts), not the number of cut lines.
    3. Write the unit fraction for one part, e.g. 1/5 for one of five equal regions.
    4. Check your reasoning on a case with three parallel cut lines: three cuts make four parts, so each is 1/4 — not 1/3.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
3.G.2

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