Shapes, partitioning, and area by counting squares
Classifying shapes by attributes, partitioning shapes into equal fractional areas, and measuring area by counting unit squares
About three lessons of 45 to 60 minutes
Shapes have a family tree, and they can be split fairly
A square, a rectangle, and a diamond-turned rhombus look different, but they all belong to the same family: quadrilaterals, because they all have exactly four straight sides. Sorting shapes by what they share, not by what they look like, is the first idea in this unit.
Once a shape is understood, it can be split into equal areas, and each equal part earns a fraction name: split a garden into 4 equal sections and each section is 1/4 of the garden. Then we go one step further and measure exactly how much space is inside a shape, by counting the unit squares that tile it.
- A square, a rectangle, and a rhombusall quadrilaterals, because they all have 4 straight sides
- A rectangular garden split into 4 equal sectionseach section is 1/4 of the garden's area
- A rectangle tiled with unit squares, 4 rows of 5count the squares: 20 square units of area
What students will be able to do
Students will sort two-dimensional shapes into categories (such as quadrilaterals) by shared attributes, partition shapes into parts with equal area and name each part as a unit fraction of the whole, and measure the area of a shape by counting unit squares.
- I can sort a set of shapes into a category, such as quadrilaterals, by checking a shared attribute.
- I can explain why a shape belongs or does not belong to a category, even if it looks unusual.
- I can split a shape into equal-area parts and name each part as a unit fraction.
- I can find the area of a shape by counting the unit squares that tile it.
Standards this unit teaches
- 3.G.A.1Common Core (US)Classify shapes by attributes
Sort shapes into categories like quadrilaterals by shared attributes such as number of sides.
- 3.G.A.2Common Core (US)Shapes and unit fractions
Divide shapes into equal areas and name each part as a unit fraction of the whole.
- 3.MD.C.6Common Core (US)Measure area by counting
Measure areas by counting unit squares in square centimetres, metres, inches, and feet.
- AC9M3M02Australian Curriculum v9 (ACARA)Perimeter and area
Measure and approximate the perimeter and area of shapes and enclosed spaces using suitable formal and informal units.
- AC9M3N02Australian Curriculum v9 (ACARA)Unit fractions and their multiples
Recognise and represent unit fractions such as halves, thirds, quarters, fifths and tenths, and combine same-denominator fractions to make a whole.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Quadrilateral
- any shape with exactly four straight sides
- Attribute
- a property a shape has, such as its number of sides or whether it has any right angles
- Unit square
- a single square, 1 unit by 1 unit, used to measure area by counting
- Area
- how much flat space is inside a shape, measured in square units
- Unit fraction
- a fraction with 1 on top, one single equal part, such as 1/4
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Classifying shapes by shared attributes
ConcreteSort a mixed pile of shape cards by asking one question at a time: how many straight sides does it have? A shape with exactly four straight sides is a quadrilateral, whatever its exact form. Squares, rectangles, rhombuses, and trapezoids are all quadrilaterals, because sorting by category looks only at the shared attribute, not the overall look.
Two traps to head off directly: rotating a shape does not change its attributes (a square turned to sit on a corner is still a square, with 4 equal sides and 4 right angles), and a category can include shapes that look quite different from each other, as long as they share the defining attribute.
- A shape has 4 straight sides but no right angles. Is it still a quadrilateral? Why?
- Does turning a rectangle so it sits on its corner change how many sides or right angles it has?
2. Partitioning a shape into equal-area unit fractions
PictorialTo name a part of a shape as a fraction, the parts must have equal area, not just look like there are the right number of pieces. Once a whole shape is split into b equal-area parts, each single part is the unit fraction 1/b of the whole, exactly the same idea as a fraction bar or circle from the fractions unit, now applied to a real shape like a garden or a field.
A rectangular garden has a total area of 20 square metres and is split into 4 equal sections, one for each type of vegetable. What is the area of each section, and what fraction of the garden is each section?
- The garden is split into 4 equal-area parts, so each part is the unit fraction 1/4 of the whole garden.
- Divide the total area by the number of equal parts: 20 ÷ 4 = 5.
Answer: Each section has an area of 5 square metres and is 1/4 of the garden.
- If the same garden were split into 5 equal sections instead, what fraction would each section be?
- Why must the sections be equal in area, not just equal in number, before you can call each one a unit fraction?
3. Measuring area by counting unit squares
PictorialArea is how much flat space is inside a shape, measured by tiling it with unit squares and counting them. A rectangle that is 4 units by 5 units can be filled with 4 rows of 5 unit squares, which is the same total as the multiplication fact 4 x 5.
The same counting idea works on an irregular shape by breaking it into rectangles first, counting the squares in each piece, then adding the pieces. An L-shaped tile made of a 4-by-2 rectangle (8 squares) joined to a 3-by-1 rectangle (3 squares) has a total area of 8 + 3 = 11 square units.
Count the area of a rectangle that is 4 units tall and 5 units wide, tiled with unit squares.
- Count the squares in one row: 5.
- Count the number of rows: 4.
- Total squares = 4 rows x 5 squares per row = 20.
Answer: The area is 20 square units.
- Why does counting rows of squares give the same answer as multiplying rows by columns?
- For the L-shaped tile, why do we add the two rectangle areas instead of multiplying them?
Common misconceptions and how to address them
MisconceptionA rotated shape is a different shape, for example a square turned to sit on one corner is called a 'diamond' and not sorted as a square.
Why it happens: Orientation is very visually striking, so students judge shape identity by how it sits on the page rather than by its actual attributes.
How to address it: Physically rotate a paper square in front of the class while counting its sides and right angles out loud each time: the attributes never change, only the picture's orientation.
Misconception'Quadrilateral' only means shapes that look like rectangles, so a rhombus or trapezoid is rejected from the category.
Why it happens: Rectangles and squares are the most familiar four-sided shapes, so they become the unconscious mental picture of the whole category.
How to address it: Sort by counting sides only, as a strict rule: exactly four straight sides earns the label quadrilateral, regardless of the angles or side lengths.
MisconceptionAny split into the right number of pieces counts as fraction parts, even when the pieces are clearly different sizes.
Why it happens: Early partitioning practice sometimes shows unequal-looking cuts, and the habit of counting pieces without checking equal area carries over from that.
How to address it: Overlay or compare the pieces directly (cut them out and stack them) before naming any fraction; unequal area means the split cannot be named as a unit fraction yet.
MisconceptionWhen counting unit squares for area, students skip partially-covered squares at the edge, or count the same square twice where two counted regions meet.
Why it happens: Counting many small identical squares accurately is a genuinely hard tracking task without a system.
How to address it: Shade or tally every square as it is counted, moving row by row in a fixed order, so no square is skipped or counted twice.
MisconceptionCounting only the squares along the outside edge of a shape (as if measuring perimeter) instead of filling and counting every square inside the shape.
Why it happens: Perimeter and area are taught close together and both involve 'going around' or 'counting' a shape, so the two ideas blur.
How to address it: Say the difference every time: perimeter walks the edge and adds lengths, area fills the inside and counts squares. Shade the whole inside of the shape before counting for area.
Guided practice (with answers)
1. Sort these by attribute: a square, a triangle, a rectangle, a rhombus, a pentagon. Which are quadrilaterals?
Answer: The square, rectangle, and rhombus are quadrilaterals, because each has exactly 4 straight sides. The triangle (3 sides) and pentagon (5 sides) are not.
2. A pizza is cut into 6 equal-area slices. What fraction is each slice?
Answer: 1/6, the unit fraction for 6 equal parts.
3. A rectangular field with area 36 square metres is split into 6 equal sections for planting. What is the area of each section?
Answer: 6 square metres, since 36 ÷ 6 = 6.
4. Find the area of a rectangle that is 3 units by 6 units, by counting unit squares.
Answer: 18 square units, since 3 rows of 6 squares is 3 x 6 = 18.
5. An L-shaped tile is made of a 5-by-2 rectangle joined to a 2-by-2 rectangle. What is its total area?
Answer: 14 square units: 5 x 2 = 10, plus 2 x 2 = 4, and 10 + 4 = 14.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. There is not yet a dedicated shape-classification printable on ChalkBee, so use the guided-practice sorting activity and the assessment below as the main practice for that skill, alongside the fraction and area worksheets for the other two standards.
Differentiation
- For classifying, sort real physical shape cutouts first, and sort by one attribute (number of sides) before adding a second (right angles or not).
- For partitioning, keep to shapes that are easy to fold in half, such as squares and rectangles, before introducing odd numbers of parts like thirds or fifths.
- For area, always tile the shape with real physical unit squares before moving to a drawn grid.
- Sort shapes by two attributes at once, such as 'quadrilateral and has a right angle', and describe what changes.
- Partition the same shape two different ways into fourths (for example, four strips versus four smaller squares) and confirm both give equal-area parts.
- Find the area of an irregular shape by splitting it into more than two rectangles.
Assessment: exit ticket
A short exit ticket sampling shape classification, partitioning, and area by counting.
1. A shape has 4 straight sides, no right angles, and all sides different lengths. Is it a quadrilateral? Why?
Answer: Yes. It has exactly 4 straight sides, which is the only requirement for the quadrilateral category.
2. A cake with area 24 square inches is cut into 8 equal-area pieces. What fraction is each piece, and what is its area?
Answer: Each piece is 1/8 of the cake, with an area of 3 square inches (24 ÷ 8 = 3).
3. Find the area of a rectangle that is 6 units by 4 units by counting unit squares.
Answer: 24 square units, since 6 rows of 4 (or 4 rows of 6) is 24.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 classifying shapes (section 1), Lesson 2 partitioning into unit fractions (section 2), Lesson 3 area by counting plus the exit ticket (section 3 and assessment).
- Language to keep saying: count the sides, equal area not just equal pieces, fill the inside and count squares. These three phrases target the unit's main misconceptions directly.
- This unit deliberately leaves the area formula (length times width) for the paired Grade 3 area and perimeter unit; here the point is the concrete act of counting unit squares, the meaning behind that formula.
- Curriculum note: ACARA's AC9M3M02 covers perimeter and area together at Year 3 using informal and formal units, matching the counting-squares half of this unit closely, and AC9M3N02 covers unit fractions at the same year as the partitioning half.
- Present mode and print both work: use Present to sort shape cards live and tile a rectangle with squares on screen, then print the worksheets for independent practice.