Fractions: Halves, Thirds, and Quarters
Warm-up
I have a chocolate bar (draw a rectangle). I want to share it equally with my friend. How should I cut it? Students suggest. If I cut it unevenly, is this fair? Why not? The intuition of fairness motivates the mathematical requirement of equal parts.
Explore
Paper folding: fold a piece of paper into halves in 3 different ways. Are all the ways valid halves? (Yes, if the two parts are equal.) Fold into quarters in 2 different ways. Are all valid quarters? Now try thirds: harder to fold evenly. How do you check if thirds are equal?
Formalize
Fair sharing context: 3 children share 2 granola bars equally. Each child gets two thirds of one bar... or does each get one third of two bars? Both descriptions give the same amount: 2 thirds. This equivalence of physical sharing situations with fraction notation is a deep connection worth dwelling on.
Fractions: Halves, Thirds, and Quarters
Comparing fractions: show a whole divided into 4 equal parts. Colour 1, then 2, then 3. Which is greatest? (3 quarters.) Arrange in order. Now divide a different whole into 3 equal parts. Is one third more or less than one quarter? This requires careful attention to the denominator: with more pieces, each piece is smaller.
Practice
Students fold paper into halves, thirds, and quarters in multiple ways, verifying equality each time. Solve 4 fair-sharing problems and draw the fraction model. Exit ticket: draw a rectangle divided into thirds and shade two thirds.
Exit ticket
Students fold paper into halves, thirds, and quarters in multiple ways, verifying equality each time. Solve 4 fair-sharing problems and draw the fraction model. Exit ticket: draw a rectangle divided into thirds and shade two thirds.
Step 1: Give partners a paper rectangle "brownie" to share equally between 2 people. Most fold once — vertically, horizontally, or diagonally — and all three folds are correct halves.
Step 2: Now the provocation: cut a rectangle into two pieces that are NOT equal, and call the bigger piece "my half." Students should protest: halves must be EQUAL shares. Unequal pieces are just… pieces.
Step 3: State the rule that survives the protest: cutting into 2 pieces isn't enough — the 2 pieces must be equal in size (amount of brownie), even if they're different SHAPES. The two triangles from a diagonal fold don't look like the two rectangles from a vertical fold, but each is exactly half the brownie.
Step 4: Test with quarters: fold a square into 4 equal parts three different ways (little squares, strips, triangles). All quarters; all different shapes. Equal AMOUNT is the test — not matching looks.
Step 1: Take predictions: "3 is more than 2, so a third is bigger" is the natural wrong guess — and worth surfacing on purpose.
Step 2: Cut two SAME-SIZE paper pizzas: one into 2 equal slices, one into 3 equal slices. Hand one slice of each to a student to hold side by side.
Step 3: The half is visibly, physically bigger. Say WHY: the same pizza got split among MORE people, so each share must get SMALLER. More sharers → smaller shares.
Step 4: The rule as students should say it: "the more pieces you cut something into, the smaller each piece is." Then stress-test: "Would you rather have half of a cookie or a third of a CAKE?" — a joyful reminder that fractions describe a piece OF SOMETHING, and the something matters.
What you see: a circle divided into 4 wildly unequal regions, 3 shaded, labelled ¾ confidently.
Step 1: Locate the correct part of the thinking: the counting is right — 3 shaded parts out of 4 total parts. The student has the out-of language.
Step 2: Expose the missing condition with money: cut a paper dollar into 4 unequal pieces and offer them "3 of the 4 pieces" — the three tiny ones. Would they take that deal for 75 cents? Suddenly equal parts MATTER.
Step 3: Repair the diagram: re-divide the circle into 4 equal parts (fold in half, then in half again), shade 3. THAT is three quarters.
Step 4: Institute the two-question check for any fraction picture: (1) Are the parts EQUAL? (2) How many of them are marked, out of how many? Question 1 must pass before question 2 means anything.