Fraction Concepts
Warm-up
Show a circle divided into 4 equal parts with 1 shaded. What fraction is shaded? (1/4.) Now show a circle divided into 4 unequal parts with 1 shaded. Can I still call this 1/4? (No: the parts are not equal.) Why does it matter that the parts are equal?
Explore
Fraction stations: (1) Region: fold paper into equal parts, shade a fraction, write the notation. (2) Set: given 12 counters, show 2/3 in a red/blue arrangement. (3) Linear: place fractions on a number line. (4) Cultural: identify what fraction of the medicine wheel represents each season and direction.
Formalize
Compare 1/2 and 1/4 using all three models. In the region model: half is bigger (more shaded). In the set model: half of 8 is 4, a quarter of 8 is 2. On the number line: 1/2 is further from 0. All three models agree. Counter-intuitive check: which is bigger, 1/3 or 1/5? The denominator is bigger for fifths, but each piece is smaller.
Fraction Concepts
Connect to the medicine wheel: the wheel has 4 directions/seasons, each a quarter. If we meet on the third day of the third season, what fraction of the year has passed? (Roughly 2.5/4 = 5/8.) This is informal reasoning that deepens fraction sense.
Practice
Students represent 6 fractions in all three models, then order 5 fractions with the same denominator from least to greatest. Exit ticket: draw and name the fraction represented by 3 shaded parts of a 5-part whole.
Exit ticket
Students represent 6 fractions in all three models, then order 5 fractions with the same denominator from least to greatest. Exit ticket: draw and name the fraction represented by 3 shaded parts of a 5-part whole.
Step 1: Draw a number line from 0 to 1. Announce the day's shift: fractions aren't only PIECES of things — they are NUMBERS with addresses on the line.
Step 2: The denominator is the cutting instruction: fourths → cut the 0-to-1 stretch into 4 EQUAL hops. Mark the cuts: 1/4, 2/4, 3/4.
Step 3: The numerator is the counting instruction: take 3 hops from 0 → land at 3/4. Circle it: that point IS the number three-fourths.
Step 4: Sanity checks that only the number line makes visible: 3/4 lives LEFT of 1 (it's less than one whole); it's exactly one hop short of 1; and 2/4 sits precisely at the halfway mark (a preview of equivalence).
Watch for: students who divide the line into 4 marks instead of 4 SPACES (they'll put 3 marks and call the line done). Hops between marks are what count — count spaces, not tick marks.
Step 1: Predict before building. Many students vote 2/5 bigger "because 5 beats 3." Log the prediction — it's about to earn its correction.
Step 2: Build both with fraction strips of the SAME whole: a strip cut into 3 equal parts (shade 2) and an identical strip cut into 5 equal parts (shade 2).
Step 3: Compare the shaded lengths directly: 2/3 clearly reaches farther. Explain with the sharing story: thirds are BIG pieces (the whole was only split 3 ways); fifths are smaller pieces (split 5 ways). Two big pieces beat two small pieces.
Step 4: State the same-numerator rule the class just discovered: when the tops match, the SMALLER bottom wins — bigger pieces, same count of them.
Step 5: One more to cement: order 3/8, 3/4, 3/6 from least to greatest without building. (3/8 < 3/6 < 3/4 — bottoms 8, 6, 4 give pieces small to large.)
The setup: Amir ate 1/2 of his granola bar; Bea ate 1/4 of her giant cookie. A student declares Amir ate more "because a half beats a quarter." Did he?
Step 1: Surface the hidden assumption: halves beat quarters OF THE SAME WHOLE. Amir's whole and Bea's whole are different objects — a slim bar versus a dinner-plate cookie.
Step 2: Draw both to rough scale: half the small bar is a small amount; a quarter of the enormous cookie is plainly bigger.
Step 3: The corrected sentence, said carefully: 1/2 > 1/4 as NUMBERS (of the same whole); but 1/2 OF A SMALL THING can be less food than 1/4 OF A HUGE THING. A fraction always answers "…of WHAT?"
Step 4: Class rule from today onward: no fraction claim is complete without naming its whole. Post it: "HALF OF WHAT?"
Quick check: "Would you rather have 1/2 of our classroom's cookies or 9/10 of MY cookie?" — argue with the rule.