Example 1 — Fractions: Building Equivalence, Addition and Comparison with Manipulatives

Learning objectives (measurable)

• Students will represent fractions with concrete fraction tiles and show equivalence (e.g., 1/2 = 2/4).
• Students will add simple fractions with like and unlike denominators using manipulatives and record symbolic answers.
• Students will explain and justify answers orally and in writing.

Materials (per pair or small group)

• Fraction tiles or fraction circles (1, 1/2, 1/3, 1/4, 1/6, 1/8)
• Plain paper plates or rectangular paper “pizza” templates
• Scissors, glue, markers
• Whiteboard or group recording sheet (OneNote page template for class)
• Optional: kitchen measuring cups for recipe scaling task (real‑life closeness)

Preparation (10–15 min)

• Prepare kits: one set fraction tiles per group; preprinted pizza templates if used.
• Create a OneNote/board page to record sample work and to display final symbolic solutions.

Step‑by‑step (60 min example)

1. Motivation (5 min)
   • Show a real object: an actual pizza or recipe for 4 servings. Ask: “If we share this between 3 people, how much does each get?”
   • State objectives and success criteria: “By the end you will show equivalent fractions, add fractions and explain your work.”
2. Mini‑teach (8–10 min)
   • Quick demonstration: model 1/2 with a circle tile and show how two 1/4 tiles fill the same area. Do this on camera or board so every student sees.
   • Pose 1 or 2 guiding questions (Why do 2/4 and 1/2 look the same? How do we record that symbolically?)
3. Hands‑on activity: Station Tasks (25–30 min)
   • Station A (Equivalence): Build 1 whole using different sets (e.g., 1/3+1/3+1/3; 1/4+1/4+1/2). Record combinations and write symbolic equations.
   • Station B (Addition with unlike denominators): Using pizza templates cut into 1/3 and 1/4 slices, combine 1/3 + 1/4, rearrange to find common denominators (use tile overlays). Translate result to symbolic form.
   • Station C (Real‑life problem): Recipe scaling — a cookie recipe uses 3/4 cup sugar for 2 people; scale to 5 people. Use measuring cups to model the fractions and sum.
   • Roles: Materials Manager, Recorder, Checker, Presenter. Rotate roles each time.
   Formative checkpoints (every 8–10 min):
   • Teacher circulates, asks diagnostic prompts: “How did you decide the common denominator? Where did you get stuck?”
   • Quick one‑minute round: each group states one result/one question.
4. Reflection & advance abstraction (10–12 min)
   • Groups present one artefact and show equations on the board. Emphasize the language: numerator, denominator, unit fraction, improper fraction, simplification.
   • Use the Information Ladder formative tool: students write 1) I know, 2) I understand, 3) I can use, 4) I noticed.
5. Summative activity / homework (5–10 min)
   • In class: quick problem: Show whether 3/8 + 1/4 is equal to 5/8 using tiles and then symbolically justify.
   • Homework: an applied problem (scale a recipe or design a 12‑slice cake with certain fraction requirements) with rubric: correct representation (40%), correct symbolic answer (40%), clear justification (20%).

Differentiation & extensions

• Struggling learners: give pre‑assembled fraction benchmarks (1/2, 1/3, 1/4) and two‑step tasks.
• Advanced learners: introduce mixed numbers and improper fractions or challenge: create three different fraction pairs that add to 1.
• Cultural closeness: use local foods (e.g., bread loaves, flatbreads) instead of pizza.

Common misconceptions & teacher prompts

• Misconception: “Add denominators” — prompt: “If one half and one quarter were lengths, how would you compare their pieces?”
• Misconception: equivalence is only about numbers, not areas — prompt students to compare areas, then translate to symbols.

Assessment tips (formative & summative)

• Observation checklist: uses manipulative correctly, explains reasoning, maps concrete to symbolic representation.
• One‑minute round or exit ticket: “Show 2/3 as tiles and write one sentence explaining why it equals 4/6.”