# Kamran's Fraction Shop ## A complete adaptive fractions lesson and classroom game **Audience:** Grades 4–7 and learners who need a concrete restart with fractions **Format:** Eight short units, approximately 8–25 minutes each **Core model:** Real kitchen press cutters + Minecraft-style cheese blocks **Story goal:** Help Kamran calculate fair cheese portions and prices for his poutine shop --- ## The story One gloomy day, Kamran's friend took him to Costco and bought him a poutine. Kamran loved the cheese curds so much that he opened his own poutine shop. There was one problem: Kamran was old-school, not very comfortable with technology, and had no idea how to price cheese fairly. Cheese blocks came in different sizes, flavours, and prices. Customers ordered different amounts. Kamran needed math students to help him answer four questions: 1. How many equal pieces will a cutter make? 2. What fraction of the original block is one piece? 3. How many pieces are needed across one or more whole blocks? 4. What should one cube—or one poutine order—cost? > Kamran: “The poutine is delicious. The bookkeeping? Uh oh—the math isn't mathing.” --- ## Learning outcomes By the end, a learner should be able to: - [ ] identify a whole and explain why fractional pieces must be equal; - [ ] connect rows × columns to factors and total pieces; - [ ] name unit fractions such as `1/2`, `1/4`, `1/6`, and `1/8`; - [ ] explain why 5 and 7 are prime using cutter arrays; - [ ] calculate a fraction of another fraction; - [ ] represent several wholes as an improper fraction; - [ ] add fractions with the same denominator; - [ ] find the cost of one equal cheese cube; - [ ] calculate the cheese cost of a poutine order; - [ ] explain their reasoning with words, pictures, or manipulatives. --- ## Adaptive starting point Grade is only a starting hint. The game uses a three-question placement round, then adjusts practice according to performance. ### Support path - Uses denominators 2, 3, 4, and 6 first. - Keeps the whole visible. - Encourages counting every piece. - Removes time pressure. - Offers sentence frames and visual hints. ### On-level path - Uses factor pairs, equivalent fractions, same-denominator addition, and simple unit cost. - Mixes images with symbolic notation. - Asks learners to explain what the numerator and denominator mean. ### Stretch path - Uses fractions of fractions, several wholes, improper fractions, and multi-step price questions. - Removes some visual support after understanding is demonstrated. - Requires a written or spoken justification. Students can raise or lower the challenge manually at any time. --- # Unit 1 — The Poutine Promise **Estimated time:** 8–12 minutes **Visual:** `/assets/kamran-poutine-story.png` ## Story > Kamran: “My friend bought me a Costco poutine. One bite and—wow—the cheese changed my life. So naturally, I opened a shop. Was that sensible? We will let the fractions decide.” ## Teach A **whole** is one complete object. A fraction describes equal parts of that whole. If one student gets a huge piece and another student gets a tiny piece, the pieces cannot both be called `1/4`. Fourths must be equal in size. ## Your turn A whole cheese block is divided into 4 equal pieces. What fraction is one piece? - [ ] `1/2` - [ ] `1/3` - [x] `1/4` - [ ] `4/1` **Reason:** The denominator 4 names all four equal pieces. The numerator 1 counts the one selected piece. ## Say it back Complete the sentence: > A fraction is fair only when the pieces are **equal**. ## Exit check - [ ] I can point to one whole. - [ ] I can explain why the pieces must be equal. - [ ] I can identify one piece out of four as `1/4`. --- # Unit 2 — One Whole Cheese **Estimated time:** 10–15 minutes **Visual:** `/assets/03-3d-voxel-cheese/00-one-whole-voxel-cheese-block.png` ## Story Kamran starts with one complete Minecraft-style cheese block. > Kamran: “This is one whole. It might be huge or small in real life, but in our math it is one complete block.” ## Teach with the cutters | Cutter | Total equal pieces | Size of one piece | |---|---:|---:| | `1 × 2` | 2 | `1/2` | | `2 × 2` | 4 | `1/4` | | `2 × 3` | 6 | `1/6` | | `2 × 4` | 8 | `1/8` | The total number of equal pieces becomes the denominator of one piece. ## Your turn A `2 × 3` cutter makes how many equal pieces? - [ ] 5 - [x] 6 - [ ] 8 - [ ] 23 Therefore one piece is `1/6` of the whole. ## Hands-on prompt 1. Set the cutter to 2 rows and 3 columns. 2. Count every cell. 3. Touch one cell. 4. Say: “This one cell is one-sixth of the whole.” ## Exit check - [ ] I can use rows × columns to count pieces. - [ ] I can name one piece as a unit fraction. --- # Unit 3 — Rows, Columns, and Factors **Estimated time:** 12–18 minutes **Visual:** `/assets/01-kitchen-press-cutters/03-2x3-six-pieces-press-cutter.png` ## Teach A cutter setting is an array. `rows × columns = total pieces` Examples: - `2 × 3 = 6` - `3 × 2 = 6` - `2 × 4 = 8` - `4 × 2 = 8` - `3 × 6 = 18` - `6 × 3 = 18` The order changes the orientation, not the total. ## Your turn Which cutter is a factor-pair match for `3 × 6`? - [ ] `2 × 8` - [ ] `1 × 17` - [x] `6 × 3` - [ ] `4 × 5` ## Explain aloud > “Three and six are factors of eighteen because three rows of six make eighteen equal pieces.” ## Support option Build the array with cubes before writing the multiplication sentence. ## Stretch option List every rectangular cutter setting for 24 pieces. **Answer:** `1 × 24`, `2 × 12`, `3 × 8`, `4 × 6` and their rotations. --- # Unit 4 — Prime Cutters **Estimated time:** 10–15 minutes **Visual:** `/assets/02-countable-voxel-cheese-blocks/06-5-blocks-matches-1x5.png` ## Story > Kamran: “I asked for a rectangular 5-piece cutter with several rows. The supplier laughed politely.” Five can be arranged as: - `1 × 5` - `5 × 1` It cannot make another whole-number rectangle. Seven behaves the same way. ## Definition A **prime number** has exactly two factors: 1 and itself. ## Your turn Why is 5 prime? - [ ] It is odd. - [x] It has only 1 and 5 as factors. - [ ] It is less than 10. - [ ] It makes better cheese. ## Compare Six is not prime because it can form `1 × 6` and `2 × 3`. ## Exit check - [ ] I can use an array to show why 5 is prime. - [ ] I can use an array to show why 7 is prime. - [ ] I can explain why 6 is composite. --- # Unit 5 — Fractions of Fractions **Estimated time:** 15–22 minutes **Visual:** `/assets/02-countable-voxel-cheese-blocks/04-8-blocks-matches-2x4.png` ## Story Kamran takes one-half of a cheese block. Then he cuts that half into four equal pieces. Each new piece is one-fourth **of** one-half. `1/4 × 1/2 = 1/8` The final piece is one-eighth of the original whole. ## Why this works Imagine the original block first cut into 2 large sections. Each section is then cut into 4 smaller sections. `2 × 4 = 8` equal small pieces in the original whole. ## Your turn What is one-half of one-fourth? - [ ] `1/2` - [ ] `1/6` - [x] `1/8` - [ ] `2/4` ## Support option Draw a rectangle. Split it into halves vertically. Split every half into fourths horizontally. Shade one final cell. ## Stretch option Find `2/3` of `3/4`. `2/3 × 3/4 = 6/12 = 1/2` ## Say it back > “The word **of** tells me to multiply the two fractional cuts.” --- # Unit 6 — More Than One Whole **Estimated time:** 14–20 minutes **Visual:** `/assets/04-grocery-scale-voxel-cheese-blocks/01-extra-large-cheddar-grocery-block.png` ## Story A real shop can process several grocery or farm blocks—not always one whole. Three whole blocks are each cut into fourths: - 1 whole makes 4 quarter-pieces. - 3 wholes make `3 × 4 = 12` quarter-pieces. - The result can be written `12/4`. - `12/4 = 3` wholes. ## Meaning of the numbers In `12/4`: - 12 counts all the pieces; - 4 names the size of every piece; - the fraction represents 3 complete wholes. ## Your turn Two whole blocks are each cut into eighths. How many eighth-sized pieces are produced? **Answer:** `2 × 8 = 16`, so the collection is `16/8` or 2 wholes. ## Stretch option If five eighth-sized pieces are used, how much remains? `16/8 − 5/8 = 11/8 = 1 3/8` --- # Unit 7 — Adding the Order **Estimated time:** 15–22 minutes **Visual:** `/assets/05-poutine-story-payoff/01-generic-canadian-food-court-poutine.png` ## Story One order needs `1/4` of a block. Another order needs `2/4`. The pieces are the same size, so count them: `1/4 + 2/4 = 3/4` The denominator remains 4 because every piece is still a fourth. ## Important mistake to avoid `1/4 + 2/4` is **not** `3/8`. Adding the denominator would pretend the pieces changed size. They did not. ## Your turn `2/8 + 3/8 = ?` - [ ] `5/16` - [x] `5/8` - [ ] `6/8` ## Unlike denominators Before adding differently sized pieces, rename them using a common denominator. `1/2 + 1/4 = 2/4 + 1/4 = 3/4` ## Drag-and-drop activity Fill an order with an exact target number of equal cheese cubes. Count the cubes before serving the tray. --- # Unit 8 — Price the Cheese **Estimated time:** 18–25 minutes **Visual:** `/assets/04-grocery-scale-voxel-cheese-blocks/03-large-marble-grocery-block.png` ## Story Cheddar, mozzarella, marble, and pepper-jack blocks can have different sizes and costs. To compare them fairly, Kamran finds the cost of one equal cube. `unit cost = total block cost ÷ number of equal cubes` ## Worked example A block costs `$24` and makes 12 equal cubes. `$24 ÷ 12 = $2.00 per cube` If one poutine uses 3 cubes: `3 × $2.00 = $6.00 of cheese` ## Your turn A block costs `$18` and makes 12 cubes. 1. What is the cost per cube? `$18 ÷ 12 = $1.50` 2. What is the cheese cost for 4 cubes? `4 × $1.50 = $6.00` ## Compare two suppliers | Supplier | Block cost | Equal cubes | Cost per cube | |---|---:|---:|---:| | A | `$24` | 12 | `$2.00` | | B | `$30` | 20 | `$1.50` | Supplier B costs more for the whole block but less for each cube. ## Final shop challenge Kamran buys 3 blocks. Each block costs `$28` and makes 16 equal cubes. A poutine uses 5 cubes. 1. Total cubes: `3 × 16 = 48` 2. Total cheese cost: `3 × $28 = $84` 3. Unit cost: `$84 ÷ 48 = $1.75` 4. Cheese cost per poutine: `5 × $1.75 = $8.75` 5. If Kamran wants `$3.25` beyond the cheese cost to cover fries, gravy, labour, and margin, the starting price would be `$12.00`. > This simplified classroom model is for learning. A real business would also include waste, taxes, utilities, rent, and other costs. --- # Unit 9 — Equivalent Cuts **Estimated time:** 12–18 minutes ## Teach One-half and two-fourths cover the same amount of one whole block: `1/2 = 2/4` Multiplying the numerator and denominator by the same nonzero number changes the name of the fraction, not its value. `2/3 × 2/2 = 4/6` ## Your turn Choose the fraction equivalent to `3/5`: - `6/10` — **correct** - `4/6` - `3/10` ## Say it back > “Equivalent fractions name the same amount with different equal-sized pieces.” --- # Unit 10 — Common Denominators and LCM **Estimated time:** 16–24 minutes ## Story Kamran must combine `1/3` of a block with `1/4` of a block. Thirds and fourths are different piece sizes, so they cannot be counted together yet. The least common multiple of 3 and 4 is 12. `1/3 = 4/12` `1/4 = 3/12` Now the pieces match: `4/12 + 3/12 = 7/12` ## LCM routine 1. List or reason through multiples of both denominators. 2. Choose the first positive multiple they share. 3. Rename each fraction using that denominator. 4. Add or subtract the numerators. 5. Simplify if possible. ## Your turn `1/2 + 1/3 = ?` The LCM of 2 and 3 is 6, so: `3/6 + 2/6 = 5/6` --- # Unit 11 — Subtract the Sold Cheese **Estimated time:** 12–18 minutes ## Matching denominators `7/8 − 3/8 = 4/8 = 1/2` The denominator remains 8 because the piece size remains eighths. Only the number of pieces changes. ## Unlike denominators `3/4 − 1/6` LCM(4, 6) = 12: `9/12 − 2/12 = 7/12` ## Mistake to avoid Do not subtract denominators. `7/8 − 3/8` is not `4/0`. A denominator of zero is not a valid piece size. --- # Unit 12 — Mixed Numbers and Improper Fractions **Estimated time:** 15–22 minutes Two whole blocks and one quarter are written as `2 1/4`. Each whole has four quarters: `2 × 4 + 1 = 9` Therefore: `2 1/4 = 9/4` To move back, divide: `9 ÷ 4 = 2 remainder 1`, so `9/4 = 2 1/4`. ## Your turn Convert `11/3` to a mixed number. `11 ÷ 3 = 3 remainder 2`, so the answer is **`3 2/3`**. --- # Unit 13 — Reciprocals and Fraction Division **Estimated time:** 16–24 minutes The reciprocal of a nonzero fraction swaps its numerator and denominator. `2/5 ↔ 5/2` To divide by a fraction, multiply by its reciprocal: `3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2` ## Meaning The question asks how many half-block orders fit inside three-fourths of a block. One full half-order fits, with half of another half-order remaining. ## Zero warning Zero has no reciprocal. A reciprocal would need to multiply by zero and make 1, but no number can do that. --- # Unit 14 — Ratios in the Poutine Recipe **Estimated time:** 12–20 minutes The shop recipe uses 2 scoops of cheese curds for every 3 scoops of fries. `curds : fries = 2 : 3` Equivalent batches scale both quantities by the same nonzero factor: | Batch | Curds | Fries | Ratio | |---|---:|---:|---:| | Test | 2 | 3 | `2:3` | | Double | 4 | 6 | `4:6` | | Triple | 6 | 9 | `6:9` | Changing only one side changes the recipe. ## Your turn If 2 scoops of curds need 3 scoops of fries, 10 scoops of curds need **15 scoops of fries**. --- # Unit 15 — Unit Rates **Estimated time:** 14–20 minutes A unit rate compares a quantity to exactly one unit. `$24 for 12 cubes = $2 for 1 cube = $2/cube` The denominator 1 may be left unwritten in an ordinary label, but the “per one” idea is still there. ## Keep the units - `$2 per cube` describes price for one cube. - `0.5 cubes per dollar` describes how much cheese one dollar buys. They are related reciprocals, but they answer different questions. ## Your turn `$21 for 14 cubes = $1.50 per cube`. --- # Unit 16 — The Invisible /1 and the Zero Trap **Estimated time:** 10–16 minutes These four rules should become automatic: | Expression | Result | Why | |---|---:|---| | `5/1` | `5` | Five groups of one remain five. Whole numbers have an invisible `/1`. | | `5/5` | `1` | Five fifths fill exactly one whole. | | `0/5` | `0` | Zero shared among five groups is still zero. | | `5/0` | undefined | No number multiplied by zero can equal five. | So `5/5` is not 5 and not 0. It is 1. `5/1` is not 0 or 1. It is 5. Division by zero is never allowed. ## Whole numbers inside fraction addition The denominator 1 is often invisible, but it can be shown when needed: `5 + 2/3 = 5/1 + 2/3` Use a common denominator of 3: `15/3 + 2/3 = 17/3 = 5 2/3` Do not add an extra `/1` when it is not useful. It is simply a way to reveal that every whole number can be written as a fraction. ## Black-hole check Ask: “What number multiplied by 0 gives 5?” There is no answer, so `5/0` is undefined. The calculator is not being stubborn; the operation has no valid value. --- # Adaptive practice bank ## Support 1. `1 × 2 = ?` — **2** 2. `2 × 2 = ?` — **4** 3. One of six equal pieces is — **1/6** 4. Which is larger, `1/2` or `1/4`? — **1/2** 5. Three fourth-sized pieces are — **3/4** 6. `5/5` — **1** 7. `5/1` — **5** 8. Which equals `1/2`? — **2/4** ## On-level 1. List a factor pair of 12 other than `1 × 12`. — **2 × 6 or 3 × 4** 2. Simplify `2/6`. — **1/3** 3. `2/8 + 3/8` — **5/8** 4. Two wholes cut into sixths produce — **12 sixth-sized pieces** 5. `$21 ÷ 14 cubes` — **$1.50 per cube** 6. `1/3 + 1/6` — **1/2** 7. `7/8 − 3/8` — **1/2** 8. An equivalent ratio to `2:3` — **4:6** 9. `5/0` — **undefined** ## Stretch 1. `2/3` of `3/4` — **1/2** 2. `3/5 + 1/10` — **7/10** 3. Three wholes cut into eighths, with five pieces used — **19/8 remain** 4. `$42` buys 24 cubes. Six cubes cost — **$10.50** 5. Design two different cutter arrays that each make 36 pieces. — **Examples: 4 × 9 and 6 × 6** 6. `2 1/4` as an improper fraction — **9/4** 7. `3/4 ÷ 1/2` — **3/2 or 1 1/2** 8. LCM(6, 8) — **24** 9. `$21 for 14 cubes` — **$1.50 per cube** --- # Motivation and reward design - Completing lesson sub-units awards cheese cookies. - Correct practice answers extend the learner's streak. - The trophy room contains 50 concealed cards: 25 cats and 25 dogs. - Locked cards show only an empty numbered slot. - Early cards use familiar internet-meme energy without copying protected meme artwork. - Rare cards use stronger borders and celebration effects. - A one-time cat-cheese heist may appear after a correct answer, but it gives the learner a bonus rather than removing earned progress. - A hidden division-by-zero Easter egg briefly opens a mathematical black hole. Rewards never replace useful feedback. The learner is told what was correct, what to reconsider, and what to try next. --- # Teacher prompts Use these questions throughout the experience: - What is the whole? - Are the pieces equal? - What does the denominator tell us here? - What does the numerator count? - How do the rows and columns predict the total? - Could the same total be built with another factor pair? - What happens when we cut a piece again? - Are we counting pieces from one whole or several wholes? - Did the piece size change when we added? - How can we find the cost of exactly one cube? - Can you show the same answer with words, an array, and a fraction? --- # Stop points The lesson is deliberately designed so a learner can stop successfully. - **Short session:** Complete Unit 1 only. - **Foundation session:** Complete Units 1–2. - **Factors session:** Complete Units 3–4. - **Fraction multiplication session:** Complete Unit 5. - **Improper fractions session:** Complete Unit 6. - **Addition session:** Complete Unit 7. - **Business application session:** Complete Unit 8. - **Equivalent and common-cut session:** Complete Units 9–10. - **Subtraction and mixed-number session:** Complete Units 11–12. - **Division and reciprocal session:** Complete Unit 13. - **Ratio and unit-rate session:** Complete Units 14–15. - **Misconception clinic:** Complete Unit 16. A pause is not a failure. Saved progress should make returning obvious and easy. --- # Completion checklist - [ ] Placement round completed or intentionally skipped - [ ] Unit 1 — The Poutine Promise - [ ] Unit 2 — One Whole Cheese - [ ] Unit 3 — Rows, Columns, and Factors - [ ] Unit 4 — Prime Cutters - [ ] Unit 5 — Fractions of Fractions - [ ] Unit 6 — More Than One Whole - [ ] Unit 7 — Adding the Order - [ ] Unit 8 — Price the Cheese - [ ] Unit 9 — Equivalent Cuts - [ ] Unit 10 — Common Denominators and LCM - [ ] Unit 11 — Subtract the Sold Cheese - [ ] Unit 12 — Mixed Numbers and Improper Fractions - [ ] Unit 13 — Reciprocals and Fraction Division - [ ] Unit 14 — Ratios in the Poutine Recipe - [ ] Unit 15 — Unit Rates - [ ] Unit 16 — The Invisible /1 and the Zero Trap - [ ] At least five adaptive drills completed - [ ] At least one note or question recorded in the math log - [ ] Final shop challenge explained aloud or in writing When these are complete, the learner earns the title **Fraction Shop Manager**.