6.AT.1 Lesson 3-1-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
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3.1 Small Group · Group 2

Tier 1 Extension · Non-Routine Synthesis · Misconception Traps · Rigorous CER Proofs

Understand Ratios · Ratio · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — What is a ratio?
1A ratio compares two quantities, and the order you write them in matters. You can write a ratio in words or with a colon, and you can draw it with a tape diagram or a double number line — and you can say why it is true, and where it would stop being true.
  • You already can do this. So we go up a level, not up a number: find what is always true, show it a second way, and be ready to defend it with a reason instead of an answer.
2Worked Example — Watch me
  1. Chef Reyes uses 3 cups of apple juice for every 2 cups of sparkling water.
  2. I am comparing two amounts: apple juice and sparkling water.
  3. I write apple juice to sparkling water, in that order, as 3 to 2, or 3:2.
  4. I can also draw the ratio. On a tape diagram I give apple juice 3 equal parts and sparkling water 2 equal parts, so one batch is 5 parts in all.
  5. Those same 5 parts show a second ratio: apple juice to the whole fruit drink is 3 to 5.
  6. On a double number line I line the two amounts up: 5 cups of fruit drink pairs with 3 cups of apple juice, and 10 cups of fruit drink pairs with 6 cups of apple juice.
3Second Model — Try it together — then prove it
  1. A bowl has 4 apples and 1 orange. What two amounts are we comparing?
  2. We write apples to oranges in order: 4 to 1, or 4:1.
  3. Now draw it: 4 equal parts for apples beside 1 part for oranges, 5 parts of fruit in all.
  4. So apples to total fruit is 4 to 5.
  5. Now prove it: say why that move had to work at all — not just that it did.
  6. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • Understand Ratios (Comprender razones) — Use a ratio to compare two quantities by division.
  • Ratio (Razón) — A way to compare two amounts, like 3 to 2.
  • Comparison (Comparación) — Looking at two or more amounts to see how they are related.
  • Part-to-part (Parte a parte) — A ratio comparing one part of a group to another part.
  • Part-to-whole (Parte a todo) — A ratio comparing one part to the whole group.
  • Colon notation (Notación con dos puntos) — Writing a ratio with two dots between the numbers, like 3:2.
5Watch out
  • A common mistake in Understand Ratios is writing the ratio in the wrong order: when asked for the ratio of apple juice to sparkling water in a recipe that uses 3 cups of apple juice for every 2 cups of sparkling water, some students write 2:3 (sparkling water to apple juice) because they list the ingredients in whatever order comes to mind rather than the order the question asks for. The two ratios describe different drinks — 3:2 means 3 cups of juice for every 2 cups of water, while 2:3 would mean only 2 cups of juice for every 3 cups of water. Always match the order of your ratio to the order named in the question: "apple juice to sparkling water" means the apple juice number comes first, giving 3:2.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 DRAG SORT

    Sort each statement — is it a part-to-part ratio or a part-to-whole ratio?

    Target Categories: Part-to-Part Part-to-Whole
    • 2 FILL TABLE

      Complete the table by writing each ratio in all three forms.

      DescriptionWord FormColon FormFraction Form
      4 cats to 7 dogs
      9 red out of 15 total
      5 wins to 3 losses
      ✏️ Scratchpad / Reasoning
    SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
    1. 3 MULTIPLE CHOICE

      Prep Bowl A: Chef Reyes stirs 7 cups of flour with 4 cups of water to make dough. What is the part-to-part ratio of flour to water?

      1. A7:4
      2. B4:7
      3. C7:11
      4. D11:4
      ✏️ Workspace & Solution Steps
    2. 4 MULTIPLE CHOICE

      A class has 5 chefs and 7 helpers (12 total). What is the part-to-whole ratio of chefs to the whole class?

      1. A5:12
      2. B5:7
      3. C7:12
      4. D12:5
      ✏️ Workspace & Solution Steps
    3. 5 MULTIPLE CHOICE

      “For every” language: Chef Reyes says, “My glaze uses 9 spoons of honey for every 4 spoons of lemon.” Written in colon notation, the ratio of honey to lemon is:

      1. A9:4
      2. B4:9
      3. C9:13
      4. D13:4
      ✏️ Workspace & Solution Steps
    SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
    1. 6 OPEN RESPONSE

      A bag of marbles has 8 red, 6 blue, and 4 green marbles. Write three different ratios using these quantities: one part-to-part ratio, one part-to-whole ratio, and explain the difference.

      ✏️ Mathematical Justification & Response
    🗣️ MATHEMATICAL DISCOURSE & TALK MOVES SMP.3 / Construct Viable Arguments

    Group Discussion Prompt: How does your visual model justify your mathematical solution?

    🗣️ Partner A:
    Partner A: "I modeled this by identifying the relationship and..."
    👂 Partner B:
    Partner B: "I agree with your step because the standard mathematical rule states..."
    ✍️ The Writing Revolution (TWR) · Sentence Expansion Because · But · So

    Complete each sentence stem to demonstrate precise mathematical reasoning:

    BECAUSE The ratio remains equivalent because both quantities are scaled by the exact same multiplicative factor.
    BUT Two ratios may look similar, but inverting the order of terms fundamentally changes the comparison.
    SO The recipe requires 3 parts flour to 2 parts water, so the unit rate is 1.5 cups of flour per cup of water.
    📐 CER MATHEMATICAL PROOF MATRIX (CER 2.0) SMP.3 / Proof & Justification

    Writing Task: Justify why your mathematical solution is accurate and complete.

    C Claim
    State your direct mathematical answer/claim with units.
    E Evidence
    Cite exact numbers, calculations, dimensions, or graph data.
    R Reasoning
    Explain the mathematical theorem, property, or definition connecting evidence to claim.
    ✍️ AUTHOR YOUR OWN EXTENSION CHALLENGE

    Create an original multi-step word problem aligned to this standard. Include constraints, and write the complete step-by-step mathematical proof below.

    ✏️ Author Workspace & Complete Solution Key
    Student Mastery Self-Assessment:
    1 · Need More Support
    2 · Getting Closer
    3 · Got It / Solid
    4 · Master / Can Teach It