Middle School Math PD – Session Notes

Four-Fours Activity

• Warm-up / extension problem with a “low floor – high ceiling”: every learner can participate, but the task still stretches advanced students.
• Goal: write every integer from $1$ to $20$ (and beyond) using exactly four 4’s and any operations the class has agreed upon.
• Strategies mentioned
  • Basic operations only: students first try without exponents, parentheses or square roots.
  • Gradual introduction of higher tools once they are needed:
  – Square root $\sqrt{4}=2$ (allows quick creation of $2,8,16,\ldots$).
  – Factorial “four-bang” $4! = 4\times3\times2\times1 = 24$.
  – Exponents: $4^2=16,\;4^3=64,\;4^4=256$.
• Sample solutions discussed
  • $\dfrac{4+4+4}{4} = 3$ (no parentheses needed if division precedes addition).
  • $(4+4+4)\div4 \div4 = 3$ (parentheses then two successive divisions).
• Pedagogical use
  • Launches Order-of-Operations conversation (natural need for PEMDAS/GEMDAS).
  • Leads organically into square roots, exponents, factorial, inverse operations.

Order of Operations & Step-by-Step Discipline

• Students perform one operation at a time, circling the part they are about to evaluate.
• After every step they copy the entire line again – builds habits for algebraic manipulation.
• Emphasise multiplication & division before addition & subtraction, and equal priority left-to-right.
• Teacher writes only one equal sign – the final one – to prevent strings of false equalities.

Worked classroom example

1. Problem: $5 + 3 \times 2 - 9 \div 3$.
2. Circle $3\times2$, compute $6$, rewrite full line.
3. Circle $9\div3$, compute $3$, rewrite full line.
4. Now only + and − remain; proceed left-to-right.

Introducing Exponents & Roots

• Connect to students’ idea of “repeated multiplication” (analogy: multiplication = repeated addition).
• Keep exponents small ($2^{\text{nd}},3^{\text{rd}},4^{\text{th}}$ power) for computational comfort.
• “Going backwards” (inverse operations) is conceptually hard; use physical metaphors (walk forward/back).

Factorial (!)

• Notation $n!$ read as “n-bang.”
• Definition, e.g.
  $4! = 4\times3\times2\times1 = 24$
• Adds another handy constant for the Four-Fours list and foreshadows permutations/combinatorics.

Evolution of the Multiplication Symbol

• From $\times$ in early grades → $\cdot$ in middle school → implied juxtaposition in algebra (e.g. $2x$).
• Classroom ritual “R.I.P. ×” helps students let go of the × and understand why • (dot) is preferred (avoids conflict with the variable $x$).

Division Notation

• Stress horizontal fraction bar only; avoid slanted bars (e.g. computer fonts) – prepares students for complex algebraic fractions.
• Reinforce that $\frac{16}{4}$, $16 \div 4$, and the long-division “house” all represent the same operation.

Rules / Phrases That Expire (Language Precision)

• “Addition & multiplication make numbers bigger” – false once fractions/negatives appear (e.g. $24\times\tfrac12 = 12$).
• “Subtraction & division make numbers smaller” – similarly limited.
  → In early grades specify whole numbers only, and revisit explicitly when new number types arise.
• “To multiply by 10 just tag on a 0” – fails for decimals ($43.5\times10 \neq 43.50$ but $435$).
  → Teach as a place-value shift.
• Equal sign misconception: many students read “= means ‘find the answer.’”
  • Use open sentences like $14+\Box = 48+\Diamond$ to emphasise “is the same as” (relational meaning).

Number-Sense Routines (Daily/Weekly)

• Which One Doesn’t Belong (WODB)
  • Strengthens mathematical vocabulary and precise justification.
  • Every option can be correct with a valid reason.
• Estimation Routine
  • Weekly exercise; research shows a high correlation between estimation skill and later math success.
  • Fold brief estimation checks into any lesson.

Differentiation & Homework Philosophy

• All students aim for the same conceptual bar; differentiation lies in supports and entry points, not in lowering the task.
• Tools: scaffolding prompts, advancing prompts, strategic grouping, optional extension sheets (e.g. harder ratio-shading page).
• Homework volume can be adjusted for students with documented accommodations, but task type remains consistent.
• Offer of pull-out math-support groups and mandatory study halls as additional layers.

Impact of COVID & Motor-Skills Observation

• Faculty noticing lingering academic and gross-motor gaps (“COVID echo”).
• Response: more movement / “zoo exercises” in lower grades, careful admissions screening for support capacity.

Ratio Foundations (6th Grade)

Concept Clarification

• Ratio: comparison of part to part (unit-less).
  Example: brown-haired : black-haired students $=5:3$.
• Fraction: comparison of part to whole or (whole + parts) to whole.
  Example: $\tfrac{5}{8}$ of class has brown hair.
• Keep ratio and fraction notations separate at first to avoid confusion.

Classroom Launch

1. Collect real-life counts (eye color, pets, etc.) and write verbal ratios.
2. Translate to numeric form $a:b$.
3. Visual link: shade parts of a rectangle so students see the whole as the sum of parts.

Ratio-Shading Task (Don Stewart “ratio shading” sheets)

• Grid (e.g. $3\times4$) represents the whole.
• Steps students discover / teacher lightly guides:
  1. Add parts $a+b$ to find total parts.
  2. Count squares in grid (e.g. $12$); divide by total parts → squares per part.
  3. Shade first part squares one colour, second part squares another.
• Order matters: $1:3 \neq 3:1$.
• Fractional parts (e.g. $2\tfrac12:1\tfrac12$) force students to subdivide rows/columns or think in half-rows.
• Extension sheet with tougher ratios available for fast finishers.

Equivalent Ratios & Double Number Lines

• Visual tool: two parallel number lines incremented identically.
  • Top line tracks first part, bottom line second part.
  • Start with anchor ratio (e.g. $2:3$ at same position).
  • Extend equal “chunks” to generate $4:6,6:9,8:12,\ldots$
• Supports shift to proportion reasoning in 7th grade.

Rates & Unit Rates (Introduced in 6th)

• Rate = ratio with attached units (e.g. $5\text{ pencils}:\$20$).
• Unit rate: divide to get “per one” (e.g. $\$4\text{ per pencil}$).
• Double number line again useful.

Preparing for 7th Grade Proportions

• Proportions (equations of equivalent ratios) delayed until after the 7th-grade algebra block for better readiness.
• Example 7th-grade tie-in: $\dfrac{5}{x}=\dfrac{20}{32}$, cross-product method grounded in earlier ratio work.

Notation, Language & Visual Consistency Reminders

• Always write the fraction bar horizontal.
• Encourage multiple representations (verbal, numeric, pictorial) of any new idea.
• Keep language precise: say “simplify” a fraction, not “reduce.”
  Fraction isn’t shrinking; it’s wearing a different “costume.”

Homework, Resources & Teacher Support

• Sharon will upload complete 6th-grade scope-and-sequence + full lesson plans, homework sets, and links on Canvas (shared drive for non-PITEP participants).
• Teachers invited to email for specific unit materials or emergency homework two months down the line.
• Recommended external site: Don Stewart’s Median blog (rich ratio visuals, assorted tasks for all grades).
• Article: “13 Rules That Expire” (will be placed in Resources folder) – highlights common sayings that later mislead.

Closing Thoughts Shared in Session

• Constantly watch students’ written work for “mathematical nonsense” (strings of equals, random operations).
• Small diagnostic prompts (e.g. $12+\Box = 5+7$) reveal misconceptions quickly.
• Flexibility with symbols, operations, and representations is cultivated deliberately – not left to chance.