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Stepwise Solvers

A 7th-grade mathematics unit providing structured graphic organizers and instructional frameworks for deconstructing and solving multi-step real-world problems.

Jjbrower

Stepwise Solvers

A 7th-grade mathematics unit providing structured graphic organizers and instructional frameworks for deconstructing and solving multi-step real-world problems.

Jjbrower

• Step 1: 2 pts • Step 3: 4 pts • Step 2: 2 pts • Step 4: 2 pts

2 Problem 2: Drone Corridor Speeds (Unit Rates & Proportions)

Target: Drone Beta (48 mph) • 90 Minutes

Step 1 Exemplar: Decode & Identify Rates

[2 Points]

Given Quantities & Times: • Drone Alpha: \(3\frac{3}{4}\) (3.75) mi in 5 min
• Drone Beta: \(4\frac{4}{5}\) (4.80) mi in 6 min
• Delivery Corridor: 72 miles total route

Target Questions to Answer: 1. Which drone has the faster cruising speed in miles per hour (mph)?
2. How many total minutes will it take to fly 72 miles?

Step 2 Exemplar: Proportional Model & Formulas

[2 Points]

Unit Rate Setup: \(\text{Rate (mi/min)} = \frac{\text{Distance}}{\text{Time}}\); \(\text{mph} = \text{mi/min} \times 60\)

Route Time Formula: \(\text{Time (min)} = \frac{\text{Distance (72 mi)}}{\text{Unit Rate (mi/min)}}\)

Step 3 Exemplar: Step-by-Step Calculations

[4 Points]

Part A: Drone Alpha

\(\frac{3.75\text{ mi}}{5\text{ min}} = 0.75\text{ mi/min}\)
\(0.75 \times 60 = \mathbf{45\text{ mph}}\)

Part B: Drone Beta

\(\frac{4.80\text{ mi}}{6\text{ min}} = 0.80\text{ mi/min}\)
\(0.80 \times 60 = \mathbf{48\text{ mph}}\)
\(\rightarrow\) Beta is faster

Part C: Delivery Route

\(\text{Time} = \frac{72\text{ mi}}{0.80\text{ mi/min}}\)
\(\text{Time} = \mathbf{90\text{ minutes}}\)
\((= 1.5\text{ hours})\)

Step 4 Exemplar: Verify & Final Statement

[2 Points]

Inverse Check: \(48\text{ mph} \times 1.5\text{ h} = 72\text{ mi}\). Equivalently, \(90\text{ min} \times 0.8\text{ mi/min} = 72\text{ mi}\). Checks out.

Complete Sentence Statement: Drone Beta is faster with a speed of 48 mph (vs. Alpha's 45 mph), and it takes exactly 90 minutes (1.5 hours) to complete the 72-mile delivery.

Common Misconceptions to Watch For:

• Inverted rate division: Dividing time by distance (\(5 \div 3.75 = 1.33\) min/mi) and assuming higher is faster.
• Unconverted time units: Calculating \(72 \div 48 = 1.5\) hours and forgetting to convert to 90 minutes.

Grade 7 Mathematics • Answer Key & Scoring Guide Page 2 of 3

Teacher Key Scoring & Exemplars

Stepwise Word Problem Solver

Scoring Rubric 10 Pts Total

• Step 1: 2 pts • Step 3: 4 pts • Step 2: 2 pts • Step 4: 2 pts

3 Problem 3: Makerspace Supply Order (Multi-Step Percents & Shortage)

Target: Total = $532.58 • Short by $32.58

Step 1 Exemplar: Decode & Identify Order Stages

[2 Points]

Catalog Data & Percentage Rates: • 1 3D Printer = $380.00; 4 Robotics Kits @ $45.00 = $180.00
• Educational Discount = 15% off item total
• Sales Tax = 8% added to discounted subtotal; Freight = $18.50 (untaxed)

Budget Limit & Target Decision: • PTA Grant Available: $500.00
• Target: Does $500 cover the total invoice? Calculate exact leftover surplus or needed shortage.

Step 2 Exemplar: Flowchart & Equations

[2 Points]

Order Sequence Flowchart: \(\text{Gross Total} \xrightarrow{-15\%} \text{Subtotal} \xrightarrow{+8\%} \text{Taxed Total} \xrightarrow{+\$18.50} \text{Final Invoice}\)

Formulas / Multipliers: \(\text{Subtotal} = (380 + 180) \times 0.85 = \$476.00\);
\(\text{Invoice} = (476 \times 1.08) + 18.50 = \$532.58\)

Step 3 Exemplar: Step-by-Step Calculations

[4 Points]

Part A: Items & 15% Off

\(\$380 + \$180 = \$560.00\)
\(\text{Discount} = 0.15 \times 560 = \$84\)
\(\text{Subtotal} = \$560 - \$84 = \mathbf{\$476.00}\)

Part B: 8% Tax & Freight

\(\text{Tax} = 0.08 \times 476 = \$38.08\)
\(\text{Taxed} = \$476 + \$38.08 = \$514.08\)
\(+\$18.50 = \mathbf{\$532.58}\)

Part C: Grant Comparison

\(\text{Grant} - \text{Invoice}:\)
\(\$500.00 - \$532.58 = -\mathbf{\$32.58}\)
\(\rightarrow\) Deficit of $32.58

Step 4 Exemplar: Verify & Final Statement

[2 Points]

Benchmark Estimate: \(\$560 - \$84 = \$476\). \(8\% \text{ of } \$476 \approx \$38\). \(\$476 + \$38 + \$19 \approx \$533\). \(\$532.58\) is reasonable.

Complete Sentence Statement: No, the \(\$500.00\) PTA grant will not cover the final invoice of $532.58; Mr. Vance is short and needs an additional $32.58.

Common Misconceptions to Watch For:

• Taxing before discount: Applying \(8\%\) tax to the pre-discounted \(\$560.00\) (\(\$44.80\) tax) instead of \(\$476.00\) (\(\$38.08\) tax).
• Taxing freight shipping: Shipping is explicitly untaxed. Taxing shipping adds \(\$1.48\) incorrectly (\(\$534.06\)).

Grade 7 Mathematics • Answer Key & Scoring Guide Page 3 of 3

#16 4 × 12 (48) = 6 × 8 (48)

#17 0 × 12 (0) = 12 × 0 (0)

#18 11 × 11 (121) > 12 × 10 (120)

#19 9 × 7 (63) < 8 × 8 (64)

#20 12 × 4 (48) < 7 × 7 (49)

Basic Multiplication Facts 0 – 12 • Missing Factors & Comparisons Key Page 2 of 2

Scenario 1: Overloaded by Sudden Deadlines & Multiple Tests

Your teacher announces a surprise quiz while you haven't finished the review. Your pulse spikes and you want to yell, "This is unfair!"

Impulse: Outburst

Your Self-Advocate Script (What you say calmly to your teacher):

Scenario 2: Partner Dismissing Your Ideas in a Group Project

A teammate rolls their eyes, cuts you off, and says your idea won't work. Your face gets hot and you want to curse or slam your book.

Impulse: Insults

Your Firm Boundary Script (What you say directly to your teammate):

Scenario 3: Accused of Talking When You Were Borrowing a Pencil

The teacher calls out across the room: "Stop talking and disrupting!" You feel humiliated and feel the urge to snap back defensively.

Impulse: Defensiveness

Your Respectful Check-In Script (What you say quietly or after class):

Two Trusted Adults at School I Can Advocate To:

My Personal Power Commitment:

"When I feel angry, I will pause for 5 seconds and speak with my calm voice because..."

Self-Advocacy & Emotional Regulation • Grade 7 Page 2 of 2

• "I don't like being called that nickname. Stop."

• "Please step back, you're in my space."

How to Receive a Boundary with Maturity

When someone says "Stop" or asks for space, earn respect by responding in 2 seconds:

• Say: "My bad, got it" or "Sorry about that."

• Immediately stop the action and step back.

• Never argue: "You're so sensitive!" (That loses respect).

Scenario Lab: What Would You Say & Do? Write authentic middle school responses

Scenario 1: Personal Property & Belongings

During science lab, a classmate reaches across your desk, grabs your phone, and begins browsing your photos without asking.

Physical Boundary

Your Calm, Direct Boundary Statement:

Scenario 2: Hurtful Teasing Masked as a "Joke"

At the lunch table, a peer makes repeated sarcastic remarks about what you brought to eat. When you frown, they smirk and say, "Can't you take a joke?"

Verbal Boundary

Your Assertive Response (Without yelling or insulting them back):

Scenario 3: When YOU Accidentally Cross Someone Else's Line

You make a joke about a classmate's art project. They turn away, cross their arms, and look hurt. You realize you crossed a line.

Accountability

Your Mature Repair Statement (What you say to make it right):

One Personal Boundary I Expect Others to Respect:

"I expect my peers to respect my space by never..."

My Commitment to Respecting Other People's Lines:

"When a peer signals discomfort or says stop, I will immediately..."

Peer Interactions & Boundaries • Grade 7 Page 2 of 2

Mocking: "You can't take a joke / You're so sensitive."

Peer Boundaries & Respectful Language • Teacher Guide Page 1 of 2

Part 2 Coaching & Roleplay Protocol

Action Protocols & Intervention

Application Lab 30:00 – 50:00

• Format: Paired Script Modeling

• Outcome: Actionable Muscle Memory

Phase 4: Tone Coaching (Setting vs. Receiving Lines)

Setting a Line: The "Flat Steel" Voice

Warn students against two common failure modes:

  • • Whining / Apologizing: "Um, sorry, can you maybe stop?" (Invites more teasing).
  • • Screaming: "GET AWAY FROM ME!" (Escalates into a spectacle).
  • • Target Tone: Eye contact, chin level, unsmiling, low volume: "Don't touch my phone. Put it back."

Receiving a Line: The "2-Second Reset"

Teach students how to take a boundary with dignity:

  • • Step back 1 full step immediately.
  • • Deliver the 4-word code: "My bad, I hear you."
  • • Do not roll eyes, scoff, or explain yourself. Moving on earns immediate peer respect.

Phase 5: Paired Practice Protocol (3 Rounds × 2 Minutes) Active Practice

Round 1: Belongings

Partner A: Reaches for Partner B's pencil case.
Partner B: Holds palm up, says: "Ask first. Don't grab my materials."
Partner A: "My bad, got it." (Pulls hands back).

Round 2: Unwanted Roasting

Partner B: Makes a sarcastic comment on shoes.
Partner A: Steady gaze: "I'm not doing that today. Drop the shoe comments."
Partner B: "Heard. Moving on."

Round 3: Inadvertent Overstep

Partner A: Leans into Partner B's screen.
Partner B: Steps back, crosses arms.
Partner A: Notices cue: "Sorry, didn't realize I was crowding you."

Teacher Interventions for Live Classroom Oversteps

When a Student Says: "I was just joking!"

Teacher Response: "Intent doesn't erase impact. Look at your peer's face right now—are they laughing? Let's use our reset script and get back to work."

When a Student Blows Up / Yells:

Teacher Response: "I see that your boundary was crossed and you are angry. But screaming loses your voice. Take 30 seconds to breathe, then tell them your boundary with your calm advocate voice."

Peer Boundaries & Respectful Language • Teacher Guide Page 2 of 2

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