MathIsimple
Back to Unit 6

Lesson 6-1: Multi-Step Problems

Community Garden Planning

Learning Objectives

By the end of this lesson, you will:

  • Break down complex problems into manageable steps
  • Select appropriate strategies for different problem types
  • Verify solutions using multiple methods
  • Apply multiple math concepts in one problem

Skills You'll Practice:

Problem DecompositionStrategy SelectionMulti-Step OperationsVerificationCritical ThinkingReal-World Application

Community Garden Challenge

Your school is planning a community garden to grow vegetables for the cafeteria. You need to help plan the garden layout, calculate materials needed, and determine the cost. This involves multiple math concepts working together!

Garden Layout

Calculate area and perimeter for different vegetable plots

Materials & Cost

Determine soil, seeds, and fencing needed with total cost

Harvest Planning

Calculate expected yields and feeding capacity

Problem 1: Garden Layout Design

The Challenge:

The school has a rectangular area that's 24 feet long and 18 feet wide for the garden. You need to divide it into 4 equal rectangular plots for different vegetables. Each plot needs a 2-foot wide path around it for walking. Calculate the area of each vegetable plot.

Step 1: Understand the Problem

Let's break this down: We have a 24 ft × 18 ft area, need 4 equal plots, with 2 ft paths around each plot.

Step 2: Plan the Layout

If we arrange plots in a 2×2 grid, we need to account for paths between plots and around the edges.

• Total area: 24 × 18 = 432 square feet
• Path width: 2 feet
• Number of plots: 4 (arranged 2×2)

Step 3: Calculate Available Space

We need to subtract the path areas from the total area to find space for the plots.

• Horizontal paths: 2 paths × 2 ft × 18 ft = 72 sq ft
• Vertical paths: 2 paths × 2 ft × 24 ft = 96 sq ft
• Corner overlaps: 4 corners × 2 ft × 2 ft = 16 sq ft
• Total path area: 72 + 96 - 16 = 152 sq ft

Step 4: Calculate Plot Area

Now we can find the area available for the 4 plots and divide by 4.

• Available for plots: 432 - 152 = 280 sq ft
• Area per plot: 280 ÷ 4 = 70 sq ft

Answer:

Each vegetable plot will have an area of 70 square feet.

Problem 2: Materials and Cost Calculation

The Challenge:

You need to calculate the total cost for the garden. Here are the costs: Soil costs $2.50 per square foot, seeds cost $0.75 per square foot, and fencing costs $8.00 per foot. The garden needs fencing around the entire perimeter plus the paths between plots.

Step 1: Calculate Soil Cost

We need soil for all 4 plots (280 sq ft total).

Soil cost = 280 sq ft × $2.50/sq ft = $700

Step 2: Calculate Seed Cost

Seeds are also needed for the plot areas.

Seed cost = 280 sq ft × $0.75/sq ft = $210

Step 3: Calculate Fencing Length

We need fencing around the perimeter and along the paths between plots.

• Perimeter: 2 × (24 + 18) = 84 feet
• Internal paths: 2 × 18 + 2 × 24 = 84 feet
• Total fencing: 84 + 84 = 168 feet

Step 4: Calculate Fencing Cost

Now calculate the cost for all the fencing needed.

Fencing cost = 168 ft × $8.00/ft = $1,344

Step 5: Calculate Total Cost

Add up all the costs to get the total project cost.

Total cost = $700 + $210 + $1,344 = $2,254

Answer:

The total cost for the community garden project is $2,254.

Problem 3: Harvest Planning

The Challenge:

The school cafeteria serves 150 students daily. Each student needs 0.25 pounds of vegetables per meal. If each square foot of garden produces 2.5 pounds of vegetables per season, will the garden produce enough vegetables for one season (90 days)?

Step 1: Calculate Daily Vegetable Need

Find out how many pounds of vegetables are needed per day.

Daily need = 150 students × 0.25 lbs/student = 37.5 lbs/day

Step 2: Calculate Season Vegetable Need

Multiply daily need by the number of days in the season.

Season need = 37.5 lbs/day × 90 days = 3,375 lbs

Step 3: Calculate Garden Production

Find out how much the garden can produce in one season.

Garden production = 280 sq ft × 2.5 lbs/sq ft = 700 lbs

Step 4: Compare Need vs Production

Compare what's needed with what can be produced.

Need: 3,375 lbs
Production: 700 lbs
Shortage: 3,375 - 700 = 2,675 lbs

Answer:

No, the garden will not produce enough vegetables. It will produce 700 lbs but the cafeteria needs 3,375 lbs, creating a shortage of 2,675 lbs. The garden would need to be about 4.8 times larger to meet the full need.

Key Multi-Step Problem Solving Strategies

Problem Decomposition

  • • Break complex problems into smaller, manageable parts
  • • Identify what information you have and what you need
  • • Determine the logical order of operations

Strategy Selection

  • • Choose appropriate methods for each step
  • • Use visual models when helpful
  • • Apply different math concepts as needed

Verification

  • • Check your work at each step
  • • Use estimation to verify reasonableness
  • • Try alternative methods when possible

Real-World Context

  • • Consider if your answer makes sense
  • • Think about practical implications
  • • Connect math to real situations

Practice Problems

Practice 1: School Fundraiser

The school is organizing a bake sale fundraiser. They need to make 200 cookies. Each batch makes 24 cookies and requires 1.5 cups of flour. Flour costs $3.50 per 5-pound bag, and there are 18 cups of flour per bag. Calculate the total cost of flour needed.

Hint: Break this into steps: batches needed → cups of flour → bags needed → total cost

Practice 2: Sports Equipment

A sports team needs to buy equipment. They need 15 basketballs at $25 each, 12 soccer balls at $18 each, and 8 volleyballs at $22 each. If they get a 10% discount on the total order, what is the final cost?

Hint: Calculate individual costs → find total → apply discount

Practice 3: Trip Planning

A class of 28 students is going on a field trip. The bus costs $150, and each student ticket costs $8.50. If the school has budgeted $400 for this trip, how much more money do they need?

Hint: Calculate total cost → compare with budget → find difference

Great Job!

You've learned how to break down complex problems into manageable steps and solve them systematically. This skill will help you tackle any multi-step problem you encounter!