Foundation Slides Grade 6: Foundation Phase
Foundation Slides
Mastering fraction operations for the ultimate construction project.
The Job Site Brief
As Blueprint Builders, every measurement must be exact. In construction, we don't just use whole numbers—we use pieces of them.
Today's Objectives
Add & Subtract fractional lengths
Multiply to find material areas
Divide to find material counts
🏗️
Project: Foundation Fractions
Operation: Addition & Subtraction
Joining Materials
To find the total length of two beams, we add their measurements. Remember: denominators must be identical before we build.
Example
\( 2\frac{1}{2} + 3\frac{3}{4} \)
Hint: Change \( 2\frac{1}{2} \) to \( 2\frac{2}{4} \) first!
Builder's Tip
Always simplify your final measurements. A master carpenter doesn't say \( \frac{4}{8} \), they say \( \frac{1}{2} \).
Quick Check
\( \frac{3}{8} + \frac{5}{8} = ? \)
Operation: Multiplication
Square Footage
Finding the area of a room layout requires multiplying side lengths.
Steps
Convert mixed to improper
Multiply numerators
Multiply denominators
Simplify results
Room Dimensions
\( 8\frac{1}{2} \text{ ft} \times 4\frac{2}{3} \text{ ft} \)
"Multiplying by a fraction smaller than 1 is like shrinking your blueprint!"
Operation: Division
Cutting to Count
How many smaller boards can we cut from one large beam? We divide the total length by the piece length.
The "Reciprocal" Rule:
\( \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} \)
"Keep the first, Change the sign, Flip the second."
Site Task
You have a \( 6 \) foot beam. How many \( \frac{3}{4} \) foot pieces can you cut?
\( 6 \div \frac{3}{4} = ? \)
Ready to Build?
Open your Blueprint Basics Worksheet. Use your pencil as your level and your mind as your tape measure.
Start Project A
Tool Check
Blueprint Basics Worksheet Blueprint Basics
Project Site: Foundation Phase | Task 01: Fraction Calibration
Contractor Name:
Date:
Site Instructions
Complete all measurements below. Show all "structural work" (steps) to ensure the building is sound. Simplify all final measurements.
Phase 1: Joining the Beams
1. Combine two steel supports measuring \( 5\frac{3}{8} \) ft and \( 4\frac{1}{4} \) ft.
SUM
2. You have a \( 12 \) ft board and cut off \( 3\frac{2}{3} \) ft. How much remains?
REMAINDER
3. A wall needs three layers: \( \frac{5}{8} \) in, \( \frac{1}{2} \) in, and \( \frac{1}{4} \) in thick. Find total depth.
TOTAL DEPTH
4. A trench was dug \( 8\frac{1}{6} \) ft deep but needs to be \( 10\frac{1}{2} \) ft deep. How much deeper must you dig?
DIFFERENCE
Phase 2: Flooring & Dividers
5. The storage closet footprint is a rectangle measuring \( 6\frac{1}{2} \) ft by \( 4\frac{3}{4} \) ft. Calculate the total area in square feet.
AREA
6. A construction worker has an \( 18 \) ft beam. How many pieces of length \( 2\frac{1}{4} \) ft can be cut from it?
PIECES
7. A bucket holds \( 5\frac{1}{3} \) gallons of paint. If each room uses \( \frac{2}{3} \) of a gallon, how many rooms can be painted with one bucket?
ROOMS
REF: DWG-6-FRAC-01 | FOUNDATION PHASE
Foundation Answer Key Answer Key
Project Site: Foundation Phase | Material: Blueprint Basics
Teacher Resource
Grade 6 Math
Phase 1: Joining the Beams
1. Combine \( 5 \frac{3}{8} \) and \( 4 \frac{1}{4} \)
\( 5 \frac{3}{8} + 4 \frac{2}{8} = 9 \frac{5}{8} \text{ ft} \)
2. \( 12 \) ft minus \( 3 \frac{2}{3} \) ft
\( 11 \frac{3}{3} - 3 \frac{2}{3} = 8 \frac{1}{3} \text{ ft} \)
3. Total Depth: \( \frac{5}{8} + \frac{1}{2} + \frac{1}{4} \)
\( \frac{5}{8} + \frac{4}{8} + \frac{2}{8} = \frac{11}{8} = 1 \frac{3}{8} \text{ in} \)
4. \( 10 \frac{1}{2} \) minus \( 8 \frac{1}{6} \)
\( 10 \frac{3}{6} - 8 \frac{1}{6} = 2 \frac{2}{6} = 2 \frac{1}{3} \text{ ft} \)
Phase 2: Flooring & Dividers
5. Area of \( 6 \frac{1}{2} \times 4 \frac{3}{4} \)
\( \frac{13}{2} \times \frac{19}{4} = \frac{247}{8} = 30 \frac{7}{8} \text{ sq ft} \)
6. \( 18 \div 2 \frac{1}{4} \)
\( 18 \div \frac{9}{4} \rightarrow 18 \times \frac{4}{9} = 2 \times 4 = 8 \text{ pieces} \)
7. \( 5 \frac{1}{3} \div \frac{2}{3} \)
\( \frac{16}{3} \div \frac{2}{3} \rightarrow \frac{16}{3} \times \frac{3}{2} = \frac{16}{2} = 8 \text{ rooms} \)
Precision Slides Grade 7: Precision Phase
Proportional Precision
Complex fractions and unit rates in large-scale design.
1:50 SCALE
Complex Fractions
"A fraction within a fraction—like a room within a house."
In structural design, we often deal with ratios that aren't simple. A complex fraction looks like this:
\( \frac{\frac{1}{2}}{\frac{3}{4}} \)
Method: Keep • Change • Flip
Design Scenario
If it takes \( \frac{3}{4} \) of a gallon to cover \( \frac{1}{8} \) of a wall, how much paint covers the whole wall?
\( \frac{\frac{3}{4}}{\frac{1}{8}} \text{ gallons/wall} \)
Work Area
Unit Rates in Construction
Labor Speed
A mason lays \( 4\frac{1}{2} \) rows of bricks in \( \frac{2}{3} \) of an hour. Rows/hour?
\( \frac{4\frac{1}{2}}{\frac{2}{3}} \)
Material Flow
Concrete pours at a rate of \( 15\frac{3}{4} \) cubic feet every \( 5 \) minutes. Rate/min?
\( \frac{15\frac{3}{4}}{5} \)
Unit Cost
\( 10\frac{1}{2} \) yards of wiring cost $42. What is the cost per yard?
\( \frac{42}{10\frac{1}{2}} \)
UNIT RATE = DIVIDE TOP BY BOTTOM
Scaling the Blueprint
Architecture relies on Scale Factors . Every inch on paper represents feet in the real world.
The Golden Rule
"Keep your ratios consistent. If the scale is \( \frac{1}{4} \text{ in} = 1 \text{ ft} \), multiply paper units to find real units."
Scale Challenge
If \( \frac{1}{2} \text{ inch} = 5 \text{ feet} \)
How many feet does 3 inches represent?
Think: How many "halves" are in 3?
Work Area
Scale Up Activity Architectural Logistics
Scale Up Activity
Project Site: Precision Phase | Task 02: Unit Rates & Complex Scales
Lead Architect:
Date:
"Greetings, Lead Architect. We have complex ratios coming in from the field. Your job is to calculate the unit rates and scale our blueprint measurements for the new skyscraper project."
Part 1: Material Logistics
1. Concrete Pouring Speed
The foundation crew poured \( 3\frac{3}{4} \) cubic yards of concrete in \( \frac{2}{3} \) of an hour. What is the unit rate in cubic yards per hour?
2. Drywalling Efficiency
A worker can sand \( 12\frac{1}{2} \) square feet of wall using \( \frac{1}{4} \) of a sanding pad. How many square feet can be sanded with one full pad?
Part 2: The Master Blueprint
Official Project Scale
\( \frac{1}{2} \text{ inch on paper} = 3\frac{1}{4} \text{ feet in reality} \)
3. Kitchen Dimensions
The blueprint shows a kitchen width of \( 4 \) inches. Use the scale above to find the actual width of the kitchen in feet.
4. Structural Height
An actual structural pillar is \( 13 \) feet tall. How many inches tall should it be drawn on the blueprint?
5. Chief Architect Challenge
Calculate the Unit Rate of the Scale . How many feet of "reality" are represented by exactly 1 inch on the blueprint?
REF: SKYSCRAPER-7-UNIT-02
Precision Answer Key Answer Key
Project Site: Precision Phase | Material: Scale Up Activity
Teacher Resource
Grade 7 Math
Part 1: Material Logistics
1. Concrete Speed (\( 3\frac{3}{4} \div \frac{2}{3} \))
\( \frac{15}{4} \div \frac{2}{3} \rightarrow \frac{15}{4} \times \frac{3}{2} = \frac{45}{8} = 5\frac{5}{8} \text{ cubic yards/hr} \)
2. Drywall Pads (\( 12\frac{1}{2} \div \frac{1}{4} \))
\( \frac{25}{2} \div \frac{1}{4} \rightarrow \frac{25}{2} \times \frac{4}{1} = 25 \times 2 = 50 \text{ sq ft/pad} \)
Part 2: The Master Blueprint
3. Width for 4 inches (\( \text{Scale: } \frac{1}{2} \text{ in} = 3\frac{1}{4} \text{ ft} \))
\( 4 \text{ inches} = 8 \text{ half-inches} \)
4. Blueprint height for 13 feet
\( 13 \div 3\frac{1}{4} = 13 \div \frac{13}{4} = 13 \times \frac{4}{13} = 4 \text{ half-inches} \)
5. Challenge: Unit Rate of Scale
\( 3\frac{1}{4} \text{ ft} \div \frac{1}{2} \text{ in} = \frac{13}{4} \times 2 = \frac{13}{2} = 6\frac{1}{2} \text{ feet per inch} \)
Structural Slides Grade 8: Analysis Phase
Structural Slopes
Rise over run, pitch calculations, and the realm of rational numbers.
COORD: 8.EE.B.5
SLOPE = \( \frac{\Delta y}{\Delta x} \)
Roof Pitch
"The steeper the slope, the faster the runoff."
Structural Eng.
In construction, slope is called Pitch . It is always a fraction: Rise over Run .
\( \frac{12}{12} \)
Steep (45°)
\( \frac{4}{12} \)
Shallow
"Every 12 inches across, how many inches up?"
The Slope Formula
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Calculate the pitch of a roof that rises 5 feet for every 10 feet of horizontal run.
\( m = \frac{5}{10} = \frac{1}{2} \)
Material Properties
Rational Numbers vs. The Irrational
Rational Measurements
Measurements that can be written as a fraction . Terminating or repeating decimals.
3/4 0.66... 5.25
Irrational Measurements
Numbers that cannot be written as fractions. Non-repeating, infinite decimals.
\( \sqrt{2} \) \( \pi \) 1.414...
Structural Impact
When calculating diagonal braces (hypotenuse), we often encounter \( \sqrt{2} \). Builders approximate these irrationals as fractions to actually make the cut.
"\( \sqrt{2} \approx 1.41 \approx 1 \frac{13}{32} \)"
Roof Pitch Problems Structural Engineering Division
Roof Pitch Problems
Project Site: Structural Phase | Task 03: Slope & Number Systems
Chief Engineer:
Date:
Part 1: Pitch Analysis
1. The Standard Peak
A roof rises \( 6 \) feet vertically for every \( 12 \) feet of horizontal distance. Calculate the slope (\( m \)) and simplify it to its lowest fractional form.
Formula: m = rise / run
2. Coordination Points
On a blueprint grid, a support beam starts at point \( (2, 3) \) and ends at \( (8, 7) \). Calculate the slope of this beam.
Part 2: Material Sorting
Identify if each site measurement is Rational or Irrational and provide a brief technical justification.
Item A
\( 5.75 \text{ ft} \)
Rational
Irrational
Item B
\( \sqrt{5} \text{ in} \)
Rational
Irrational
Item C
\( 4.\overline{3} \text{ yd} \)
Rational
Irrational
Part 3: The Steepest Challenge
3. A wheelchair ramp must have a slope of exactly \( \frac{1}{12} \). If the ramp needs to rise \( 3 \) feet to reach the entrance, what must be the horizontal run of the ramp?
Show your algebraic steps below to verify the design meet codes.
STRUCTURAL-8-SLOPE-03
G8 MATH
Structural Answer Key Answer Key
Project Site: Structural Phase | Material: Roof Pitch Problems
Teacher Resource
Grade 8 Math
Part 1: Pitch Analysis
1. Standard Peak (\( 6 \text{ ft rise, } 12 \text{ ft run} \))
\( m = \frac{\text{Rise}}{\text{Run}} = \frac{6}{12} = \frac{1}{2} \)
2. Coordination Points (\( (2,3) \) to \( (8,7) \))
\( m = \frac{7 - 3}{8 - 2} = \frac{4}{6} = \frac{2}{3} \)
Part 2: Material Sorting
A. \( 5.75 \)
Rational
Can be written as \( 5\frac{3}{4} \) or \( \frac{23}{4} \). Terminating decimal.
B. \( \sqrt{5} \)
Irrational
Non-repeating, non-terminating decimal. \( 5 \) is not a perfect square.
C. \( 4.\overline{3} \)
Rational
Can be written as \( 4\frac{1}{3} \) or \( \frac{13}{3} \). Repeating decimal.
Part 3: The Steepest Challenge
Wheelchair Ramp Solution
\( \frac{1}{12} = \frac{3}{x} \)
Cross-multiply to solve for horizontal run:
\( 1 \times x = 12 \times 3 \)
\( x = 36 \text{ feet} \)
