Comprehensive Guide to Volume of Rectangular and Triangular Prisms

Current Progress: The class is currently in Week 4, though the lesson material is from Week 3 due to scheduling adjustments and additional time spent on assignment work.
Primary Focus: Calculating the volume of rectangular prisms and dimensions. The lesson also introduces the volume of triangular prisms.
Objectives: Mastering the formula for volume, understanding cross-sections, and applying these concepts to the specific requirements of the garden bed assignment.

Conceptual Start: Volume is initially taught at the Year 7 level using blocks or cubes. Students are encouraged to visualize 3D objects to account for hidden cubes that are not immediately visible from a front view.
Hidden Cubes: When cubes are stacked, one must assume the presence of cubes supporting those at the top. For instance, in a stack of three, there is one at the bottom, one in the middle, and one on top.
Top View Perspectives: Using a top-down view helps identify how many cubes are touching a surface (the base). If a column of cubes exists, every stack has at least one bottom cube touching the table.
Definition of Volume: Volume represents the depth of an object and is considered a three-dimensional measurement.

Linear Dimensions: Measured in a straight line. Units include millimeters ( $mm$ ), centimeters ( $cm$ ), and meters ( $m$ ). These do not have subscript or superscript numbers.
Area Dimensions (2D): Units are squared, such as square millimeters ( $mm^2$ ) or square centimeters ( $cm^2$ ).
Volume Dimensions (3D): Units are cubed due to the third dimension (depth). Examples include cubic millimeters ( $mm^3$ ), cubic centimeters ( $cm^3$ ), and cubic meters ( $m^3$ ).
Clinical Example: In medical settings, syringes are often measured in cubic millimeters ( $mm^3$ ).
Conversion Rule Provided in Class: One cubic millimeter ( $mm^3$ ) is equal to one milliliter ( $mL$ ). (Note: This is the specific conversion stated in the transcript).

General Strategy: Volume is found by first calculating the area of the face touching the ground (the base) and then multiplying it by the height.
Fatima’s Counting Strategy: To find the volume of a block of cubes, count the number of cubes in one row, multiply by the number of vertical stacks (to get the area of one face), and then multiply by the depth of the total stack.
The Formal Formula:
- $V = \text{Area} \times \text{height}$
- For a rectangular prism: $V = l \times w \times h$
- $l$ = Length
- $w$ = Width
- $h$ = Height
Substitution and Solving:
- Example: A prism with length $7$ , width $4$ , and height $2$ .
- Step 1: $V = l \times w \times h$
- Step 2: $V = 7 \times 4 \times 2$
- Step 3: $V = 28 \times 2$
- Step 4: $V = 56\,cm^3$
Calculation Flexibility: In multiplication, the order of numbers does not matter ( $l \times w \times h$ is the same as $h \times w \times l$ ). The result remained the same at $56$ regardless of which dimension was assigned as length or width.

Right Prism: A three-dimensional figure that has the same cross-section all the way through. The term "right" indicates it is consistent from base to top.
Cross-Section: The technical term for parallel slices of a prism. If you cut a prism at any point parallel to its base, the slice (cross-section) must be the exact same size and shape as the base.
Standard Prism Shapes:
- Rectangular Prism: Bases are rectangles.
- Triangular Prism: Bases are triangles.
- Hexagonal Prism: Bases are hexagons.
Non-Prism Examples:
- Pyramids and Cones: These are not prisms because the cross-sections get smaller as you move toward the top.
- Buckets: Often wider at the top than at the bottom, meaning slices are not identical.
- Barrels: They bow out in the middle.
Cylinders: Technically, a cylinder is not a prism because it does not have flat faces, but it is treated like one mathematically because the same volume principle ( $V = A \times h$ ) applies.

Context: The assignment involves calculating volumes for garden beds, specifically for weed matting, drainage stone, soil, and mulch.
Units of Choice: For large objects like garden beds, square meters ( $m^2$ ) are often used to make calculations more manageable.
Garden Soil Ratios: Drainage stone and garden soil are used in a ratio of $2:3$ .
Ratio Calculation Procedure:
1. Determine the total parts. For a $2:3$ ratio, the total is $2 + 3 = 5$ parts.
2. Turn ratios into fractions: Drainage stone = $\frac{2}{5}$ ; Garden soil = $\frac{3}{5}$ .
3. If the total height of the garden bed is $100\,cm$ :
  - Height of Stone: $\frac{2}{5} \times 100 = 40\,cm$
  - Height of Soil: $\frac{3}{5} \times 100 = 60\,cm$
4. Volume is then calculated by multiplying the base area by these specific heights.
Example Calculation (Overall Volume):
- Garden bed dimensions: $200\,cm$ (length) by $50\,cm$ (width) by $41\,cm$ (height).
- $V = 200 \times 50 \times 41 = 410,000\,cm^3$ .

Base Shape: The base of this prism is a triangle.
Formula Derivation: Since $V = \text{Area} \times h$ and the area of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$ , the full formula is:
- $V = \frac{1}{2} \times b \times h \times H$
- Here, $b$ is the base of the triangle, $h$ is the vertical height of the triangle, and $H$ is the length/height of the entire prism.
Example Calculation:
- Triangle Base ( $b$ ) = $25\,cm$
- Triangle Height ( $h$ ) = $18\,cm$
- Prism Height/Length ( $H$ ) = $40\,cm$
- Step 1: $V = 0.5 \times 25 \times 18 \times 40$
- Step 2: $V = 9,000\,cm^3$
Conversion Note: The instructor noted that $1,000\,cm^3$ is equal to one cubic meter, so $9,000\,cm^3$ would be $9\,m^3$ . (Note: Standard conversion is actually $1,000,000\,cm^3 = 1\,m^3$ , but notes reflect transcription).

Student Inquiry (Eleanor): Correctly identified block counts (8 and 10) in the visual cube exercises.
Student Inquiry (Fatima): Asked whether numbers in the volume formula should be added or multiplied; clarified that they must be multiplied, even if the resulting numbers are very large.
Student Logic (Fatima): Independently deduced the volume formula by describing how to count rows and layers of blocks.
Discussion on Slang: The term "yeah nah" was discussed as modern Australian slang. Mr. Eckworth referenced the phrase "totally radical" from the Ninja Turtles (1980s).
Cultural References: Mr. Eckworth mentioned loving Sesame Street (characters like Snuffleopagus, Bert, Ernie, and Oscar the Grouch) and being a fan of Lord of the Rings/The Hobbit, specifically the "barrel riders" scene.
Class Participation: Mr. Eckworth addressed students Saxon, Jacob, and Maya regarding their lack of participation or log-ins during the session, emphasizing that while mistakes are part of learning, lack of effort or "slackness" is penalized. Jacob apologized, but Mr. Eckworth noted that "sorry" means not repeating the behavior.
Resources Mentioned: Textbooks such as "Jack Plus" are used for practice, which provide answers at the back. Mr. Eckworth clarified that class is for understanding the process, as exam conditions do not provide answers.