Gr 11 Math Lit: November Mix P(2)

Maps, Plans, and Other Representations

National Road Maps

• Show major routes, roads, and distances between major cities and towns in South Africa
• Skills required:
  • Deciding on the most appropriate route between destinations
  • Estimating traveling costs and times
  • Planning stopping points for accommodation, breaks, or refuelling
  • Using distance tables to determine distances

Strip Charts

• Show locations of towns, other roads, and points of interest on a specific route
• Skills required:
  • Determining more accurate traveling distances on specific sections of a route
  • Deciding on appropriate stopping points for refuelling or breaks
  • Estimating traveling costs and times
  • Using distance markers and indicators on the chart

Street Maps

• Show positions and names of streets and other locations within a city
• Skills required:
  • Identifying locations using symbols
  • Following and providing directions using street names and directional indicators (e.g., left, right)
  • Using a grid reference system with the street index to find specific streets

Housing Complex Maps

• Show positions of houses within a small housing complex
• Skills required:
  • Working with a map that shows a small area with limited detail
  • Following and describing directions to a specific location within the complex

Using Scales

• Essential to use the given scale to estimate traveling distances when a distance table or distance markers are not provided
• Two types of scale calculations:
  • Calculation Type 1: Map Measurement to Actual Distance
    • Measure a distance on a map
    • Use the given scale to convert this measurement to the actual distance
    • Formula: Actual distance = Map measurement × Scale factor
  • Calculation Type 2: Actual Distance to Map Measurement
    • Start with a known actual distance
    • Use the given scale to convert this actual distance to the map measurement
    • Formula: Map measurement = Actual distance ÷ Scale factor

Types of Scales

Number Scale

• Expressed in the format 1:50,000
• Shows the relationship between the size of the area on the map and the actual size of that area
• Conversion is straightforward: enlarge or reduce by the scale factor

Bar Scale

• Shows the relationship between measured length and actual distance
• Measuring with a ruler on the bar scale shows how many measured units are equal to actual distance
• Different from a number scale in that it does not state how many times smaller the map is from the actual size

Building Plans

Key Concepts

• Floor Plan: shows a 2-dimensional perspective of a building or structure as seen from above, without the roof
• Elevation Plan: shows a 2-dimensional perspective of a building or structure as seen from the side
• Design Drawing: shows the design, layout, and components of an item to be constructed or manufactured

Key Concepts for Working with Plans

• Describe features of a building/structure
• Match features between plans
• Understand compass directions
• Use scale to determine dimensions

Section 2: Design Drawings

• Describe the item/object shown on the design plan
• Identify the dimensions of the item/object
• Describe the different component parts required to make the item/object
• If necessary, identify a cutting list of parts needed to make the item/object

Section 3: Drawing Scaled Plans

• Calculate plan measure from actual dimension
• Scale calculation: when converting from an actual dimension to a plan measure, use the given scale to reduce the actual measurement proportionately
• Formula: Plan Measure = Actual Dimension ÷ Scale Factor

Section 4: Models

• Build 3D models from 2D plans/nets
• Draw 2D plans/nets of 3D models
• Use models to investigate various problems
• Purpose: to facilitate the movement between 2D and 3D representations of objects
• Process:
  • Understanding 2D plans/nets
  • Building 3D models
  • Drawing 2D plans/nets

Section 5: Assembling

• Understand instructions
• Gather materials and tools
• Measure and cut materials
• Estimate quantities and costs
• Calculate time
• Follow steps in sequence
• Check and adjust
• Use mathematical tools and techniques
• Steps to assemble components:
  • Interpret instructions
  • Prepare materials and tools
  • Measure accurately
  • Cut materials as needed
  • Estimate quantities and costs
  • Estimate time
  • Follow the sequence
  • Check and adjust
  • Complete the assembly### Probability in Predictions
• Historical trends: probability values are determined by observing past trends to predict future events
• Advertisements: examples of probability in practice, such as car insurance rates based on historical accident data
• Uncertainty in prediction: probability values do not guarantee outcomes

Patterns and Relationships

• Constant ratio relationship: a value increases by a factor, and then this new value is increased by the same factor, and so on
  • Characterized by a graph that is not a straight line, but becomes steeper at an increasing rate
  • Example: monthly cost of food purchases increasing at a rate of 7.4% per year
• Combination of relationships: graphs can be made up of different types of relationships
  • Example: cost of making calls on a cell phone contract with a flat portion and an increasing portion
• Step-function relationships: different fees or values are charged for different time intervals
  • Example: parking fees at a supermarket
  • Characterized by a graph that looks like a series of steps

Working with 2 Relationships

• Point of intersection: where two graphs intersect, and the values represented by the graphs are exactly equal
• Estimating points of intersection: a method to determine the values at which two relationships are equal
  • Identify the two graphs, locate the intersection point, read the horizontal and vertical values
• Trial and improvement method: another method to determine the values at which two relationships are equal
  • Write the equations, substitute values, and refine estimates

Measurements

• Conversions: conversion factors are used to switch between different units of measurement
• Converting weight: from kilograms to pounds, and from pounds to kilograms
• Converting temperature: from Celsius to Fahrenheit, and from Fahrenheit to Celsius
• Converting between solid and liquid quantities: using conversion factors specific to ingredients or materials
• Converting from m² to liters to determine paint quantities: calculating the amount of paint needed based on the surface area and the spread rate of the paint

Practical Tips for Performing Conversions

• Always use the provided conversion factors
• Convert units step-by-step
• Double-check calculations
• Understand the context

Time

• Interpreting time recording values: converting time values into a single unit, such as seconds
• Performing calculations involving time recording values: calculating the difference between two time values
• Designing and making sense of timetables: identifying the week, date, and time of an event, and the venue

Formulas Summary

• Time conversions: converting minutes to seconds, hours to seconds, minutes to hours, and seconds to hours
• Calculating time differences: converting to a single unit, subtracting the values, and converting back to the desired time format

Practical Application

• Interpreting time: converting and understanding stopwatch values and other time recordings
• Calculating differences in time: determining the time elapsed between events accurately
• Using timetables: extracting key information to plan and organize activities effectively

Measuring and Estimating

• Understanding units of measurement: standard units, conversion of units, and measurement techniques
• Measuring techniques: measuring length, area, and volume, and measuring mass and weight
• Estimation techniques: rounding and approximating, estimating quantities, and using benchmarks
• Application of measurement in real-life contexts: scaling, perimeter, area, and volume calculations, and rate and proportion
• Accuracy and precision: using significant figures, and error analysis

Steps to Measure and Estimate

• Identify the quantity to be measured or estimated
• Choose the appropriate tool or method
• Perform the measurement or estimation
• Convert units if necessary
• Record the measurement or estimation
• Analyze and apply the results

Calculate

• Surface area: the total area of all the surfaces of a 3-dimensional object

  • Formulae: surface area of a rectangular box, and surface area of a cylinder
  • Appropriate units: square units such as mm², cm², or m²

• Volume: the amount of space inside a hollow 3-dimensional object, or the amount of space that a solid 3-dimensional object takes up

  • Formulae: general formula for volume of a rectangular-based container, volume of a rectangular box, and volume of a cylindrical container
  • Appropriate units: cubic units such as mm³, cm³, or m³

• Conversion between units: often necessary when performing surface area and volume calculations### Converting between Units for Surface Area and Volume

• When painting a wall, paint is sold in liters, not in square units, so a conversion method is needed to convert from square units to liters.

Volume Conversion Ratios

• 1 cubic meter (m³) can hold 1000 liters (L) of water.
• 1 cubic centimeter (cm³) equals 1 milliliter (mL) of water.

Important Conversion Considerations

• Even though 1 meter (m) equals 100 centimeters (cm), this does not mean that 1 square meter (m²) equals 100 square centimeters (cm²) or that 1 cubic meter (m³) equals 100 cubic centimeters (cm³).
• Converting between units involves multiplying or dividing by the correct conversion factor: for example, 1 m³ = 1,000,000 cm³.
• Similarly, converting between units involves multiplying or dividing by the correct conversion factor: for example, 1 cm² = 100 mm².

Maps, Plans, and Other Representations

National Road Maps

• Show major routes, roads, and distances between major cities and towns in South Africa
• Skills required:
  • Deciding on the most appropriate route between destinations
  • Estimating traveling costs and times
  • Planning stopping points for accommodation, breaks, or refuelling
  • Using distance tables to determine distances

Strip Charts

• Show locations of towns, other roads, and points of interest on a specific route
• Skills required:
  • Determining more accurate traveling distances on specific sections of a route
  • Deciding on appropriate stopping points for refuelling or breaks
  • Estimating traveling costs and times
  • Using distance markers and indicators on the chart

Street Maps

• Show positions and names of streets and other locations within a city
• Skills required:
  • Identifying locations using symbols
  • Following and providing directions using street names and directional indicators (e.g., left, right)
  • Using a grid reference system with the street index to find specific streets

Housing Complex Maps

• Show positions of houses within a small housing complex
• Skills required:
  • Working with a map that shows a small area with limited detail
  • Following and describing directions to a specific location within the complex

Using Scales

• Essential to use the given scale to estimate traveling distances when a distance table or distance markers are not provided
• Two types of scale calculations:
  • Calculation Type 1: Map Measurement to Actual Distance
    • Measure a distance on a map
    • Use the given scale to convert this measurement to the actual distance
    • Formula: Actual distance = Map measurement × Scale factor
  • Calculation Type 2: Actual Distance to Map Measurement
    • Start with a known actual distance
    • Use the given scale to convert this actual distance to the map measurement
    • Formula: Map measurement = Actual distance ÷ Scale factor

Types of Scales

Number Scale

• Expressed in the format 1:50,000
• Shows the relationship between the size of the area on the map and the actual size of that area
• Conversion is straightforward: enlarge or reduce by the scale factor

Bar Scale

• Shows the relationship between measured length and actual distance
• Measuring with a ruler on the bar scale shows how many measured units are equal to actual distance
• Different from a number scale in that it does not state how many times smaller the map is from the actual size

Building Plans

Key Concepts

• Floor Plan: shows a 2-dimensional perspective of a building or structure as seen from above, without the roof
• Elevation Plan: shows a 2-dimensional perspective of a building or structure as seen from the side
• Design Drawing: shows the design, layout, and components of an item to be constructed or manufactured

Key Concepts for Working with Plans

• Describe features of a building/structure
• Match features between plans
• Understand compass directions
• Use scale to determine dimensions

Section 2: Design Drawings

• Describe the item/object shown on the design plan
• Identify the dimensions of the item/object
• Describe the different component parts required to make the item/object
• If necessary, identify a cutting list of parts needed to make the item/object

Section 3: Drawing Scaled Plans

• Calculate plan measure from actual dimension
• Scale calculation: when converting from an actual dimension to a plan measure, use the given scale to reduce the actual measurement proportionately
• Formula: Plan Measure = Actual Dimension ÷ Scale Factor

Section 4: Models

• Build 3D models from 2D plans/nets
• Draw 2D plans/nets of 3D models
• Use models to investigate various problems
• Purpose: to facilitate the movement between 2D and 3D representations of objects
• Process:
  • Understanding 2D plans/nets
  • Building 3D models
  • Drawing 2D plans/nets

Section 5: Assembling

• Understand instructions
• Gather materials and tools
• Measure and cut materials
• Estimate quantities and costs
• Calculate time
• Follow steps in sequence
• Check and adjust
• Use mathematical tools and techniques
• Steps to assemble components:
  • Interpret instructions
  • Prepare materials and tools
  • Measure accurately
  • Cut materials as needed
  • Estimate quantities and costs
  • Estimate time
  • Follow the sequence
  • Check and adjust
  • Complete the assembly### Probability in Predictions
• Historical trends: probability values are determined by observing past trends to predict future events
• Advertisements: examples of probability in practice, such as car insurance rates based on historical accident data
• Uncertainty in prediction: probability values do not guarantee outcomes

Patterns and Relationships

• Constant ratio relationship: a value increases by a factor, and then this new value is increased by the same factor, and so on
  • Characterized by a graph that is not a straight line, but becomes steeper at an increasing rate
  • Example: monthly cost of food purchases increasing at a rate of 7.4% per year
• Combination of relationships: graphs can be made up of different types of relationships
  • Example: cost of making calls on a cell phone contract with a flat portion and an increasing portion
• Step-function relationships: different fees or values are charged for different time intervals
  • Example: parking fees at a supermarket
  • Characterized by a graph that looks like a series of steps

Working with 2 Relationships

• Point of intersection: where two graphs intersect, and the values represented by the graphs are exactly equal
• Estimating points of intersection: a method to determine the values at which two relationships are equal
  • Identify the two graphs, locate the intersection point, read the horizontal and vertical values
• Trial and improvement method: another method to determine the values at which two relationships are equal
  • Write the equations, substitute values, and refine estimates

Measurements

• Conversions: conversion factors are used to switch between different units of measurement
• Converting weight: from kilograms to pounds, and from pounds to kilograms
• Converting temperature: from Celsius to Fahrenheit, and from Fahrenheit to Celsius
• Converting between solid and liquid quantities: using conversion factors specific to ingredients or materials
• Converting from m² to liters to determine paint quantities: calculating the amount of paint needed based on the surface area and the spread rate of the paint

Practical Tips for Performing Conversions

• Always use the provided conversion factors
• Convert units step-by-step
• Double-check calculations
• Understand the context

Time

• Interpreting time recording values: converting time values into a single unit, such as seconds
• Performing calculations involving time recording values: calculating the difference between two time values
• Designing and making sense of timetables: identifying the week, date, and time of an event, and the venue

Formulas Summary

• Time conversions: converting minutes to seconds, hours to seconds, minutes to hours, and seconds to hours
• Calculating time differences: converting to a single unit, subtracting the values, and converting back to the desired time format

Practical Application

• Interpreting time: converting and understanding stopwatch values and other time recordings
• Calculating differences in time: determining the time elapsed between events accurately
• Using timetables: extracting key information to plan and organize activities effectively

Measuring and Estimating

• Understanding units of measurement: standard units, conversion of units, and measurement techniques
• Measuring techniques: measuring length, area, and volume, and measuring mass and weight
• Estimation techniques: rounding and approximating, estimating quantities, and using benchmarks
• Application of measurement in real-life contexts: scaling, perimeter, area, and volume calculations, and rate and proportion
• Accuracy and precision: using significant figures, and error analysis

Steps to Measure and Estimate

• Identify the quantity to be measured or estimated
• Choose the appropriate tool or method
• Perform the measurement or estimation
• Convert units if necessary
• Record the measurement or estimation
• Analyze and apply the results

Calculate

• Surface area: the total area of all the surfaces of a 3-dimensional object

  • Formulae: surface area of a rectangular box, and surface area of a cylinder
  • Appropriate units: square units such as mm², cm², or m²

• Volume: the amount of space inside a hollow 3-dimensional object, or the amount of space that a solid 3-dimensional object takes up

  • Formulae: general formula for volume of a rectangular-based container, volume of a rectangular box, and volume of a cylindrical container
  • Appropriate units: cubic units such as mm³, cm³, or m³

• Conversion between units: often necessary when performing surface area and volume calculations### Converting between Units for Surface Area and Volume

• When painting a wall, paint is sold in liters, not in square units, so a conversion method is needed to convert from square units to liters.

Volume Conversion Ratios

• 1 cubic meter (m³) can hold 1000 liters (L) of water.
• 1 cubic centimeter (cm³) equals 1 milliliter (mL) of water.

Important Conversion Considerations

• Even though 1 meter (m) equals 100 centimeters (cm), this does not mean that 1 square meter (m²) equals 100 square centimeters (cm²) or that 1 cubic meter (m³) equals 100 cubic centimeters (cm³).
• Converting between units involves multiplying or dividing by the correct conversion factor: for example, 1 m³ = 1,000,000 cm³.
• Similarly, converting between units involves multiplying or dividing by the correct conversion factor: for example, 1 cm² = 100 mm².

Maps, Plans, and Other Representations

National Road Maps

• Show major routes, roads, and distances between major cities and towns in South Africa
• Skills required:
  • Deciding on the most appropriate route between destinations
  • Estimating traveling costs and times
  • Planning stopping points for accommodation, breaks, or refuelling
  • Using distance tables to determine distances

Strip Charts

• Show locations of towns, other roads, and points of interest on a specific route
• Skills required:
  • Determining more accurate traveling distances on specific sections of a route
  • Deciding on appropriate stopping points for refuelling or breaks
  • Estimating traveling costs and times
  • Using distance markers and indicators on the chart

Street Maps

• Show positions and names of streets and other locations within a city
• Skills required:
  • Identifying locations using symbols
  • Following and providing directions using street names and directional indicators (e.g., left, right)
  • Using a grid reference system with the street index to find specific streets

Housing Complex Maps

• Show positions of houses within a small housing complex
• Skills required:
  • Working with a map that shows a small area with limited detail
  • Following and describing directions to a specific location within the complex

Using Scales

• Essential to use the given scale to estimate traveling distances when a distance table or distance markers are not provided
• Two types of scale calculations:
  • Calculation Type 1: Map Measurement to Actual Distance
    • Measure a distance on a map
    • Use the given scale to convert this measurement to the actual distance
    • Formula: Actual distance = Map measurement × Scale factor
  • Calculation Type 2: Actual Distance to Map Measurement
    • Start with a known actual distance
    • Use the given scale to convert this actual distance to the map measurement
    • Formula: Map measurement = Actual distance ÷ Scale factor

Types of Scales

Number Scale

• Expressed in the format 1:50,000
• Shows the relationship between the size of the area on the map and the actual size of that area
• Conversion is straightforward: enlarge or reduce by the scale factor

Bar Scale

• Shows the relationship between measured length and actual distance
• Measuring with a ruler on the bar scale shows how many measured units are equal to actual distance
• Different from a number scale in that it does not state how many times smaller the map is from the actual size

Building Plans

Key Concepts

• Floor Plan: shows a 2-dimensional perspective of a building or structure as seen from above, without the roof
• Elevation Plan: shows a 2-dimensional perspective of a building or structure as seen from the side
• Design Drawing: shows the design, layout, and components of an item to be constructed or manufactured

Key Concepts for Working with Plans

• Describe features of a building/structure
• Match features between plans
• Understand compass directions
• Use scale to determine dimensions

Section 2: Design Drawings

• Describe the item/object shown on the design plan
• Identify the dimensions of the item/object
• Describe the different component parts required to make the item/object
• If necessary, identify a cutting list of parts needed to make the item/object

Section 3: Drawing Scaled Plans

• Calculate plan measure from actual dimension
• Scale calculation: when converting from an actual dimension to a plan measure, use the given scale to reduce the actual measurement proportionately
• Formula: Plan Measure = Actual Dimension ÷ Scale Factor

Section 4: Models

• Build 3D models from 2D plans/nets
• Draw 2D plans/nets of 3D models
• Use models to investigate various problems
• Purpose: to facilitate the movement between 2D and 3D representations of objects
• Process:
  • Understanding 2D plans/nets
  • Building 3D models
  • Drawing 2D plans/nets

Section 5: Assembling

• Understand instructions
• Gather materials and tools
• Measure and cut materials
• Estimate quantities and costs
• Calculate time
• Follow steps in sequence
• Check and adjust
• Use mathematical tools and techniques
• Steps to assemble components:
  • Interpret instructions
  • Prepare materials and tools
  • Measure accurately
  • Cut materials as needed
  • Estimate quantities and costs
  • Estimate time
  • Follow the sequence
  • Check and adjust
  • Complete the assembly### Probability in Predictions
• Historical trends: probability values are determined by observing past trends to predict future events
• Advertisements: examples of probability in practice, such as car insurance rates based on historical accident data
• Uncertainty in prediction: probability values do not guarantee outcomes

Patterns and Relationships

• Constant ratio relationship: a value increases by a factor, and then this new value is increased by the same factor, and so on
  • Characterized by a graph that is not a straight line, but becomes steeper at an increasing rate
  • Example: monthly cost of food purchases increasing at a rate of 7.4% per year
• Combination of relationships: graphs can be made up of different types of relationships
  • Example: cost of making calls on a cell phone contract with a flat portion and an increasing portion
• Step-function relationships: different fees or values are charged for different time intervals
  • Example: parking fees at a supermarket
  • Characterized by a graph that looks like a series of steps

Working with 2 Relationships

• Point of intersection: where two graphs intersect, and the values represented by the graphs are exactly equal
• Estimating points of intersection: a method to determine the values at which two relationships are equal
  • Identify the two graphs, locate the intersection point, read the horizontal and vertical values
• Trial and improvement method: another method to determine the values at which two relationships are equal
  • Write the equations, substitute values, and refine estimates

Measurements

• Conversions: conversion factors are used to switch between different units of measurement
• Converting weight: from kilograms to pounds, and from pounds to kilograms
• Converting temperature: from Celsius to Fahrenheit, and from Fahrenheit to Celsius
• Converting between solid and liquid quantities: using conversion factors specific to ingredients or materials
• Converting from m² to liters to determine paint quantities: calculating the amount of paint needed based on the surface area and the spread rate of the paint

Practical Tips for Performing Conversions

• Always use the provided conversion factors
• Convert units step-by-step
• Double-check calculations
• Understand the context

Time

• Interpreting time recording values: converting time values into a single unit, such as seconds
• Performing calculations involving time recording values: calculating the difference between two time values
• Designing and making sense of timetables: identifying the week, date, and time of an event, and the venue

Formulas Summary

• Time conversions: converting minutes to seconds, hours to seconds, minutes to hours, and seconds to hours
• Calculating time differences: converting to a single unit, subtracting the values, and converting back to the desired time format

Practical Application

• Interpreting time: converting and understanding stopwatch values and other time recordings
• Calculating differences in time: determining the time elapsed between events accurately
• Using timetables: extracting key information to plan and organize activities effectively

Measuring and Estimating

• Understanding units of measurement: standard units, conversion of units, and measurement techniques
• Measuring techniques: measuring length, area, and volume, and measuring mass and weight
• Estimation techniques: rounding and approximating, estimating quantities, and using benchmarks
• Application of measurement in real-life contexts: scaling, perimeter, area, and volume calculations, and rate and proportion
• Accuracy and precision: using significant figures, and error analysis

Steps to Measure and Estimate

• Identify the quantity to be measured or estimated
• Choose the appropriate tool or method
• Perform the measurement or estimation
• Convert units if necessary
• Record the measurement or estimation
• Analyze and apply the results

Calculate

• Surface area: the total area of all the surfaces of a 3-dimensional object

  • Formulae: surface area of a rectangular box, and surface area of a cylinder
  • Appropriate units: square units such as mm², cm², or m²

• Volume: the amount of space inside a hollow 3-dimensional object, or the amount of space that a solid 3-dimensional object takes up

  • Formulae: general formula for volume of a rectangular-based container, volume of a rectangular box, and volume of a cylindrical container
  • Appropriate units: cubic units such as mm³, cm³, or m³

• Conversion between units: often necessary when performing surface area and volume calculations### Converting between Units for Surface Area and Volume

• When painting a wall, paint is sold in liters, not in square units, so a conversion method is needed to convert from square units to liters.

Volume Conversion Ratios

• 1 cubic meter (m³) can hold 1000 liters (L) of water.
• 1 cubic centimeter (cm³) equals 1 milliliter (mL) of water.

Important Conversion Considerations

• Even though 1 meter (m) equals 100 centimeters (cm), this does not mean that 1 square meter (m²) equals 100 square centimeters (cm²) or that 1 cubic meter (m³) equals 100 cubic centimeters (cm³).
• Converting between units involves multiplying or dividing by the correct conversion factor: for example, 1 m³ = 1,000,000 cm³.
• Similarly, converting between units involves multiplying or dividing by the correct conversion factor: for example, 1 cm² = 100 mm².

Maps, Plans, and Other Representations

National Road Maps

• Show major routes, roads, and distances between major cities and towns in South Africa
• Skills required:
  • Deciding on the most appropriate route between destinations
  • Estimating traveling costs and times
  • Planning stopping points for accommodation, breaks, or refuelling
  • Using distance tables to determine distances

Strip Charts

• Show locations of towns, other roads, and points of interest on a specific route
• Skills required:
  • Determining more accurate traveling distances on specific sections of a route
  • Deciding on appropriate stopping points for refuelling or breaks
  • Estimating traveling costs and times
  • Using distance markers and indicators on the chart

Street Maps

• Show positions and names of streets and other locations within a city
• Skills required:
  • Identifying locations using symbols
  • Following and providing directions using street names and directional indicators (e.g., left, right)
  • Using a grid reference system with the street index to find specific streets

Housing Complex Maps

• Show positions of houses within a small housing complex
• Skills required:
  • Working with a map that shows a small area with limited detail
  • Following and describing directions to a specific location within the complex

Using Scales

• Essential to use the given scale to estimate traveling distances when a distance table or distance markers are not provided
• Two types of scale calculations:
  • Calculation Type 1: Map Measurement to Actual Distance
    • Measure a distance on a map
    • Use the given scale to convert this measurement to the actual distance
    • Formula: Actual distance = Map measurement × Scale factor
  • Calculation Type 2: Actual Distance to Map Measurement
    • Start with a known actual distance
    • Use the given scale to convert this actual distance to the map measurement
    • Formula: Map measurement = Actual distance ÷ Scale factor

Types of Scales

Number Scale

• Expressed in the format 1:50,000
• Shows the relationship between the size of the area on the map and the actual size of that area
• Conversion is straightforward: enlarge or reduce by the scale factor

Bar Scale

• Shows the relationship between measured length and actual distance
• Measuring with a ruler on the bar scale shows how many measured units are equal to actual distance
• Different from a number scale in that it does not state how many times smaller the map is from the actual size

Building Plans

Key Concepts

• Floor Plan: shows a 2-dimensional perspective of a building or structure as seen from above, without the roof
• Elevation Plan: shows a 2-dimensional perspective of a building or structure as seen from the side
• Design Drawing: shows the design, layout, and components of an item to be constructed or manufactured

Key Concepts for Working with Plans

• Describe features of a building/structure
• Match features between plans
• Understand compass directions
• Use scale to determine dimensions

Section 2: Design Drawings

• Describe the item/object shown on the design plan
• Identify the dimensions of the item/object
• Describe the different component parts required to make the item/object
• If necessary, identify a cutting list of parts needed to make the item/object

Section 3: Drawing Scaled Plans

• Calculate plan measure from actual dimension
• Scale calculation: when converting from an actual dimension to a plan measure, use the given scale to reduce the actual measurement proportionately
• Formula: Plan Measure = Actual Dimension ÷ Scale Factor

Section 4: Models

• Build 3D models from 2D plans/nets
• Draw 2D plans/nets of 3D models
• Use models to investigate various problems
• Purpose: to facilitate the movement between 2D and 3D representations of objects
• Process:
  • Understanding 2D plans/nets
  • Building 3D models
  • Drawing 2D plans/nets

Section 5: Assembling

• Understand instructions
• Gather materials and tools
• Measure and cut materials
• Estimate quantities and costs
• Calculate time
• Follow steps in sequence
• Check and adjust
• Use mathematical tools and techniques
• Steps to assemble components:
  • Interpret instructions
  • Prepare materials and tools
  • Measure accurately
  • Cut materials as needed
  • Estimate quantities and costs
  • Estimate time
  • Follow the sequence
  • Check and adjust
  • Complete the assembly### Probability in Predictions
• Historical trends: probability values are determined by observing past trends to predict future events
• Advertisements: examples of probability in practice, such as car insurance rates based on historical accident data
• Uncertainty in prediction: probability values do not guarantee outcomes

Patterns and Relationships

• Constant ratio relationship: a value increases by a factor, and then this new value is increased by the same factor, and so on
  • Characterized by a graph that is not a straight line, but becomes steeper at an increasing rate
  • Example: monthly cost of food purchases increasing at a rate of 7.4% per year
• Combination of relationships: graphs can be made up of different types of relationships
  • Example: cost of making calls on a cell phone contract with a flat portion and an increasing portion
• Step-function relationships: different fees or values are charged for different time intervals
  • Example: parking fees at a supermarket
  • Characterized by a graph that looks like a series of steps

Working with 2 Relationships

• Point of intersection: where two graphs intersect, and the values represented by the graphs are exactly equal
• Estimating points of intersection: a method to determine the values at which two relationships are equal
  • Identify the two graphs, locate the intersection point, read the horizontal and vertical values
• Trial and improvement method: another method to determine the values at which two relationships are equal
  • Write the equations, substitute values, and refine estimates

Measurements

• Conversions: conversion factors are used to switch between different units of measurement
• Converting weight: from kilograms to pounds, and from pounds to kilograms
• Converting temperature: from Celsius to Fahrenheit, and from Fahrenheit to Celsius
• Converting between solid and liquid quantities: using conversion factors specific to ingredients or materials
• Converting from m² to liters to determine paint quantities: calculating the amount of paint needed based on the surface area and the spread rate of the paint

Practical Tips for Performing Conversions

• Always use the provided conversion factors
• Convert units step-by-step
• Double-check calculations
• Understand the context

Time

• Interpreting time recording values: converting time values into a single unit, such as seconds
• Performing calculations involving time recording values: calculating the difference between two time values
• Designing and making sense of timetables: identifying the week, date, and time of an event, and the venue

Formulas Summary

• Time conversions: converting minutes to seconds, hours to seconds, minutes to hours, and seconds to hours
• Calculating time differences: converting to a single unit, subtracting the values, and converting back to the desired time format

Practical Application

• Interpreting time: converting and understanding stopwatch values and other time recordings
• Calculating differences in time: determining the time elapsed between events accurately
• Using timetables: extracting key information to plan and organize activities effectively

Measuring and Estimating

• Understanding units of measurement: standard units, conversion of units, and measurement techniques
• Measuring techniques: measuring length, area, and volume, and measuring mass and weight
• Estimation techniques: rounding and approximating, estimating quantities, and using benchmarks
• Application of measurement in real-life contexts: scaling, perimeter, area, and volume calculations, and rate and proportion
• Accuracy and precision: using significant figures, and error analysis

Steps to Measure and Estimate

• Identify the quantity to be measured or estimated
• Choose the appropriate tool or method
• Perform the measurement or estimation
• Convert units if necessary
• Record the measurement or estimation
• Analyze and apply the results

Calculate

• Surface area: the total area of all the surfaces of a 3-dimensional object

  • Formulae: surface area of a rectangular box, and surface area of a cylinder
  • Appropriate units: square units such as mm², cm², or m²

• Volume: the amount of space inside a hollow 3-dimensional object, or the amount of space that a solid 3-dimensional object takes up

  • Formulae: general formula for volume of a rectangular-based container, volume of a rectangular box, and volume of a cylindrical container
  • Appropriate units: cubic units such as mm³, cm³, or m³

• Conversion between units: often necessary when performing surface area and volume calculations### Converting between Units for Surface Area and Volume

• When painting a wall, paint is sold in liters, not in square units, so a conversion method is needed to convert from square units to liters.

Volume Conversion Ratios

• 1 cubic meter (m³) can hold 1000 liters (L) of water.
• 1 cubic centimeter (cm³) equals 1 milliliter (mL) of water.

Important Conversion Considerations

• Even though 1 meter (m) equals 100 centimeters (cm), this does not mean that 1 square meter (m²) equals 100 square centimeters (cm²) or that 1 cubic meter (m³) equals 100 cubic centimeters (cm³).
• Converting between units involves multiplying or dividing by the correct conversion factor: for example, 1 m³ = 1,000,000 cm³.
• Similarly, converting between units involves multiplying or dividing by the correct conversion factor: for example, 1 cm² = 100 mm².