00.4 - Maths Recs

All Diploma Programme physics students should be able to:

 IB Mathematical Requirement

Online Resource:  

1.  perform the basic arithmetic functions: addition, subtraction, multiplication and division

   1. Khan Academy: Arithmetic Basics 

   2. Khan Academy: Order of Operations

2. carry out calculations involving means, decimals, fractions, percentages, ratios, approximations and reciprocals

   1. Mean: 

      • Khan Academy: Measures of Central Tendency

   2. Decimals:

      • Khan Academy: Converting Between Decimals and Fractions

   3. Basic Fractions:

      • Khan Academy: Calculations with Fractions

3. carry out manipulations with trigonometric functions

   1. Khan Academy: Trigonometry 

4. carry out manipulations with logarithmic and exponential functions (HL only)

   1. KhanAcademy: Logarithmic Equations

5. use standard notation (for example, 3.6 × 10^6)

6. use direct and inverse proportion 

   1. KhanAcademy: Direct and Inverse Relationships

7. solve simple algebraic equations

   1. KhanAcademy: Working with Units

8. solve linear simultaneous equations

   1. Khan Academy: System of Equations

9. plot graphs (with suitable scales and axes) including two variables that show linear and non-linear relationships

   1. Excel Graphing: Basics of Graphing

   2. Excel Graphing: Error Bars

10. interpret graphs, including the significance of gradients, changes in gradients, intercepts and areas (see some common examples below). 

11. Linear Graphs: 

    1. Slope of v-t graph (but it applies to all graphs)

    2. Comparison of v-t Graphs (still applies to all graphs) 

    3. Areas of a Graph 

12. Quadratics: 

13. Inverse: 

    1. draw lines (either curves or linear) of best fit on a scatter plot graph

14. Excel Graphing: Trendlines

15.   on a best-fit linear graph, construct linear lines of maximum and minimum gradients with relative accuracy (by eye) taking into account all uncertainty bars

16. Excel Graphing: Error Bars

17. Excel Graphing: Trendlines (Part 1 and Part 2)

    1. interpret data presented in various forms (for example, bar charts, histograms and pie charts)

    2. represent arithmetic mean using x-bar notation (for example, x)

    3. express uncertainties to one or two significant figures, with justification.

18. Linearize graphs of data   

    1. Lineariaze Relationships

    2. Inverse Relationships

    3. Quadratic Relationships

    4. Square Root Relationships

Valuable information can be obtained by analyzing graphs. 

From any graph three common quantities can be found:

1. Values - 

   1. values at a given coordinate,

   2. y-axis Intercepts,

   3. x-axis intercepts,

   4. maximums or minimums of quadratics equations

   5. Most common intercepts:

      1. For two resistors in parallel, the intercepts of the 1/R1 v. 1/R2 graph give the effective resistance.

      2. For mirrors or lenses, the intercepts of the  1/v-1/u  graph give the focal length.

      3. The x-intercept of a kinetic energy maximum-frequency graph gives the threshold frequency.

      4. The y-intercept of a kinetic energy maximum-frequency graph gives the work function.

2. The slope or gradient of a line- 

   1. Determining the relationship between two variables is at the core of DP Physics.

   2. Can also be described as the derivative of a function or the tangent line.

   3. Most common slopes:

      1. The slope of a position-time graph gives the velocity. 

      2. The slope of a velocity-time graph gives the acceleration. 

      3. The slope of a force-elongation graph gives the spring constant.

      4. For Ohmic materials the slope of a voltage-current graph gives the resistance.

      5. The slope of a sine incident angle-sine refracted angle graph gives the index of refraction.

      6. The slope of a kinetic energy maximum-frequency graph gives Planck's constant.

3. The area under the curve

   1. Typically only done with linear expressions

   2. Can be found using area of a rectangle, triangle, or trapezoid.

   3. Most common areas under the curve:

      1. The area under the curve of a velocity-time graph gives the displacement. 

      2. The area under the curve of an acceleration-time graph gives the change in velocity.

      3. The area under the curve of a force-time graph gives the impulse (change in momentum).

      4. The area under the curve of a force-distance graph gives the work done.

      5. The area under the curve of a pressure-volume graph gives the work done by the gas.

      6. The area under a charge-voltage graph gives the work required to charge the capacitor i.e. energy stored.