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Achievement objective S7-4

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

  • Investigate situations that involve elements of chance:
    • A. comparing theoretical continuous distributions, such as the normal distribution, with experimental distributions
    • B. calculating probabilities, using such tools as two-way tables, tree diagrams, simulations, and technology.

Indicators

A. Comparing theoretical continuous distributions, such as the normal distribution, with experimental distributions:

  • Describes and compares distributions and recognises when distributions have similar and different characteristics.
  • Carries out experimental investigations of probability situations, understanding the ways a sample is likely to be representative of a population.
  • Is beginning to use mean and standard deviation as sample statistics or as population parameters.
  • Chooses an appropriate model to solve a problem.

B. Calculating probabilities, using such tools as two-way tables, tree diagrams, simulations, and technology:

  • Uses two-way frequency tables to solve simple probability problems, including starting to work with simple conditional probabilities.
  • Constructs and interprets probability trees with probabilities and outcomes on branches to solve probability problems representing either a series of events in time and/or a series of decision-making points, as well as probability problems with and without replacement.
  • Uses simple simulations to represent probability situations (using appropriate technology).
  • Students learn that situations involving real data from statistical investigations can be investigated from a probabilistic perspective.

Progression

S7-4 links from S6-3 and to S7-3, S8-4.

What is new/changed?

  • Students first meet probability trees as a calculation tool.
  • Use of two-way tables to solve probability problems.
  • Simulations remain an important focus.
  • The normal distribution as a useful model for explaining and exploring many situations.

Possible context elaborations

  • CensusAtSchool is a valuable website for classroom activities and information for teachers on all things statistics.
  • Play Rice Bombing (drop rice grains on an A3 sheet with a target on it and count grains in each region), draw the distribution and describe it.
  • Measure hand spans of students, plot and describe.
  • Compare graphs of data from real-life measurement variables, for example, heights, leaf length, birth weight, Census at School measurements, and note similarities and differences.
  • Compare several different symmetrical distributions with same mean and range but different distributions and discuss/develop rationale for using standard deviation. Begin to estimate mean and standard deviation from the plot of a distribution.
  • Investigate the number of cereal packets required to obtain a full set of animal cards using appropriate technology.
  • Two-way frequency tables can be introduced by asking students simple questions such as: “How many of you have at least one brother?”, “How many of you have at least one sister?” and “How many of you have at least one brother and at least one sister?”. Organise the results into a two-way frequency table.

    Two-way frequency table
      Sisters No_Sisters Total
    Brother(s) 10 5 15
    No_Brothers 8 4 12
    Total 18 9 27

    Students can then answer questions such as:
    What is the probability that the student … has at least one brother? … has at least one sister? … is an only child? … has a sister but no brothers?
    If a student has at least one sister, what is the probability that she also has a brother? If a student has at least one brother, what is the probability that he also has a sister? If a student does not have any brothers, what is the probability that he does have a sister? If a student does not have any sisters, what is the probability that he has at least one brother?

  • To introduce tree diagrams, it can be useful to start with a concrete situation that the students have previously encountered, to facilitate understanding how probability trees can simplify representation of complex situations. For example, Play Two Spot (a player pays to play and rolls three dice. If there is at least one 2 rolled, the player doubles his money, if not, he loses it) and calculate the probability of rolling at least one 2 when rolling two dice, then three dice.
  • Sampling with and without replacement should be introduced with probability trees, starting with simple concrete situations, such as drawing counters from a bag.
  • Investigate the number of cereal packets required to obtain a full set of animal cards using appropriate technology. Simulate

Assessment for qualifications

NCEA achievement standards at level 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the NZQA website.

The following achievement standard(s) could assess learning outcomes from this AO:

  • AS91267 Mathematics and statistics 2.12 Apply probability methods in solving problems
  • AS91268 Mathematics and statistics 2.13 Investigate a situation involving elements of chance using a simulation

Refer to the mathematics and statistics matrix.

Last updated September 26, 2013



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