Stage 2 - patterns and algebra – problem solving

Pose problems based on number patterns; solve a variety of problems using problem – solving strategies, including:

  • trial and error;
  • drawing a diagram
  • working backwards
  • looking for patterns
  • ask questions about how number patterns have been created and how they can be continued

Supporting students with learning difficulties

Strategy

Students can:

  1. pose problems based on number patterns
  2. solve a variety of problems using problem-solving strategies, including:
  • trial and error
  • drawing a diagram
  • working backwards
  • looking for patterns
  • ask questions about how number patterns have been created and how they can be continued

Activities to support the strategy

At this stage, students need to be able express their ideas and understanding by communicating and providing reasons for their thinking. Students are required to be able to use words to describe patterns and apply their knowledge in problem solving situations.

The mathematical language required needs to be modelled by the teacher and students need to be provided with opportunities to talk through their thinking. Students need experiences in both solving problems and posing their own problems.

  1. Using question prompts such as Newman’s prompts (PDF 41.18KB)
  2. Read the question
  3. What is the question asking you to do?
  4. How are you going to find the answer?
  5. Do the working out
  6. Write the answer to the question

At this stage, patterns and algebra problems often involve working backwards, using a process of elimination, trial and error and using inverse operations. These strategies need to be explicitly taught to students. The problems below provide students with opportunities to explore these problem solving strategies and share their strategies with others.

For further information please see Newman’s error analysis

Activity 1 – elimination

  • Bronte is thinking of a number.
  • The number is less than 12
  • When the number is divided by 2 the answer is an odd number
  • When it is divided by 3 the answer is an even number
  • What number is Bronte thinking of?

Use Newman’s prompts as a whole class to model how to start thinking about solving the problem.

When you reach questions 3 and 4 that deal with how to work the problem out, ask students what problem solving approach they might take. Some may suggest guess, or trial and error. Discuss with the students how they could use the process of elimination to assist in solving this problem.

  • “We know the number is less than 12, let’s write those numbers on the board.
  • It says the number can be divided by 2, let’s circle all those numbers and eliminate (cross out) the others.
  • It says the number can be divided by 3, out of the numbers we already have circled, which one(s) can also be divided by 3.
  • That only leaves 6.
  • Let’s test the number 6 to make sure we are correct.
  • When we divide 6 by 2, is the answer odd? Yes.
  • When we divide 6 by 3, is the answer even? Yes.
  • 6 is our answer.”

Activity 2 – working backwards

  • Ben started with a number
  • He added 3 to it
  • He doubled the answer, then subtracted 4
  • He then had 26
  • What number did Ben start with?

Use Newman’s prompts (PDF 41.18KB) as a whole class to model how to start thinking about solving the problem.

When you reach questions 3 and 4 that deal with how to work the problem out, ask students what problem solving approach they might take. Some may suggest guess, or trial and error. Present the information from the question stem in number sentence form.

Activity 3 – Looking for a pattern

Use Newman’s prompts as a whole class to model how to start thinking about solving the problem. Continue with the same process as before, only this time introduce students to the strategy of looking for a pattern.

Next Steps

Provide lots of different examples where students can talk through the different problem solving strategies they could use. Past NAPLAN papers provide teachers with a good bank of questions to use for this process.

Always ask students if there is another way to solve the problem. Some strategies may be more efficient than others or may be more appropriate, either way,  students need to develop a bank of problem solving strategies to use.

References

Australian curriculum reference: ACMNA060

Describe, continue, and create number patterns resulting from performing addition or subtraction.

NSW syllabus reference: MA2-8NA

Generalises properties of odd and even numbers, generates number patterns, and completes simple number sentences by calculating missing values.

NSW numeracy continuum reference:

Aspect 3: Pattern and number Structure – Number Properties.

NSW literacy Continuum reference: VOCC10M1

Vocabulary knowledge, Cluster 10, Marker 1: Demonstrates understanding that words can have different meanings in different contexts.

Other literacy continuum markers: SPEC9M4

Aspects of speaking, Cluster 9, Marker 4: Contributes relevant ideas to discussions, asks questions and re-phrases to clarify meaning.  WRIC9M4: Aspects of writing, Cluster 9, Marker 4: Structures texts using paragraphs composed of logically grouped sentences that deal with a particular aspect of a topic.

Resources

Nrich website - Maths problems: Patterns activities for Stage 2

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