2. Create a 5 Whys. Think about the purpose of maths and what type of mathematical learners you wish to create in the classroom.

3. What is 2/4 divided by 1/8? Solve on mini-whiteboards independently.

4.  If so, WHY did you solve it in this way?  Can you explain the process to some- one else? (Imagine if throughout your explanation they continually asked “why?” )  Think about the prior knowledge/ understandings a person would need in order to effectively understand this process.  Try to list at least 5 prior understandings relating to division, fractions and any other mathematical concepts you see as relevant.

5.  List any current knowledge you have about division and fractions.  Can any of this help you?  What would make this equation easier?  Can you use a drawing or concrete materials to help you?

6.  Teaching begins with a lesson that mentally engages students with an activity or question.  It captures their interest and provides an opportunity for them to express what they know about the concept or skill being developed.  It helps them to make connections between what they know and the new ideas.

7.  Students carry out activities in which they can explore the concept or skill. They grapple with the problem or phenomenon and attempt to describe it in their own words.  This phase allows students to acquire a common set of experiences that they can use to help each other make sense of the new concept or skill.

8.  Only after students have explored the concept or skill does the teacher provide the concepts and terms used by the students to develop explanations for the phenomenon they have experienced. Question for reflection.....  Why does explain follow explore, rather than the other way around?  What do you generally do in your classroom?

9.  Why do we use the reciprocal process? 2/4 divided by 1/8 = 2/4 x 8/1= 16/4 = 4 We need to recognise that 2/4 divided by 1/8 is 2/4 / 1/8 If we can change the denominator to 1, then we are left with a simpler fraction. But we then need to multiply the numerator also to keep the fraction equivalent. 1/8 x 8/1 = 8/8 = 1 2/4 x 8/1 = 16/4 = 4

10.  This requires an understanding of the inverse properties of multiplication and division. Whole numbers 8 divided by 1 = 8 8 divided by 8 = 1 8 x 1 = 8 1 x 8 = 8 Using fractions 1 divided by 8 = 1/8 1 divided by 1/8 = 8 8 x 1/8 = 1 8/1 x 1/8 = 1

11.  This phase provides opportunities for students to apply what they have learned to new situations and develop a deeper understanding of the concept or greater use of the skill.  It is important for students to discuss and compare their ideas with each other during this phase.

12.  Create a model/ drawing.  Can you see a pattern?  Explore and manipulate the models/ drawings.  Can you create statements of understandings, recognise mathematical rules/ structures or develop questions to test or investigate?  Take the time to construct personal understandings (requires repeated opportunity for exploration, manipulation, questioning, application in a range of contexts etc.)  Do less better !

13. 1 divided by 1/4 =4 1 divided by 2/4 = 2 1 divided by 3/4 = ? 1 divided by 4/4 = 1 3/4 divided by 1/4 = 3 3/4 divided by 2/4 = ? 3/4 divided by 4/4= 3/4 3/4 divided by 5/4 = ? Use the inverse/ reciprocal process ONLY to check your answers. How else might you realise that your answer is not correct?

14.  That multiplication is repeated addition/ groups of/ of... eg 3 x 2 is 3 groups of 2 = 6 1/2 x 4 is 1/2 a group of 4 = 2  That a fraction is a part of a whole.  That the fraction depends on what we define the whole as.  That dividing means sharing between/ how many sets/ groups of/ how many times the divisor fits into the dividend/  That division is the inverse of multiplication.  That the same part of a whole can be represented in different ways (1/2 = 2/4... equivalent fractions) What else?

15.  The final phase provides an opportunity for students to review and reflect on their own learning and new understanding and skills.  It is also when students provide evidence for changes to their understanding, beliefs and skills.

16. 1. What are some of the misconceptions/ mistakes that students may make if they were taught the formula without understanding: explain what they may be likely to do using the first example. (2/4 divided by 1/8) 2. Can you create a question to test your current knowledge that you could offer to some-one else in the room? 3. How could you evaluate a child’s understanding? 4. What conclusions/ statements can you make about division of fractions?

17.  Explore the concepts/understanding s involved in solving these equations.  Think about how you might structure a lesson to explore the concept, using the E5 model.

18.  Engage  Explore  Explain  Elaborate  Evaluate

19.  What are you doing in your classroom/planning that supports this type of maths learning? OR  What could you be doing in your classroom/ planning that supports this type of maths learning?  Discuss and develop strategies.