1. E22 ReSolve: Maths by Inquiry engaging classroom resources
Helen Haralambous
Mathematics Education Consultant

2. Warm Up - 24 Game Integers
Materials: 12 cards per pair, including some of 1, 2 or 3 points – one white dot, 2 red dots or 3 yellow dots. These are pre – stacked, starting with white dot cards.

3. Warm Up - The 24 game Integers
The Rules:
Game for two players
Cards are worth 1, 2 or 3 points, in order of difficulty
Play in pairs, (9’s filled in red). Start with 1 point cards(have one white dot)
Place cards on table, between two players. Both players are playing at the same time for the same top card.
Use all 4 numbers on the card to make 24.
You can add, subtract, multiply or divide.
Win a card by being the first to touch the card and give a correct solution
Once you take your card the next card is in play.
Aim:
To make 24 with all four numbers on a card. You can add, subtract, multiply or divide.
Winner is the player with the most points after all cards are claimed.

4. 24 Game Integers
Solution
( ) x (8x1)

5. Content strand - Number and Algebra
Sub – strands
Number and Place Value
Real Numbers
Money and Financial Matters
Patterns and Algebra
Linear and Non-Linear Relationships
Focus –

6. An example - Year 8 Scope and sequence chart
Content strand - Number and Algebra
Copies provided with
Year 7 focus
Year 8 focus
Year 9 focus
Year 10 focus
Key concepts to be referred to during various tasks
Be aware of the level above (level 9) and the level below (level 7)

7. Mathematics Scope and Sequence chart (Victorian Curriculum & AC)
Mathematics Scope and Sequence chart (Victorian Curriculum & AC). Example: Number & Algebra
Strand
Sub-strand
Level 7
Level 8
Level 9
Number and Algebra
Patterns and algebra
Introduce the concept of variables as a way of representing numbers using letters.
Create algebraic expressions and evaluate them by substituting a given value for each variable.
Extend and apply the laws and properties of arithmetic to algebraic terms and expressions
Extend and apply the distributive law to the expansion of algebraic expressions.
Factorise algebraic expressions by identifying numerical factors.
Simplify algebraic expressions involving the four operations.
Extend and apply the index laws to variables, using positive integer indices and the zero index.
Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate.

8. Mathematics Scope and Sequence chart (Victorian Curriculum & AC)
Mathematics Scope and Sequence chart (Victorian Curriculum & AC). Example: Number & Algebra
Strand
Sub-strand
Level 7
Level 8
Level 9
Number and Algebra
Patterns and algebra
Introduce the concept of variables as a way of representing numbers using letters.
Create algebraic expressions and evaluate them by substituting a given value for each variable.
Extend and apply the laws and properties of arithmetic to algebraic terms and expressions.
Design and implement mathematical algorithms using a simple general purpose programming language
Extend and apply the distributive law to the expansion of algebraic expressions.
Factorise algebraic expressions by identifying numerical factors.
Simplify algebraic expressions involving the four operations.
Use algorithms and related testing procedures to identify and correct errors
Extend and apply the index laws to variables, using positive integer indices and the zero index.
Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate.
Apply set structures to solve real-world problems

9. Patterns and Algebra – examples
Keep the Key Concepts in mind (highlight) as you work through the various activities.
Addition Chain ( Year 7 /8)

10. Algebra – (resource) ReSolve
Lesson Resource - Addition Chain
Lesson Abstract
This lesson resource prompts students to examine and generalise numerical relationships arising from “addition chains”. These are recursive sequences, such as the Fibonacci sequence, generated by summing the previous two terms.
The teacher demonstrates a calculation shortcut to create a sense of curiosity about how it is done and why it works.
This leads to a student search for similar relationships, using spreadsheets. The resource requires students to use algebra to justify the conditions for which these relationships hold true.

11. Lesson Resource - Addition Chain
Demonstrating a mathematical magic trick
Challenge to students: So - called mathematical magic tricks have a basis in logic – they can be explained. Challenge is to go beyond, understand & invent our own magic tricks.
Look at one example, understand why it works, use understanding to invent a similar one.

12. Lesson Resource - Addition Chain
Each person to write down two numbers vertically aligned ( their starting or ‘seed numbers’)
Add them
Write the answer under the second number
Add the last two numbers in the list
Repeat until there are 10 numbers in the list
Add the total of all 10 numbers in the list

13. Lesson Resource - Addition Chain Example1 7 2 5 3 12 4 17 5 29 6 46 7 75 8
121 9
196 10
317
TOTAL =
825

14. Lesson Resource - Addition Chain
Volunteer to write their seed numbers on board & list of 10 numbers
2nd volunteer
How did I find a total so quickly
Can you find a hypothesis?
Discuss
Total is 11 times the 7th number

15. Lesson Resource - Addition Chain
To assist students to come to a hypothesis provide an
Enabling prompt
Subtract each of the numbers in your list from the total (in this case 825)
Compare results in your table group, what do you notice about the 7th term?
Term
NUMBER
Number subtracted from total 1 7
=818 2 5
= 820 3 12
825 – 12=813 4 17
825 – 17 = 808 29
825 – 29 = 796 6 46
825 – 46 = 779 75
825 – 75 = 750 8
121
825 – 121 = 704

16. Lesson Resource - Addition Chain: The relationship of numbers in addition chains
The 7th subtraction always provides an interesting result: the answer is 10 times the number being subtracted.
What does this mean about the relationship between the 7th number and the total of the 10 numbers?
Total sum - T 7 = 10 T7
hence Total sum = 11T7
Check relationship generalizes: A couple, students record 1st, 2nd and 7th numbers
Teacher gives totals, either add a few more on board for students to predict or students predict totals of other group members
STUDENT 1 2
FIRST NUMBER
SECOND NUMBER …
SEVENTH NUMBER

17. Lesson Resource - Addition Chain: Investigating other relationships
What happens if we go beyond 10 numbers?
Is there a quick way to calculate the sum of the first
11 numbers?
12 numbers?
etc
Without actually doing the addition?
Setting up a spreadsheet (one supplied with teachers notes) - demonstrate

18. Lesson Resource - Addition Chain: Investigating other relationships
First integer factor appears when 14 numbers are summed
What would be the(trick) relationship here?
Sum = 29 x 9th number

19. Mathematics Scope and Sequence chart (Victorian Curriculum & AC)
Mathematics Scope and Sequence chart (Victorian Curriculum & AC). Example: Measurement and Geometry
Strand
Sub-strand
Level 7
Level 8
Level 9
Measurement & Geometry
Location and transformation
Describe translations, reflections in an axis, and rotations of multiples of 90° on the Cartesian plane using coordinates. Identify line and rotational symmetries
This sequence ends at level 7

20. Mathematics Scope and Sequence chart (Victorian Curriculum & AC)
Mathematics Scope and Sequence chart (Victorian Curriculum & AC). Example: Measurement and Geometry
Strand
Sub-strand
Level 7
Level 8
Level 9
Measurement & Geometry
Geometric
Reasoning
Identify corresponding, alternate and co-interior angles when the two straight lines are crossed by a traversal.
Investigate conditions for two lines to be parallel and solve simple numerical problems using reasoning
Demonstrate that the angle sum of a triangle is 180° and use this to find the angle sum of a quadrilateral
Classify triangles according to their side and angle properties and describe quadrilaterals.
Define congruence of plane shapes using transformations of congruent shapes to produce regular patterns in the plane including tessellations with and without the use of digital technology
Develop the conditions for congruence of triangles
Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning.
Use the enlargement transformation to explain similarity and develop the conditions for triangles to be similar.
Solve problems using ratio and scale factors in similar figures.

21. 90 – 270 POLYGONS
Draw at least three different polygons, with different number of sides, that consist of only 90o and 270o angles
How many 90o angles and 270o angles would there be in a polygon with 100 sides?
How many 90o angles and 270o angles would there be in a polygon with 2s sides?

22. 90 – 270 POLYGONS Feedback Would you use this task? How to teach this?
Discuss in table groups.
How would you use the task?
What year level?
How to teach this?
Start with big question
Scaffold
Enabling prompts - as a group write down some enabling prompts eg. can you use an organiser?
Challenging prompts - as a group write down some challenging prompts. e.g. generalising or finding a rule

23. Your exploration
Explore

24. Websites referred to RESOURCE WEBSITE 24 Game Integers
Available from MAV shop
Victorian Curriculum F-10 (home page)
Scope & sequence (downloadable word docs)
ReSolve (Maths by Inquiry)

25. Thank you and Feedback Thursday 6 December 2018