FORM: What are protractors like?

We have been exploring the properties of 2D shapes and the types of angles they have and why. 

To help us explore their connections, we used the Key Concept: Form to examine a protractor.

We used the think-pair-share routine to record what we notice when we look at a protractor. 

Some fascinating observations and wonderings were made:

Look at and listening in to what each child observed was also a great informal assessment to get a picture of who might need support in using them to measure angles in polygons. 

As PYP teachers, we often overlook the power of the key concept 'Form' especially when they get to Year 6, but 'Form' is really powerful and can be used for important understandings to emerge in children's minds.


After sharing with our partner, we then shared our observations as a whole class:

We then shared some wonderings we had and want to find out about:

We then drew some different sorts of angles (acute, obtuse, reflex etc) and experimented with different ways we can use a protractor to measure their sizes.

Giving children the opportunity to draw and measure their own angles rather than being fed a worksheet of angles to measure allows their thinking to deepen and allows them to explore what curiosities they might be harbouring. It also allows them to explore and make their own discoveries. Giving children ownership of their own learning, as we know, is key to authentic and meaningful learning to occur. 

Some samples:

In sharing, some of us shared our how they were trying to create a strategy to measure reflex angles.  They felt it was difficult with a semi-circle protractor.

We used this wondering as a whole class to find out.  After a few minutes of partners thinking of strategies, we shared two.

The first, the child felt would work best if the reflex angle was closer to a straight angle:

So what if the angle is not close to the straight angle? What other strategy could we use?

We drew reflex angles with our partners and had some time coming up with possible strategies. Of those created, we felt the following was the most effective:

Investigating Relationships and Connections of Angles

To help enquire further into our central idea:

              When angles co-exist, connections and relationships form.


I placed provocations along the hallway.  The children chose ones they found interesting to enquire into and using protractors etc used the taped shapes or angles to investigate connections or relationships between them.

Children discovered how all the angles of all quadrilaterals add up to 360°!!!

Why?   Hmmmmm.......good question!

How are the angles of a triangle connected with the angles of a quadrilateral?

Some children discovered that all the interior angles added up to 180°!!

"But why is that?" A student overhearing this asked.

We often discuss how understanding the 'whys' in maths is actually more important than the 'hows' as it deepens our understandings, so it's great when I hear the children now asking themselves and others these questions. 

What relationships exist when angles add to a straight angle of 180°?

How does knowing that a circle is 360° help us to find connections or relationships with these angles?

When two sets of parallel lines intercept, what connections or relationships amongst the angles exist and why is that?

What connections exist with the interior or exterior angles of pentagons?

A lot of excited discussions took place as the children investigated.  Theories were formed and tested and they recorded their discoveries and wonderings to share later in small group discussions.

 

 

Whilst doing this towards the end of our unit, in retrospect I think this could have been a successful lead in provocation engagement at the beginning of our unit to get the children thinking about angle relationships.

As it is, they felt it was great to help deepen their understandings of their central idea and always appreciate being able to choose what interests them rather than being told what to do on a worksheet.

Fun, engaging and deep thinking took place with this simple enquiry-based experience :)

Progressively unnecessary. Therefore, a teacher - Cristina Milos @surreallyno

Today during maths learning I thought of Cristina Milos's @surreallyno Twitter page quote: "Making myself progressively unnecessary- Therefore a teacher."

(I highly recommend you follow her on Twitter for her insights into learning or read her blog: momentssnippetsspirals.wordpress.com She will challenge your pedagogical thinking in a great way leading you out-of-the-box and then back in because everyone is thinking outside of it only to lead you back out again. She is the teacher that you wish you could be a fly on the wall of her classroom for a few weeks at least.)

I think about her quote quite often actually in class lately as a barometer of when I should step in to assist children when they are doing mathematical enquiries and when I should keep a step back and give them the opportunity to go where they want to go and to also solve problems they might be in a state of confusion over. 

We are having an amazing time lately continuing to enquire into our central idea:

The children have identified curiosities they want to explore and have either buddied up or chosen to investigate alone.  Some of their investigations stem from wonderings we have posted on our central idea wonder wall throughout our unit and some have stemmed from this poster provocation I created:

From examining and discussing with tables we were able to create some interesting investigation questions (or lines of inquiry):

The children have had a few days to investigate one or two questions and are ever so proud whenever they make a discovery of connections or relationships.

For much of this, I stand back and contemplate Cristina's quote.  

If a student needs some inspiration of where to go next, I step in to discuss where they are at where their wonderings are leading them to help them find a new direction. But essentially, they are coming up with amazing discoveries without a teacher's assistance.  

Today one of the many interesting chats that was generated amongst groups was whether a two-sided shape existed (eventually a semi-circle was suggested) and whether a line is actually a shape and therefore a 2 sided shape or not.  These wonderings didn't directed connect with our central idea , but they were wonderings that sparked interest in those that were in ear-shot of them and so some students debated their own theories and tried to support those theories with examples.

Other groups were proudly declaring that they had disproven their theories. Yes,- disproved. Proudly. 

That is a pretty amazing mindshift we have managed to create amongst us. that maths isn't about being correct and there is just as merit is disproving as well as proving our own maths theories we formulate.  

Whilst one group has been investigating if there is a pattern with the total interior angles of polygons- triangle, quadrilateral, pentagon, hexagon etc they themselves felt it would be great to buddy up with another group who are investigating the same but with exterior angles.  Both groups were amazed at what the other had discovered and are now in the process and gaining even deeper understandings of how angles have relationships and connect.

I like to think that when we can get to a point with our students when they are challenging their own and each other's thinking without a teacher guiding them, then some mathematical magic is happening.  This is what enquiry-based learning is all about. You can't get this sort of deep and self-directed learning from textbooks, teacher-directed instruction or worksheets. When we can create a learning environment where the children are driving their own enquiries from their own chosen wonderings, then self-pride as mathematicians and the wonders of mathematical thinking are planted and grow.

And for us, as Cristina says so profoundly: 

"Making myself progressively unnecessary. Therefore, a teacher."

I love this being my recent mantra in the classroom......

Different Format for a PYP Maths Planner

I used to be a strong advocate for creating PYP planners for stand alone maths and can still see a lot of merit in them. 

But, lately I feel slightly dissatisfied because with the PYP planner format for maths stand alone because it makes a unit too teacher-driven and this makes inquiry-based maths learning difficult to achieve. For maths stand alone, unless you are willing to go the extra mile and complete all the profiles etc etc, it becomes a mammoth amount of documenting when it shouldn't be for maths.



So, I've been playing around with different ways we could create a PYP planner for maths that is perhaps more practical for teachers to use and more importantly to make maths unit more student driven and inquiry-based. 

The sample below isn't completed as we are still in the midst of our enquiries. 

What I like about it is that it ensures the provocations and pre-assessment wonderings give the students voice in what they will enquire into.  It values the student wonderings and ensures those wonderings drive our unit.

Tapping into student curiosities is the bedrock of inquiry-based classrooms and this should also happen with mathematical learning.  When the children explore what they really want to know rather than what a scope & sequence document dicates what they have to know, the learning instantly becomes authentic, inspiring and meaningful. I like to think those conceptual and skill understandings will have far more longevity in their memories too for when they revisit those maths concepts in later years.



Sample format:

Google doc link to planner

                               Maths Planner Year Level:  6

                              Strand /Topic:  2D Shapes & Measuring Angles      

                                 (Needs to be done post-area unit due need to understand quadrilaterals can be 2 triangles)                          

                               Duration:        3 weeks approximately depending on student-initiated enquiries

                               Links to UOI etc:  Visual Arts- Cubist art making: angles used

              

Central Idea:	PYP Phase 4 Conceptual Understanding:
When angles co-exist, connections and relationships are formed.	Geometric tools and methods can be used to solve problems relating to shape and space.

Provocation/s

Pre-Assessment:

° Which are angles?   Draw types of lines on board:  Discuss which are angles- why/why not? Establish understanding angles are formed when straight lines intercept ° Can an angle exist alone?      Look at an acute and obtuse angle on board.   Write theory on post it note and share with class on continuum establish understanding that angles always co-exist ° Angles hunt at home Students predict the types of angles they would find the most and the least at home.  Home learning: find examples of types of angles in objects at home and draw.  Think why right angles are most commonly found and revolution least. ° Why is a circle 360°? Discussion Establish understanding that 360 was chosen because it has many factors and therefore easy to divide.  What would a 480° circle look like?  A metric circle to align with our base 10 number system?  Students design and share findings. ° Give assorted triangles, quadrilaterals and polygons.   Explore types of angles and connections / relationships found.

° 10 minute open-ended pre-assessment. Record your understandings of angles- What are angles, types of angles and their sizes, angles in shapes etc. ° With partner, draw and discuss the sizes of different angles known (acute? obtuse? right? straight? reflex? revolution?) Observation assessment. ° Ability to measure angles with a protractor Draw 3 lines across an A4 page at different angles vertically and another 3 horizontally to create angles. Observe students ability to measure the angles. Extension: naming the shapes and types of angles created.

Student Wonderings From Provocation/s & Pre-Assessments to Explore:

Student Wonderings During Unit That Were Explored:

° FORM: Are there other types of angles other than acute, obtuse, right, straight, reflex and revolution? ° CAUSATION: Why are right angles the most commonly found? ° FORM: Can an angle be larger than a revolution (360°)? ° CONNECTION: What connections exist between the angles of triangles and quadrilaterals? ° CONNECTION:What patterns exists when we add all the angles of 2D shapes?   ° CONNECTION: What relationships exist between the angles of regular and irregular polygons? ° CONNECTION: When lines or parallel lines intercept, what connections between the angles form? ° CONNECTION: Do relationships and patterns exist between interior and exterior angles?

° Is there a pattern for the exterior angles of polygons? ° Is a 100 sided shape called a centagon? ° If we make one angle in a triangle larger, what effect does it have on the other angles? How do they co-exist? With quadrilaterals? ° What do all the angles of a hectagon equal? ° What do all the angles of 3D shapes equal? Is there a pattern? ° Why are angles of  a triangle 180° and a straight angle also 180°?- Connection? ° What is the connection between quadrilaterals adding to 360° and a circle being 360°? ° If the 3 angles on a corner of a cube equal 270°, will the angles of a pyramid also equal 270? Why or why not?

Outcomes:

Lines of Inquiry & Learning Experiences:

° Describe, measure and construct types of angles: obtuse, acute, straight, reflex, right ° Understand that geometric ideas and relationships can be used to solve problems in other areas of mathematics and real life. identifying and naming right-angled triangles • manipulating, identifying and naming isosceles, equilateral and scalene triangles • comparing and describing side properties of isosceles, equilateral and scalene triangles • exploring by measurement angle properties of isosceles,equilateral and scalene triangles by measuring • exploring by measurement angle properties of squares, rectangles, parallelograms and rhombuses

° FORM: Are there other types of angles other than acute, obtuse, right, straight, reflex and revolution? Some students discovered angle connections such as complimentary angles exist ° CAUSATION: Why are right angles the most commonly found? Discussion after home learning investigation Discovered that right angles provide strength in objects compared to other objects. Used books to support theories ° FORM: Can an angle be larger than a revolution (360°)? Group used a clock to enquire into types of angles and discovered angles larger than 360° can exist. ° CONNECTION: What connections exist between the angles of triangles and quadrilaterals? Some groups enquired into relationships of interior and exterior angles of shapes Discovering connection between triangle adding to 180° and a straight angle being 180° Discovering connection between quadrilaterals adding to 360° and a circle being 360° Can every quadrilateral be divided into two triangles? When we split a quadrilateral into two triangles, what is the relationship with the angles? Used this Nrich activity to help with enquiry You might decide to try this activity. The circles are printed for you. A straight angle is 180° and all the angles of a triangle are 180°.  Why does this connection exist? A circle is 360° and all the angles of a quadrilateral are 360°. Why does this connection exist?      Going further: - Can we use this new understanding with pentagons or hexagons etc?                     ° CONNECTION:What patterns exists when we add all the angles of 2D shapes?   Discovery that all the triangles add up to 180° and that all the angles of a quadrilateral add up to 360°.   Some created a theory of the number pattern that might continue with pentagons, hexagons etc.   Tested theory & changed  theory as needed.  NOTED:  The shapes do NOT have to be regular polygons.                      ° CONNECTION: What relationships exist between the angles of regular and irregular polygons? Some groups discovered number pattern between regular and irregular polygons’ interior and exterior angles ° CONNECTION: When lines or parallel lines intercept, what connections between the angles form?      Some groups discovered complimentary angles and how intersecting lines create a 360° circle What connections exist with the angles? How can we find the size of the angles without using a protector for all of them?                 ° CONNECTION: Do relationships and patterns exist between interior and exterior angles? What can you discover?  Is there a number pattern with exterior angles of regular and irregular polygons?  Why or why not?

Summative Assessment:

PYP Learner Profile or Attitudes used to Assess Maths Learning

Students continually reflect throughout the unit about their new discoveries and understandings of our central idea. Reflection on learning:

Teacher & Self Reflection Feedback Criteria: ° Thinker / Reflective: Connecting learning experiences to deepened understanding of our central idea ° Communicator: Effectively explaining understandings visually and /or in writing ° Curiosity: Takes learning further by asking questions to enquire into ° Committed: Making good use of learning time, participating actively in discussions, showing evidence of taking learning further

Resources:

Reflection: What worked well for next time? What didn’t work well for next time? Ways to improve how to differentiate for next time?

Book: What’s Your Angle Pythagoras? studyladder protractor activities to display on data screen           Lower/mid group: large printed copies of triangles, quadrilaterals and regular polygons for enquiries

° Maintaining wonder wall of questions helped raise curiosity and allowed for easier student-owned investigations ° Not leading students to cut angles off triangles to find connection with straight angle and ditto for quadrilaterals with circle at own pace of discovery made it more authentic and meaningful for those students ° open enquiry into the central idea- amazing! ° doing studyladder protractor on data screen helped a lot in showing how to use a protractor for those who didn’t know!

What do you think?

Is it missing something?

Should some sections be moved around?

I'd love to hear your thoughts so please do post below: