a quote “teachers spend much more time worrying about what they are going to tell students than...

A Quote

“Teachers spend much more time worrying about what they are going to tell students than thinking about what experiences they are going to providefor students. To ensure that students learn in class requires carefully designed experiences that keep them engaged and make them think” Weisman

© Project Maths Development Team 2014 www.projectmaths.ie

Chief Inspector’s Report

Teach for Understanding

Mathematical Rigour

Students Collaborating

Making Connections

Challenge Able Students

Justify Reasoning


A Different Type of Lesson 1

Watch the videoand try to guessthe question I’m going to ask!!


Guesstimation!

Without any calculations, guesstimate how many Post-its are needed to cover all sides of the file cabinet apart from the base?

(P.S. you are allowed to get it wrong!!!)


What information would be useful to know?


A Different Type of Lesson

7.618cm


http://www.google.ie/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=ZP1eI4y4Kpid_M&tbnid=kKyMV7DXfqbTiM:&ved=0CAUQjRw&url=http://iwanttowalkinbeauty.blogspot.com/2012/01/self-confidence-boosting-post-its.html&ei=rJ6MUputIcWP7AabmIHwAg&bvm=bv.56643336,d.ZGU&psig=AFQjCNGBhk7S6qUKwCJvSPU4K2jpmhfjVg&ust=1385033767379880

A Different Type of Lesson


A Hint!!


Actual Dimensions

Height183cm

Width91 cmWidth91 cm

Depth46 cm


Do the Math!!!


http://www.google.ie/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=LpZUDNtFr4cPWM&tbnid=USDIJBNYKgWhWM:&ved=0CAUQjRw&url=http://www.bluefocusmarketing.com/2012/03/13/do-the-math-to-calculate-the-roi-of-business-blogs/&ei=oZGMUryrFumN7AawiIDwBg&psig=AFQjCNE8TLMRS_YR8j0amvS3DeTP31L0ww&ust=1385030388221277

Were you right????


Were you right ?


Can you come up with further questions?

If the WIDTH of the cabinet was doubled, how many more post-its would be needed?

If the HEIGHT of the cabinet was doubled, how many more post-its would be needed?

If the DEPTH of the cabinet was doubled, how many more post-its would be needed?

How long would it take to cover if it took 40 seconds for every 5 Post-its?

If you had 1,000,000 Post-its, what kind of file cabinet could you cover?


Teaching this Way

Engaging for students

Covers the Learning Outcome(s). (3.4 Applied Measure)

Accessible to most abilities

“Realistic”

Differentiation: Challenging extension questions


A Different Type of Lesson 2


Challenge 1

2

Draw a square on the dotted grid paper

encompassing exactly 25 units .

: All vertices must be on grid points.Note


Challenge 1


Challenge 2

2

Now, draw a square on the same dotted grid

paper encompassing exactly 29 units .

: All vertices must be on grid points.Note

2Verify that the area is 29 units in different ways.three© Project Maths Development Team 2014 www.projectmaths.ie

Method 1

5

5

5

5111

1 1

11 1

1


Method 2

55

5 5


Method 3


Student Misconception


Class Investigation

2

Investigate how many separate squares (regular or tilted)

with areas between 1 and 16 units inclusive can possibly be

drawn on dotted grid paper. Again, all vertices must be

on grid points.


Possible Areas

1

2

4

5

8

9

10

13

16


Searching for Patterns: Regular Squares

1 4 9 16

Square 1 2 3 4 ……………. n

Area 1 4 9 16 ……………. 2n


Searching for Patterns: “1-up” Tilted Squares

2 5 10 17

Square 1 2 3 4 ……………. n

Area 2 5 10 17 ……………. 2 1n © Project Maths Development Team 2014 www.projectmaths.ie

Your Turn: “2-up” Tilted Squares

Square 1 2 3 4 ……………. n

Area …………….

Square 1 2 3 4 ……………. n

Area 5 8 13 20 ……………. 2 4n

5 8 13 20


Searching for Patterns

Square 1 2 3 4 ……………. n

Regular

1 4 9 16 …………….

1-Up 2 5 10 17 …………….

2-Up 5 8 13 20 …………….

3-Up 10

13 18 25 …………….

4-Up 17

20 25 32 …………….

2n

2 1n 2 4n 2 9n 2 16n

What does the formula for the area of tilted squares look like? © Project Maths Development Team 2014 www.projectmaths.ie

Proof: Area of a Tilted Square

a

b2 2 a b

2 2a b


Extension Question2Draw a square of area 25 units tilted

What are the areas of

the next squares which

can be represented as both

regular and tilted squares

on the grid paper? Explain

how you got your answer.

three


Further Investigation 1

2

5



10

8



How many tilted squares will fit inside a square with area 25?

How can I count the number of tilted squares

which will fit inside regular square ? any

1713



Tilted Equilateral Triangles


Extension 1: Pushing Brighter Students

The opposite vertices of a

tilted square have coordinates

(a,b) and (c,d). What are the

coordinates of the other two

vertices?



1. Can you prove that numbers of the form 4n+3 are not possible areas of tilted squares?

2. When is a number expressible as the sum of two squares?


http://www.google.ie/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=xK6RzQuZ2KyHVM&tbnid=43oP9ysS9dwzQM:&ved=0CAUQjRw&url=http://www.mid-day.com/news/2012/may/080512-Mugging-turns-college-dropout-into-math-genius.htm&ei=7_3MUvyMDYmO4ATKpYHIAw&bvm=bv.58187178,d.bGE&psig=AFQjCNFyIudDSEyNDi0Xe52Z096KHLtn1Q&ust=1389252417329413

http://www.google.ie/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=xK6RzQuZ2KyHVM&tbnid=43oP9ysS9dwzQM:&ved=0CAUQjRw&url=http://www.mid-day.com/news/2012/may/080512-Mugging-turns-college-dropout-into-math-genius.htm&ei=7_3MUvyMDYmO4ATKpYHIAw&bvm=bv.58187178,d.bGE&psig=AFQjCNFyIudDSEyNDi0Xe52Z096KHLtn1Q&ust=1389252417329413

Content Slopes of Perpendicular and Parallel lines

Pythagoras’s Theorem

Area of Squares and Right Angled Triangles

Finding areas by “dissection” methods

Surds/ Number Theory

Investigating and Collecting Data

Searching for Patterns

Generalising to a method

Proof

Reasoning, Problem Solving, Persevering.© Project Maths Development Team 2014 www.projectmaths.ie

Teaching this Way

Connects to other areas of the syllabus

Similar to doing “real” mathematics

Can be adapted to all levels

Level Playing Pitch (entry point is accessible for all)

Enjoyable for Students/Teacher as “guide on the side”

Preparation for exams e.g. Jigsaw question


a quote “teachers spend much more time worrying about what they are going to tell students than...

Documents

mathematical rigour

kind of file cabinet

designed experiences

actual dimensions height

chief inspectors report

applied measure accessible

learning outcomes

quote teachers