CSDCEO
Marian Small
decembre 2015
Ordre du jour
•  Some “leftover issues”
•  Discussing how your lessons went
•  Answering questions based on the last
session or your lessons
•  Thinking more about classroom diagnostics
•  Planning lesssons
Some issues to clarify
•  The model we used is not a rule; it’s an
idea and a useful approach.
•  It is not for EVERY lesson, but for many
lessons and certainly for concept lessons.
•  It is important to teach concept lessons.
Some issues to clarify
•  Concept lessons can happen early in a topic
or later in the topic.
For example
•  Early in trigonometry
For example
•  Objectif:
•  When angles are similar, the ratios:
opposite/hypotenuse, adjacent/hypotenuse
and opposite/adjacent are the same.
•  They are different if triangles are not
similar.
Exploration
Exploration
•  Calculate the three side lengths for each of
the triangles.
Exploration
•  Do the same with triangles with a 50°
angle.
Exploration
•  What do you notice?
Consolidation
•  In which triangle were the opposites bigger?
Why does that make sense?
•  In which triangles were the adjacents
bigger? Why does that make sense?
Definitions
•  You have to just tell them the definitions
for sine, cosine and tangent, either before
or after the consolidation.
Or
•  Objectif:
Sines of angles change faster near
0° than 90°. Cosines are the opposite.
Exploration
You look at the sines and cosines for various
angles.
•  How much does the sine change if the angle
increases by 1°?
•  How much does the cosine change in those
situations?
Some info
•  Sin 0
•  Sin 1
•  Sin 2
0
sin 80
0.985
0.017
sin 81
0.988
0.035
sin 82
0.990
Consolidate
•  What did you notice?
•  If you double angles, do you double sines?
•  How many degrees might an angle change
for the sine to double? Etc.
Or consider ÷ of fract
•  Objectif: a/b ÷ c/d means how many sets of
c/d fit in a/b. You can answer by getting a
common denominator and figuring out how
many of numerator 2 fits in numerator 1
(e.g. 3/4 ÷ 3/8 = 6/8 ÷ 3/8 = 6÷ 3)
Mise en train
•  Why would you figure out 24 ÷ 4 to figure
out how many 4s there are in 24?
•  What do you think 12/15 ÷ 4/15 means?
Exploration
Use your fraction tower to answer these
questions.
•  How many of fraction 2 fits into fraction 1
each time?
Exploration
•  Fraction 2
Fraction 1
3/8
6/8
3/10
6/10
3/5
6/5
Exploration
•  Fraction 2
Fraction 1
2/5
7/10
2/9
1/3
3/10
2/5
Consolidation
•  What would the answer to 4/[] ÷ 2/[] be?
Why?
•  Does the denominator matter?
Consolidation
•  What would the answer to 5/[] ÷ 2/[] be?
Why?
•  Does the denominator matter?
Consolidation
•  How is solving 13/4 ÷ 2/4 like solving how
many pairs of socks are there if you have
13 socks?
•  How are the answers to 4/5 ÷ 2/5 and 2/5
÷ 4/5 related?
How did your lessons
go?
•  Sit with other teachers at your grade level.
•  Talk about the strengths and weaknesses of
the lesson you tried.
•  Talk, as a group, about how to change it
before someone else tries it.
Questions you have
•  It would be nice if you raised some
question related to what we did last time
that you want an answer to.
•  Then you’ll ask it and together we will try
to answer.
Task 1
3 compréhension (or just better) questions
than connaissance
•  Grade 7
•  Grade 8
•  Grade 9
•  Grade 10
Subtracting (entiers)
Multiplying fractions
Proportional Thinking
Quadratic Equations/Formula
Subtracting entiers
•  You subtracted two integers and the
answer was –5.
•  What could they have been if both were
positive?
•  What could they have been if both were
negative?
Subtracting entiers
•  Explain why 4 – (–3) is not the same
answer as –3 – 4 without just saying a rule
or what the answers are.
Subtracting entiers
•  If you know that [] – * = –4, then what is
[] – (* +4)?
Multiplying fractions
•  You multiply 3/5 by another fraction. The
answer is more than 3/5, but less than 1.
•  What could the other fraction be?
Multiplying fractions
•  Draw a picture to show why 2/3 x 4/7 =
4/3 x 2/7.
Multiplying fractions
•  Why does 5/3 x a/b have to be more than
a/b?
Proportional thinking
•  If 14/30 = 3/[], how do you know [] has to
be more than than 6 WITHOUT
SOLVING?
Proportional thinking
•  Why might you solve 2/3 = 14/x differently
than 3/6 = 137/x?
Proportional thinking
•  You have a recipe that uses 2 ¼ cups of
flour for every ¾ cup of sugar.
•  You fill an unmarked cup with sugar.
•  How much flour should you put in?
Quadratic equations
•  How do you know that x
have roots?
2+
8 = 0 cannot
Quadratic equations
•  Write a quadratic equation that you can
quickly tell has no solutions.
•  Write another one that you can quickly tell
has only one solution.
Quadratic equations
•  Suppose you know that the discriminant of
x2 + bx + c = 38. How far apart are the
roots?
Task 2
•  Objectifs for:
•  Grade 7 Subtracting (entiers)
•  Grade 8 Multiplying fractions
•  Grade 9 Proportional Thinking
•  Grade 10 Quadratic Equations/Formula
Teacher diagnostics
Choose one of these topics:
•  Grade 7
•  Grade 8
•  Grade 9
•  Grade 10
Area of trapezoid
Multiplying/dividing integers
Proportional reasoning
Systems of linear equations
Now
•  Make a list of three conaissance and three
compréhension questions you could ask to
see if students are for the topic.
•  Create a diagnostic.
•  We will share.
My example
I will use a different topic.
algebraic expressions and equations in Grade
9
My three skills
•  Evaluating expressions
•  Simplifying expressions
•  Solving very simple equations
My three ideas
•  Any equation or expression describes a
variety of “similar” situations.
•  Equations describe relationships or
generalizations
•  Some equations can be solved but some
cannot.
Evaluating expressions
•  What is the value of 3x – 2y + 8 if x = –3
and y = –4?
•  How does the value change if x = –2?
If I wanted more
concept
•  The expression 3x – 8 is worth 30 more
for one value of x than another. What
might the two values have been?
Simplifying expressions
•  Simplify: 2 –3x + 4x + 5
•  Simplify: 4(2x – 8)
•  Simplify: 3x – 8 – (2x – 5)
Solving very simple
equations
Show your work in solving each equation:
•  3x – 2 = –4x + 12
•  4x + 3 = –8x – 9
My three ideas
•  Any equation or expression describes a
variety of “similar” situations.
•  Equations describe relationships or
generalizations
•  Some equations can be solved but some
cannot.
Any equation describes
a variety of situations
•  Write two different stories that go with
this equation:
3x – 8 = 12.
Equations describe
relationships
How can you use algebra to say that:
•  the value of y is 1 less than the value of x?
•  the value of y is 3 more than double the
value of x?
Equations describe
generalizations
How can you use algebra to say:
•  multiplying by 2/3 is the same as dividing by
3 and then multiplying by 2?
•  You can multiply the sum of two numbers
by 5 by multiplying each one by 5 and
adding them.
Can you solve these
equations?
•  3x + 5 = 2x – 8
•  3x + 5 = 2x + 6 + x – 1
Now
•  It’s your turn.
Your diagnostic
•  Make a list of three ideas and three skills
you think kids need to be ready for the
topic.
•  Create a diagnostic.
•  We will share.
Creating new lessons
•  The first and maybe biggest issue is:
•  What will be your objectif?
Our focus should be
•  Objectifs that are about ideas, not just
about skills
•  That are not contenus, but are related to
the contenus
Quadratics
•  Une même régularité ou généralisation
peut être représentée d’une variété de
manières équivalentes. Chacune de ces
représentations peut permettre de mieux
comprendre certaines caractéristiques de
cette régularité ou de cette généralisation.
Quadratics
•  Je comprends le rôle des paramètres a, h et
k dans les relations y = a (x– h)2 + k et leur
effet dans un graphique.
Linear
•  Je comprends comment le changement de
la condition initiale d’une situation linéaire
affecte le graphique, la table de valeurs et
l’équation.
Views of solids
•  Je comprends que lorsque je connais
seulement deux plans de vues d'un solide, il
y a beaucoup de possibilités de
construction d’un solide, mais moins que
quand on connaît un seul plan de vue et
plus que quand on connaît trois plans de
vues.
Percentage
•  Je peux expliquer la raison pour laquelle si
on connaît seulement le %, on connaît la
relation qui existe, mais pas les valeurs
exactes.
Solving equations
•  Je comprends que je peux résoudre une
équation linéaire complexe sous une forme
plus simple mais équivalente.
Multiple
representations
•  Une même régularité ou généralisation
peut être représentée d’une variété de
manières équivalentes. Chacune de ces
représentations peut permettre de mieux
comprendre certaines caractéristiques de
cette régularité ou de cette généralisation.
So now you try
•  Choose a topic.
•  List two or three important objectives.
Once you have that
•  After figuring out the objectif, you want to
think of a problem with its consolidation
that will lead there.
•  Only later will we work on the mise en
train.
So let’s start with, e.g.
•  Views of solids
Exploration
•  Cacher un solide mystère sous une boîte,
montrer et dessiner au tableau la vue de
face du solide, ensuite laisser les élèves
construire un solide possible (il y a une
infinité de possibilités)
•  vue du haut du solide, ensuite laisser les
élèves réarranger leur solide (il y a moins
de possibilités)
Exploration
•  Vue de côté droite, ensuite laisser les
élèves réarranger leur solide (il y a une
possibilité)
Consolidation
•  Est-ce que les vues de face et de derrière
sont pareilles? Devaient-elles être pareilles?
•  Est-ce que les vues de côtés droite et
gauche sont pareilles? Devaient-elles être
pareilles?
Consolidation
•  Est-ce que les vues du haut et de bas sont
pareilles? Devaien-telles être pareilles?
•  Quand tu regardes à la vue du haut, peux-
tu connaître sans faute à quoi ressemble la
vue des côtés ou la vue de face et de
derrière? Explique.
Consolidation
•  Pourquoi avons-nous tant de possibilités?
•  Combien de plan du vue aurions-nous
besoin pour que tout le monde finisse avec
le même objet?
Mise en train
•  Montrer un arrangement 3 x 3 de cubes
emboîtables. Dites que cela représente la
vue du haut d’un objet.
•  À quoi peut ressembler cet objet?
•  Cet objet contient combien de cubes?
•  Est-ce que la vue du côté est la même?
Solving systems of
equations
•  Getting a simpler form
Exploration
•  Adam dit que chaque équation avec des
fractions a la même solution qu’une autre
équation sans fractions.
•  Son exemple était celuici :
2/3x – ¼ = x/5
28x – 15 = 0
•  Es tu d’accord avec lui? Explique.
Consolidation
•  Comment sais-tu que les deux équations
ont la même solution?
•  Peut-on dire que les deux équations sont
équivalentes? Explique
Consolidation
•  Que pourrait être la première étape pour
simplifier l’équation avec des fractions?
Explique
Mise en train
•  Faire un pense-parle-partage,
•  Explique ce que tu ferais pour simplifier
½ – 3/4 ? Simplifie.
•  Que pourrait être la première étape pour
simplifier ¾ x = ½. Simplifie.
There are more
•  There are more lessons to look at in the
google doc:
•  https://docs.google.com/document/d/
157w_brtwXRjH5YPEBQyxMgFGgC3jmAb
GCVHX7Y8NgOs/edit?ts=564a0e41 .
Your job
•  To create more lessons with strong
objectifs and strong consolidations based
on good problems.
I am here
•  To help, but most of your feedback will
come from your colleagues.
Other questions
•  What other questions do you have?
Download
•  www.onetwoinfinity.ca
•  CSDCEO2