UNIDAD DIDÁCTICA: NUMERICAL PROPORTIONALITY (2º ESO):
He elegido
la Unidad Didáctica "Proporcionalidad Numérica" de 2º de ESO los
siguientes (que aparecen en la segunda diapositiva de la presentación de los
materiales que adjunto), y que se resumen en un Diccionario ilustrado de
Matemáticas online, para consultar palabras en todas las sesiones, aunque
se usará fundamentalmente en la primera para repasar las palabras más comunes
relativas a la Unidad Didáctica e ir estableciendo así el andamiaje; los
materiales del Escritorio Virtual AICLE Symbaloo que recopilé para la
actividad 2.3, que incluyen teoría, videos cortos y actividades interactivos;
una actividad interactiva propia de My LearningApps y una presentación con
ejemplos resueltos y explicados compartida por Google Drive; así como también
el pdf con estos ejemplos, todo esto para trabajarlo en las sesiones 2 a 5 (se
pasarían por Classroom y los alumnos los harían junto con otros planteados por
ellos mismos en la primera sesión), siendo la sesión 5 para repaso y dudas y
para que los alumnos entresaquen de los textos el vocabulario tipo de esta
Unidad Didáctica para presentarlo en el trabajo por grupos de la Sesión 6. En
la Sesión 7 se hará un examen escrito final con algunas preguntas elegidas de
los ejemplos resueltos explicados en clase, ya que el fin de la educación es
que los niños aprendan, no que aprueben o suspendan. A continuación paso a
describir los materiales usados, y después la Unidad Didáctica:
LIST OF MATERIALS AVAILABLE FOR THE
DU “Numeric Proportionality” (2º ESO):
-
An
illustrated Maths Dictionary to look up words and build up vocabulary:
https://www.mathsisfun.com/definitions/
My Symbaloo Webmix (the name should´ve been numerical-proportionality, but err
is human!)
https://www.symbaloo.com/mix/numerical-proporcionality
- My LearningApps Interactive
Activity: https://learningapps.org/display?v=pc9y97xrn21
- This bilingüal presentation (Theory + Activities for the classroom)
with the first eleven slides in L2 (English) + the last seven slides in L1
(Spanish)
based on made up worked examples of proportionality, percents and proportional
shares:
https://drive.google.com/file/d/1Vn3jS4wdi5jfz5QKlRBnE4J_iYKTd_b8/view?usp=sharing
- The link to the slides with
the worked examples in pdf format:
https://drive.google.com/file/d/1hRJAQ0EfVz_R7GBY46rYe44bdYTbu8YD/view?usp=sharing
- & a Webgraphy with
collected links to Open Educational Resources (OER) for Content Language
Integrated Learning (CLIL) used to teach this Didactic Unit (DU):
https://www.khanacademy.org/test-prep/praxis-math/praxis-math-lessons/praxis-math-number-and-quantity/a/gtp--praxis-math--article--ratios-and-proportions--lesson
https://www.youtube.com/watch?v=r97sw6ecdEk
https://www.youtube.com/watch?v=rj7DweP8e58
https://www.ixl.com/math/grade-7/find-the-constant-of-proportionality-from-a-table
https://math.wonderhowto.com/how-to/apply-fundamental-rule-proportions-302393/
http://recursostic.educacion.es/descartes/web/materiales_didacticos/Ratio_&_proportional_division/Proporcion.htm
https://www.youtube.com/watch?v=kuvdMCDqmKg
https://mathsmadeeasy.co.uk/gcse-maths-revision/direct-and-inverse-proportion-gcse-revision-and-worksheets/
https://www.youtube.com/watch?v=USmit5zUGas
https://www.youtube.com/watch?v=RQ2nYUBVvqI
https://www.youtube.com/watch?v=USKmhLeQkEU
https://www.youtube.com/watch?v=1U9qAaKQ7Ds
https://www.youtube.com/watch?v=aX2o9OYqHFI
https://www.youtube.com/watch?v=3Iz4GpNSMj4
https://www.youtube.com/watch?v=PRQT06rfGtI
https://www.futurelearn.com/info/courses/maths-linear-quadratic/0/steps/12093
https://www.sangakoo.com/en/unit/proportional-distributions-direct-and-inverse
https://www.mathsisfun.com/percentage.html
https://www.youtube.com/watch?v=WYWPuG-8U5Q
https://www.mathsisfun.com/numbers/percentage-change.html
https://www.youtube.com/watch?v=TpZXX-GsmB0
https://www.youtube.com/watch?v=FCObY53fM-g
https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-ratio-proportion/cc-7th-write-and-solve-proportions/e/constructing-proportions-to-solve-application-problems
https://www.math10.com/tests/percent-percentage-test.html
https://www.math-only-math.com/worksheet-on-inverse-variation.html
https://virtualnerd.com/common-core/grade-7/7_RP-ratios-proportional-relationships/A/3/percent-proportion-word-problem-example
UNIDAD DIDÁCTICA NUMERICAL PROPORTIONALITY:
-
Está compartida por Drive en:
https://drive.google.com/file/d/1Z9TcGa3LM95BR9EA-b6hn04apb4R3sir/view?usp=sharing
Teacher: José Carlos Hernán Cabanillas
Course/ Level: 2º ESO
Subject: Mathematics
DU Title: Numerical Proportionality
Observations: May be harder to learn if lacking previos knowledge in
fractions
1.
Learning outcomes / Evaluation criteria
* Know and manage the concepts of ratio and proportion
between two magnitudes.
* Recognize direct and inversely proportional magnitudes
* Solve problems of Proporcionality by either using The Rule
of Three or computing the Unit Ratio.
* Understand the concept of Direct and Inverse Proportional
Distributions.
* Solve problems of both Direct and Inverse Proportional
Distributions.
* Understand and manage the concepts relative to Percentages.
2. Subject Content
* Concepts of Magnitude, Ratio and Proportion.
* Concepts of Direct and Inversely Proportional Magnitudes.
* Relationship of Direct Proportion with Equivalent
Fractions. Fundamental Theorem of Proportions.
* Proportionality Tables.
* Calculation of Proportionality Constants.
* Methods to solve Problems on Simple Direct and Inverse
Proportionality.
* Concepts of Direct and Inverse Proportional Distributions.
* Methods to solve Problems on Direct and Inverse
Proportional Distributions.
* Concept of Percentage.
* Methods to solve Problems
on Percentages.
3. Language Content / Communication
3.1 Vocabulary
Essential vocabulary:
Nouns:
Magnitude, Ratio, Proportion, Quote, Rate, Percentage, Distribution, tap. sink, fuel, worker, building.
Adjectives:
Proportional, Equivalent, Direct, Inverse
Adverbs:
Directly, Inversely, Proportionally.
Prepositions: per
Verbs:
Keep (proportionality), Distribute (proporcionally), To be to (one magnitude
relative to another): Six liters of fuel are to one-hundred kilometers like
three liters to fifty kilometers.
Reusable vocabulary:
Nouns:
Problem, Solution, Fraction, Question, Volume, Distance, Capacity, Speed, time,
factor, number, integer.
Adjectives:
unknown, easy, difficult.
Adverbs:
Faster, Slower, Bigger, Smaller, Longer, more, less, thus. So.
Prepositions:
like, as.
Verbs:
Solve, Work out, Think, Given, Let be, Explain, Identify, Find, Check out,
Multiply, Divide, Review.
Class routines:
Any
volunteer?
Be
silent!
Help
your partner
Go
to the blackboard
Listen
attentively
Note
down
Now
I´ll deliver you the exams
Let´s
review what you already know
Do
your homework
3.2. Structures
Routines:
Let´s
begin with the theme on Proportions
Are
you familiar with Proportions?
What
do you understand by Ratio?
Let´s
talk about everyday situations on this theme
Put
yourself an example about two directly proportional magnitudes
Grammatical content:
Comparatives: One
car spends three times more fuel on travelling three hundred kilometers than on
travelling one hundred kilometers
Conditional: If
a tap throws/spout water in a 60 liter-capacity sink at a ratio of 5 liters in
10 seconds, then: how long would take to fill it in minutes?, ¿and what is the
ratio in liters per minute?
Prepositions: at
a ratio of, per minute
Classroom management:
Are
you happy with it, or shall I explain it again?
Listen
and pay attention to the explanation = Escuchar y atended a la explicación.
Understood?
Any
doubt? or Doubts?
Tell
me when you´re done with the exercises
3.3.
Discourse type
We´ll
use a mix of Descriptive and Argumentative Discourse to explain the
different forms of Proportionality between two magnitudes, posing examples
of situations of everyday life, like the proportions of ingredients in the
making of recipes, the calculation of the price in euros per kilo when buying
groceries at the supermarket, or the more or less time spent in making a
particular building relative to the number of workers, while at the same time depicting
on the blackboard the logic of the proportions explained and its
relationships.
3.4.
Language Skills
Writing:
when copying the definitions or concepts and solving the problems.
Reading: by
reading the exercises they´ll learn the translated vocabulary in them.
Speaking: for
they will we asked to explain to the rest the different subjects treated
Listening:
while looking at videos and the board, they´ll listen to the explanation
Social
Interaction: when asking doubts and explaining them to
their classmates.
4. Contextual
(cultural) elements
-
The
subject will be linked with daily life activities known to the students, such
as:
*
Calculating the cost they will be charged for a shopping list
based on known prices of quantities of the different products or items, that
will be applicable, for example, when going to the grocery store or to
any store in general, to reckon if they will have money enough to buy it all
and how much will be left to spare.
* Reckoning how much diesel their parents´s car will have to refuel to go on a vacation
trip of a given distance from their village to a beach town knowing how much fuel the car
spends per 100 kilometers.
* Knowing how much time it will take them to go from their homes to the school according
to their paces.
* Knowing how much water will be spent to water the plants at their backyards based on
the tap flow rate.
* Calculating the amounts of ingredients needed for several guests knowing the quantities
used at home, either per person or per group, when making recipes with their family,
whether it´s the mean meal or a dessert, such as a paella rice, an ajoblanco cold soup or
some specific dish typical of their village/s.
5. Cognitive
(thinking) processes
The
students will be asked to think of situations of their background or to
recall some memories they may have, to identify elements of the matter
which they already know, such as types of magnitudes, rates and ratios beween
them, when they form a direct proportion and when an inverse one, percentages
of discount in sales, etc., as to ascertain their initial level of knowledge to
be able to construct new and upper levels of the cognitive process within the
scaffolding strategy of teaching-learning, so that from this first step (remembering)
we´ll be slowly advancing into more complex tasks by explaining the
relationship of a direct proportion between ratios with the equivalent
fractions, and deducing the way to solve the new equivalent algebraic fractions
where the missing value of the magnitude is replaced by the symbol X or
“unknown”, and keeping on explaining the rest of the DU samewise (percents,
proportional shares, etc.). After they have understood these contents,
by means of several examples, they´ll have to apply their new acquired
knowledge to some problems the teacher will put for them to solve in pairs
or in groups, having to do with their daily routines, such as explained above,
and posed in their contextual or cultural background, thus easying the
discussion and analysis of the activities, required to acquire
knowledge, check and evaluate each other´s solutions and recognise and
discards errors. Finally, some of them will be asked to “be” the teacher, and
this way create themselves some problems and explain how to solve
them.
6a.
Task/s
*
Search for web information on the doses of ingredients in a
particular recipe for several guests (in pairs)
* Listen to short videos that will
help them understand the matter whilst repeating aloud to improve diction.
* Listen to and note down the
problem statements dictated by the teacher.
* Search for information on
compsumtion rates in cars and make up several problems of rule of threes
and inverse rule of threes with varying distances, speeds, and times, based on
previously seen ones (in pairs)
* Invent a problem of discount sale
prices to understand percent increases and decreases.
* Make a short presentation
with slides in a computer program about the words and sentences learned in the
L2 and their translation in the L1 (in groups)
6b.
Activities
-
We´ll
begin little by little with an activity of brainstorming for all, where
any one may be able to remember and tell aloud to the rest of the Class specific
situations of their lives where they identify the concepts of magnitudes,
rates, ratio, direct and inverse proportion, percent variation and distribution
of shares.
-
The
sessions will intermix with projections of lessons and videos in the L2 with
subtitles on also in the L2, where the teacher will be explaining in the
blackboard the same concepts, but in the L1, while translating aloud alongside
the running of the video, thus allowing the student to learn the matter and the
same time the second language needed. These videos may be found in the Symbaloo
webmix specifically created for this DU: https://www.symbaloo.com/mix/numerical-proporcionality
-
We´ll
follow on by making exercises the traditional way, related to the
context of the students as explained before, that will help fix the
mathematical concepts while the students copy on their notebooks.
-
Later,
some students in the Class will be asked for to assume the role of teachers,
making up some problems that we´ll solve altogether afterwards, promoting the participation
of all the rest of the classmates, who will be free to ask for doubts and also
to invent their own problems.
-
Finally,
the students will be asked to make a Final Project in groups of threes or fours
consistent of a short presentation with slides where to depict words and
phrases learned in L2, and their meaning in L1.
7. Methodology
7.1 Organization
and class distribution / timing
- The Didactic Unit will be
scheduled to last for 7 sessions, 55 minutes each, where the teacher will
use several methods involving the use of TIC and a “compulsory” participation
of the pupils to fix their attention and allow them to attain knowledge by the
feedback received in the teaching-learning process:
- According to this, Session 1 will
be to learn new vocabulary and make a Brainstorm game with all the Class to
pose problems of everyday life on magnitudes, rates, ratios, direct and inverse
proportions and percents, out of which the most interesting ones will be noted
down to solve them in later sessions.
- Sessions 2 to 4 will deal with the
explanation of the main concepts in the blackboard alongside the occasional
projection of short videos in the L2 (English) with subtitles on, in the same L2, for the students to
keep attention, stopping the video at different points to give them time to
copy the contents of the blackboard on their notebooks. At the same time, the
teacher will translate the unknown words and phrases to the students, who will
note them down too. Besides, some exercises will be made on the
blackboard and digital board and some others proposed from the textbook as
homework for the next day.
- Session 5 will be used to work in pairs or in
groups of threes or fours to search the web for information on
consumption rates in cars, amounts of ingredients in recipes, discount
percents, and vocabulary related to the DU. In that very session they will
propose several problems of everyday life they had found and they will
prepare their Final Project: a presentation on vocabulary learned to be
exposed to the rest.
- During Session 6 several
randomly selected presentations will be read by their creators, and the
time left will be used in making some exercises from the textbook and
solving doubts.
- Session 7 is reserved for the Final Written
Exam on the DU.
7.2 Resources
/ Materials
-
Textbook.
-
Notebooks.
-
Short
Videos and lessons collected in the following Symbaloo webmix: https://www.symbaloo.com/mix/numerical-proporcionality
-
Laptops
to find information for the Final Project (Short presentation on vocabulary
learned)
-
Blackboard
and white chalk.
-
Digital
Whiteboard and markers.
-
Projector
and canvas screen.
7.3 Key
Competences
* The abbreviations of the spanish names of the Key
Competences according to LOMCE shown between brackets at the end of each one
have been taken from the official web of the Spanish Ministry of Education: https://www.educacionyfp.gob.es/educacion/mc/lomce/curriculo/competencias-clave/competencias-clave.html
-
Competence
in Linguistic Communication (CCL)
-
Mathematical
Competence, and Basic Competences in Science and Technology (CMCT)
-
Digital
Competence (CD)
-
Learning
to learn (CPAA)
-
Social
and civid competences (CSC)
-
Sense
of iniciative and entrepreneurship Competence (SIE)
8. Evaluation
(criteria and instruments)
* Evaluation criteria:
- Use fractional
numbers, decimals, their operations and properties to collect, transform and
exchange information and solve problems related to daily life (CCL, CMCT, CSC)
- Choose the appropriate form of calculation (mental, written or with a
calculator), using different strategies that allow simplifying operations with
whole numbers, fractions, decimals and percentages and estimating the coherence
and precision of the results obtained (CMCT, CD, CPAA, SIE)
- Use different strategies (use of tables, obtaining and using the constant
of proportionality, reduction to unity, etc.) to obtain unknown elements in a
problem from known elements in real life situations in which there are
percentage variations and directly or inversely proportional magnitudes (CMCT,
CSC, SIE)
- Use correctly the specific
vocabulary of the subject matter in L2, both orally and written (CCL)
- Participate in class both
individually and in group (CSC)
- Be respectful with the teacher and
with the rest of classmates (CSC)
- Keep the notebook neat and complete
with the lesson & exercises copied and corrected in L2 & L1 (CLC)
* Evaluation instruments:
- Behaviour in class (respect to
the teacher and the classmates and collaborative attitude): 5%
- Class Work
(Notebook neat, complete and clean, Final Project and homework done): 15%
- Final Written
Exam: 80%
Bibliografical
notes and Credits:
This template has been taken from the Course “Use of open
educational resources for CLIL” from the Distance Training Platform of the
Extremadura Education Council at https://moodle.educarex.es/formprof, upon a former model in Pérez
Torres, I. 2009. Apuntes sobre los principios y características de la
metodología AICLE, en V. Pavón, J. Ávila (eds.), Aplicaciones didácticas
para la enseñanza integrada de lengua y contenidos. Sevilla: Consejería de
Educación de la Junta de Andalucía-Universidad de Córdoba.171-180.
It is based above all upon
practical experience at designing DU and the conversations with teaching partners
and experts, and the theory of the 4 Cs from Do Coyle has also been taken into
account, as related in numerous publications, such as: Coyle, D., Hood, P. and
Marsh, D., 2010. Content and Language Integrated Learning. Cambridge
University Press.
