Mathematics Support

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Mathematics Support

• Differentiated Instruction

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Differentiating Instruction

• differentiating instruction means that
  students have multiple options for taking in
  information, making sense of ideas, and
  expressing what they learn. In other words, a
  differentiated classroom provides different
  avenues to acquiring content, to processing or
  making sense of ideas, and to developing products
  so that each student can learn effectively.
• Tomlinson 2001

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Supporters of differentiation believe

• All students have areas of strength
• All students have areas that can be strengthened
• Students bring prior knowledge and experience to
  learning
• Emotions, feelings and attitudes affect learning
• All students can learn
• Students learn in different ways at different
  times
• Gregory and Chapman 2006

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Diversity in the Classroom

• Using differentiated tasks is one way to attend
  to the diversity of learners in your classroom.

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Differentiating Instruction

• Some ways to differentiate instruction in
  mathematics class
• Common Task with Multiple Variations
• Open-ended Questions
• Differentiation Using Multiple Entry Points

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Common Tasks with Multiple Variations

• A common problem-solving task, and adjust it for
  different levels.
• Students tend to select the question or the
  numbers that are challenging enough for them
  while giving them the chance to be successful in
  finding a solution.

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• Choose three consecutive numbers, square them,
  and add the squares.
• Divide by 3 and record the whole number
  remainder.
• What happened? Why?

B12 calculate products and quotients in relevant
contexts by using the most appropriate method C7
represent square and triangular numbers
concretely, pictorially, and symbolically
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So why was the remainder 2?

• (n-1)2 n2 (n1)2 3n2 2
• But its also true that
• n2 (n1)2 (n2)2 3n2 6n 5
• 3n2 6n 3
  2

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Or use a model
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Plan Common Tasks with Multiple Variations

• The approach is to plan an activity with multiple
  variations.
• For many problems involving computations, you can
  insert multiple sets of numbers or have students
  select their own numbers.
• You may also opt to give two or more choices of
  activities which relate to a common topic or
  outcome.
• Common questions are carefully constructed so
  that students can contribute to the conversation
  no matter which variation they chose to explore.

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A proportional example

• You used 240 g of rice. What was the total mass
  of the rice if
• Task A It was 1/3 of the total mass of the rice.
• Task B It was 2/3 of the total mass of the rice.
• Task C It was 40 of the total mass of the rice.

A4 demonstrate an understanding of equivalent
ratios A5 demonstrate an understanding of the
concept of percent as a ratio
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Common Tasks with Multiple Variations

• When using tasks of this nature all students
  benefit and feel as though they worked on the
  same task.
• Class discussion can involve all students.
• Questions should be phrased so that all students
  can offer comments and answers.
• There is additional work prepared for any early
  finishers

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What are some questions you could ask of students

• Choose a topic or an outcome(s) from your grade
  level curriculum and create a differentiated
  activity for your students.

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Open-ended Questions

• Open-ended questions have more than one
  acceptable answer and can be approached by more
  than one way of thinking.

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Open-ended Questions

• Well designed open-ended problems provide most
  students with an obtainable yet challenging task.
  
• Open-ended tasks allow for differentiation of
  product.
• Products vary in quantity and complexity
  depending on the students understanding.

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Open-ended Questions

• An Open-Ended Question
• should elicit a range of responses
• requires the student not just to give an answer,
  but to explain why the answer makes sense
• may allow students to communicate their
  understanding of connections across mathematical
  topics
• should be accessible to most students and offer
  students an opportunity to engage in the
  problem-solving process
• should draw students to think deeply about a
  concept and to select strategies or procedures
  that make sense to them
• can create an open invitation for interest-based
  student work

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Open-ended Questions Adjusting an Existing
Question

• Identify a topic.
• Think of a typical question.
• Adjust it to make an open question.
• Example Ratios
• The ratio of cats to dogs in a neighbourhood is
  exactly 2 to 1. There are 15 dogs. How many cats
  are there?
• The ratio of cats to dogs in a neighbourhood is
  exactly 2 to 1. How many cats and how many dogs
  might there be in the neighbourhood?

A3 write and interpret ratios, comparing
part-to-part and part-to-whole
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Sample question

• Describe 25 as a percent of a number in as many
  ways as you can. Make sure some percents are big
  and some are little.

A5 demonstrate an understanding of the concept of
percent as a ratio
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• 100 of 25
• 50 of 50
• 1 of 2500
• 10 of 250
• 25 of 100
• 5 of 500

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Other possibilities

• Name a fraction that is a bit less than 0.6.
  Explain how you know. Can you name another
  fraction that is between 0.6 and your suggestion?
  
• The mean of some numbers is 14. What are the
  numbers?
• The data can be shown using a coordinate graph
  with 4 quadrants. What might the data be?

A9 relate fractional and decimal forms of
numbers F8 demonstrate an understanding of the
differences among mean, median, and mode F3 plot
coordinates in four quadrants
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Open-ended Questions

• Use your curriculum document or other resource to
  find examples of open-ended questions.
• Find two closed-questions from your curriculum
  document (or think of a typical question).
• Change them to open-ended questions.
• Be prepared to share both versions of your
  questions.

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Differentiation Using Multiple Entry Points

• Van de Walle (2006) recommends using multiple
  entry points, so that all students are able to
  gain access to a given concept.
• diverse activities that tap students particular
  inclinations and favoured way of representing
  knowledge.

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Multiple Entry Points

• Multiple Entry Points are diverse activities that
  tap into students particular inclinations and
  favoured way of representing knowledge.

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Multiple Entry Points

• Based on Five Representations

• Based on Multiple Intelligences

• Concrete
• Real world (context)
• Pictures
• Oral and written
• Symbols

• Logical-mathematical
• Bodily kinesthetic
• Linguistic
• Spatial

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Sample - Ratio
A3 Write and interpret ratios, comparing
part-to-part and part-to-wholeA4 Demonstrate
understanding of equivalent ratios
Conduct a survey of classmates on a subject of your choosing. Then, write ratio comparisons between/ among the results. Variation of 6A3.4 Real world (context) Symbols Logical-mathematical Linguistic Use an addition or a multiplication chart and explain how it could be used to describe ratios. Using pictures Oral and written Symbols Logical-mathematical Linguistic Find examples of ratios in Sobeys or Superstores weekly grocery ads. Make a display of the pictures for the bulletin board. Be sure to label your work. Page 6-6 Real world (context) Using pictures
Represent the following ratios using tiles or other manipulatives. Record what you have built with drawings 42 35 16 48 Concrete Using pictures Find the following body ratios by measuring Wrist size ankle size Wrist size neck size Head height full height 6A3.2 Real world (context) Symbols Spatial Model two situations that could be described by the ratio 34. The second situation must involve a different number of items than the first. 6A3.1 Real world (context) Bodily kinesthetic or Pictures
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Sample Rotational Symmetry
6E8 Students will be expected to make
generalizations about the rotational symmetry
property of all members of the quadrilateral
family and of regular polygons
Explain how you can make a teaching aid that demonstrates symmetry using paper, scissors and a thumb tack. Create a prototype for the class. Concrete models Logical mathematical Escher used many types of symmetries in his art. Visit the following site and prepare comments about symmetry you see in his artwork. http//www.mcescher.com/ Using pictures Written language Make patterns on the geoboard that have rotational symmetry of order 2. Record your patterns on geopaper. Concrete models
Examine a pack of playing cards. Do the cards have reflective or rotational symmetry? Is your answer true for all cards? Use words and pictures to explain your answer. Using pictures Oral and written language Complete a Frayer Model about rotational symmetry. (Appendix V page 107, Mathematics Grade 6 A Teaching Resource) Logical mathematical Using pictures Oral and written language Linguistic Identify logos that have rotational symmetry. Select four and write short descriptions of their symmetry, including comments on their reflective symmetry, if they have it. (E8.6 page 6-89) Real world situations Oral and written language Linguistic
All activities demand some level of spatial
sense.
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Creating Tasks With Multiple Entry Points
Using the outcomes for decimals, create tasks
with multiple entry points. Take into
consideration the five representations real
world (context), concrete, pictures,
oral/written, and symbolic and multiple
intelligences logical/mathematical, bodily
kinesthetic, linguistic, spatial.



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Possible Uses for the Grid

1. Introduce some of the activities to students
   being careful to select a range of entry points.
   Ask students to choose a small number of
   activities. Other activities can be used for
   reinforcement or assessment tasks.
2. Arrange 9 activities on a student grid 3 rows of
   3 squares. Ask the students to select any 3
   activities to complete, as long as they create a
   Tic-Tac-Toe pattern.
3. Other ideas?

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Differentiating Instruction

• When might you use each of these types of
  differentiation? Why would you select one rather
  than another type?
• Common Task with Multiple Variations
• Open-ended Questions
• Differentiation Using Multiple Entry Points