Mathematics Support

Differentiated Instruction

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

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

Diversity in the Classroom

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

Differentiating Instruction Some ways to differentiate instruction

in mathematics class

Common Task with Multiple Variations

Open-ended Questions

Differentiation Using Multiple Entry Points

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.

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 methodC7 represent square and triangular numbers concretely, pictorially, and symbolically

So why was the remainder 2? (n-1)2 + n2 + (n+1)2 = 3n2 +2

But it’s also true that n2 + (n+1)2 + (n+2)2 = 3n2 + 6n + 5

= 3n2 + 6n + 3 + 2

Or use a model

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.

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 ratiosA5 demonstrate an understanding of the concept of percent as a ratio

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”

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.

Open-ended Questions

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

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 student’s understanding.

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

Open-ended Questions Adjusting an Existing Question

1. Identify a topic.2. Think of a typical question.3. 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?

3

1

4

13

1

4

13

1

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

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

100% of 25 50% of 50 1% of 2500 10% of 250 25% of 100 5% of 500

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 numbersF8 demonstrate an understanding of the differences among mean, median, and modeF3 plot coordinates in four quadrants

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.

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.

Multiple Entry Points

Multiple Entry Points are diverse activities that tap into students’

particular inclinations and favoured way of representing knowledge.

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

Sample - Ratio

Conduct a survey of classmates on a subject of your choosing. Then, write ratio comparisons between/ among the results.Variation of 6A3.4Real world (context)SymbolsLogical-mathematicalLinguistic

Use an addition or a multiplication chart and explain how it could be used to describe ratios.Using picturesOral and writtenSymbolsLogical-mathematicalLinguistic

Find examples of ratios in Sobey’s or Superstore’s weekly grocery ads. Make a display of the pictures for the bulletin board. Be sure to label your work.Page 6-6Real world (context)Using pictures

Represent the following ratios using tiles or other manipulatives. Record what you have built with drawings:4:2 3:5 1:6 4:8 ConcreteUsing pictures

Find the following body ratios by measuring:Wrist size: ankle sizeWrist size: neck sizeHead height: full height6A3.2Real world (context)SymbolsSpatial

Model two situations that could be described by the ratio 3:4. * The second situation must involve a different number of items than the first.6A3.1Real world (context)Bodily kinesthetic or Pictures

A3 Write and interpret ratios, comparing part-to-part and part-to-wholeA4 Demonstrate understanding of equivalent ratios

Sample – Rotational Symmetry

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 modelsLogical 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 picturesWritten 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 picturesOral and written language

Complete a Frayer Model about rotational symmetry. (Appendix V page 107, Mathematics Grade 6: A Teaching Resource)Logical mathematicalUsing picturesOral and written languageLinguistic

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 situationsOral and written languageLinguistic

6E8 Students will be expected to make generalizations about the rotational symmetry property of all members of the quadrilateral “family” and of regular polygons

* All activities demand some level of spatial sense.

Creating Tasks With Multiple Entry PointsUsing 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.

Possible Uses for the Grid1. 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?

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