Mathematics 5: 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

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 numbers that are challenging enough for them while giving them the chance to be successful in finding a solution.

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.

An Example of a Common Task with Multiple Variations

Marian has a new job. The distance she travels to work each day is {5, 94, or 114} kilometers. How many kilometers does she travel to work in {6, 7, or 9} days?

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.

Outcomes C7 Manipulate the dimensions of a rectangle so that the area remains the same

D5 Develop formulas for areas and perimeters of squares and rectangles

Materials: Coloured tiles and centimetre grid paper

Differentiation: the choice will be the number of tiles they select.

Choose 12, 20 or 36 tiles. Make as many different rectangles as you can with all tiles and record the perimeter of each rectangle in a table. What do you notice about the areas and perimeters?

Additional extensions: Determine the greatest possible perimeter. Determine the least possible perimeter.

Follow-up Activity 2

Now You Try:

Choose 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: Money How much change would you

get back if you used a ten dollar bill to buy Caesar salad and juice?

I bought lunch at the cafeteria and got a few coins back in change. How much did I start with and what did I buy?

Today’s Specials

Green Salad $4.15Caesar Salad $5.20Veggies and Dip

$6.75Fruit Plate $5.99Macaroni $6.35Muffin

$1.85Milk

$1.75Juice

$2.25Water

$1.85

Open-ended Questions Use your curriculum document or Math Makes

Sense to find examples of open-ended questions.

Find two closed-questions from Math Makes Sense or from your curriculum document.

Change them to Open-ended Questions.

Be prepared to share one 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 – Area and Perimeter

1. Using grid paper, draw a square that has side lengths of two units. Determine the perimeter and area. Try again with squares of different side lengths. Write about the relationship between the side length of a square and its perimeter and its area.

(E4.1 page 5-81)

2. Explore the area and perimeter of squares and rectangles using the website below*. Make two observations about what you notice about changes in sizes of squares or changes in height or width of rectangles. Record one question you have about area or perimeter.

3. A farmer has 100m of fencing to make a pen for his pigs. He decides that a square or a rectangle would be the best shape. What are some possible sizes he could make? What would the area be of each shape? Which pen would you recommend?

(E4.2 page 5-81)

4. Read the book, Spaghetti and Meatballs for All! (Burns, Marilyn. Spaghetti and Meatballs for All! Brainy Day Books, 1995) Retell the story in your own words to a classmate using tiles to illustrate the main ideas.

5. Make a display to show when area and perimeter are used in the real world. You may cut out pictures or draw them yourself or make a graphic organizer using words or pictures.

6. Use string or masking tape to define a rectangle on the floor with an area large enough for four students to sit comfortably during silent reading. Record the perimeter of the rectangle.

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?