Differentiating Mathematics

Instruction

Jane Silva, Mathematics Instructional Leader SW

Four Corners

When I think about teaching mathematics, I feel like…

Our Differentiation

Framework• With your group, organize the terms in order to show a relationship among the words.

•Using the provided blank cards, label your groups according to the relationship you identified.

TDSBs Differentiation Framework

1 Readiness 2 Interest 3 Learning Profile

4 Content 5 Product 6 Process 7 Environment

Strategies I Use in Mathematics

Describe a strategy you use to differentiate according to the number you roll.

Record your strategies on sticky notes.

*Identifying similarities and differences (e.g., Venn diagram)

*Summarizing and note taking (e.g., mind maps, concept maps)

*Reinforcing effort and providing recognition (e.g., goal- setting)

*Non-linguistic representations (e.g., graphic organizers)

*Cooperative learning (e.g., jigsaw, think-pair-share)

*Setting objectives and providing feedback (e.g., exit card, rubrics)

*Questions, cues and advance organizers (e.g., anticipation guides)

Differentiation Strategies

“The teacher does not try to differentiate everything for everyone

every day.”

Tomlinson

Individualized instruction

for each student

Providing instruction to meet the range of

student needs

Differentiating Instruction…

IS… IS NOT…

*Cubing

*Menus

*Choice Boards

*RAFTs

*Tiering

*Learning Centers

Structures forDifferentiating Instruction

IMPOSSIBLE

LIKELY

CERTAIN

Describe probability as a measure of the likelihood that an event will occur, using mathematical language

Structure: CubingProbability

Structure: CubingGeometry & Spatial Sense

Compare & Contrast

Face 1: I understand…

Face 2: I don’t understand…

Face 3: I find it easy to…

Face 4: I find it difficult to…

Face 5: I learned…

Face 6: I still want to know…

Structure: CubingJournal Prompts

Appetizer (Everyone):

*What is a pattern?

Main dish (Choose 1):

*Create a repeating pattern using pattern blocks.

*Create a growing pattern using pattern blocks?

Side dishes (Choose 2):

*Describe a pattern that results from repeating an action.

*Describe a pattern that results from repeating an operation.

*Describe a pattern that results from using a transformation.

Dessert (If you wish)

*Create a growing pattern. How is it the same as a repeating pattern? How is it different?

Structure: MenuPatterning

*With a partner collect 1 basket of pattern blocks. Take turns sorting blocks into two different groups and ask your partner to guess your sorting rule? (Some rules could be shapes that stack, roll, slide, or shapes with 3 edges, 4 vertices, 6 faces)

*Look around our classroom, draw: 2 things that are rectangles, 3 things that are square, 1 thing that is a triangle, and 4 things that are circles.

*Use a set of tangrams to create a design. Trace around the outside of each shape.

*Choose 2 different shapes. Write about how they are different and about how they are the same.

Structure: MenuGeometry

Explain how you would add 23 and 17 using

mental math

Use base ten blocks to show how you would add

23 and 17

Use base ten materials to represent the relationship between a decade and a

centuryExplain how 3 groups of 2

is equal to 2 + 2 + 2Use manipulative to show that 3 groups of 2 is equal

to 3 x 2

Give a real-life example of when you might need to

know that 3 groups of 2 is 3 x 2

Explain how 6 divided by 3 is similar to 3 x 2

Use manipulative to show that 6 divided by 3 is 2

Give a real-life example of when you might need to know that 6 divided by 3

is 2

Structure: Choice BoardNumber Sense & Numeration

Complete question # …. on page …. in your text.

Make up a jingle that would help someone remember the steps for subtracting mixed numbers.

Think of a situation where you would add fractions in your everyday life.

Choose the pro or con side and make your argument:The best way to add mixed numbers is to make them into equivalent improper fractions.

Someone asks you why you have to get a common denominator when you add and subtract fractions. What would you say?

Create a subtraction of fractions question where the difference is 3/5.

Describe your strategy.

Replace the blanks with the digits 1, 2, 3, 4, 5, and 6 and add these fractions:[]/[] + []/[] + []/[]

Draw a picture to show how to add 3/5 and 4/6.

Find or create three fraction subtraction problems. Solve them and show your work.

Structure: Choice BoardFractions

Role Audience Format Topic 

Exponent 

Jury 

Instructions 

Laws of Exponents

  

Acute Triangle 

 Obtuse Triangle

 Dear John Letter

 Our Differences

 Parts of a Graph

 

 TV Audience

 Script

 Which of Us Is

Most Important? 

 Plus Sign

 

 Multiplication

Sign

 Romantic Card

 Why We Go

Together 

Structure: R.A.F.TVarious

Role Audience Format Topiclength Teacher Pictures How I help you

find the perimeter of a square

height Principal Words How I help you find the perimeter of a rectangle

distance Student Numbers How I help you find the perimeter of a circle

Structure: R.A.F.TMeasurement

Tier 1: all fractions are proper; have common denominators; and can be

modeled

Tier 2: fractions are proper and improper; have different denominators, but all can be modeled

Tier 3: fractions are proper and improper and not all can easily be modeled

Structure: TIERINGFractions

Find the surface of the following shapes: 

Activity A: Provide simple rectangular prisms and cylinders with measurements provided.  Activity B: Provide simple rectangular prisms and cylinders where students must first measure.  Activity C: Provide pictures of simple rectangular prisms and cylinders with measurements provided. Activity D: Ask students to find examples of cylinders or rectangular prisms.

Structure: TIERINGMeasurement

Describe what type of data could be represented by this graph (an image of the graph is depicted).

Station 1: PictographStation 2: Bar GraphStation 3: Stem-and-Leaf PlotStation 4: Broken Line GraphStation 5: Histogram

Structure: Learning Centres

Data Management

Strategies I May Use

To Differentiate Think about one strategies you can use to

differentiate mathematics.

Discuss your thoughts on implementing this strategy in pairs.

Be prepared to share.

Two Core Strategies for Differentiating Mathematics

Instruction

Strategy: Open Question

Allows for different students to approach it by using different processes or strategies.

Allows students at different stages of mathematical development to benefit and grow from attention to the task.

Is framed in such a way that a variety of responses or approaches is possible.

Strategy: Open Questions

You add fractions and the sum is 1.What could the fractions be?

1/2 + 1/2 = 1

1/4 + 1/4 + 1/4 + 1/4 = 1

2/8 + 6/8 = 1

1/3 + 2/6 + 4/12 = 1

Strategy: Open Questions

What makes it open?

Assessment: To reveal what students understand about fractions.

Big Idea:There are many ways to represent numbers when adding fractions.

Choice:It allows students at different levels of readiness to respond to the question.

Visit at least 2 stations. Stations are organized by strand.

Centres: Open Questions

Number Sense &

NumerationGeometry &

Spatial Sense

Measurement

Data Management & Probability

Patterning & Algebra

Select and try an open question.

Questions are grouped by grade: K-2, 3-5, and 6-8.

Centres: Open Questions

Discuss with others how you might use this in your teaching practice.

Decide which big idea most appropriately reflects the task.

Centres: Open Questions

Visit at least 2 stations (stations are organized by strand).

Select and try an open question (questions are grouped by grade clusters)

Discuss with others how you might use this in your teaching practice.

Decide which big idea most appropriately reflects the task.

Centres: Open Questions

K-2: Show the number 7 in as many ways as you can.

3-5: The sum is 42. What is the question?

6-8: Create a sentence that uses each of the following words and numbers. Other words and numbers can be used.

40, percent, most, 80

Big Idea: There are many ways to represent numbers

Centre: Number Sense and Numeration

K-2: Draw a design or shape made up of three shapes. The design should have symmetry.

3-5: How many different shapes can you make by using five green pattern block triangles? Triangles must match along full sides.

6-8: Show how to put together squares to create shapes with eight sides.

Big Idea: New shapes can be created by either combining or dissecting existing shapes.

Centre: Geometry and Spatial Sense

K-2: Two shapes are the same size. What could they be?

3-5: A polygon has a perimeter of 44 units. Draw five possible shapes.

6-8: A shape has an area of 200 square metres. What could its length and width be?

Big Idea: The same object can be described by using different measurements.

Centre: Measurement

K-2: The forth picture in a pattern consists of five squares as shown. What could the first, second, third and fifth pictures look like?

3-5: A pattern begins like this: 2, 6, … How might it continue?

6-8: A pattern is built by adding pairs of terms to get the next term. There is a 10 somewhere between the forth term and the tenth term. What could the pattern be? Think of as many possibilities as you can.

Big Idea: A group of items form a pattern only if there is an element of repetition, or regularity, that can be described with a pattern

rule.

Centre: Patterning and Algebra

? ? ? ?

K-2: What might this graph be about?

3-5: Select a graph type you would use to display these data. Why is your choice a good way to show the data?

6-8: Create a set of data that can be appropriately described by using a histogram. Create that histogram.

Big Idea: Graphs are powerful data displays because they quickly reveal a great deal of information.

Centre: Data Analysis and Probability

Favourite DinosaursT-Rex 25Triceratops 3Stegosaurus 8Brachiosaurus 2

Turn around a question

Closed: What is the sum of 2, 4, and 6?

Open: The sum of three numbers is 12. What can these numbers be?

Strategies for Creating Open Questions

Ask for similarities and differences

Closed: What is perimeter? What is area?

Open: How are perimeter and area the same? How are they different?

Strategies for Creating Open Questions

Change the question

Closed: What number has 3 hundreds, 2 tens, 2 thousands, and 4 ones?

Open: You can model a number with 11 base ten blocks. What could the number be?

Strategies for Creating Open Questions

Leave the Values Open

Closed: Use 3 triangles to make a trapezoid. Draw the lines of symmetry.

Open: Draw a design or shape made up of three shapes. The design should have symmetry.

Strategies for Creating Open Questions

Ask for a number sentence

Create a sentence that includes the numbers 3 and 4 as well as the words “and” and “more”.Ex: the sum of 3 and 4 is more than 6; 4 is more than 3 and more than 1

Strategies for Creating Open Questions

1. Select a curriculum expectation.2. Determine the big idea (or learning

goal).3. Consider student readiness, learning

profiles, and interests.4. Use a “opening up” strategy to develop

a question.

Make your OwnOpen Questions

Strategy: Parallel Tasks

Sets of tasks, usually two or three, that are designed to meet the needs of students at different developmental levels, but get at the same big idea and are close enough in context that they can be discussed simultaneously.

Create a repeating pattern that begins with 3, 5,…

Create an increasing pattern that begins with 3, 5,…

Strategy: Parallel Tasks

Strategy: Parallel Tasks

What makes it parallel?

Assessment: Differ in sophistication.

Big Idea:They both posses the same big idea: a group of items form a pattern only if there is an element of repetition, or regularity, that can be described with a pattern rule.

Choice:It allows students at different levels of readiness to select a question.

Strategy: Parallel Tasks

What makes it parallel?

Consolidation Questions: They both possess the same set of consolidation questions:

What is your pattern?What makes it a pattern?What would be your 10th number?

Resources

Guides to Effective Instruction (K-6) – www.eworkshop.on.ca

Edugains (7-12) – www.edugains.ca

National Library of Virtual Manipulatives – Google “nlvm”