More ways to Differentiate Instruction in Mathematics Heather Hardin Arkansas Department of Education Professional Development Anthony Owen Arkansas Department

More ways to Differentiate Instruction in Mathematics

Heather Hardin

Arkansas Department of Education

Professional Development

Anthony Owen


Curriculum & Instruction

Elements needed to effectively differentiate

instruction:

Big Ideas: the fundamental principals that link together the

specifics

Prior Assessment: formal or informal data collecting that determines a

students abilities or deficiencies

Choice: students are provided options for selecting tasks,

learning opportunities, etc…

Two Core Strategies for Differentiating Mathematics Instruction

Open Questionsquestions framed in way that a variety of

approaches and responses are possible

Parallel Tasksusually two or three tasks that are designed to

meet the needs of students at different developmental levels but address the same big idea and are similar enough in context to be discussed at the same time

Open QuestionsNon-Example:

Question 2:Draw a graph of y=3x2-12x+17. Tell what you notice.

(In this question, students create the graph and state whatever it is they notice, whether that is the vertex, its parabolic shape, or its concavity.)

Question 1: Write the quadratic y=3x2-12x+17 in vertex form.

(In this question, if the student does not know what vertex form is, there is no chance they will answer it correctly.)

Example:

Strategies for Creating Open Questions:

1.Turning around a question

2.Asking for similarities and differences

3.Replacing a number, shape, measurement unit, etc… with a blank

4.Asking for a number sentence

1. Turning around a question

Instead of giving the question, give the answer.

Example: What is 75% of

20?

15 is a percent of a number.

What percent of what number is

it?

What is the hypotenuse of a

right triangle if the legs are 3 units and

4 units long?

One side of a right triangle is 5 units long? What could

the other side lengths be?

2. Asking for similarities and difference

Choose two items and ask students how they are alike and how they are different

Example: How are √2 and √5 are alike? Different?

√2 √5

3. Replacing a number with a blank

Replace a number (or numbers) with a blank and allow students to choose which

number(s) to use

Example:

Find the volume of a cylinder with a

radius of 7 in and a height of 12 in.

Choose a radius and height for a

cylinder and calculate the

volume.

4. Asking for a sentence

Ask students to create a sentence that includes certain words and numbers

Example: Create a sentence that includes

the words “linear” and “increasing” as well as the

number 4 and 9.

An increasing linear pattern could include

the numbers 4 and 9.

In a linear pattern

starting at 4 and increasing by 9, the tenth

term will be 85.

A linear pattern that is increasing by 9

and grows faster than one

that is increasing by

4.

Prompt:

Possible Responses:

Strategies for Creating Open Questions:

1.Turning around a question

2.Asking for similarities and differences

3.Replacing a number, shape, measurement unit, and so forth with a blank

4.Asking for a number sentence

Shortcuts for Creating Open Questions:

Examples of ways to quickly create an open question using questions from a text resource:

Graph and solve this linear system of

equations: 0.5x + 0.6y = 5.4

-x+y = 9

Write two equations involving both x an y. Determine the values

that make both of them true.

Solve for m:

4m – 1 = -25 5 2 2

The solution to an equation is m=-15. The equation involves a

fraction. What might the equation be?

Matthew has 20 ounces of a 40% salt solution. How much salt should he add to

make it a 45% solution?

Matthew has 20 ounces of a 40% salt solution. He wants a solution with a greater percentage of salt. Decide on the percentage of salt you want. Tell how much salt to

add.

What to Avoid in an Open Question:

Avoid making questions too vagueThis can discourage thinkingExample: What is infinity? (A better question

would be: How do you know that there are an infinite number of decimals between 0 and 1?)

Avoid making questions that are too specificThis targets a narrow level of understanding

and will not engage students who are not at that level

Example: What is the period of the sine function?

Teaching Tips for Open Questions

Providing a set of items and asking how they are similar is often an easy way to create an open question. It is valuable, however, if there is more than one way to “sort” the items.

Allowing students some choice in which equations they represent provides easier access for some students than suggesting what equation might be represented.


Using a phrase like “at least 30” instead of a specific number makes a question more open than a question using a specific number. It is also useful to ask what students know rather than asking for a specific piece of information.

Giving the answer and asking for the question is a “fail-safe” strategy for creating open questions.


By removing the numbers and labels from a graph, a question becomes open to students and encourages them to think more conceptually. The strategy of not labeling graphs and axes can be used to open up many other tasks as well.

By not giving values in a problem and asking students to choose the values, a question is opened up and they can start with numbers they are comfortable with.


Teachers often give two pieces of information in a problem and ask for a third. The simple change of offering only one piece of information and asking for two open up the question.

Starting with the answer and asking the students for the question, as is done when giving the slope and asking for points, is a useful generic strategy for creating open questions.


A question that asks students to come up with an equation for certain roots is more accessible to students because they can work “backward” rather than having to address a question for which they do not feel they have an obvious starting point.

Questions that allow students to choose to agree or disagree are useful for many students because the question gives them a starting point. Rather than having to originate an entire response, all they have to do is come up with a reason for their choice.

You try…

Benefits of Open Questions

All students should be able to contribute to the discussions

The questions allow for differentiation of responses based on each student’s understanding

There are many “good answers”

They provide opportunity for effective follow-up discussions

Two Core Strategies for Differentiating Mathematics Instruction

Open Questionsquestions framed in way that a variety of

approaches and responses are possible

Parallel Tasksusually two or three tasks that are designed to

meet the needs of students at different developmental levels but address the same big idea and are similar enough in context to be discussed at the same time

Parallel TasksExample:

Option 1: Draw a triangle ABC in Quadrant II of a coordinate grid. Reflect it so that the image is in Quadrant IV. Describe your reflection line.

Option 2: Draw a triangle ABC in Quadrant II of a coordinate grid. Reflect it so that the image is in Quadrant IV. Determine the matrix that describes the transformation.

Common follow-up questions:

-How did you know that what you had performed was a reflection?-Did the line of reflection have a positive or negative slope? Why?-Why can you describe the image of any point using just one piece of information in addition to the point’s coordinates (either the line or the matrix)?

Strategies for Creating Parallel Tasks

It is important to make use of prior assessment data to determine how students might differ developmentally in approaching the bid idea

Develop similar enough tasks that students could answer common questions about both

Parallel TasksExample:

Option 1: Someone suggests that the school driveway is 4,000,000 mm long? is it a long driveway?

Option 2: Someone suggests that a shopping mall might be 4,000,000 cm2 in area. Do you think that’s reasonable?

Common follow-up questions: -Is is easy to imagine how big our measurement actually is?

-Why does it help to think of it in terms of other units?

-What other units did you choose? How did you re-write the measurement in those units?

-How can you tell whether your answer is reasonable?

Shortcuts for Creating Parallel Tasks:

Examples of ways to quickly create parallel tasks using questions from a text resource:

Original Task (e.g., from a text): 486 students voted in the school election. That was about 53% of the student body. How many students are in the school?

Option 1:486 students voted in the school election. That was about 53% of the student body. How many students are in the school?

Option 2: 486 students voted in the school election. That was about 60% of the student body. How many students are in the school?

Common follow-up questions:

-How do you know that there are more than 500students?

-How do you know that there are fewer that 1000students?

-How might you estimate your answer?

-What does the question have to do with lookingfor equivalent ratios orfractions?

Teaching tips for Parallel Tasks

By asking students to create and item like an existing one, they are likely to consider many attributes of the provided item. The openness is in allowing them to choose what attribute to consider.

It is always useful to explore “what if” questions-where one assumption or constraint in a situation is altered to see its effect on other aspects of the situation.


It is often a good idea to ask about what something might be but also what it might not be.

One of the ways to create a set of parallel tasks is to provide either more or less information about a similar situation in one of the options.

Sometimes it makes sense to add a simple challenge to the “easier” option to ensure that students are always moving forward.

Sharing of student work and strategies should be encouraged, especially when students are responding to questions that give them the opportunity to be creative in their answers.


When creating parallel tasks, it is often helpful to ensure that calculations involving both options lead to similar results. This makes it easier to ask questions related to estimating the answer.

One way to create parallel tasks is to provide extraneous information in one tasks that is not provided in the other.


When creating parallel tasks, it is often helpful to ensure that calculations involving both options lead to similar results. This makes it easier to ask questions related to estimating the answer.

One way to create parallel tasks is to provide extraneous information in one tasks that is not provided in the other.

Fundamental principals for developing new questions and

tasks:All open questions must allow for correct

responses at a variety of levels.

Parallel tasks need to be created with variations that allow struggling students to be successful and proficient students to be challenged.

Questions and tasks should be constructed in such a way that all students can participate together in the follow-up discussion.

All material in this presentation came from this resource. NCTM Stock #: 13782


Heather HardinOffice of Professional Development

Director: Kevin Beaumont501-682-4232

Anthony OwenOffice of Curriculum & Instruction

Director: Stacy Smith501-682-7442

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More ways to Differentiate Instruction in Mathematics Heather Hardin Arkansas Department of Education Professional Development Anthony Owen Arkansas Department