More ways to Differentiate Instruction in Mathematics Heather Hardin Arkansas Department of Education Professional Development Anthony Owen Arkansas Department of Education 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 Questions questions framed in way that a variety of approaches and responses are possible Parallel Tasks usually 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 Questions Non-Example: Question 2: Draw a graph of y=3x 2 -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=3x 2 -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 = 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 vague This can discourage thinking Example: 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 specific This 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.

Teaching Tips for Open Questions 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.

Teaching Tips for 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.

Teaching Tips for Open Questions 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.

Teaching Tips for 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.

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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 Questions questions framed in way that a variety of approaches and responses are possible Parallel Tasks usually 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 Tasks Example: 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 Tasks Example: 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 cm 2 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 500 students? -How do you know that there are fewer that 1000 students? -How might you estimate your answer? -What does the question have to do with looking for equivalent ratios or fractions?

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.

Teaching tips for Parallel Tasks 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.

Teaching tips for Parallel Tasks 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.

Teaching tips for Parallel Tasks 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

Arkansas Department of Education Heather Hardin Office of Professional Development Director: Kevin Beaumont Anthony Owen Office of Curriculum & Instruction Director: Stacy Smith