CLEF Conference August 13, 2012

1. Differentiated Instruction
Making Mathematics Accessible and
Challenging to All Students
CLEF - August 13, 2012

2. What is differentiation?
• Modifying learning tasks to better fit the ability levels,
interests, and learning styles of all students.
• Providing high-quality, engaging, and challenging
mathematics activities for all students.

3. Goals for this Session
• Define differentiation
• Examine several differentiation strategies with
examples aligned to the CCSS for Mathematics
• Create a mathematical task in grade level groups with
at least 2 modifications to differentiate instruction
• Examine strategies for differentiating assessment and
tracking student growth.
• This presentation can be accessed online at www.***

4. How do I start?

5. Grade 1-OA- Operations
and Algebraic Thinking
• 1 - Use addition and subtraction within 20 to solve word
problems involving situations of adding to, taking from,
putting together, taking apart, and comparing, with
unknowns in all positions, e.g., by using objects,
drawings, and equations with a symbol for the unknown
number to represent the problem.
• 4 - Understand subtraction as an unknown-addend
problem. For example, subtract 10 – 8 by finding the
number that makes 10 when added to 8.

6. Differentiated Problem Task

7. 7 children were playing, ___ more came, now there
are 11

8. 7 children were playing, ___ more came, now there
are 11

9. 6 children were playing, ___ more came, now there
are 14

10. Grades 3-5 Geometry
• 3-G-1- Understand that shapes in different categories (e.g., rhombuses,
rectangles, and others) may share attributes (e.g., having four sides), and
that the shared attributes can define a larger category (e.g.,
quadrilaterals). Recognize rhombuses, rectangles, and squares as
examples of quadrilaterals, and draw examples of quadrilaterals that do
not belong to any of these subcategories.
• 4-G-2 - Classify two-dimensional figures based on the presence or
absence of parallel or perpendicular lines, or the presence or absence of
angles of a specified size. Recognize right triangles as a category, and
identify right triangles.
• 5-G-3 - Understand that attributes belonging to a category of two-
dimensional figures also belong to all subcategories of that category. For
example, all rectangles have four right angles and squares are rectangles,
so all squares have four right angles.

11. Student Choice Activities
Entry level - 4th grade

12. Student Choice Activities
Entry Level Task

13. Student Choice Activities
Multiple Level Task

14. Student Choice Activities
High Level Task

15. Student Choice Activities
High Level Task

16. Grade 5NF 3 - Number and
Operations - Fractions
• Interpret a fraction as division of the numerator by
the denominator (a/b = a ÷ b). Solve word problems
involving division of whole numbers leading to
answers in the form of fractions or mixed numbers,
e.g., by using visual fraction models or equations to
represent the problem. For example, interpret 3/4 as
the result of dividing 3 by 4, noting that 3/4
multiplied by 4 equals 3, and that when 3 wholes are
shared equally among 4 people each person has a
share of size 3/4.

17. Sharing Brownies

18. Questions to Consider When
Designing Modifications

19. What makes the third question more difficult
than the first two?
What makes the last question the most difficult?

20. What makes the second question more difficult
than the first one?
What makes the last question the most difficult?

23. Guiding Questions When
Guiding Questions When
Designing Tasks
Designing Tasks

24. HS-A-SSE-2-3-Seeing
Structure in Expressions
• 2. Use the structure of an expression to identify ways
to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2,
thus recognizing it as a difference of squares that can
be factored as (x2 – y2)(x2 + y2).
• 3. Choose and produce an equivalent form of an
expression to reveal and explain properties of the
quantity represented by the expression.
• a. Factor a quadratic expression to reveal the zeros of
the function it defines.

25. Examples From Algebra

26. Example - High School Algebra

27. HS - A-CED-1 - Creating
Expressions
• Create equations and inequalities in one variable and
use them to solve problems. Include equations arising
from linear and quadratic functions, and simple
rational and exponential functions.

28. questions
you
could ask to
help students
struggling
with this
task.
Extensions?

29. 5-MD-2 Measurement and Data - Make a line plot to display a data
set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use
operations on fractions for this grade to solve problems involving
information presented in line plots. For example, given different
measurements of liquid in identical beakers, find the amount of
liquid each beaker would contain if the total amount in all the
beakers were redistributed equally.

30. 6-SP-5 - Statistics and
Probability
• Summarize numerical data sets in relation to their context, such as
by:
• c - Giving quantitative measures of center (median and/or
mean) and variability (interquartile range and/or mean absolute
deviation), as well as describing any overall pattern and any
striking deviations from the overall pattern with reference to the
context in which the data were gathered.
• d - Relating the choice of measures of center and variability to
the shape of the data distribution and the context in which the data
were gathered.

32. Your Turn
• Each group will choose one of two standards. If your group
finishes early, you may either think about other modifications
or consider the other standard.
• Design an engaging task for students to complete to help them
develop or assess the level of proficiency in the standard
• Create at least 2 modifications of the task to differentiate
instruction. You might differentiate for ability levels, learning
styles, interests, or problem solving strategy. Consider these
questions when designing your task.

33. Grades K-2
• 2-NBT-5 - Fluently add and subtract within 100 using
strategies based on place value, properties of operations,
and/or the relationship between addition and subtraction.
• 1-MD-2 - Express the length of an object as a whole number
of length units, by laying multiple copies of a shorter object
(the length unit) end to end; understand that the length
measurement of an object is the number of same-size length
units that span it with no gaps or overlaps. Limit to contexts
where the object being measured is spanned by a whole
number of length units with no gaps or overlaps

34. Grades 3-5
• 4-NF-2 - Compare two fractions with different numerators and different
denominators, e.g., by creating common denominators or numerators, or
by comparing to a benchmark fraction such as 1/2. Recognize that
comparisons are valid only when the two fractions refer to the same whole.
Record the results of comparisons with symbols >, =, or <, and justify the
conclusions, e.g., by using a visual fraction model.
• 5-MD-3 - Recognize volume as an attribute of solid figures and
understand concepts of volume measurement.
• - A cube with side length 1 unit, called a “unit cube,” is said to have “one
cubic unit” of volume, and can be used to measure volume.
• - A solid figure which can be packed without gaps or overlaps using n
unit cubes is said to have a volume of n cubic units.

35. Grades 6-8
• 6-RP-3 - Use ratio and rate reasoning to solve real-world and mathematical problems,
e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line
diagrams, or equations.
• b - Solve unit rate problems including those involving unit pricing and constant
speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns
could be mowed in 35 hours? At what rate were lawns being mowed?
• c - Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means
30/100 times the quantity); solve problems involving finding the whole, given a part and
the percent.
• 8-NS-2 - Use rational approximations of irrational numbers to compare the size of
irrational numbers, locate them approximately on a number line diagram, and estimate
the value of expressions (e.g., π^2). For example, by truncating the decimal expansion of
√2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to
continue on to get better approximations.

36. High School
• F-BF3- Building Functions- Identify the effect on the graph of
replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific
values of k (both positive and negative); find the value of k given
the graphs. Experiment with cases and illustrate an explanation
of the effects on the graph using technology. Include recognizing
even and odd functions from their graphs and algebraic
expressions for them.
• G-SRT-7 - Similarity, Right Triangles, and Trigonometry-
Explain and use the relationship between the sine and cosine of
complementary angles

38. Sample Lesson Plan Format for Differentiation

40. Sample Tic-Tac-Toe Assessment
Choices

41. Choice Activities
• Math/Science-experiment or research project with mathematics embedded into it
• Math/Social Studies- problem embedded into a cultural, historical, or social
context
• Writing about mathematics - problem solving strategies, explaining
relationships, etc...
• Math/Literature connection - set the stage for a problem or introduce a concept
• Game - practice skills or problem solving, analyze strategies
• Logic problem - puzzles, Venn diagrams
• Building project- Kinesthetic learners, paper folding, block building, models with
manipulatives
• Problem solving - May target a specific strategy, ex. look for a pattern, start
w/simple example, work backwards
• Data Project - collect, analyze, display data to explore a topic

42. Criteria for Selecting Choice Activities
• Activity requires investigation
• Can be solved in multiple ways
• Complex, requires a variety of skills, develops perseverance
• Designed so students of a variety of abilities can begin the
problem at their own level; some may need more guidance
• Activity may provide practice or fresh insight into
skills/concepts in unit
• Engaging activity
• Can be completed individually or in small groups
• Encourages reflection and communication about key
mathematical ideas
• Activities reflect a variety of learning styles
• Activities reflect a variety of levels on Bloom's Taxonomy

43. Sample Chart to Record Student Progress
Toward Meeting the Standard

44. Other Examples of Activities
Grade 6-EE-2 Expressions and
Equations
• c - Evaluate expressions at specific values of their
variables. Include expressions that arise from
formulas used in real-world problems. Perform
arithmetic operations, including those involving
whole-number exponents, in the conventional order
when there are no parentheses to specify a particular
order (Order of Operations). For example, use the
formulas V = s3 and A = 6 s2 to find the volume and
surface area of a cube with sides of length s = 1/2.

45. Order of Operations

46. Teacher Reflection
Order of Operations Activity

48. 1-OA-6 Operations and Algebraic
Thinking
• Add and subtract within 20, demonstrating fluency for
addition and subtraction within 10. Use strategies such
as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 +
4 = 14); decomposing a number leading to a ten (e.g., 13
– 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship
between addition and subtraction (e.g., knowing that 8 +
4 = 12, one knows 12 – 8 = 4); and creating equivalent
but easier or known sums (e.g., adding 6 + 7 by creating
the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

50. References
• Bray, W. S. (2009, October). The power of choice. Teaching Children
Mathematics, 178-183.
• Chval, K. B., & Davis, J. A. (2009, January). The gifted student. Mathematics
Teaching in the Middle School, 14(5), 267-274.
• Little, C. A., Hauser, S., & Corbishley, J. (2009, August) Constructing
complexity for differentiated learning. Mathematics Teaching in the Middle
School, 15(1). 34-42.
• Pierce, R. L., & Adams, C. L. (2005, October). Using tiered lessons in
mathematics. Mathematics Teaching in the Middle School, 11(3). 143-149.
• Wilkins, M. M., Wilkins, J. M., & Oliver, T. (2006, August). Differentiating the
curriculum for elementary gifted mathematics students. Teaching Children
Mathematics, 6-13.