Two Ideas
to Help Bring the
Standards for Mathematical Practice
into Your HS Math Classroom
 Focus on Algebraic Structure
Deborah Hughes Hallett
 Use Images and Short Videos and ask “What’s the Question?”
Dan Meyer
Algebraic Structure
What do we mean by Algebraic Structure?
Seeing the forest, not the trees
Algebraic Structure addressed in the CCSS-M Standards for Mathematical
Practice:
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young
students, for example, might notice that three and seven more is the same amount as
seven and three more, or they may sort a collection of shapes according to how many
sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 +
7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x
+ 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing
an auxiliary line for solving problems. They also can step back for an overview and shift
perspective. They can see complicated things, such as some algebraic expressions, as
single objects or as being composed of several objects. For example, they can see 5 –
3(x – y)2 as 5 minus a positive number times a square and use that to realize that its
value cannot be more than 5 for any real numbers x and y.
Algebraic Structure addressed in the CCSS-M Content Standards for K – 5
4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a
statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal
statements of multiplicative comparisons as multiplication equations.
5.NF.5. Interpret multiplication as scaling (resizing), by:
•
Comparing the size of a product to the size of one factor on the basis of the size
of the other factor, without performing the indicated multiplication.
Explaining why multiplying a given number by a fraction greater than 1 results in a product
greater than the given number (recognizing multiplication by whole numbers greater than 1 as a
familiar case); explaining why multiplying a given number by a fraction less than 1 results in a
product smaller than the given number; and relating the principle of fraction equivalence a/b = (n
× a)/(n × b) to the effect of multiplying a/b by 1.
Algebraic Structure addressed in the CCSS-M Content Standards for HS
Algebra (Seeing Structure in Expressions)
Interpret the structure of expressions.
A-SSE.1. Interpret expressions that represent a quantity in terms of its context. ★
◦ Interpret parts of an expression, such as terms, factors, and coefficients.
•
Interpret complicated expressions by viewing one or more of their parts as a
single entity. For example, interpret P(1+r)n as the product of P and a factor not
depending on P.
A-SSE.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 (x 2 – y2)(x2
+ y2).
Functions (Interpreting Functions)
Analyze Functions Using Different Representations
F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show
zeros, extreme values, and symmetry of the graph, and interpret these in terms of a
context.
b. Use the properties of exponents to interpret expressions for exponential functions. For
example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y =
(1.01)^12t, y = (1.2)^t/10, and classify them as representing exponential growth or decay.
F-IF.9. Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one
quadratic function and an algebraic expression for another, say which has the larger maximum.
▪ A-SSE.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).
▪ F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal
and explain different properties of the function.
◦
a. Use the process of factoring and completing the square in a quadratic function to
show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context
b. Use the properties of exponents to interpret expressions for exponential functions. For
example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y =
(1.01)^12t, y = (1.2)^t/10, and classify them as representing exponential growth or decay.
The last statement above (in part b) is an extension of the previous example from Grade 5, where
each factor of a product is a scalar of the other. Each of the examples are exponential functions of
the form y = ab^t. In these examples a = 1, so each expression represents a product.
The first expression (1.02)^t represents the product (1)(1.02) or [(1)(1.02)](1.02) or
[(1)(1.02)(1.02)](1.02) and on and on...Each iteration of the product will be greater than the
product of the earlier iterations, because 1.02 is larger than 1. Therefore, as t increases, we will
see exponential growth that approaches positive infinity.
On the other hand, (0.97)^t represents the product (1)(.97) or [(1)(.97)](.97) or [(1)(.97)(.97)](.97)
and so on...Each iteration of the product will be less than the product of the earlier iterations
because .97 is less than 1. So as t increases, the product, which has a value of 1 when t = 0, will get
smaller than 1. Each iteration will result in 1(0.97)^t to get closer to zero, but the product will
never be zero or negative, so we will see exponential decay that approaches zero.
In the Functions conceptual category under "Interpreting Functions" (IF), the Rule of Four is
mentioned in the context of being able to look at two representations and compare and, knowing
their structure within that representation, are able to mentally "picture" the other representation
in order to compare and contrast.
F-IF.9. Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one
quadratic function and an algebraic expression for another, say which has the larger
maximum.
Below are some examples from the problem set "Problems in Algebraic Structure, 2011" that Dr.
Hughes Hallett sent me in February, 2011. The problems approach the idea of looking at the
structure from different perspectives. For example, some of the problems rely on looking at the
structure of one expression and algebraically manipulating another expression into the same
form. Here's an example:
The student first needs to be able to interpret the original expression as a particular structure. It
is the difference of a constant k and a fraction. The numerator of the fraction has variable x with a
coefficient of a constant m. The denominator of the fraction is the difference of the variable x and
a constant n.
Then the student needs to be able to algebraically manipulate the given expressions in #9 and #10
into the above structure. Once they succeed in the algebraic manipulation, they can identify the
values of k, m, and n.
Problem # 11 introduces a bit more rigor in the sense that students need to translate between the
verbal and symbolic representations, as well as keep track of units.
In # 11a above, the student needs to interpret the given situation algebraically. They need to
recognize that some of the letters are constant in the situation and some are variable in the
situation. They also need to understand what the units of measure are for the variables and the
constants. Only by doing this can they interpret what the expressions represent in practical
terms.
The second part to # 11 continues to challenge the student to make sense of the expressions by
thinking about the units of the variables and constants and how the operation of multiplication in
xS and yT act on the units, giving the expressions xS and xT yet another unit . When the variable
x cubic yards is multiplied by S trips to a job site by Truck # 1, then the expression xS represents
the total number of cubic yards of sand carried to the job site by Truck # 1. Similarly,
yT represents the total number of cubic yards of sand carried to the job site by Truck # 2. So, the
expression "xS > yT" can be interpreted verbally as "The total number of cubic yards of sand
delivered to the job site by Truck # 1 is greater than the total number of cubic yards of sand
delivered to the job site by Truck # 2." This understanding is then coupled with the knowledge
that y > x, which is interpreted as "Truck # 2 holds more sand each trip than Truck # 1". The
student is asked to verbally explain what that suggests about the situation. If Truck # 1 holds less
than Truck # 2, but Truck # 1 carried more sand to the job site than Truck # 2, then Truck # 1
must have made many more trips to the job site...the little truck that could! Therefore, we can
conclude that S > T.
Not stopping at the verbal explanation, the problem then asks the student to verify this assertion
using an algebraic representation. Since we are using inequalities, the direction of the arrows will
remain the same if we multiply or divide both sides by a positive #.
Given xS > yT and y > x, Prove S > T
y > x and T > 0, we can multiply both sides of the inequality by T and keep the direction of the
arrow, so yT > xT
xS > yT and yT > xT, so xS > xT
Since x > 0, we can divide both sides of xS > xT by x and keep the direction of the inequality
arrow, which results in S >T
The above situation may seem complicated, but in reality, the actual situation can be interpreted
by students in Grade 8 or Algebra I. The rigor in this problem does not lie in the complexity of
the model. The rigor lies in the more abstract thinking based on the algebraic structure. If
students are taught from the early stages of working with real world problems...as early as
Kindergarten... to always include the units of the numbers being considered, they will create a
habit of mind that will serve them later in algebra. Once letters are introduced to represent
numbers in Grade 6 (Grade 6 Expressions and Equations 6.EE.2a & c, 6.EE.6, 7, 8), students also
need to define the units represented by the letters (variables). If units are stressed along the way,
the above problem will not be a stretch for Algebra I students.
Problem # 11 reinforces several of the Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
4. Model with mathematics.
6. Attend to precision.
7. Look for and make use of structure.
For the Hughes Hallett Problems and Solutions on Algebraic Structure, go to the following link:
http://www.watsonmath.com/wp-admin/post.php?post=150&action=edit&message=1
Dan Meyer: Use Images and Short Videos and ask “What’s
the Question?”
Check out Dan Meyer’s
websites:
http://blog.mrmeyer.com/
http://101qs.com/
Three Acts can be accessed at
http://blog.mrmeyer.com/?p=10285
Three Acts to a Mathematical Story
Storytelling gives us a framework for certain mathematical tasks that
is both prescriptive enough to be useful and flexible enough to be
usable. Many stories divide into three acts, each of which maps neatly
onto these mathematical tasks.
Act One
Introduce the central conflict of your story/task clearly,
visually, viscerally, using as few words as possible.
Show them the first half of the video of shooting the basket.
Act Two
The protagonist/student overcomes obstacles, looks for
resources, and develops new tools.
What tools do they have already? What tools can you help them
develop? They'll need quadratics, for instance. Help them with that.
Act Three
Resolve the conflict and set up a sequel/extension.
Show them the ball going into the basket.
Conclusion
Many math teachers take act two as their job description. Hit the
board, offer students three worked examples and twenty practice
problems. As the ALEKS algorithm gets better and Bill Gates throws
more gold bricks at Sal Khan and more people flip their classrooms,
though, it's clear to me that the second act isn't our job anymore. Not
the biggest part of it, anyway. You are only one of many people your
students can access as they look for resources and tools. Going
forward, the value you bring to your math classroom increasingly will
be tied up in the first and third acts of mathematical storytelling, your
ability to motivate the second act and then pay off on that hard work.
Dan Meyer’s “Ten Design Principles for Engaging Math Tasks”
 Perplexity is the goal of engagement. We can go ten rounds
debating eggs, broccoli, or candy bars. [references a debate, long since
settled — dm] What matters most is the question, “Is the student
perplexed?” Our goal is to induce in the student a perplexed, curious
state, a question in her head that math can help answer.
• Concise questions are more engaging than lengthy ones, all
other things being equal. Engaging movies perplex and interest you in
their first ten minutes. No movie on this list took more than twenty
minutes to set up its context, characters, and conflict. The same is
true of engaging math problems, either pure or applied. Use a short
sentence or simple visual to “hook” the student into the space of the
problem. Use later sentences to expand on it. This order is often
inverted in problems that fail to engage students.
• Pure math can be engaging. Applied math can be boring. The
engagement riddle isn’t solved by taking pure math problems and
shoehorning them into contexts that don’t want them. It’s hard to
argue that two trains traveling in opposite directions from Philadelphia
at different speeds is more engaging than “How many ways can you
think to turn 20 into 10?”
• Use photos and video to establish context, rather than words,
whenever possible. Rather than describing the world’s largest
coffee cup in words, show a photo or a video of it. Not only because
our words fail to capture what’s so engaging about the coffee cup but
because we should find ways to lower the language demand of our
math problems whenever possible.
• Use stock photography and stock illustrations sparingly. The
world of stock art is glossy, well-lit, and hyper-saturated and looks
nothing like the world our students live in. It is hard to feel engaged in
or perplexed by a world that looks like a distortion of your own.
• Set a low floor for entry, a high ceiling for exit. Write problems
that require a simple first step but which stretch for miles. Consider
asking students to evaluate a model for a simple case before
generalizing. Once they’ve generalized, considered reversing the
question and answer of the problem.
• Use progressive disclosure to lower the extraneous load of
your tasks. This is one of the greatest affordances of our digital
platform: you don’t have to write everything at once on the same
page. While students work on one part of a problem, there’s no need
to distract them by including every other part of the problem in the
same visual space. Once they answer the first part of the problem,
progressively disclose the next. This technique has far-reaching
applications.
• Ask for guesses. People like to guess, speculate, and hypothesize.
Guessing is engaging. Before disclosing all the abstractions of
parabolic motion on the basketball court, just show a video and ask
the question, “Do you think the ball will go in?” Once they’ve
answered, continue the rest of your unit, lesson, or problem, now with
more engaged learners. They’ll want to know if they’re right or not so
be sure to pay off on that engagement later by showing them.)
• Make math social. More engaging than having a student guess whether
or not the ball goes in is showing her how all of her classmates
guessed also. Summarize the class’ aggregate responses with a bar
chart. Students will enjoy seeing each others’ short answers and
opinions but we can also use the same social interactions to engage
them in pure math. Have your students a) select three x-y pairs and
b) check if they’re solutions of x + y < 5. If everyone in the class sees
the results of everyone else’s investigation, a visualization of linear
inequalities will emerge on the class’ composite graph.
• Highlight the limits of a student’s existing skills and
knowledge. New mathematical tools are often developed to account
for the limitations of the old ones. You can’t model the path of a
basketball with linear equations — we need quadratics. You can’t
model the growth of bacteria with a quadratic equations — we need
exponentials. Offer students a challenge for which their old skills look
useful but turn out to be ineffective. That moment of cognitive conflict
can engage students in a discussion of new tools and counter the
perception that math is a disjointed set of rules and procedures, each
bearing no relationship to the one preceding it.