Standards for Mathematical Practice

Two Ideas
to Help Bring the

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 (x2 – 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

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, yTrepresents 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 > yTand 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 andyT > xT, soxS >xT

Since x > 0, we can divide both sides ofxS >xT by xand 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.