Session 4: Final Algorithms in A Story of Units
Sequence of Sessions
Overarching Objectives of this July 2013 Network Team Institute
● Implementation: Participants will be able to articulate and model the instructional approaches to teaching the content of the lessons of the first module.
● Standards alignment and focus: Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade.
● Coherence: Participants will be able to articulate connections from the content of previous grade levels to the content of this module.
● Participants will be able to articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment and end of module assessment.
● G6-7: Participants build the foundation to develop tape diagram skills for themselves and for students and fellow teachers.
● G8&10: Participants experience content from Grade 8 geometry and relate those experiences to the content of Grade 10 geometry, building instructional capacity to bridge gaps for incoming 10th graders.
● G9: Participants can articulate the coherence across the grade 9 curriculum, describing the focus of each module and relating the modules to each other.
High-Level Purpose of this Session
· To examine and practice the algorithms employed in A Story of Units.
· Understand the coherence within and across grades in order to promote conceptual understanding.
Related Learning Experiences
● In Session 1, a curriculum overview, participants will gain an understanding of how each module contributes to the overall progression of concepts throughout the grade-level.
● Session 3, Exploration of Models, will prepare participants to utilize models appropriately in promoting conceptual understanding throughout A Story of Units.
● Session 4, Algorithms, will prepare participants to utilize algorithms appropriately in promoting conceptual understanding throughout
A Story of Units.
● Session 6, Leadership to Support A Story of Units, will help participants articulate examples of how to support implementation of the curriculum and draw connections to the Evidence Guide for Planning and Practice in a Single Lesson.
Key Points
· All algorithms involve the manipulation of units.
· Each algorithm builds towards the next, culminating in the long division algorithm.
· The long division algorithm is foundational to an understanding of the real number system and advanced mathematics.
Session Outcomes
What do we want participants to be able to do as a result of this session? / How will we know that they are able to do this?· Examine and practice the algorithms employed in A Story of Units.
· Understand the coherence within and across grades in order to promote conceptual understanding. / · Participants will be able to successfully work the sample algorithms and explain how the algorithms build to long division
Session Overview 9-10:30
Section / Time / Overview / Prepared Resources / Facilitator PreparationIntroduction to the Algorithms / 23 min / · Explanation of algorithms / · PowerPoint
· Facilitator’s Guide / · Review PowerPoint and Session Notes
Addition and Multiplication / 33 min / · Demonstration of use of algorithms and the development of the Numbers base Ten Progression
· Practice using the algorithms / · PowerPoint
· Facilitator’s Guide / · Review PowerPoint and Session Notes
Subtraction and Division / 31 min / · Demonstration of use of algorithms and the development of the Numbers base Ten Progression
· Practice using the algorithms / · PowerPoint
· Facilitator’s Guide / · Review PowerPoint and Session Notes
Session Roadmap
Section: Introduction to the Algorithms / Time: 23 min 9:00- 9:23[minutes] In this section, you will
· Examine the use of algorithms in preparation for using this information to teach students and colleagues / Materials used include:
·
Time / Slide # / Slide #/ Pic of Slide / Script/ Activity directions / GROUP
X / 1 / / NOTE THAT THIS SESSION IS DESIGNED TO BE 90 MINUTES IN LENGTH.
This session is the beginning of Day 2 of the NTI. Day 1 provided an overview of the curriculum, grade-level Module Focus sessions, and demonstrations of instructional models that support implementation of A Story of Units. Additional grade-level Module Focus sessions will follow this session on algorithms and the final session will address the role of principals and other school leaders.
5 min / 2 / / Put these problems up as participants come into the session.
Get started with mental math.
Take 3 minutes to work a few different mental math strategies to solve the problems on the board. We’ll only check the ones in black. If you finish early tell your neighbor how you solved.
Highlight that each solution employs the associative property as a compensation strategy.
Addition
Associative Property 298 + 357 = 298 + 2 + 355 (A compensation strategy)
Associative Property 4527 + 3219 = (45 hundreds + 32 hundreds) + (27 ones + 19 ones) (place value strategy)
Subtraction
Compensation 658- 298 = 658 – 300
Count up from $3.68 to $4 to $10 (Cashiers’ method)
Multiplication
Associative property. 5 x 248 = 5 x 2 x 124 (Compensation for multiplication)
Distributive property 25 x 34 = (32 x 25) + (2 x 25) = 8 x (4 x 25) + (2 x 25) = 850
Associative property 6 x 24 = 12 x 12 (Compensation for multiplication)
Division
1240 ÷ 5 = 2 x (1240 ÷ 10) = 248 (Compensation)
850 ÷ 25 = (800 ÷ 25) + (50 ÷ 25)
4281 ÷ 3 = 42 hundreds ÷ 3 + 81 ones ÷ 3 = 14 hundreds + 27 ones = 1427
10 seconds / 3 / / Our objectives for this session are to:
• Examine and practice the algorithms employed in A Story of Units.
• Understand the coherence within and across grades in order to promote conceptual understanding.
10 seconds / 4 / / We will start with a rationale for the session.
1 min / 5 / / Looking out over the room I see a lot of laptops, tablets, and smartphones. So I feel confident in saying that we all use algorithms every day. For example, simplistic spell checking algorithm might read in one word at a time from a document, check it against a database of “correctly spelled” words, and if it finds it, great! If not, it’s marked as a potentially misspelled word. But such an algorithm, put simply is just a step by step procedure for solving a class of problems, whether we’re talking about math or not. You feed in the data, execute the steps and out pops a result (hopefully). It could be a spell-checker, a computer game, or a guidance system on a cruise missile, there’s an algorithm guiding the process. So we don’t want to give students the idea that “use the algorithm” just means “record b.” Instead, “algorithm” is more about the process, even if we do adopt a particular way of recording it.
2 min / 6 / / In elementary school we have what we call the “standard algorithms.” The term even appears in the CCLS. These are cyclic procedures for solving any arithmetic problem. The things is, we have all sorts of algorithms to choose from. For example one could say that counting up, one at a time, is an algorithm for addition. You can certainly solve any whole number addition problem that way (and even decimal addition with some tweaking). But while this works fine for some cases: 5 + 2 (fiiiiive, 6, 7), in other cases, it’s not so great: try counting up for 437 + 875. But there are numerous, efficient algorithms. On top of that students learn all sorts of mental math strategies, so why target a specific one as “standard?”
Having a standard algorithm provides a fall-back position. When they don’t see a quick mental math option, they know they can use the algorithm (they don’t have to choose from a menu at that point). This frees them up, when reading a word problem for example, to recognize an embedded arithmetic problem for what it is, know that they have a way to solve it when the time comes to do so, and focus on the higher level relationships within the problem. In that way, the algorithm serves as a support for their ability to contextualize and decontextualize, which is the content of MP.2—they read the word problem, analyze the contextual relationships, and then strip away the context, and solve abstractly. They then reinterpret the result based on the context of the problem.
1 min / 7 / / For examples, students might say, “The answer is 5 kilograms 790 grams!”
They have decontextualized the arithmetic effectively but have not contextualized it back into the story.
The teacher asks them to make a statement answering the question.
“The potatoes and the onions weigh 5 kilograms 790 grams together.”
Now they have contextualized the problem again.
10 min / 8 / / Importance of the System of Algorithms
There are really strong relationships between the operations. Addition and subtraction are two sides of the same coin. Multiplication and division are similarly related. Multiplication can be solved by addition, and actually embeds addition within the standard algorithm. And as for long division? Think for a moment about the different operations required to carry it out. All of these conceptual and even procedural ties mean that we don’t want to think of the algorithms in isolation. They are intimately connected.
It’s really easy when you’re tasked with preparing something for a particular grade to put on your blinders and focus only on that one thing. The problem with that is that choices we make early on impact what happens later. Something as simple as how we show regrouping can either support your work in building from addition to multiplication, or…not. What we’ve done is to think about the progression of the entire system of algorithms, so that it is consistent within itself, and each algorithm builds toward the next, culminating in long division.
Importance of Long Division
Let’s talk about long division. Long division is one of those topics that a lot of teachers and students dread. Some have even gone so far as to say that long division is outdated, and not really needed anymore. After all, if students understand what division means and how to use it, then for the division problems that are computationally very difficult, why not just use a calculator? We all carry cell phones now, and can whip out a calculator at the drop of a hat. The thing is, they have a point. In fact, you could make that argument with all of the algorithms. If all you’re interested in is the ability to get the answer and apply it, then there may be some truth to that. So, I get the feeling that some think of mathematicians as being just like grumpy cat here when we say it’s important to do it. But I don’t think of myself as a grumpy mathematician. So, why do we say it’s important? There are lots of reasons. For one, the algorithms present a rich opportunity to make use of and teach the arithmetic properties. These are the things that carry on to algebra, trig, calculus, and beyond. For long division in particular there’s a really cool reason:
Once students move beyond whole number division, they use long division to relate fractions to decimal numbers. Now, that might sound a little boring at first, but what it does is open the door for fractions like 1/3 or 1/7 to be converted into a decimal number (happens by grade 7), not just the “easy” ones like 1/10 = 0.1.
Actually, that might’ve sounded boring too. So let me explain. Woven throughout the K-12 curriculum there’s a story unfolding, and the idea of what constitutes a number continues to expand as they build out the real line. They begin right away with counting (using whole numbers), they’re introduced to 0, and place value units. All of these numbers being neatly organized into a cute little number line that, incidentally, hints at something more (after all, it’s continuous…). In grade 3 students come to the realization that there are more numbers than they realized: there are fractions, and they’re on the number line too, between the whole numbers they know and love! But here’s the thing, any interval can be subdivided as many times as you want, and it appears that fractions “fill up” the number line. In a sense, they do. However, in another sense, they don’t even come close.
You see, every fraction, every one of these numbers that seems, on the surface, to fill up the number line can be converted into a decimal number. Sometimes that decimal number contains an infinite number of digits, e.g. 1/3 = 0.33333….., 1/7 = 0.142857. But no matter what, if we start with a fraction, and convert it to a decimal number, the decimal will either terminate in a finite number of digits, e.g. ¼ = 0.25, or it will have a finite period (7.NS.2d it repeats!). But what about a decimal number with an infinite number of digits, and NO period? Can it be written as a fraction of the form a/b? The answer is NO WAY! Now, to the ancient Pythagoreans, the rational numbers were IT, and they were pretty passionate about that. According to one legend, Hippasus was at sea with some fellow Pythagoreans, when he discovered the existence of irrational numbers, and when he informed his shipmates, they had him thrown overboard! Growing up in Texas people had bumper stickers that said “Don’t mess with Texas.” I like to think these guys had the same mentality, “Don’t mess with rational numbers”—they weren’t about to accept that there was anything else out there.
Thus students see that there are irrational numbers and this allows them to finally fill in the entire number line for the first time in their lives, and it’s division that opens that door. This is where numbers like Pi, e, and most square roots live.
There are other reasons of course. Division represents the first time students see a truly cyclic algorithm that may not always terminate (e.g. 1 divided by 3) and they have to come to grips with that. The same ideas show up when students work with polynomials in algebra, and partial fractions in calculus, and without them, high school and college level mathematics can’t be fully realized. The concepts are not limited to just doing division. They continue to show up in mathematics, computer science, and science in general.