Session 4: Final Algorithms in A Story of UnitsSequence 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 Preparation

Introduction 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:

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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)SubtractionCompensation 658- 298 = 658 – 300Count up from $3.68 to $4 to $10 (Cashiers’ method)MultiplicationAssociative 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 AlgorithmsThere 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 DivisionLet’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.

3 min 9 Last NTI we talked about the wall between elementary school and the middle/high schools. Getting them started in the world of real numbers is what we are doing in kindergarten through fifth grade. Without long division, the complete number system stays closed.

Section: Addition and Multiplication Time: 33 min 9:23-9:56

[minutes] In this section, you will Practice using the addition and multiplication algorithms in

preparation for redelivering this matetrial

Materials used include: Review PowerPoint and Session Notes

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1 min 10 • We are starting with addition and multiplication in which parts join to make a whole. Smaller units are composed into larger units.

• We follow with subtraction and division in which the whole is separated into parts. Larger units are decomposed.

2 min 11 8 and 4. The most basic algorithm is to count fingers. (Demonstrate) There are 12 in all.As math advances, students find short cuts and are also taught valuable short cuts such as this process, begun in first grade, of completing a unit. Students have a process, a strategy, an algorithm. Decompose one addend to complete a unit of ten.Two components, one a thought process and the other a way of recording.Counting to ten and then counting four more on the fingers is the process and recording system for the demonstrated “counting all.”The completion of the unit is the thought process and the two equations are the recording system for the “make ten” strategy.“Make ten’ is an algorithm in Grade 1 just as ‘counting all while tracking on fingers ‘is an algorithm, though highly inefficient, in kindergarten.This G1 algorithm sets the foundation for the addition algorithm by making 10 and adding on the remainder.

2 min 12 The addition algorithm from Grade 1 moves forward into the standard algorithm for addition in Grade 2. Each addend is broken down, decomposed by place value just as the 4 was decomposed in Grade 1.Just as we saw the ‘4’ decomposed in 8 + 4, we now see the addends decomposed so that we add like units.Cycle 1:Add the ones.Change 10 ones for 1 ten. Cycle 2:Add the tens.Standard algorithm, a place value strategy, is taught in grade 2, though we don’t have to call it the standard algorithm. We started the presentation with mental math because we want anyone doing mathematics to carefully choose their strategies. Mental math is one, the algorithm is another when no obvious mental strategy is evident.

Model on the place value chart how you add 24 + 58. We’re going to add like units. Make a 10 strategy on place value chart, then record it on the expression. Explain that new units are recorded below because within the system of algorithms, for multi digit multiplication, without this recording, new units are put in the incorrect place as will be evidenced in slide 24.

2 min 13 Units change in Grades 4 and 5 as can be seen in the slide. We want to see the relationship of the different units and the relevance of their earlier work. 24 has mixed units (2 tens 4 ones) just as does 2 feet 4 inches or 3 boys and 2 girls.Walk through the language of units with the tenths.2.4 + 5.8The presenter engages teachers in place value language while solving the decimal addition problem.Cycle 1T: 4 tenths + 8 tenths is?P: 12 tenths.T: Can I make a larger unit?P: Yes.T: What unit?T: A oneCycle 2T: 2 ones + 5 ones + 1 one is?S: 8 ones.T: The answer is?S: 8 and 2tenths.

2 min 14 Give participants time to practice. Refer back to slide 12 to remind them they have the language of units from the student’s earlier example and can use it to support their words.Remind to use the language of units when referring to the ones. This is the most neglected place value.

2 min 15 The focus of G3 work with multiplication is developing the ability to treat any number as a unit, and hence, counted.Give each finger a value of 6 and count sixes on the fingers. 1 six, 2 sixes…. Early on we give them the count by process or repeated addition process which at some point becomes the fact they don’t have to think about any more.Now we’re going to focus on multiplication. Addition is the mother of all the operations. Beginning in G3 we start with the distributive property, which really moves the addition into the realm of multiplication. 2 sixes and 3 sixes is 5 sixes. That’s the distributive property.

2 min 16 By adding the units you can very clearly illustrate the distributive property.Just as 2 apples + 3 apples is 5 apples, so too is 2 sixes + 3 sixes equal to 5 sixes. We have 5 units of six that we decompose into 2 sixes and 3 sixes. We have decomposed the number of groups. We also on the right have 5 units of six but this time we have decomposed the groups of six into 5 fives and 1 five. We can also think of the decompositions as that on the left being the decomposition of the number of rows or the width; that on the right being the decomposition of the number of columns or length.

3 min 17 How does this play out in G4?Show the 6 units of twenty four on the place value chart. Relate this to slide before with the arrays. Here you see the unit of 24, but you see the units partitioned in Grade 4 (tens on one side, ones on the other.) We multiply the unit twenty four six times. We compose 2 units. We can change 20 ones for 2 tens (does this look familiar from the addition algorithm?) Show this on the place value chart. Relate the multiplication to repeated addition. Students see that the composition is the same, and is recorded the same. This is a cyclic algorithm. Cycle 16 times 4 ones is 24 ones, 2 tens 4 ones.Cycle 26 times 2 tens is 12 tens, 1 hundred 2 tens (or 12 tens since it is the final place value.Same with addition. Changing smaller units for larger units with addition.Point out that the ‘2’ is the same regrouping that was done in addition.Alternate way of saying:This is six units of 24.6 twenty fours.;We use place value to break that down.6 x 4 ones + 6 x 2 tens.Developing the standard algorithm from the partial products.

1 min 18 Use this as an exemplar for the language of units.

2 min 19 Alternate algorithms (array model shown now as an area model). The array transitions into the area model. Explain that the 24 has been partitioned, and relates to the distributive property. We want students to be able to manipulate numbers in unit from as well as in number form. This flows right out of Grade 3 distributive property.The partial products are listed from largest to smallest in the algorithm to the left, assuming that is the way the person moved through the area model, starting with the largest units and moving to the smallest. The unit form is used on the right in the order of the standard algorithm, starting by multiplying the ones and then multiplying the tens.

3 min coupled with next slide

20 30 units of 24.Show the partitioned area model and explain how the model shows the equation. 3 tens, or thirty, times 2 tens, or twenty equals 6 hundred.10 x 10 is 100 (you get a new unit), if you do that 6 times you’ve got 600. Translate into a much simpler model (next slide). We want ours students to understand how the units are behaving. A length times a length gives a square unit. 1 cm x 1 cm = 1 cm squared. 1 ten x 1 ten = 1 hundred square units. We largely omit the language of square units until returning to word problem situations involving actual measurements. Otherwise, the language is too complex.Notice, that in anticipation of the algorithm, just like on the place value chart, we are starting with the ones and then multiplying the tens.

X 21 This is the simplified version of the prior slide. Point out that when recording the “one” should not go in the tens place but in the hundreds place (Relates to the recording of the standard algorithm in slide 24.)

1 min 22 Use this slide to support practice with the language of units.

1 min 23 30 x 24 = 3 x (10 x 24)Understand what’s going on the place value chart. Explains the model slowly step by step. Show the 24 shifting one place to the left when being multiplied by 10. Once the unit has shifted over once more, it’s triple.30 x 24 = 10 x (3 x 24)We start by tripling the unit of twenty four.The entire product then shifts one place value to the left as it is multiplied by 10.

2 min 24 36 units of twenty-four.Here the only difference is that instead of 30 twenty fours we have 36 twenty fours. We’re putting them together. There’s no new mathematical complexity. The two partial products have been modeled previously, 6 x 24 and 30 x 24.Explain the connections between the model, the arithmetic, and the distributive property as necessary. Point out that the regrouped unit, the product of 3 tens and 2 tens is correctly recorded in the hundreds place in super script above the second partial product. In the “traditional” algorithm, this would have been incorrectly recorded above the “2” of 24 in the tens place.

2 min 25 How many partial products now? 4. Students see the product that matches each rectangle/part on the area model. Relate the partial products to the both algorithms. Return to slide 24 if necessary to clarify the relationship of the standard algorithm to the right, to the area model.

1 min 26 How does this relate to Grade 5 with decimals?If students are multiplying 2.4 times 36, then don’t have to think of it as 2 and 4 tenths. They can think of it as 24 tenths, multiply the whole numbers of tenth units and then reinterpret the answer in standard form with decimal notation.

4 min 27

Section: Subtraction and Division Time: 31 min 9:56-10:30

[minutes] In this section, you will .. Practice using subtraction and division algorithms in

preparation for redelivery of this training

Materials used include: Session PowerPoint Session Notes

1 min 28 Now we’re going to move into algorithms that go from whole to part, subtraction and division. Decomposing, changing larger units into smaller units, renaming, unbundling.

2 min 29 Walk the participants through the slide and how it relates to what the students are thinking and recording.We could use the really old fashioned way of counting down, but we’re trying to teach more efficient processes for subtraction. Show how to decompose 12 to subtract 12 – 8. Break 12 into 10 and 2. Can’t take 8 from 2 ones. Take it from the ten. That leaves me 2 and 2 more. This simple algorithm, strategy, process, pays off in Grade 2 as Grade 1 students become accustomed to analyzing if there are enough ones to subtract the ones or is it better to go to the tens.

2 min 30 Walk through the slide step by step.Notice that we model only the whole amount, the total, on the place value chart.In grade 2 we again break the numbers apart by units on the place value chart to subtract. We want to extract the 24 that’s within 82 and find out what’s left. If I take the part, 24, out of the whole, 82, we find the missing part. Do I have enough ones or not? Take 1 ten and change it for 10 ones. Why use the magnifying glass? Helps us avoid the misconception of saying 4 – 2. Gets students to focus on the whole number rather than inverting the ones. A cyclic operation. Systematically say do I have enough ones to subtract? Tens? Change all units until the entire total or whole, is ready for subtraction. The magnifying glass is a mnemonic.

1 min 31 This is the language of units that we want to hear the students speaking as they learn the algorithms. The ones are the most neglected. Identify those ones!

3 min 32 Walk through the slide.6 hundreds, no tens and ones. Go all the way to the hundreds to get some units of one. Two methods are shown: Method 1(Place value chart in the middle)The place value chart models decomposing 1 hundred for 10 tens and then decomposing 1 ten for 10 ones. Method 2 (Place value chart to the far right)Models decomposing 1 hundred as 9 tens and 10 ones. Much cleaner and avoids the mistakes that come in stacks of regrouped numbers.

1 min 33 An exemplar of the language of units.

3 min 34 Circulate and support teachers in using their words.

1 min 35 The focus of G3 is the study of basic division facts as relates to multiplication without remainders.Long division. I have heard so many Grade 5 teachers saying that long division is obsolete. You might reiterate that long division matters. When we’re looking at 6s one more time, we can count the number of units of 6 in 24. How many sixes in 24? 4. That’s the measurement way of looking at division. We’re not partitioning. We’re accustomed to thinking about sharing! We can interpret the divisor to be telling us the size of the unit rather than the number of units.

1 min 36 Analyze the number bonds relationship to the array.This is not a situation we would readily use the distributive property to subtract since most everyone knows that 30 ÷ 6 is 5. However, 56 ÷ 8 might be since 56 decomposes nicely into 40 and 16, or 5 eights and 2 eights.The point is that like with multiplication, where we find partial products, we can find partial quotients and add them up.

3 min 37 Like subtraction, starting with the whole amount.Unlike all the other standard algorithms, we are distributing the largest units first, moving from left to right on the place value chart.. Are students are going to distribute the largest bills first?Cycle 14 tens distributed into 3 groups, One ten in each group.I made 3 groups of 1 ten.One ten is remaining.I can change 1 ten for 10 ones.Ten ones + 2 ones is 12 ones. Cycle 2

12 ones distributed into 3 groups,Four ones in each group.I made 3 groups of 4 ones.No ones are remaining.My quotient is 14.Notice the notation that is analogous in the third place value chart of the 3 tens distributed with the 12 ones to be distributed.

2 min 38 What is the largest area I can make that is easy to divide by 3? We make a rectangle with an area of 30 and a width of 3. The side length is unknown. Just like we had previously partial products, we now have a partial quotient. Show how to record that information on the algorithm. I have an area of 12 remaining and I want to find its unknown length. Show how this relates to the area model in multiplication.

2 min 39 Give participants the opportunity to go through the thinking of each of the children as they solve 420 divided by 30.(The reason 42 is not subdivided in the place value chart is because it was done on the prior pages.)

3 min 40 You can see immediately that this is related to 42 divided by 3. Always round the divisor first therefore round 29 to 30. Cycle 1T: Estimate to find 42 hundreds divided by 30. (Show first estimate.)S: 1 hundred. T: 1 hundred times 29?S: 29 hundreds (Record and find the remainder in the hundreds place, image #2)T: How many hundreds remaining? S: 13 hundreds.T: Decompose those units into tens and add the tens from the whole.S: 138 tens.Cycle 2. T: Estimate to find 138 tens divided by 30. (Show second estimate.)S: 4 tens. T: 4 tens times 29?S: 116 tens (Record and find the remainder in the tens place, image #3)T: How many tens remaining?S: 22 tens.T: Decompose those units into ones and add the ones from the whole.S: 227 ones.Cycle 3.T: Estimate to find 227 ones divided by 30. (Show first estimate.)S: 7 ones.T: 7 ones times 29?S: 203 ones (Record and find the remainder in the ones place, image #4)T: How many ones remaining? S: 24 ones.T: Our remainder is 24 ones.Note: In G5, we could certainly continue into the decimal place

values. Cycling through the process. We know from the overview that because this is a rational number, the decimal expansion will either terminate or result in a repeating decimal.This one really shows how the operations and the algorithms tie together.

5 min 41

2 min 42 Take 2 minutes to turn and talk with others at your table. Share your observations and ask them to do the same.

1 min 43 Read the key points

Use the following icons in the script to indicate different learning modes.

Video Reflect on a prompt Active learning Turn and talk

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