unit 1 – patterns and problem web viewpatterns and the real number line. ... they know that...

Patterns and the Real Number LineAligned to the Common Core Standards

Written by Dr. Jonathan Katz ISA Mathematics Coach

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Dear Math Teacher,

What is mathematics and why do we teach it? This question drives the work of the math coaches at ISA. We love mathematics and want students to have the opportunity to begin to have a similar emotion. We hope this unit will bring some new excitement to students.

This unit was originally written three years ago and has gone through several iterations. It is a unit that revisits concepts and procedures students experienced in middle school but with an expectation that students will leave with deeper understanding. This unit has been redesigned to align with the Common Core Standards. Essential to this work is an inquiry approach to teaching mathematics where students are given multiple opportunities to reason, discover and create. Problem solving is the catalyst to the inquiry process so as you look closely at this unit you will see students constantly placed into problem solving situations where they are asked to think for themselves and with their classmates.

The first four Common Core Standards of Practice are central to this unit. Through the constant use of problematic situations students are being asked to develop perseverance and independent thought, to reason abstractly and quantitatively, and to critique the reasoning of others.

Mathematical modeling is present throughout the unit as students are asked to describe and analyze different bare number problems and real world situations and represent them mathematically. Students are also asked to create models including the final project, which is to create a real number line with precise placement of different real numbers and understanding of different number systems.

The other four Standards of Practice are also present in this unit. Two of them are central to the inquiry approach. You will see these two statements in the last two standards.

Mathematically proficient students look closely to discern a pattern or structure. Mathematically proficient students notice if calculations are repeated, and look

both for general methods and for shortcuts.

We believe, as do many mathematicians, that mathematics is the science of patterns. This underlying principle is present in all the work we do with teachers and studentsIn this unit you will see that students are often asked to discern a pattern within a particular situation. This leads students to making conjectures and possibly generalizations that are both conceptual and procedural.

Thank you for looking at this unit and we welcome feedback and comments.

Sincerely,

Dr. Jonathan Katz(For the ISA math coaches)

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Unit 1 – Patterns and the Real Number Line

Essential Questions: Why do mathematicians describe mathematics as the science of patterns? How are different types of numbers related?

Interim Assessments/Performance TasksWhere do I place these numbers? – Lesson 1Where do I place these fractions? – Lesson 2What is the 100th digit in the decimal expansion of 1/7? – Lesson 3Critiquing the Thinking of Others – Lesson 4Is Juanita Correct? – Lesson 5The Locker Problem – Lesson 6Can I use patterns to describe multiplication of signed numbers? – Lesson 9

Final Assessment: Creating a Real Number Line

What will students be able to do at the end of the unit? Students will be able to compare and contrast rational and irrational

numbers (all real numbers). Students will understand where different numbers are located on the

number line. Students will be able to talk about the relationship between the finite and the

infinite within number systems. (How many numbers are there between any two integers?)

Students will be able to describe the patterns within decimals and fractions. (e.g. Can you figure out the 100th digit in the decimal expansion of 1/7?)

Students will understand how the discovery of patterns can help us make conjectures.

Students will understand how to compute with signed numbers and why the approach they use makes sense.

What enduring understanding will students have?

There is an infinite amount of numbers between any two numbers and they include both rational and irrational numbers.

Rulers based on different fractional units are the same and different. Numbers can be thought about in different ways with different number

systems describing different sets of numbers. Since patterns are the structure of mathematics, pattern recognition is

essential for understanding rational and irrational numbers in all their forms.

Common Core Content Standards in the Unit

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8.NS.1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion, which repeats eventually into a rational number.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., π2). 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.

Common Core Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

1. Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving

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complex problems and identify correspondences between different approaches.

2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

3. Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

4. Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results

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in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

5. Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

6. Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

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.

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8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

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Patterns and the Real Number Line UnitLesson 1

Teacher GuideNumerical Pattern Introduction

(To the Teacher: Since mathematical problem solving requires strong observational skills and having the ability to be a pattern hunter you might begin the class looking at a clip from a Sherlock Holmes film showing the power of observation in solving problems. A question you might ask is, “How is the way Sherlock Holmes thinks related to what we do in mathematics?”)

Opening ActivityIn your groups, study the patterns in the following sequences and write the next three terms. Explain how you came up with your responses.

2, 4, 3, 5, 4, 6, 5, 7, 6,…

2000, 200, 20, 2, …

116 ,

18 ,

14

12 ,…

(Note to Teacher: It’s important that students get a chance to explain their thinking, even if their reasoning is wrong so that: 1) you can learn about how they think. 2) see how you can begin to use errors to develop greater inquiry in the class. You need to be prepared to let the students’ ideas and questions lead the class.)

(Note to teacher: Give out a number line with integers from -10 to 10. This line should have space for adding fractions and decimals today.)

Second ActivityLook at the number line and discuss the following four questions in your group. Be prepared to share your ideas with the rest of the class.

1. What different types of numbers can you represent on this number line?

2. How many different numbers are there between zero and one? Give examples to explain your thinking.

3. Where would you place a number like 249 ? Why?

4. Which number is bigger √35 or 6.2? Give evidence for your thinking.

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5. How are fractions and decimals related to each other?

[To the Teacher: List on chart paper all the ideas that students come up with. (Post on the wall for future reference.) This will lead to the content of everything else discussed in this unit. Students might share a false idea. These ideas can and should be argued about in the coming days. Big ideas to think about in this lesson and the rest of the unit:

1) In a finite space, there are an infinite amount of numbers.

2) Between any two real numbers (no matter how close they are to each other on the number line) there are an infinite amount of numbers.

3) All rational numbers can be represented in an infinite amount of ways including as decimals and fractions.

4) The decimal expansion of all rational fractions either terminates or is infinite with a predictable repeating pattern.

5) Irrational numbers have a decimal expansion that is infinite with no pattern and therefore not predictable. Mathematicians have been fascinated with these numbers including pi.

6) Equal quantities exist in the same location on the number line.

7) A number is either positive or negative except for zero.

Performance Task: Where do I place these numbers?

Place these numbers on the number line as close as possible to where you think they may belong. Use the number line handed to you by the teacher. Explain why you placed them where you did.

23 , 1.8,

−229 , and √29

(To the Teacher: Look over this student work to assess their understanding of different types of numbers and their place on a number line. This should inform your work for the upcoming lessons. You can treat the performance task like a diagnostic. It should inform the rest of the work in this unit.)

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Patterns and the Real Number Line UnitStudent Activity Sheet

Lesson 1Name_______________________Date________________________

Numerical Pattern Introduction

Opening ActivityIn your groups, study the patterns in the following sequences and write the next three terms. Explain how you came up with your responses.

2, 4, 3, 5, 4, 6, 5, 7, 6,…

2000, 200, 20, 2,…

116 , 1

8 , 14 1

2,…

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Second ActivityYou will be given a number line with integers from -10 to 10. This line will have space for adding fractions and decimals today.

Look at the number line and discuss the following four questions in your group. Be prepared to share your ideas with the rest of the class.

1. What different types of numbers can you represent on this number line?

2. How many different numbers are there between zero and one? Give examples to explain your thinking.

3. Where would you place a number like 249 ? Why?

4. Which number is bigger √35or 6.2? Give evidence for your thinking.

5. How are fractions and decimals related to each other?

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Performance Task: Where do I place these numbers?

Place these numbers on your number line as close as possible to where you think they may belong. Use the number line handed to you by the teacher. Explain why you placed them where you did.

23 , 1.8,

−229 , and √29

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Teacher GuideHow Do I Make Sense of a Ruler?

Opening Activity: What do I understand about fractions?Look at the picture below. Write down all your observations and questions. Include in your observations ideas about size (larger versus smaller), equivalence (fractions that have the same value), and anything else you find interesting.

(To the Teacher: Many ideas can come out of this observation activity. Be ready to talk

about ideas represented visually, such as 12=2

4=3

6 or 111

< 19 or the meaning of 1.)

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Second Activity:

[To the Teacher: Distribute four different rulers of the same length marked (without units) but broken up into fourths, eighths, twelfths, and sixteenths.]

Look closely at the four rulers and discuss these questions in your group.

What is the same?What is different?How do I read points on these rulers?(To the Teacher: Have students share their thoughts. Make sure at the end of this discussion that all students know how to read each ruler. Let them come up with methods to do this).

Third Activity:

Locate 12 on all of the rulers. Describe each point numerically on the four rulers and

compare them to each other.

(To the Teacher: Have a discussion on how students came up with a ½ on each of the rulers with a focus on the notion of equivalence.

Locate 34 on each of the rulers. Describe each point numerically on the four rulers and


How many different numbers can you place at any point on the number line? Explain your thinking.

Performance Task: Where do I place these fractions?

Create a number line from 0 to 1 and make sure it is broken into twelfths. After you have done that place with as much precision as you can the following two numbers:

56 and

68

Explain why you placed them where you did.

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Homework: Part 1 of the ProjectIn this unit we are developing a deeper understanding of the number line and the different types of numbers that we can place on the number line. Today you are going to start the project and work on it throughout the unit. You are going to be given a piece of oak tag that you will use to create a number line. You can divide it up in any way you want. Over the next couple of weeks you will be adding numbers to this number line.

Along with the number line you are going to produce an Explanation Document. You will use this document to explain the work you did in each part of the project and answer different questions. You will have special title for each section of the document. The title for this section is How Do I Make Sense of a Ruler.

Today you will do the first phase of the project which is to locate a set of given fractions on your created number line. In the Explanation Document explain why you placed the numbers where you did.

Here they are: 14

56 1

49

3550

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Lesson 2Name_______________________Date________________________

How Do I Make Sense of a Ruler?

Opening Activity: What do I understand about fractions?

Look at the picture below. Write down all your observations and questions. Include in your observations ideas about size (larger versus smaller), equivalence (fractions that have the same value), and anything else you find interesting.

Second Activity

Second Activity:

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You are going to be given four different rulers…Look closely at the four rulers and discuss these questions in your group.

What is the same?What is different?How do I read points on these rulers?

Be ready to discuss your ideas with the rest of the class

Third Activity:

Locate 12 on all of the rulers. Describe each point numerically on the four rulers and


Locate 34 on each of the rulers. Describe each point numerically on the four rulers and


How many different numbers can you place at any point on the number line? Explain your thinking.

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Performance Task: Where do I place these fractions?

Create a number line from 0 to 1 and make sure it is broken into twelfths. After you have done that place with as much precision as you can the following two numbers: 5

6 and 68

Explain why you placed them where you did.

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Homework: Part 1 of the Project

In this unit we are developing a deeper understanding of the number line and the different types of numbers that we can place on the number line. Today you are going to start the project and work on it throughout the unit. You are going to be given a piece of oak tag that you will use to create a number line. You can divide it up in any way you want. Over the next couple of weeks you will be adding numbers to this number line.

Along with the number line you are going to produce an Explanation Document. You will use this document to explain the work you did in each part of the project and answer different questions. You will have special title for each section of the document. The title for this section is How Do I Make Sense of a Ruler.

Today you will do the first phase of the project which is to locate a set of given fractions on your created number line. In the Explanation Document explain why you placed the numbers where you did.

Here they are: 14

56 1

49

3550

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Teacher GuideDecimals, Decimals, Decimals

Opening ActivityWhat do these fractions look like as decimals?

12

34

23

912

You may use a calculator but first try to do it using pencil and paper. Explain how you got your answers. Why did you do what you did? Compare these different fractions and decimals. What do you notice?

[To the Teacher: You want students to differentiate between terminating and repeating decimals. (You might want to help students realize that these are the same by noting that a terminating decimal can be thought of as an infinitely repeating decimal where the repeated part is all zeros.) Be sure to show students that they indicate repeating infinite decimals by putting a bar above the first series that repeats. As you know the calculator

can cause a problem since it rounds the last digit. So 23 will be written as 0.6666667.

This could make for a valuable discussion. This issue will come up in the next activity. For the next activity you can have each group do all five sets or have them do just one set and have them place their results on chart paper which you would place on the board. Each group can now make observations of all five sets and begin to discuss the different patterns and answer the questions.]

Second Activity You will be given a calculator and asked to change one or more of the following sets of fractions into decimals.

Set 1: 17

, 27

, 37

, 47

, 57

, 67

Set 2: 19

, 29

, 39

, 49

, 59

, 69

, 79

, 89

Set 3: 111

, 211

, 311

, 411

, 511

, 611

, 711

, 811

, 911

, 1011

Set 4: 16

, 26

, 36

, 46

, 56

,

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Set 5: 18

, 28

, 38

, 48

, 58

, 68

, 78

When your group has the results from all 5 sets, you should look them over and think about the following questions:What patterns do you notice?

Which sets seem to belong together? Why?

What do you notice about the infinite and the finite?

Third Activity:

This will be a challenge…Take your time. Try to use both paper and pencil and the calculator to find the decimals

Can you find a pattern for the fractions with a denominator of 14? Be ready to describe what you have found? How is related to the pattern when the denominator is 7?

Performance Task

What is the 100th digit in the decimal expansion of 17? Show all work and give evidence

for your solution.

Homework: Part 2 of the Project:In this part of the project you will have four things to do. You will place a new set of numbers on the number line based on today’s lesson. Then you will have three questions to ponder. All the work you do must be placed in your Explanation Document with the title, Decimals, Decimals, Decimals.

1) On your created number line, you are going to be given fractions which you place on the number line in decimal form. If they’re repeating decimals, they must be in blue, if terminating they must be in red. Show all your work and explain why you placed the decimal in a particular location.

35 1

23

712

439

2) Construct a number between 1 and 2 using all the digits from 0 to 9 only once, which is as close to 1.5 as possible. Why do you think this is the closest possible number to 1.5? Now place this number on the number line.

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3) If you were free to use the digits as many times as you want construct a number that you think would be even closer to 1.5 but not equal to 1.5. Explain your thinking. Place this number on the number line.

4) Now write both of these number as fractions. Explain what you did.

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Lesson 3Name_______________________Date________________________

Decimals, Decimals, Decimals

Opening Activity

What do these fractions look like as decimals?

12

34

23

912

You may use a calculator but first try to do it using pen and paper. Explain how you got your answers. Why did you do what you did? Compare these different fractions and decimals. What do you notice?

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Second Activity

You will be given a calculator and asked to change one or more of the following sets of fractions into decimals.

Set 1: 17

, 27

, 37

, 47

, 57

, 67

Set 2: 19

, 29

, 39

, 49

, 59

, 69

, 79

, 89

Set 3: 111

, 211

, 311

, 411

, 511

, 611

, 711

, 811

, 911

, 1011

Set 4: 16

, 26

, 36

, 46

, 56

,

Set 5: 18

, 28

, 38

, 48

, 58

, 68

, 78

When your group has the results from all 5 sets, you should look them over and think about the following questions:What patterns do you notice?

Which sets seem to belong together? Why?

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What do you notice about the infinite and the finite?

Third Activity:

This will be a challenge…Take your time. Try to use both paper and pencil and the calculator to find the decimals

Can you find a pattern for the fractions with a denominator of 14? Be ready to describe what you have found? How is related to the pattern when the denominator is 7?

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Performance Task

What is the 100th digit in the decimal expansion of 17?

Show all work and give evidence for your solution.

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Homework: Part 2 of the Project:

In this part of the project you will have four things to do. You will place a new set of numbers on the number line based on today’s lesson. Then you will have three questions to ponder. All the work you do must be placed in your Explanation Document with the title, Decimals, Decimals, Decimals.

1) On your created number line, you are going to be given fractions which you place on the number line in decimal form. If they’re repeating decimals, they must be in blue, if terminating they must be in red. Show all your work and explain why you placed the decimal in a particular location.

35 1

23

712

439

2) Construct a number between 1 and 2 using all the digits from 0 to 9 only once, which is as close to 1.5 as possible. Why do you think this is the closest possible number to 1.5? Now place this number on the number line.

3) If you were free to use the digits as many times as you want construct a number that you think would be even closer to 1.5 but not equal to 1.5. Explain your thinking. Place this number on the number line.

4) Now write both of these number as fractions. Explain what you did.

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Teacher GuidePoints Between Points

Group Opening Activity:

How many numbers are there between 13 and

12? How do you know? Give Examples.

[To the Teacher: For the interval between any two points, as small as it may be, there will be an infinite amount of points within the interval. There will be many misconceptions that you will have to help students resolve by supporting them to think

differently. Many students might think that there are no numbers between 13 and

12 . What

question might you ask them to have them rethink that notion?

Two ways that we might help students think about this question are:

1) Find the equivalent fractions of 13 and

12 with denominators that are multiples,

(e.g. since 13= 4

12 and 12= 6

12 then 512would fall in between. If

13= 8

24 and 12=12

24

then 924

,

1024

∧11

24would fall in between. This can be repeated indefinitely.)

2) Change each fraction to a decimal and find some of the fractions that fall in

between. (e.g. since 13 = .333… and

12= .500

so .34,.35,.36, .365, .37, .37521453765, etc. all fall in between.)]

Second ActivityIn your groups, find at least two points that lie on the number line between the given points:

14 and

58

16 and

26

316 and

14

Performance Task: The Midpoint Dilemma

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Jose says that the midpoint between 14 and

18 is

16 . Is he correct? Justify your answer.

Note: If he is incorrect, find the true midpoint and explain your thinking.

Can you come up with a strategy for finding the midpoint between any two fractions?

Homework: Part 3 of the ProjectIn this part of the project you will show your understanding of how many numbers there are between any two numbers. Remember you are writing about your thinking on the Explanation Document. Title this section, Points between Points.

1) Pick any two fractions that haven’t been discussed before. Locate them on your created number line with one color. In a different color, locate the midpoint. Also locate two other points that are between the two fractions

2) Now take the midpoint you found and one of the original fractions you chose and find the midpoint of these two points and one other point. Now place these two new points on the number line.

3) If we continued with this process could we keep finding more points, including the midpoint? Explain in your own word the answer to the question:

How many points are there between any two points?

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Lesson 4Name_______________________Date________________________

Points between Points

Group Opening Activity:

How many numbers are there between 13 and

12? How do you know? Give examples.

Be ready to defend your answer.

Activity 2

In your groups, find at least two points that lie on the number line between the given points:

14 and

58

16 and

26

316 and

14

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Performance Task: The Midpoint Dilemma

Jose says that the midpoint between 14 and 18 is 1

6 . Is he correct? Justify your answer. Note: If he is incorrect, find the true midpoint and explain your thinking.

Can you come up with a strategy for finding the midpoint between any two fractions?

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In this part of the project you will show your understanding of how many numbers there are between any two numbers. Remember you are writing about your thinking on the Explanation Document. Title this section, Points between Points.

1) Pick any two fractions that haven’t been discussed before. Locate them on your created number line with one color. In a different color, locate the midpoint. Also locate two other points that are between the two fractions

2) Now take the midpoint you found and one of the original fractions you chose and find the midpoint of these two points and one other point. Now place these two new points on the number line.

3) If we continued with this process could we keep finding more points, including the midpoint? Explain in your own word the answer to the question:

How many points are there between any two points?

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Teacher Guide

Adding Fractions

First Activity

Juanita wrote the following on the board:

12 +

13 =

25

Trevor says that this is not true. Who is right? Give evidence to support your opinion.(To the Teacher: If students don’t remember how to add fractions you might recommend

to them to convert the fractions to decimals and show that 25 falls between

12∧1

3 . So

the answer is not plausible. You can build on the previous lesson when they thought about points between points to help students think about how to add fractions.)

Second ActivityYou are going to walk to the door. In your first move you walk half way to the door. In your second move you walk half way from your new spot. In your third move you walk half way from the newest spot.

How many moves will it take you to reach the door if you keep repeating the process of walking half way?(To the Teacher: You are going to have students think about this problem which does not seem to be a problem about summing fractions. Have students discuss/argue/defend their thinking before proceeding.)

Third ActivityThink about the following questions. How are they related to the fraction sums below?

After your first step, how much of the distance to the door have you walked?After your second step, how much of the distance to the door have you walked?After your third step, how much of the distance to the door have you walked?After your fourth step, how much of the distance to the door have you walked?After your fifth step, how much of the distance to the door have you walked?

Write down the sums1) 12+ 1

42) 12+ 1

4+ 1

8

33

3) 12+ 1

4+ 1

8+ 1

16

4)12+ 1

4+ 1

8+ 1

16 + 1

32

Look at all your results and write down all your observations.

Describe the patterns you have observed.

What can the pattern of the sums tell us about the finite and infinite?

Can you predict the sum of: 12+ 1

4+ 1

8+ 1

16 + 1

32 + 1

64 ? Explain your thinking.

With your new understandings answer the question again. How many steps will it take you to reach the door?

(To the Teacher: Notice that sums are approaching 1 but will never reach it. Also, that the sum has the same denominator as the last fraction being added with a numerator that is one less. For a series of n terms, the sum is (2n – 1)/ 2 n Thus we will never reach the door.)

34


In this part of the project you will show your understanding of adding fractions. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Adding Fractions.

1) Kasim bought a pizza. His friend Jose wanted 38 of the pizza. Kathy wanted

wanted 14 of the pizza and Kasim wanted

12 of the pizza. Did he buy enough

pizza? Show your work and place the amount of pizza Kasim needed to buy on the number line.

2) CHALLENGE: Find the sums of each of the following

13+ 1

9=¿

13+ 1

9+ 1

27=¿

13+ 1

9+ 1

27+ 1

81=¿

Describe the patterns that you see in the result

Can you predict what will be the result for

13+ 1

9+ 1

27+ 1

81+ 1

243=¿

35


Lesson 5Name_______________________Date________________________

Adding Fractions

Group Performance Task: Is Juanita Correct?

Juanita wrote the following on the board:

12 + 1

3 = 25

Trevor says that this is not true. Who is right? Give evidence to support your opinion.

36

Second ActivityYou are going to walk to the door. In your first move you walk half way to the door. In your second move you walk half way from your new spot. In your third move you walk half way from the newest spot.

How many moves will it take you to reach the door if you keep repeating the process of walking half way?

Be ready to discuss your reasoning

37

Third ActivityThink about the following questions. How are they related to the fraction sums below?

After your first step, how much of the distance to the door have you walked?After your second step, how much of the distance to the door have you walked?After your third step, how much of the distance to the door have you walked?After your fourth step, how much of the distance to the door have you walked?After your fifth step, how much of the distance to the door have you walked?

Write down the sums5) 12+ 1

4

6) 12+ 1

4+ 1

8

7) 12+ 1

4+ 1

8+ 1

16

8)12+ 1

4+ 1

8+ 1

16 + 1

32

Look at all your results and write down all your observations.

Describe the patterns you have observed.

What can the pattern of the sums tell us about the finite and infinite?

Can you predict the sum of: 12+ 1

4+ 1

8+ 1

16 + 1

32 + 1

64 ? Explain your thinking.

38

With your new understandings answer the question again. How many steps will it take you to reach the door?

Journal Writing: Describe in your words what you learned from today’s activity. Is it surprising? Why?

39


In this part of the project you will show your understanding of adding fractions. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Adding Fractions.

3) Kasim bought a pizza. His friend Jose wanted 38 of the pizza. Kathy wanted

wanted 14 of the pizza and Kasim wanted

12 of the pizza. Did he buy enough

pizza? Show your work and place the amount of pizza Kasim needed to buy on the number line.

4) CHALLENGE: Find the sum of each of the following

13+ 1

9=¿

13+ 1

9+ 1

27=¿

13+ 1

9+ 1

27+ 1

81=¿

Describe the patterns that you see in the result

Can you predict what will be the result for

13+ 1

9+ 1

27+ 1

81+ 1

243=¿

40


Teacher GuideRadicals

Students will work on the following problem in groups of two or three.

The Locker Problem

There are 1000 lockers lined up numbered 1 to 1000 and 1000 students. The lockers are all closed. The first student, Jasmine, walks by and opens all the lockers. Then the second student, Al, walks by and goes to every second locker starting at #2 and closes it. Then Mary walks by and goes to every third locker starting at #3 closing the opened lockers and opening the closed lockers. The 4th student walks by and goes to every fourth locker starting at #4 closing the opened lockers and opening the closed lockers. This routine goes on until student 1000, Elvina, goes to locker #1000 and either closes it or opens it. After this is finished which lockers will be open? Why?

[To the Teacher: If the students struggle, you can recommend that they simplify the problem by looking at 10 or 20 lockers. Let them try any approach they think will help them find the solution Another suggestion is that a couple of groups could get together and act it out (with actual lockers if possible) or by having human lockers who turn one way for open and the other for closed. Have students share the different ways they approach the problem If students try only 10 some will see that lockers 1, 4 and 9 will be open. What might they think will be the next locker that will be opened?)

After a good deal of discussion and solutions have been talked about you can hand out a blank number line on an 81/2 by 11 page that can be labeled from 1 to 20. Have students mark all the numbers whose lockers remained open. Underneath all the integers from 1 to 20, have students list all the factors of that number. Have them observe and discover why certain lockers remained open.

FYI: The result you will find that is very interesting is the only lockers that will remain open are the lockers numbered with square numbers, e.g. 1,4,9,16, etc. The reason is that that these numbers have an odd number of factors, e.g. the factors of 9 are 1,3,and 9. the factors of 16 are 1,2,4,8,and 16. whereas the factors of 6 (a non square number) are 1,2,3,6.

To get ready for the second activity write on the board the numbers, 1,4,9,16,… Ask the students what is unique about these numbers and why? We know that they have and odd number of factors because they are the only numbers that result from a whole number multiplying by itself.

41

Write 1 = 12, 4 = 22, 9 = 32, 16 = 42, etc. on the board.

Ask: since 1,4,9,16, etc are the results of a whole number squared, can 2, 3, 5, 6,7, 14, etc. be the result of a whole number squared? Have kids discuss this. Hear their ideas on this. This should lead to another way of expressing this square concept, i.e. square roots. Show students that the inverse operation for 22 = 4 is √4 = 2, 32 = 9 therefore √9 = 3. Now you can ask, “What happens when you square root any whole number?”]

Second Activity Using your calculator look at the results for

√3

√7

√15

Write down all your observations.

How do these compare to √4, √9, √16?

How do they compare to repeating decimals?(To the Teacher: The results will be similar to what we saw with repeating decimals, but in this case there is no pattern in the decimals that result from these square roots.)

Third ActivityHere are four groups of numbers. Observe each group and describe how they are similar and how they are different

Group 1√2≈1.4142135623730950488016887242097…

√7≈2.645751311064590716171096573817...

√12≈3.464101615137754587054892683012...

Group 2√16 = 4.0000000000000000000000000000…

√25= 5.0000000000000000000000000000…

Group 3

42

17= .142857142857142857142857142857…

111= .0909090909090909090909090909…

Group 435 = 0.600000000000000000…

58 = 0.625000000000000000…

(To the Teacher: You can use this as an opportunity to introduce the term irrational numbers for the first group of decimals. Remind the students about the characteristics of rational numbers that they can be expressed as a fraction resulting in terminating decimals or infinitely repeating decimals. Can √16∨√9 be written as a fraction? What about √6∨15? Closing idea to discuss with students: Irrational numbers cannot be written as a patterned infinite decimal and thus cannot be written as a fraction.

For the Next Activity: Another thing for students to know is that you can estimate the value of a radical by knowing the perfect squares on each side of it.)

Fourth Activity: Who is the Better Estimator?In teams of 2, without using your calculator, estimate the value of the following radical to the nearest tenth. Your team must be ready to explain how you came up with your estimate. The team that gets closest gets a point.

√17

(To the Teacher: Write students’ results for the first problem on the board and ask some of the students to explain how they reached their result. Students should be prompted to listen and come up with a good method to be able to estimate the value of a radical. You could use your calculator on the white board to show the first set of digits in the irrational number. If you want to get more challenging you can eliminate rounding to the nearest tenth and see which group can get more precise. This could make for interesting discussions.)

√30

√53

√91

43


In this part of the project you will show your understanding of radicals. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Radicals.

1) On your created number line, locate the following irrational numbers in a color not used before.

√45

√101

√89

√29

√3

2) Now go to your calculator and press the key for Pi (π). What kind of number do you think it is? Why? Locate it on your number line.

3) Pi has a fascinating history. Go on the internet and research what you can find out about Pi (π). Tell some of the story of pi in a short essay of at least two paragraphs.

44


Lesson 6Name_______________________Date________________________

Radicals

Group Performance Task:

The Locker Problem

There are 1000 lockers lined up numbered 1 to 1000 and 1000 students. The lockers are all closed. The first student, Jasmine, walks by and opens all the lockers. Then the second student, Al, walks by and goes to every second locker starting at #2 and closes it. Then Mary walks by and goes to every third locker starting at #3 closing the opened lockers and opening the closed lockers. The 4th student walks by and goes to every fourth locker starting at #4 closing the opened lockers and opening the closed lockers. This routine goes on until student 1000, Elvina, goes to locker # 1000 and either closes it or opens it. After this is finished which lockers will be open? Why?

45

Second Activity

Using your calculator look at the results for

√3

√7

√15

Write down all your observations.

How do these compare to √4, √9, √16?

How do they compare to repeating decimals?

46

Third ActivityHere are four groups of numbers. Observe each group and describe how they are similar and how they are different

Group 1√2≈1.4142135623730950488016887242097…

√7≈2.645751311064590716171096573817...

√12≈3.464101615137754587054892683012...

Group 2√16 = 4.0000000000000000000000000000…

√25= 5.0000000000000000000000000000…

Group 317= .142857142857142857142857142857…

111= .0909090909090909090909090909…

Group 435 = 0.600000000000000000…

58 = 0.625000000000000000…

47

Fourth Activity: Who is the Better Estimator? In teams of 2, without using your calculator, estimate the value of the following radical to the nearest tenth. Your team must be ready to explain how you came up with your estimate. The team that gets closest gets a point.√17

√30

√53

√91

48


In this part of the project you will show your understanding of radicals. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Radicals.

4) On your created number line, locate the following irrational numbers in a color not used before.

√45

√101

√89

√29

√3

5) Now go to your calculator and press the key for Pi (π). What kind of number do you think it is? Why? Locate it on your number line.

6) Pi has a fascinating history. Go on the internet and research what you can find out about Pi (π). Tell some of the story of pi in a short essay of at least two paragraphs.

49


Teacher Guide

Working with Signed NumbersFive Approaches to Adding Signed Numbers

Opening ActivityWe have focused mainly on the number between the whole numbers. Let’s spend a few minutes on the following sets of numbers. Look at them closely. Compare and contrast them. How are they similar and how are they different?{1, 2, 3, 4, …}{0, 1, 2, 3, …}{…-2, -1, 0, 1, 2, …}(To the Teacher: Ideas you might want to bring out are: these are infinite sets, the relation of sets and subsets, in the first two sets we know the smallest number but not in the third set, and we cannot name the largest number in any of these sets.Finally you should give names to these sets so students will be able to use the mathematical language.)

Main Activity:(To the Teacher: Now we will focus on computing with the numbers on the number line with a focus on signed numbers. Form groups of three or four. Each group will get one of the following tasks. Each group is responsible for every group members learning. Each of the groups will present their findings to the class at large explaining in detail the process they went through. Please have Group 2 present tomorrow as we will use Hot and Cold Cubes to understand subtraction. You want to help students see how the different approaches are related and to find an approach that makes sense to them. Since this unit has focused on the number line you might let the number line group go first. The following presenting groups might be asked if they see a connection between their approach and the number line approach.)Group 1: Algebra TilesYou will be given ten (single unit) algebra cubes of one color and ten of another. The group’s job is to decide which color represents positive numbers and which represents negative numbers. Then you have to come up with a strategy of how to combine them to answer the following questions and ultimately develop a general rule. Be ready to present your findings to the class.

Q1. -6 + (+6)

Q2. -3 + (-6)

Q3. +3 + (-6)

Q4. +3 + (+6)

Q5. -3 + (+6)

50

Group 2: Hot and Cold CubesNow you will read “Hot and Cold Cubes” excerpted from the Interactive Mathematics Program1. Answer the following questions and develop a general rule. Be ready to present your findings to the class.

Q1. -6 + (+6)

Q2. -3 + (-6)

Q3. +3 + (-6)

Q4. +3 + (+6)

Q5. -3 + (+6)HOT AND COLD CUBES

In a far-off place, there was once a team of amazing chefs who cooked up the most marvelous food ever imagined.

They prepared their meals over a huge cauldron, and their work was very delicate and complex. During the cooking process, they frequently had to change the temperature of the cauldron in order to bring out the flavors and cook the food to perfection.

They adjusted the temperature of the cooking either by adding special hot cubes or cold cubes to the cauldron or by removing some of the hot or cold cubes that were already in the cauldron. The cold cubes were similar to ice cubes except that they didn’t melt, and the hot cubes were similar to charcoal briquettes, except they didn’t lose their heat.

If the number of cold cubes in the cauldron was the same as the number of hot cubes, the temperature of the cauldron was 0 degrees on their temperature scale. For each hot cube that was put in the cauldron, the temperature went up one degree; for each hot cube removed, the temperature went down one degree. Similarly, each cold cube put in lowered the temperature one degree and each cold cube removed raised it one degree.

The chefs used positive and negative numbers to keep track of their changes they were making to the temperature. For example, suppose 4 hot cubes and 10 cold cubes were dumped into the cauldron. Then the temperature would be lowered by 6 degrees altogether, since 4 of the 10 cold cubes would balance out the 4 hot cubes, leaving 6 cold cubes to lower the temperature 6 degrees. They would write +4 + -10 = -6 to represent these actions and their overall result.

1 Fendel, D., Resek, D., Alper, L., & Fraser, S. (1999). Interactive mathematics program. Emeryville, CA: Key Curriculum Press.

51

Group 3: Money, Money, money

A. Jose loves money!! He also loves buying things. He has ten dollars and he owes his sister fifteen dollars. What is his financial situation? (A little suggestion: Represent having money as positive and owing money as negative.) Write a mathematical statement to express this situation.

B. Jose owes his sister fifteen dollars and then borrows twenty dollars more from her. What is his financial situation now? Write a mathematical statement to express this situation.

C. This time let Jose, have more money than he owes. Write this situation in words and then write a mathematical statement to express the situation.

Now use what you have observed in the previous three problems to answer the following questions and develop a general rule. Be ready to present your findings to the class.

Q1. -6 + (+6)

Q2. -3 + (-6)

Q3. +3 + (-6)

Q4. +3 + (+6)

Q5. -3 + (+6) Group 4: The Number Line Draw a number line with negative and positive numbers from -10 to +10. Now use this number line to answer the following questions.

Q1. -6 + (+6)

Q2. -3 + (-6)

Q3. +3 + (-6)

Q4. +3 + (+6)

Q5. -3 + (+6)

Can you come up with a general rule? Be ready to present your findings to the class.

52

Group 5: Patterns and More Patterns Look at the table of completed additions. Observing the patterns what would you predict the answers to be for the following additions?

+4 +4 +4 +4 +4 +4 +4 +4 +4+3 +2 +1 0 -1 -2 -3 -4 -5+7 +6 +5 +4

Use what you observed above, to complete the next set of additions.

+4 +3 +2 +1 0 -1 -2 -3-3 -3 -3 -3 -3 -3 -3 -3

-3 -3 -3 -3 -3-2 -1 0 +1 +2

Now, see if you can develop general statements about adding positives with positives with positives, negatives with negatives and positives with negatives. Be ready to present your findings to the class

Now answer the following questions based on your observations of patterns.

Q1. -6 + (+6)

Q2. -3 + (-6)

Q3. +3 + (-6)

Q4. +3 + (+6)

Q5. -3 + (+6)

(To the Teacher: You should go around the room supporting the groups where needed. If it is possible, support through questioning. Each group should present except group 2. It is more important for the groups to be able to talk about their process, what they observed, and their understandings. Why do all these approaches work to explain the addition of signed numbers? While each student is asked to be an authority on their approach they should also look to see if there is a particular approach that works best for them.)

53


In this part of the project you will show your understanding of adding signed numbers. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Adding Signed Numbers.

1) Explain which approach to adding fraction makes the most sense to you. Why does it make sense? How will you use it to add signed numbers?

2) Add the following: (-3) + (-8) + (6) + (-9) + (12) then place your result on the number line.

3) Add: (−23

¿+( 34) then place your results on the number line.

4) What would you have to add to -0.4 to get the result of 1.75? Place the result on the number line.

54


Lesson 7Name_______________________Date________________________

Working with Signed NumbersFive Approaches to Adding Signed Numbers

Opening ActivityWe have focused mainly on the number between the whole numbers. Let’s spend a few minutes on the following sets of numbers. Look at them closely. Compare and contrast them. How are they similar and how are they different?{1, 2, 3, 4, …}{0, 1, 2, 3, …}{…-2, -1, 0, 1, 2, …}

55

Main Activity: Adding Signed Numbers

Each group will be asked to work on one of the following tasks. They are each different ways of thinking about adding signed numbers. Your job is to become an authority on your approach. Why does your approach make sense? Why does it help to explain how we add signed numbers? Be prepared to explain your approach to the rest of the class

Group 1: Algebra Tiles You will be given ten (single unit) algebra cubes of one color and ten of another. The group’s job is to decide which color represents positive numbers and which represents negative numbers. Then you have to come up with a strategy of how to combine them to answer the following questions and ultimately develop a general rule. Be ready to present your findings to the class.

Q1. -6 + (+6)

Q2. -3 + (-6)

Q3. +3 + (-6)

Q4. +3 + (+6)

Q5. -3 + (+6)

56

Group 2: Hot and Cold Cubes

Now you will read “Hot and Cold Cubes” excerpted from IMP. Answer the following questions and develop a general rule based on the reading. Be ready to present your findings to the class.

Q1. -6 + (+6)

Q2. -3 + (-6)

Q3. +3 + (-6)

Q4. +3 + (+6)

Q5. -3 + (+6)HOT AND COLD CUBES

In a far-off place, there was once a team of amazing chefs who cooked up the most marvelous food ever imagined.

They prepared their meals over a huge cauldron, and their work was very delicate and complex. During the cooking process, they frequently had to change the temperature of the cauldron in order to bring out the flavors and cook the food to perfection.

They adjusted the temperature of the cooking either by adding special hot cubes or cold cubes to the cauldron or by removing some of the hot or cold cubes that were already in the cauldron.

The cold cubes were similar to ice cubes except that they didn’t melt, and the hot cubes were similar to charcoal briquettes, except they didn’t lose their heat.

If the number of cold cubes in the cauldron was the same as the number of hot cubes, the temperature of the cauldron was 0 degrees on their temperature scale.

For each hot cube that was put in the cauldron, the temperature went up one degree; for each hot cube removed, the temperature went down one degree. Similarly, each cold cube put in lowered the temperature one degree and each cold cube removed raised it one degree.

The chefs used positive and negative numbers to keep track of their changes they were making to the temperature.

For example, suppose 4 hot cubes and 10 cold cubes were dumped into the cauldron. Then the temperature would be lowered by 6 degrees altogether, since 4 of the 10 cold cubes would balance out the 4 hot cubes, leaving 6 cold cubes to lower the temperature 6 degrees. They would write +4 + -10 = -6 to represent these actions and their overall result.

57

Group 3: Money, Money, money

A. Jose loves money!! He also loves buying things. He has ten dollars and he owes his sister fifteen dollars. What is his financial situation? (A little suggestion: Represent having money as positive and owing money as negative.) Write a mathematical statement to express this situation.

B. Jose owes his sister fifteen dollars and then borrows twenty dollars more from her. What is his financial situation now? Write a mathematical statement to express this situation.

C. This time let Jose, have more money than he owes. Write this situation in words and then write a mathematical statement to express the situation.

Now use what you have observed in the previous three problems to answer the following questions and develop a general rule. Be ready to present your findings to the class.

Q1. -6 + (+6)

Q2. -3 + (-6)

Q3. +3 + (-6)

Q4. +3 + (+6)

Q5. -3 + (+6)

58

Group 4: The Number Line Draw a number line with negative and positive numbers from -10 to +10. Now use this number line to answer the following questions.

Q1. -6 + (+6)

Q2. -3 + (-6)

Q3. +3 + (-6)

Q4. +3 + (+6)

Q5. -3 + (+6)

Can you come up with a general rule? Be ready to present your findings to the class.

59

Group 5: Patterns and More Patterns Look at the table of completed additions. Observing the patterns what would you predict the answers to be for the following additions?

+4 +4 +4 +4 +4 +4 +4 +4 +4+3 +2 +1 0 -1 -2 -3 -4 -5+7 +6 +5 +4

Use what you observed above, to complete the next set of additions.

+4 +3 +2 +1 0 -1 -2 -3-3 -3 -3 -3 -3 -3 -3 -3

-3 -3 -3 -3 -3-2 -1 0 +1 +2

Now, see if you can develop general statements about adding positives with positives with positives, negatives with negatives and positives with negatives. Be ready to present your findings to the class

Now answer the following questions based on your observations of patterns.

Q1. -6 + (+6)

Q2. -3 + (-6)

Q3. +3 + (-6)

Q4. +3 + (+6)

Q5. -3 + (+6)

60


In this part of the project you will show your understanding of adding signed numbers. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Adding Signed Numbers.

1) Explain which approach to adding fraction makes the most sense to you. Why does it make sense? How will you use it to add signed numbers?

2) Add the following: (-3) + (-8) + (6) + (-9) + (12) then place your result on the number line.

3) Add: (−23

¿+( 34) then place your results on the number line.

4) What would you have to add to -0.4 to get the result of 1.75? Place the result on the number line.

61


Teacher Guide

Working with Signed NumbersSubtracting Signed Numbers

Opening Activity: Journal WritingPrompt: How do you add signed numbers? Does your method work all the time? Why?

Second Activity (To the Teacher: Once students have written in their journals you should begin this lesson by having the group that read “Hot and Cold Cubes” present today, explaining the story that they read and how addition can be understood through this story. Make sure the group’s explanation was clear enough so that all understand the following activity.)

In the same groups as yesterday, you will read and respond to this section from “Hot and Cold Cubes.” Before you begin, be sure you fully understand what the group presented about adding using hot and cold cubes.

If the chefs added 3 hot cubes and then removed 2 cold cubes, the combined result would be to raise the temperature 5 degrees. The mathematical representation for this would be:

+3 – (-2) = +5

And if they wrote -5 - +6 = -11 it would mean that 5 cold cubes were added and then we removed 6 hot cubes. The combined result would be to lower the temperature 11 degrees.

(To the Teacher: This might be a good time to discuss what students understood from this short story. Does it make sense to them? Students have been taught this before and will often come in with a method such as “keep change, change.” Because students haven’t made sense of subtraction they often make mistakes or remember the rule and use it incorrectly. We are trying to give them a basis for subtraction. A question they might think about is how is this approach related to the number line. Can they transfer this approach to the number line approach?)

Now let’s see if you follow this by writing the mathematical representation of each of the following sentences.

1. 5 hot cubes were added, and then 6 cold cubes were removed.2. 6 hot cubes were removed, and then 4 cold cubes were removed.3. 4 cold cubes were added and then 8 cold cubes were removed.

62

Third ActivityDescribe using hot and cold cubes, the action represented by the mathematical expressions below and give the result of the action.

1. +4 – (-3)2. -6 – (+3)3. -8 – (-12)

Can you transfer this approach to what to do using a number line? Be ready to explain to the rest of the class?

(To the Teacher: Have kids explain their understanding of subtraction using the examples above. A question you might ask is, “Are you sure you are correct? Why?”)

Fourth Activity: Creating a Problem

Create three subtraction problems using signed numbers and explain how to get the answer.

Homework: Part 7 of the ProjectIn this part of the project you will show your understanding of subtracting signed numbers. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Subtraction of Signed Numbers.

1) Explain how you subtract signed numbers and why the approach you use makes sense. Use examples to support your explanation.

2) Find the result of the following then place it on a number line.

( 23 )−(−1

4 )3) Find the result of the following then place it on the number line.

(-3 – -4) + (2 – 6)

63


Lesson 8Name_______________________Date________________________

Working with Signed NumbersSubtracting Signed Numbers

Opening Activity: Journal WritingPrompt: How do you add signed numbers? Does your method work all the time? Why?

Second ActivityIn the same groups as yesterday, you will read and respond to this section from “Hot and Cold Cubes.” Before you begin, be sure you fully understand what the group presented about adding using hot and cold cubes.

Hot and Cold CubesIf the chefs added 3 hot cubes and then removed 2 cold cubes, the combined result would be to raise the temperature 5 degrees. The mathematical representation for this would be:

+3 – (-2) = +5

And if they wrote -5 - +6 = -11 it would mean that 5 cold cubes were added and then we removed 6 hot cubes. The combined result would be to lower the temperature 11 degrees.

Now let’s see if you follow this by writing the mathematical representation of each of the following sentences then find the solution.

1. 5 hot cubes were added, and then 6 cold cubes were removed.2. 6 hot cubes were removed, and then 4 cold cubes were removed.3. 4 cold cubes were added and then 8 cold cubes were removed.

64

Third ActivityDescribe using hot and cold cubes, the action represented by the mathematical expressions below and give the result of the action.

1. +4 – (-3)2. -6 – (+3)3. -8 – (-12)

Can you transfer this approach to what to do using a number line? Be ready to explain to the rest of the class?

Fourth Activity: Creating a Problem

Create three subtraction problems using signed numbers and explain how to get the answer.

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Homework: Part 7 of the ProjectIn this part of the project you will show your understanding of subtracting signed numbers. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Subtraction of Signed Numbers.

4) Explain how you subtract signed numbers and why the approach you use makes sense. Use examples to support your explanation.

5) Find the result of the following then place it on a number line.

( 23 )−(−1

4 )6) Find the result of the following then place it on the number line.

(-3 – -4) + (2 – 6)

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Teacher Guide

Working with Signed NumbersMultiplication and Division of Signed numbers

(To the Teacher: Here are two approaches to multiplication, one will be using the idea of patterns and the other will use the idea of hot and cold cubes. Everyone independently will work on the pattern approach as a performance task. There should be a discussion of what students observed and what conjectures they made based on the performance task and you should write it on the board. The class will test out the conjectures using Hot and Cold Cubes. Form groups of three or four students and have them read the section from Hot and Cold Cubes with the purpose of seeing if they got the same results as they did in the performance task.)

Opening Activity: Performance Task Can I use patterns to describe multiplication of signed numbers?

You already know the answer to +4 times +3, since you know 4 times 3. Observe the given products and use the patterns to predict the other results.

These are multiplication problems:

+4 +4 +4 +4 +4 +4 +4 +4 +4 +3 +2 +1 0 -1 -2 -3 -4 -5+12 +8 +4 0

Observe your results, using what you learned, and predict the following products.

+4 +3 +2 +1 0 -1 -2 -3 -4 -3 -3 -3 -3 -3 -3 -3 -3 -3

Write down your observations about the patterns you see and what you notice about the multiplication of signed numbers.

Write down your conjectures about what happens when you multiply signed numbers

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Second Activity: Remember the story about chefs using hot and cold cubes. Make sure everyone in the group understands this story before you read the following.

HOT and COLD CUBESSometimes the chefs wanted to raise or lower the temperature by a large amount but did not want to put the cubes into the cauldron one at a time. So for large jumps in temperature, they would put in and take out bunches of cubes. For instance if the chefs wanted to raise the temperature 100 degrees, they might toss, 5 bunches of 20 hot cubes into the cauldron, instead of putting 100 hot cubes, one at a time.

When the chefs used bunches of cubes to change the temperature, they used a multiplication sign to record their activity. For example, to describe tossing 5 bunches of 20 hot cubes into the cauldron, they would write:

+5 . +20 = +100

The chefs could also change the temperature by removing bunches, e.g. if they removed 3 bunches of 5 hot cubes each, the result was to lower the temperature 15 degrees, because each time a bunch of 5 hot cubes was removed, the temperature went down 5 degrees. They would write:

-3 . +5 = -15

How would you write the following sentences, mathematically with result?1. 2 bunches of 6 cold cubes were added.2. 4 bunches of 7 hot cubes were removed.3. 3 bunches of 6 cold cubes were removed.

Write down your observations about the patterns you see and what you notice about the multiplication of signed numbers. How do your results compare with the results from the pattern activity? Can you make a set of statements of what to do when you multiply signed numbers?

Be ready to present your process and findings to the rest of the class.

Third Activity: What is true about division of signed numbers?Remember back in elementary when you learned about multiplication and division. The teacher taught you that if 4 times 3 equal 12, then 12 divided by 3 equals 4. Look at the table below.

IF 4 . 3 = 12THEN 12 ÷ 3 = 4

Use this knowledge to find the results in each of the following.

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IF +4 . -3 = ____THEN ___÷ -3 = ____

IF -4 . -3 = ____THEN ___÷ -3 = ____

IF +4 . +3 = ____THEN ___÷ +3 = ____

IF -4 . +3 = ____THEN ___÷ +3 = ____

Write down your observations about the patterns you see and what you notice about the relationship between multiplication and division of signed numbers.(To the Teacher: You want the kids to see that the same rules govern both multiplication and division, and also that the divisor could have been the first number from the multiplication example rather than the second yielding the same results.)

Homework: Part 8 of the ProjectIn this part of the project you will show your understanding of multiplication and division signed numbers. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Multiplication and Division of Signed Numbers.

1) Explain how to multiply signed numbers and why your ideas make sense. Use examples to support your ideas.

2) Find the product of (-2) (-3) (12) (-1). Place your result on the number line.

3) Multiply the following and observe the results.

a) (1) * (1) * (1) * (1) * (1) b) (-1) * (1) * (1) * (1) * (1) c) (-1) * (-1) * (1) * (1) * (1) d) (-1) * (-1) * (-1) * (1) * (1) e) (-1) * (-1) * (-1) * (-1) * (1) f) (-1) * (-1) * (-1) * (-1) * (-1)

What patterns do you see? Can you make a conjecture about multiplying signed numbers based on the number of negatives?

What do you predict (-1)221 will equal? Why?

Do you think this is a true statement? Why?

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(-2) 221 = (2) 221


Lesson 9Name_______________________Date________________________

Working with Signed NumbersMultiplication and Division of Signed Numbers

Opening Activity: Performance Task Can I use patterns to describe multiplication of signed numbers?

You already know the answer to +4 times +3, since you know 4 times 3. Observe the given products and use the patterns to predict the other results.

These are multiplication problems

+4 +4 +4 +4 +4 +4 +4 +4 +4 +3 +2 +1 0 -1 -2 -3 -4 -5+12 +8 +4 0

Observe your results, using what you learned, and predict the following multiplications.

+4 +3 +2 +1 0 -1 -2 -3 -4 -3 -3 -3 -3 -3 -3 -3 -3 -3

Write down your observations about the patterns you see and what you notice about the multiplication of signed numbers.

Write down your conjectures about what happens when you multiply signed numbers.

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Second Activity: Remember the story about chefs using hot and cold cubes. Make sure everyone in the group understands this story before you read the following.

HOT and COLD CUBESSometimes the chefs wanted to raise or lower the temperature by a large amount but did not want to put the cubes into the cauldron one at a time. So for large jumps in temperature, they would put in and take out bunches of cubes. For instance if the chefs wanted to raise the temperature 100 degrees, they might toss, 5 bunches of 20 hot cubes into the cauldron, instead of putting 100 hot cubes, one at a time.

When the chefs used bunches of cubes to change the temperature, they used a multiplication sign to record their activity. For example, to describe tossing 5 bunches of 20 hot cubes into the cauldron, they would write:

+5 . +20 = +100

The chefs could also change the temperature by removing bunches, e.g. if they removed 3 bunches of 5 hot cubes each, the result was to lower the temperature 15 degrees, because each time a bunch of 5 hot cubes was removed, the temperature went down 5 degrees. They would write:

-3 . +5 = -15

How would you write the following sentences, mathematically with result?1. 2 bunches of 6 cold cubes were added.2. 4 bunches of 7 hot cubes were removed.3. 3 bunches of 6 cold cubes were removed.

Write down your observations about the patterns you see and what you notice about the multiplication of signed numbers. How do your results compare with the results from the pattern activity? Can you make a set of statements of what to do when you multiply signed numbers? Be ready to present your process and findings to the rest of the class.

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Third Activity: What is true about division of signed numbers?Remember back in elementary when you learned about multiplication and division. The teacher taught you that if 4 times 3 equal 12, then 12 divided by 3 equals 4. Look at the table below.

IF 4 . 3 = 12THEN 12 ÷ 3 = 4

Use this knowledge to find the results in each of the following.

IF +4 . -3 = ____THEN ___÷ -3 = ____

IF -4 . -3 = ____THEN ___÷ -3 = ____

IF +4 . +3 = ____THEN ___÷ +3 = ____

IF -4 . +3 = ____THEN ___÷ +3 = ____

Write down your observations about the patterns you see and what you notice about the relationship between multiplication and division of signed numbers.

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In this part of the project you will show your understanding of multiplication and division signed numbers. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Multiplication and Division of Signed Numbers.

4) Explain how to multiply signed numbers and why your ideas make sense. Use examples to support your ideas.

5) Find the product of (-2) (-3) (12) (-1). Place your result on the number line.

6) Multiply the following and observe the results.

g) (1) * (1) * (1) * (1) * (1) h) (-1) * (1) * (1) * (1) * (1) i) (-1) * (-1) * (1) * (1) * (1) j) (-1) * (-1) * (-1) * (1) * (1) k) (-1) * (-1) * (-1) * (-1) * (1) l) (-1) * (-1) * (-1) * (-1) * (-1)

What patterns do you see? Can you make a conjecture about multiplying signed numbers based on the number of negatives?

What do you predict (-1)221 will equal? Why?

Do you think this is a true statement? Why? (-2) 221 = (2) 221

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Teacher Guide

Working with Signed NumbersOrder of Operations

Opening ActivityIn your group, find the solutions to each of the following. Make sure your group agrees on the answers.

3 + 4 × 5 4 – 2 + 8 -3(2 – 5) (-3)2 + (-2) 3

2 + 3(4 – 2) 2

Now that you’ve all agreed on your answers, ask the teacher for calculators and use the calculator to check to see if your answers are correct. If you don’t have the same answer as the calculator, try and figure out why.

After you’ve seen the results, can you explain the process that the calculator went through to get its answers?

Be ready to present your thinking to the class.

(To the Teacher: The purpose of this activity is for students to see that there is an agreed upon order of operations. This is necessary so people from all over the world can communicate mathematically. They should come to see through this investigation the accepted order. Students have learned this before but will probably still make some mistakes. There will also be the problem of how to write some of these expressions on the calculator. They will probably need your help.)

Second Activity: Game of Order of Operations

(To the Teacher: Make up an order of operations problem. On separate pieces of construction paper, write each symbol of the problem. For example, if the problem were

-4(3 + 1) 2 you would do the following

- 4 ( 3 + 1 )2

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Each of these boxes, representing a separate piece of paper, should be shuffled before giving to a group. You give the group the answer to the problem and their job is to arrange the papers in an order that will yield this answer.

Make up a set for each group. Give them five minutes to solve, and then have them pass their set of papers to the next group. Continue until all groups have seen all the problems.

Finally, each group creates a problem that they place on construction paper. Have the groups try to solve each others’ problems.

Take a look at the Student Activity Sheet to see how the game is explained to them)

Homework: Part 9 of the ProjectIn this part of the project you will show your understanding of multiplication and division signed numbers. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Order of Operations.

1) What would happen if there was no order of operations? Explain using examples.2) Using the order of operations find the result of the following. Place the results on

the number line

3 – (-2) * (3) + (-2)

3 – 5 2 + (-6)

-2(-2+4)2

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Patterns and the Real Number Line Unit

Student Activity SheetLesson 10

Name_______________________Date________________________

Working with Signed NumbersOrder of Operations

Opening ActivityIn your group, find the solutions to each of the following. Make sure your group agrees on the answers.

3 + 4 × 5 4 – 2 + 8 -3(2 – 5) (-3)2 + (-2) 3

2 + 3(4 – 2) 2

Now that you’ve all agreed on your answers, ask the teacher for calculators and use the calculator to check to see if your answers are correct. If you don’t have the same answer as the calculator, try and figure out why.

After you’ve seen the results, can you explain the process that the calculator went through to get its answers?

Be ready to present your thinking to the class.

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Second Activity: Game of Order of OperationsYou and your partners will be given a set of papers with a number or operation sign including a negative sign, or parenthesis on the paper. You will also be given a calculator. The teacher will give you a number which is the answer to an order of operation problem. Your job is to put the papers in order so if we follow the order of operations we would get the number the teacher gave you. Here is an example.

The teacher said the answer is -64. She gave you a negative sign, a 4, a set of parenthesis, a 3 , a plus sign, a 1 and a power of 2. You have to arrange it correctly to get -64 The correct answer would look like:

- 4 ( 3 + 1 )2

You will have five minutes to get the answer.

Each group will get four different problems. How many can you get?

Third Activity:

You are going to now create a problem just like the ones you just worked on and try to stump the other groups. You will be given construction paper. Be ready to give it to the other groups. Then each group will try to solve each others’ problems.

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Homework: Part 9 of the ProjectIn this part of the project you will show your understanding of multiplication and division signed numbers. You will answer the following questions. Remember you are showing all your work and thinking in the Explanation Document. Title this section, Order of Operations.

3) What would happen if there was no order of operations? Explain using examples.4) Using the order of operations find the result of the following. Place the results on

the number line

3 – (-2) * (3) + (-2)

3 – 5 2 + (-6)

-2(-2+4)2

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Teacher Guide

Bringing Everything TogetherCompleting the Project

Opening Activity:We have looked at many different sets of numbers in this unit. We have worked to understand them, know their place on the number line and even compute with them. Now we will look at them all together and see what we can further understand about them. Look at all these sets below. Mathematicians call these groupings of numbers, sets of numbers, with each set made up of a finite or infinite quantity of that type of number.

{1, 2, 3, 4, …}{ set of rational numbers}{ set of fractions between 0 and 1}{…-2, -1, 0, 1, 2, …}{ set of all numbers on the number line or real number line} {set of irrational numbers} {0, 1, 2, 3, …}{set of whole numbers greater than 15 and less than 30}

Be ready to describe each of these setsCan any of these sets be included in any of the other sets? List as many groupings as you can in which all the elements of one set are also elements of the other set.

Is there a set in which all the other sets are contained? Explain

(To the Teacher: Here will be an opportunity to develop further language. You might talk about mathematicians calling many of these subsets of the larger set. This means that all of the numbers in the subset are also in the larger set of which they are a subset. You might introduce the term real numbers, if you haven’t already, as we have been working with a real number line throughout the unit.)

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Second Activity: Questions to ponder:State if they are true or false and give evidence to support your thinking. Be prepared to discuss your reasoning.

1) True or False: All whole numbers are integers but all integers are not whole numbers

2) True or False: Irrational numbers and rational numbers have a zero in common.3) True or False: There are an infinite amount of rational radicals and there are an

infinite amount of irrational radicals.4) True or False: The smallest possible positive rational number is unnamable.5) True or False: Addition is the only operation that when you combine two

negatives the answer will always be negative.

(To the Teacher: This will be a chance to learn about the depth of your students’ understanding of the many concepts discussed in this unit. Students would benefit from a whole class discussion about these statements. It will be important to stress that students need to support their answers with evidence from what they learned in this unit.)

Third Activity: Completing the Project

Now you will have chance to put your finishing touches on the project. The closing section will be your answers to the two essential questions. In the Explanation Document title this section, Closing Statement. You are going to use the different experiences you had in this unit to talk about the following two questions:

Why do mathematicians describe mathematics as the science of patterns?

How are different types of numbers related?

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Lesson 11Name_______________________Date________________________

Bringing Everything TogetherCompleting the Project

Opening Activity:We have looked at many different sets of numbers in this unit. We have worked to understand them, know their place on the number line and even compute with them. Now we will look at them all together and see what we can further understand about them. Look at all these sets below. Mathematicians call these groupings of numbers, sets of numbers, with each set made up of a finite or infinite quantity of that type of number.

{1, 2, 3, 4, …}{ set of rational numbers}{ set of fractions between 0 and 1}{…-2, -1, 0, 1, 2, …}{ set of all numbers on the number line or real number line} {set of irrational numbers} {0, 1, 2, 3, …}{set of whole numbers greater than 15 and less than 30}

Be ready to describe each of these setsCan any of these sets be included in any of the other sets? List as many groupings as you can in which all the elements of one set are also elements of the other set.

Is there a set in which all the other sets are contained? Explain

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Second Activity: Questions to Ponder:State if they are true or false and give evidence to support your thinking. Be prepared to discuss your reasoning.

1) True or False: All whole numbers are integers but all integers are not whole numbers

2) True or False: Irrational numbers and rational numbers have a zero in common.

3) True or False: There are an infinite amount of rational radicals and there are an infinite amount of irrational radicals.

4) True or False: The smallest possible positive rational number is unnamable.

5) True or False: Addition is the only operation that when you combine two negatives the answer will always be negative.

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Third Activity: Completing the Project

Now you will have chance to put your finishing touches on the project. The closing section will be your answers to the two essential questions. In the Explanation Document title this section, Closing Statement. You are going to use the different experiences you had in this unit to talk about the following two questions:

Why do mathematicians describe mathematics as the science of patterns?

How are different types of numbers related?

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unit 1 – patterns and problem web viewpatterns and the real number line. ... they know that...

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