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|Time for Unit: 2-3 weeks |

|Pre Assessment: Pretest on integers, simplifying expressions, and solving two-step equations. |

|Common Core Standards to be addressed: |

|CCSS - 8NS1. 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. |

|CCSS - 8NS2. 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 |

|CCSS -8EE2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. |

|Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. |

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|The following SC 2007 State Standards are addressed: |

|SC standard - N/O 8-2.6: Apply strategies and procedures to approximate between two whole numbers the square roots or cube roots of numbers less than 1,000. |

|SC standard - N/O 8-2.4: Compare rational and irrational numbers by using the symbols and =. |

|SC standard - N/O 8-2.3: Represent the approximate location of irrational numbers on a number line. |

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|Strategies for Teaching the Math CCSS |

|Create “worthwhile” problems as a foundation for daily instruction. |

|Use real data and current events to make mathematics more engaging and relevant. |

|Ask quality questions and promote discourse |

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|Skills – What students need to be able to do |

|Write (expressions and equations) |

|Read (expressions) |

|Evaluate (expressions) |

|Identify (mathematical terms) |

|Perform (order of operations) |

|Apply (properties of operations) |

|Generate (equivalent expressions) |

|Solve (equations) |

|Solve (real-world and mathematical problems) |

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|Concepts – What students need to be able to do |

|KNOW (rational and irrational numbers) |

|UNDERSTAND (decimal expansion) |

|SHOW (decimal expansion repeats) |

|CONVERT (repeating decimal expansion to a rational number) |

|USE (rational approximations of irrational numbers) |

|COMPARE (sizes of rational numbers) |

|LOCATE (rational numbers approximately on a number line) |

|ESTIMATE (value of expressions) |

|(square root and cube root symbols) |

|REPRESENT (solutions to equations) |

|EVALUATE |

|(square roots of perfect squares) |

|(cube roots of perfect cubes) |

|KNOW/APPLY (properties of integer exponents) |

|GENERATE (equivalent numerical expressions) |

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|Big Ideas for Unit |

|- Describe the difference between rational and irrational numbers. |

|- Understand the relationship between squares, square roots, cubes, and cube roots. |

|Students must first have an understanding of: |

|- changing fractions to decimals |

|- exponents (squares and cubes) |

|- terminating and repeating decimals |

|- plotting decimals on a number line |

|Lesson 1: Changing Fractions to Decimals |

|Time for lesson: Lessons should not be limited to a certain number of days or stretched out for a certain number of days. Lesson 1 is an |

|outline to help the teacher ensure that the students understand the lesson and all of its content. It is important to monitor and adjust the |

|pace of the lesson as needed by the students. The teacher should determine the proper amount of time to spend on the lesson. |

| |

|Common Core standard addressed: |

|8NS1. 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. |

|SC NO 8-2.3 Represent the approximate location of irrational numbers on a number line. |

| |

|Common Core Standard in plain language: Convert between fractions and decimals and tell whether it is rational or irrational from the |

|quotient. |

| |

|Mathematical Practices: |

|8.MP.2. Reason abstractly and quantitatively. |

|8.MP.6. Attend to precision. |

|8.MP.7. Look for and make use of structure. |

| |

|Key Vocabulary: |

|numerator, denominator, rational numbers, irrational numbers, repeating decimals, terminating decimals, |

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|- Students must know how to change a fraction to a decimal. Give students the following fraction, ¼. Ask students to do a quick write to |

|explain how to change the fraction to a decimal. Students should be allowed to discuss their answers with each other. Have a class |

|discussion about the different ways to change the fraction to a decimal. Some students may think with quarters, some students may divide, and|

|some may convert to a percent then a decimal etc. After the discussion have student work with a partner and give them the following fractions|

|and ask them to convert them to decimals. 5/8, 1/3, 2/3, -2/5, 8/5. Do not assist the students as they are working. The teacher should just|

|observe and look for common errors that are being made. Use these observations to provide further instruction whole class as to what some |

|common errors are being made. |

| |

|- Have students do a quick write about what they noticed when converting 8/5 to a decimal? Choose students to share their answers. This |

|should lead into a class discussion about improper fractions and mixed numbers. Ask students how they think you convert mixed numbers and |

|improper fractions to decimals. Have students complete the following examples: 4 1/2 , -7 1/8, |

| |

|- For each of the above examples, have students graph each of the decimals on a number line. Have students compare their answers with a |

|partner. Go over the number line with the whole class and have a class discussion about how they knew where to place the decimals. |

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|- Pose students with the following question: How many different types of decimals are there? Give students time to think and write about |

|their answer to this question. Have students share what they wrote. If the terms, terminating decimal and repeating decimal are shared, use |

|this as a time to teach about these two types of decimals. If the terms are not brought up during discussion the teacher should introduce |

|those words to the class. |

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|- Ask the students to convert certain fractions to a decimal. Ask them to write whether the decimal is a repeating or terminating and explain|

|how they know. |

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|- Students should be given the opportunity to explore converting fractions to decimals with a calculator. Use the following activity: |

|“Repeating and Terminating Decimals” to allow students the opportunity to explore these two concepts further. Make sure to point out to |

|students that when writing a repeating decimal, the repeating bar on the decimal must only cover the numbers that are actually repeating. |

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|- Since students have been converting fractions to decimals, ask the class to discuss with a partner how to do the opposite, and convert a |

|decimal to a fraction. Give students the opportunity to brainstorm ideas. (This is not a new concept for them. They have done this in 6th |

|and 7th grade). Use the following activity so students can explore how to convert decimals to fractions. |

|- Activity: Write on the board this list of terminating decimals: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9. Use a 3-column format with the|

|columns labeled Decimal, Fraction, and Simplified Fraction. |

|Say: Writing a terminating decimal as a fraction is simple. You're almost there when you read the decimal. Read the decimal numbers I've |

|written on the board. Tell me whether the fraction form can be simplified. |

|Students should have no trouble reading these decimals and identifying which fraction can be written with a lesser denominator. Have a student|

|record the fractional forms of these decimals. |

|Ask: Think about the table we've just made. Is there something special about the decimals that have a simpler fractional form? |

|Students should notice that what look like even numbers all simplify, while the odd numbers don't. This isn't exactly what's going on here, so|

|add 0.25 and 0.75 to the list and lead a discussion about what's really happening: When you look at the numerals that make up a decimal |

|number, you're looking at the numerator of a fraction. The denominator is deduced by the placement of the decimal point. If the numerator is a|

|factor of the denominator (as in the case of 0.2, 0.25, 0.5) or if the numerator has a common factor with the denominator (as in the case of |

|0.4, 0.6, 0.75, 0.8), then there is a simplified fractional form of the decimal. |

|Now erase the table of terminating decimals and replace it with a 2-column table of repeating decimals ([pic] ,[pic] ,[pic] ,[pic] ). Label |

|the columns Decimal and Fraction. |

|Ask: Can anyone fill in the second column of this table? |

|Most of your students should recognize [pic]as [pic]and [pic] as[pic]. Some may recognize [pic]as[pic], but it's unlikely that anyone will |

|recognize [pic] as[pic]. |

|- Once going through this activity give students the opportunity to try converting some repeating decimals into fractions. Remind students to|

|simplify or reduce when possible. A) 0.666666, B) 0.25252525, C) 0.8888, D) 3.1212121212 |

|Other resources that can be used during this lesson: |

|AIMS Activity notebook |

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|Formative Assessment: Give students a chart with two columns on it. One column labeled fraction and the other column labeled decimal. Fill |

|in a couple of items in the chart and have the students convert from one to the other. |

|Lesson 2: The Real Number System |

|Time for lesson: Lessons should not be limited to a certain number of days or stretched out for a certain number of days. Lesson 2 is an |

|outline to help the teacher ensure that the students understand the lesson and all of its content. It is important to monitor and adjust the |

|pace of the lesson as needed by the students. The teacher should determine the proper amount of time to spend on the lesson. |

| |

|Common Core standard addressed: |

|8NS1. 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. |

|8NS2. 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 |

|South Carolina State standard addressed: |

|N/O 8-2.4: Compare rational and irrational numbers by using the symbols and =. |

|N/O 8-2.3: Represent the approximate location of irrational numbers on a number line. |

| |

|Common Core Standard in plain language: Convert between fractions and decimals and tell whether it is rational or irrational from the |

|quotient. Estimate irrational numbers to compare them, locate on a number line, and estimate expression values. |

| |

|Mathematical Practices: |

|8.MP.2. Reason abstractly and quantitatively. |

|8.MP.4. Model with mathematics. |

|8.MP.6. Attend to precision. |

|8.MP.7. Look for and make use of structure. |

|8.MP.8. Look for and express regularity in repeated reasoning. |

| |

|Key Vocabulary: |

|integers, rational numbers, irrational numbers, real numbers, natural numbers, |

| |

|- Have students create a Real Numbers “window pane” in their math notebooks. The window pane should have two sections labeled rational |

|numbers and irrational numbers. Students should do a close reading of the two articles titled, “What are Rational Numbers?” and “What are |

|Irrational Numbers?” both by Jason Marshall. As students do their close reading, students should use their window pane to write down any |

|important information they find about rational and irrational numbers. Students should be allowed to write examples in their boxes as well. |

|Have students share their window panes with a partner and discuss what they found important. Go over whole class and make sure a discussion |

|of integers, whole numbers, and natural numbers are included in the discussion. Discuss which category of numbers these would fall into. |

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|- Once students have completed the window pane activity, students should answer the following questions: |

|1) What is the difference between rational and irrational numbers? |

|2) Why are integers rational? |

|3) What are some examples of irrational numbers? |

|4) The fraction 1/5 is what kind of number? How do you know? |

| |

|- Real Number Sort Activity – Students can play this game on the white board with magnets or sticky notes. Students should be given a sticky |

|note with a real number on it. Time the students as they go to the white board and place it in the correct category. After each student has |

|placed their sticky note, have the students examine each of the numbers and their placement. Students will discuss with a partner if any of |

|the numbers have been placed in the wrong category. Once students have had the opportunity to discuss, have a whole class discussion about |

|numbers that were incorrectly placed and why. For each incorrect answer, add 5 seconds to the time. Do the activity once more with a |

|different set of numbers and have them try to beat their original time. Make this a competition between other classes, to see who can get the|

|fastest time. |

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|Other resources that can be used during this lesson: |

| username and password is bufsd |

|Type rational and irrational numbers in the search box and a 2 minute video will play. There are also other activities and quizzes that can |

|be used to go along with this lesson. |

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|The following video is a good video that explains rational and irrational numbers: |

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|The video is about 10 minutes and you should pick and choose which segments to show. |

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|Formative Assessment: Ask students to describe the difference between rational and irrational numbers. Give students something similar to the|

|activity directly above and have them correctly place the numbers in the correct location. |

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|Lesson 3: Comparing Rational Numbers |

|(this lesson is a continuation of the previous lesson) |

|Time for lesson: Lessons should not be limited to a certain number of days or stretched out for a certain number of days. Lesson 3 is an |

|outline to help the teacher ensure that the students understand the lesson and all of its content. It is important to monitor and adjust the |

|pace of the lesson as needed by the students. The teacher should determine the proper amount of time to spend on the lesson. |

| |

|Common Core standard addressed: |

|8NS1. 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. |

|8NS2. 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 |

|South Carolina State standard addressed: |

|N/O 8-2.4: Compare rational and irrational numbers by using the symbols and =. |

|N/O 8-2.3: Represent the approximate location of irrational numbers on a number line. |

| |

|Common Core Standard in plain language: Convert between fractions and decimals and tell whether it is rational or irrational from the |

|quotient. Estimate irrational numbers to compare them, locate on a number line, and estimate expression values. |

| |

|Mathematical Practices: |

|8.MP.2. Reason abstractly and quantitatively. |

|8.MP.4. Model with mathematics. |

|8.MP.6. Attend to precision. |

|8.MP.7. Look for and make use of structure. |

|8.MP.8. Look for and express regularity in repeated reasoning. |

| |

|Key Vocabulary: |

|integers, rational numbers, irrational numbers, real numbers, natural number |

| |

|- In the last lesson, students learned what rational and irrational numbers are. In this lesson the focus is going to be on comparing |

|rational numbers. Separate students into groups of 4 or 5. Each group with receive 4 or 5 cards (depending on the number of students in the |

|group) with a rational number written on the card. A different number should be on each card. For example, use: -5/4, -0.2, 4.31, -3, 5/2, |

|-13/3 etc. Tell the students that their job is to place the rational numbers in the order from least to greatest. Students should discuss |

|with each other the best way to go about completing the task and work together in order to complete the activity. |

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|- Once students have completed the activity, have a whole class discussion about the best ways they found to complete the task. Students |

|should now have the rational numbers in the correct order. Next give each group a sheet of chart paper and have them place the numbers on a |

|number line. Do not give the students any other instructions. It will be important for you to see what the students discuss and do in order |

|to complete the task. Students will have a hard time understanding where to place the rational numbers on the number line. Have the groups |

|present their number lines to the class and have them explain how they came up with where to place the numbers. After all groups have |

|presented, have a whole class discussion about how to place numbers on a number line. |

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|- Students should be given the opportunity to practice this more. Use the textbook page 222 (#’s 1 – 19) to check for |

|student understanding. Students should also complete Finish Line page 33. |

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|Other resources that can be used during this lesson: |

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|This is a jeopardy format game where students compare various rational numbers. |

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|Formative Assessment: Give students a similar activity like the one above. Students should place the numbers in order from least to greatest,|

|place the numbers in the correct order, and place the numbers on a number line. |

|Lesson 4: Square Roots |

|Time for lesson: Lessons should not be limited to a certain number of days or stretched out for a certain number of days. Lesson 4 is an |

|outline to help the teacher ensure that the students understand the lesson and all of its content. It is important to monitor and adjust the |

|pace of the lesson as needed by the students. The teacher should determine the proper amount of time to spend on the lesson. |

| |

|Common Core standard addressed: |

|8EE2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive |

|rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. |

| |

| |

|Mathematical Practices: |

|8.MP.2. Reason abstractly and quantitatively. |

|8.MP.5. Use appropriate tools strategically. |

|8.MP.6. Attend to precision. |

|8.MP.7. Look for and make use of structure. |

| |

|Key Vocabulary: |

|integers, rational numbers, irrational numbers, real numbers, natural number, square number, perfect square, square roots, cubed roots |

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|- Give students the following headline: “Square Root Day is rarer than Leap Day because it only occurs nine times every century!” Students |

|should have heard of Pi Day growing up in their math classes. Ask students if they have ever heard of “Square Root Day”. Give students the |

|list of square root days. Ask students what they can infer based on the days. |

|March 3, 2009 (3/3/09) |

|April 4, 2016 (4/4/16) |

|May 5, 2025 (5/5/25) |

|June 6, 2036 (6/6/36) |

|July 7, 2049 (7/7/49) |

|August 8, 2064 (8/8/64 ) |

|September 9, 2081 (9/9/81) |

|- Have students do a close reading of the article, “What are Square Roots?” by Jason Marshall. As they read students should write down any |

|important information they find in the article. The students should have a discussion with a partner about what they found important in the |

|article. |

|- Ask students if you were to show them a square with an area of 25 inches, if they could figure out the length of each side? What if you |

|were to give them a square with an area of 9 inches, what would each side’s length be? Continue with questions in this manner until students |

|understand the concept. |

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|- Ask students based on this information, what they think it means to square a number and how did it come from finding the area of squares. |

|Also ask students to write down what they think it means to be a perfect square. Give students the opportunity to write on this topic. After|

|writing students should discuss their thoughts with a partner. Have a whole class discussion to ensure that students understand what it means|

|to square a number. |

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|- Ask students what would happen if we wanted to work backwards? What would that be called? Just like we talked about the squares area and |

|you found one side. Square roots are one of two equal factors of a number (What number squared equals the number?). Students should have the|

|opportunity to work with positive and negative answers. Have students practice finding the square roots of perfect squares, both positive and|

|negative, and graph them on a number line. |

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|- Students should begin seeing the relationship between squaring numbers and taking the square root. |

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|- Have students look back at the list of “Square Root Days”. Ask students to find the next three days that will be a “Square Root Day” |

|answer: October 10, 2100 November 11, 2132 December 12, 2144 |

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|- The sheets found at the following website will be used for the next few activities. This activity will help students learn how to estimate |

|the square root of non perfect squares. The task found in the link below, is an excellent way for students to explore and develop an |

|understanding of non perfect squares. |

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|- Once students have an understanding of square roots and non perfect squares, they should be introduced to cubed roots. Have students do a |

|quick write about what it means to cube something. Have a class discussion about what they have written down. Once you have had the |

|discussion about what it means to cube something, ask students to use their prior knowledge of square roots and determine what they think it |

|means to take the cubed root of something. Have a classroom discussion about what they think may happen. Give them the number 27 and ask |

|them to find the cubed root of it. |

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|- Use the example of volume of a square to explain using l x w x h. This will help students understand that the number has to be the same. |

|Ask students to come up with as many numbers who have a cubed root in 2 minutes. Have students share some of the numbers that they were able |

|to come up with. |

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|- Students should make a list of 1 -10 and cube each of the numbers. This will help students estimate between cube roots when it is not a |

|whole number answer. Students should be given examples of numbers that do not have a perfect cubed root and ask them to estimate their |

|answers and place them on a number line. |

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|Other resources that can be used during this lesson: |

|AIMS Activity Notebook |

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|Formative Assessment: |

|The link above is to Root Jeopardy. Play the game with students and have all students write their answers on mini white boards to check for |

|understanding. |

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|Problem Solving: Rational or Irrational Reasoning? (Implement the teaching strategies from Navigating the Mathematics Common Core State |

|Standards) |

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|Learning Task – Students will analyze the reasoning to the answer of a test question about rational or irrational numbers. The students will |

|have to create a study guide with explanations, examples, and graphics to help clear up any misconceptions. Activity is attached and can also|

|be found on the Georgia DOE website. |

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|Lesson 5: Comparing rational and irrational numbers |

|Time for lesson: Lessons should not be limited to a certain number of days or stretched out for a certain number of days. Lesson 5 is an |

|outline to help the teacher ensure that the students understand the lesson and all of its content. It is important to monitor and adjust the |

|pace of the lesson as needed by the students. The teacher should determine the proper amount of time to spend on the lesson. |

| |

|Common Core standard addressed: |

|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., (2). For example, by truncating the decimal expansion of √2, show that √2 is |

|between 1and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. |

| |

|Mathematical Practices: |

|8.MP.2. Reason abstractly and quantitatively. |

|8.MP.5. Use appropriate tools strategically. |

|8.MP.6. Attend to precision. |

|8.MP.7. Look for and make use of structure. |

| |

|Key Vocabulary: |

|integers, rational numbers, irrational numbers, real numbers, natural number, square number, perfect square, square roots, cubed roots |

| |

|- In the past lessons students have learned what rational and irrational numbers are. They have also learned about square numbers, square |

|roots, cubed numbers, and cubed roots. Students should now be able to compare the numbers by placing them on a number line and using >, ................
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