Mathematics Success Grade 8

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1 T92 Mathematics Success Grade 8 [OBJECTIVE] The student will create rational approximations of irrational numbers in order to compare and order them on a number line. [PREREQUISITE SKILLS] rational numbers, plotting rational numbers on a number line [MATERIALS] Student pages S41 S54 Algebra tiles (red and yellow units 25 yellow and 5 red per student pair) Number Line Calculator Sticky Notes Ordering Number Cards Pages (1 3) (T123 - T125) [ESSENTIAL QUESTIONS] 1. Explain how to find the rational approximation for irrational numbers in the form of square roots. 2. Why is it helpful to know how to find rational approximations for irrational numbers? Justify your thinking. 3. How can you compare irrational values that are written in different forms? Explain your thinking. [WORDS FOR WORD WALL] square root, rational approximation, radical, irrational numbers, approximate, terminating decimal, repeating decimal, perfect squares [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) *For Cooperative Pairs (CP) activities, assign the roles of Partner A or Partner B to students. This allows each student to be responsible for designated tasks within the lesson. [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer [WARM-UP] (IP, WG) S41 (Answers on T109.) Have students turn to S41 in their books to begin the Warm-Up. Students will determine the square roots of perfect squares, identify decimal equivalents of fractions and mixed numbers and categorize them as terminating or repeating. Monitor students to see if any of them need help during the Warm-Up. After students have completed the warm-up, review the solutions as a group. {Graphic Organizer, Verbal Description}

2 Mathematics Success Grade 8 T93 [HOMEWORK] Take time to go over the homework from the previous night. [LESSON] [2-3 days (1 day = 80 minutes) - (M, GP, WG, CP, IP)] SOLVE Problem MODELING (WG, GP) S42 (Answers on T110.) Have students turn to S42 in their books. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to use rational approximations of irrational numbers to compare and order the values and plot them on a number line. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description, Graphic Organizer} Square Roots of Irrational Numbers Concrete and Pictorial (M, IP, CP, GP, WG) S42, S43, S44 (Answers on T110, T111, T112.) GP, M, CP, WG: Pass out the red and yellow algebra unit tiles. Make sure students know their designation as Partner A or Partner B. Use the following activity to model the concept of square roots of irrational numbers. {Concrete Representation, Pictorial Representation, Verbal Description, Graphic Organizer} Square Roots of Irrational Numbers Concrete and Pictorial Step 1: Have student pairs complete Question 1 4 on S42 to review the process of how to determine the square root of perfect squares. Review the answers as a whole group. Step 2: Have students turn to page S43. Have student pairs try to make a square using 12 yellow algebra tiles. Partner A, were we able to make a perfect square? (No) Partner B, what is the closest shape to a square that we can create? (3 by 4 rectangle) Model for students how to fill in the area below the rectangle with red tiles to make a perfect square. Partner A, what are the dimensions of the new square? (4 by 4) Step 3: Model how to draw a square around the largest square that is completely yellow. Partner B, how many tiles are in the square? (9) Partner A, what is the square root of your perfect square? (3)

3 T94 Mathematics Success Grade 8 Partner B, explain how you determined the square root. (The measure of one side of the square is 3. 3 times 3 is equal to 9.) Partner A, how many tiles are there that are not part of your perfect square? (7) Partner B, how many of these are yellow? (3) Step 4: We can approximate the square root of 12 using a mixed number. Partner A, what is the whole number of the square root of the largest perfect square? (3) The fraction is the number of yellow tiles over the total number of tiles outside of the perfect square. So, the approximate value of 12 = Partner B, what is this mixed number in decimal form? Use the calculator to divide 3 by 7 to find the decimal portion of the number. ( ) Partner A, using a calculator, find 12. ( ) Partner B, how does your decimal from the tiles compare to the calculator result? (They are very close and differ in the hundredths column. Each is a bit less than 3.5.) Partner A, explain why the two values are different. (When we divide 3 by 7 it is an approximate value and the calculator gives a more exact value when we enter 12.) Step 5: Have students turn to page S44. Direct students attention to Question 1. Partner A, how many yellow chips do you have to use to try to create a square? (6) Partner B, are you able to make a square using 6 tiles? (No) Draw the six tiles in the box for Question 1. Box in the largest perfect square that you can make. Partner A, how many tiles are in the largest perfect square? (4). Partner B, what is the square root of this perfect square? (2) Explain how you know this. (The square root is the measure of one side of the square.) Partner A, explain how we use the value of the square root of the perfect square. (The square root of the perfect square will be the whole number in our mixed number approximation.) Partner B, add red tiles to complete the next largest perfect square. Partner A, how many total tiles are outside of the perfect square? (5) Partner B, how many of the tiles outside are yellow? (2) Partner A, what is the fraction of yellow tiles outside of the box over total tiles outside of the box. 2 5 Partner B, what is the mixed number that approximates the square root of 6? 2 2 5

4 Mathematics Success Grade 8 T95 Step 6: Direct students to Question 3. Partner B, how many yellow tiles do you have to use to try to create a square? (13) Partner A, are you able to make a square using 13 counters? (No) Model for students how to draw the thirteen counters in the box for Question 3. Partner B, what is the largest perfect square you can make? (3 by 3). Model how to box in the largest perfect square. Partner A, how many tiles are in the largest perfect square? (9). Partner B, what is the square root of this perfect square? (3) Partner A, explain how we use the value of the square root of the perfect square. (The square root of the perfect square will be the whole number in our mixed number approximation.) Partner B, add red tiles to complete the next largest perfect square. Partner B, how many total tiles are outside of the perfect square? (7) Partner A, how many of the tiles outside are yellow? (4) Partner B, what is the fraction of yellow tiles outside of the box over total tiles outside of the box. 4 7 Partner A, what is the mixed number that approximates the square root of 13? IP, CP, WG: Have students complete Questions 2, 4, 5 and 6 on S44. Students will follow the same process explained above in the modeling. Allow students time to go through the process and model how to find the greatest perfect square and then create a fraction. Be sure that students are using the concrete tiles to create the squares and that they are drawing the pictorial representation to model the concrete. {Concrete Representation, Verbal Description, Pictorial Representation, Graphic Organizer} Rational Approximations with Number Lines (M, GP, CP, WG, IP) S45, S46, S47 (Answers on T113, T114, T115.) M, GP, CP, WG: Have students turn to S45 in their books. In this activity students will complete the chart to help them find the rational approximation of irrational values. This extends the activity that students have completed with the concrete and pictorial representations of the square roots. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Graphic Organizer, Graph}

5 T96 Mathematics Success Grade 8 MODELING Rational Approximations with Number Lines Step 1: Direct students attention to the top of S45. You can estimate the square root of a number that is not a perfect square without the algebra tiles. For example, if you were asked to find the square root of 94 you would not be able to find a whole number that when multiplied by itself equals 94. You could, however, find the two closest perfect squares that are less than 94 and greater than 94. Partner A, what is a perfect square that is close to, but less than 94? ( 81 = 9) Partner B, what is a perfect square that is close to, but greater than 94? ( 100 = 10) Partner A, what can we conclude about 94? (It is between 9 and 10.) Partner B, is 94 closer to 100 or 81? (100) Partner A, explain what this means. ( 94 is closer to 10.) Step 2: Sometimes we need a more exact answer than simply the range. In this case we can apply our understanding of the tiles. Number Perfect square and its square root close to, but less than the number Perfect square and its square root close to, but greater than the number Difference between perfect squares Difference between number and lower perfect square Rational approximation in the form of a mixed number = = = = By completing this chart, we will be able to find an approximation without using the algebra tiles. Partner B, the second column asks for the perfect square and its square root close to, but less than the number. What is this number? ( 81=9) Partner A, the third column asks for the perfect square and its square root close to, but greater than 94. What is this number? ( 100 = 10) Partner B, when we were using the tiles to find an approximation, explain how we created the denominator of the fraction. (We found the total number of tiles that were necessary to build the next largest square.)

6 Mathematics Success Grade 8 T97 Partner A, identify what operation we can use to find the number of tiles necessary to build the next largest perfect square. (Subtraction: perfect square above number perfect square below) Partner B, in the fourth column, find the difference between the perfect squares above and below the number. ( = 19) Step 3: Have students turn to S46. Partner A, when using the tiles to find an approximation, explain how we created the numerator of the fraction? (We found the number of yellow tiles that were outside of the perfect square that contributed to building the next square.) Partner B, identify the operation we can use to find how far away the number is from the lower perfect square. Explain. (Subtraction: Subtract the lower perfect square from the number.) Partner A, what is the difference? (94 81 = 13) Partner B, explain how we wrote the fraction part of the mixed number which represented the approximation of our square root? (We wrote the numerator as the difference between the number and lower perfect square and the denominator as the difference between perfect squares.) = 13 Partner A, what is the fraction we create? = = Partner B, what is the whole number that should accompany the fraction? (9) Step 4: By creating a mixed number, we are creating a rational number that is approximately the same value as the irrational number that we started with. We created a (rational approximation). We can also plot the rational approximation on a number line. Partner A, explain how we can create a decimal from our rational approximation. (Divide the numerator by the denominator and add the whole number to the decimal.) Partner B, what is the decimal form of this rational approximation? ( ) With your partner, use the number line below to plot a point to show the location of the rational approximation of the number. Label it with the number, the rational approximation and the decimal. Partner A, where is the 94 located on the number line? (between 9.6 and 9.7)

7 T98 Mathematics Success Grade 8 Step 5: Direct students attention to the top of S47. Explain to students that they will be using the same process from the table on S45 to complete this chart on S47. Guide students through the questions to complete the first example. Then, model how to plot the point with its decimal approximation on the number line. Partner B, what is the square root that we are using to find the rational approximation? 22 Partner A, what is the square root and perfect square that is closest to but less than 22? ( 16 = 4) Partner B, what is the square root and perfect square that is closest to but greater than 22? ( 25 = 5) Partner A, explain how to find the difference between the perfect squares. (25 16 = 9) Partner B, explain how to find the difference between the number and the lower perfect square. (22 16 = 6) Partner A, explain how to find the rational approximation of 22 using the information from the chart. (We can create a fraction that represents the difference between the number and the lower perfect square, which is the numerator, over the difference between the perfect squares, which is the denominator.) Write your fraction in the final column. 6 9 Partner B, what do we need to write with the fraction when we write the rational approximation in the last column? (The whole number 4) Partner A, explain how you know it is 4? (The largest perfect square that is less than the irrational number is 16 and its square root is 4.) Partner B, What is the rational approximation for the form of the mixed number? Partner A, explain how we know where to plot the point on the number line. (Divide the numerator by the denominator in the fraction, then add the whole number.) Plot the point on the number line. *Teacher Note: Students may use the calculator here and round to two decimal places.

8 Mathematics Success Grade 8 T99 IP, CP, WG: Have students complete the rest of the page by filling in the table and plotting the points. The table will guide them through the questioning, but remind students that the fourth column re flects finding the total number of tiles outside of the perfect square and then fifth column re flects the number of yellow tiles outside of the perfect square. Take time to review solutions after students have worked on completing this page. {Verbal Description, Graphic Organizer, Graph} Categorizing Irrational Numbers and Decimal Expansions (M, GP, CP, WG) S48, S49 (Answers on T116, T117.) M, WG, GP, CP: Students created a graphic organizer for rational numbers in Lesson 4. In this activity, we will be adding the category of irrational numbers and working with decimal approximations of rational and irrational numbers. Make sure students know their designation as Partner A or Partner B.{Verbal Description, Graphic Organizer} MODELING Categorizing Irrational Numbers and Decimal Expansions Step 1: Have students look at the graphic organizer on S48. Partner A, there is one category in the graphic organizer on the bottom of the page that has not been identi fied. What are the two values in that section? ( 2 and π) Partner B, what do you notice about those two values? Explain your thinking. (Neither value can be written as a ratio in the form a b with both the numerator and denominator as integers.) Have partners discuss Question 3. Partner A, if a number is not rational, what term could we use to describe the opposite of rational? (Irrational) Add the label of Irrational Numbers to the graphic organizer on S48. Partner B, why is this section separate from counting numbers, whole numbers and integers? Explain your thinking. (Counting numbers, whole numbers and integers are all part of the group known as rational numbers because they can be written as ratios. Irrational numbers are values that cannot be written as ratios.) Have students suggest other irrational values that can be added to the wall chart.

9 T100 Mathematics Success Grade 8 Step 2: Direct students attention to the top of S49. Partner A, what is the first number in the graphic organizer? (π) Partner B, using a calculator, type the pi key and hit ENTER to find the decimal form of this number. ( ) Have students record this number in the table. Partner A, do you think pi is a rational or irrational number? (Irrational) Partner B, explain Partner A s choice and tell whether the value is approximate or exact. (Pi does not fit into any of the categories for rational numbers that we created with our graphic organizer. The value is approximate because the decimal does not terminate.) Partner B, identify the next number. 4 6 Partner A, explain how we find the decimal form of this fraction. (Divide the numerator by the denominator.) *Teacher Note: The division of the fractions that are in the graphic organizer can be modeled in a variety of ways based upon your students needs. Some students may only need to see the first fraction divided out to see that it repeats and then they can work in student pairs to determine the decimal for the other two fractions in the chart. If there are students who need more support, you can model all three division problems that are in the chart. Step 3: Partner B, what is the quotient decimal when you divide 4 by 6? ( ) Partner B, is this number rational or irrational? (rational) Partner A, explain Partner B s choice and tell whether the value is approximate or exact. (The number 4 6 is in the a form. The fraction is b approximate because the decimal repeats and must be rounded.) Partner A, what is the third number in the table? ( 2) Partner B, using your calculator, enter the square root of 2 to find the decimal form. ( ) Partner A, is this number rational or irrational? (Irrational) Partner B, explain Partner A s choice and whether the number is approximate or exact. (This number does not fit into any of the categories for rational numbers that we created with our graphic organizer. The number is approximate because the decimal does not terminate and must be rounded)

10 Mathematics Success Grade 8 T101 Have partners complete the last two rows of the graphic organizers on their own. They can use a calculator to find the decimals or complete the long division to determine the decimal. Be sure to review the solutions of the organizer before moving on to drawing conclusions and exploring patterns. Step 4: Direct students attention to Question 1 below the graphic organizer. Partner A, what do you notice about the decimals of the rational numbers? (They either stop, or they continue on with a repeating pattern.) Partner B, what do you notice about the decimals of the irrational numbers? (They continue on with no repeating pattern.) Partner A, what type of decimal is the equivalent of 3 8? (terminating decimal) Partner B, explain this. (There is a point where the quotient comes out evenly with no remainder.) Partner A, what type of decimal is the equivalent of 4 6? (repeating decimal) Partner B, explain your thinking. (When you divide 4 and 6, it repeats the same number over and over in the quotient. We write the repeating portion of the decimal with a bar over it. Therefore, 4 should be written 6 as 0.66 because the 66 will repeat continuously.) Partner A, take a look at 3 in the table on S49. What do you notice 82 about its decimal? (After a while, it begins to repeat.) Partner B, how would we write the decimal using the bar notation? ( ) Remember that we know that all ratios in the form of a b are rational numbers. We also concluded that terminating and repeating decimals represent rational numbers. Therefore, if a number is rational and its decimal does not terminate, then it must repeat. *Teacher Note: Be sure to discuss that sometimes using a calculator to determine decimals can be confusing. If students divide a fraction to find a decimal and notice it does not terminate and also does not repeat, it may just be the view of the calculator. While calculators are very helpful, they may not extend the decimal far enough for us to find the repeating pattern or identify where the decimal terminates. Therefore, it is important to recognize the a over b form to know immediately that a number is rational.

11 T102 Mathematics Success Grade 8 Ordering Irrational Numbers Using the Number Line (M, GP, CP, WG, IP) S50 (Answers on T118.) M, GP, CP, WG: Have students turn to S50 in their books. In this activity students will apply what they have learned about determining rational values of irrational numbers to order irrational values on a number line. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Graphic Organizer, Graph} MODELING Ordering Irrational Numbers Using the Number Line Step 1: Direct students to the top of S50. Partner A, explain how this activity is different than the activity on S47. (In this activity we will be plotting all of the numbers on one number line.) Partner B, what is the irrational number in Question 1? ( 12). Partner A, what is the square root and perfect square that is closest to but less than 12? ( 9 = 3) Partner B, what is the square root and perfect square that is closest to but greater than 12? ( 16 = 4) Partner A, explain how to find the rational approximation of 12 using the information from the chart. (We can create a fraction that represents the difference between the number and the lower perfect square - which is the numerator - over the difference between the perfect squares - which is the denominator.) Write your fraction. 3 7 Partner B, What is the rational approximation in the form of a mixed number? Partner A, explain how we know where to plot the point on the number line. (Divide the numerator by the denominator in the fraction, then add the whole number.) Partner B, what is the decimal form? (approximately 3.43) Partner A, where should the point be plotted? (A bit before the halfway mark between 3 and 4.) Have students plot the point and label the point as 12. Step 2: Direct students attention to Question 2. Partner A, what is the irrational number in Question 2? ( 35). Partner B, what is the square root and perfect square that is closest to but less than 35? ( 25 = 5) Partner A, what is the square root and perfect square that is closest to but greater than 35? ( 36 = 6)

12 Mathematics Success Grade 8 T103 Partner B, explain how to find the rational approximation of 35 using the information from the chart. (We can create a fraction that represents the difference between the number and the lower perfect square - which is the numerator - over the difference between the perfect squares - which is the denominator.) Write your fraction Partner A, What is the rational approximation in the form of a mixed number? Partner B, explain how we know where to plot the point on the number line. (Divide the numerator by the denominator in the fraction, then add the whole number.) Partner A, what is the decimal form? (5.91) Partner B, where should the point be plotted? (The point should be very close to 6 but directly before it.) Model how to plot the point and label it as 35. Step 3: Have student pairs complete Questions 3 and 4 and then review the answers as a whole group. Step 4: Partner A, now that we ve plotted our four points, what do you notice about the points? (They are in order from least to greatest.) Partner B, why was it helpful to convert the square roots to rational values before plotting them on the number line? Explain your thinking. (Converting the square roots to a rational value gave us a more accurate value to plot on the number line.) Partner A, why was it valuable to see them all on a number line instead of simply ordering them from least to greatest? Justify your thinking (It s very simple to order the square roots from least to greatest because we are looking at the whole numbers. Looking at the values on the number line helps us to see how far apart the decimals really are.) IP, CP, WG: Have students complete the rest of S50 by writing rational approximations and plotting points for Questions 5 8. Be sure to take a moment to review the students solutions as a whole group. {Verbal Description, Graphic Organizer, Graph}

13 T104 Mathematics Success Grade 8 Comparing Irrational Numbers (M, GP, CP, WG, IP) S51 (Answers on T119.) M, GP, CP, WG: Have students turn to S51 in their books. In this activity students will complete the chart to help them understand how to compare irrational numbers abstractly. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Graphic Organizer} MODELING Comparing Irrational Numbers Step 1: Direct students attention to Question 1. Partner A, identify what type of number is on the left of Question 1. (Irrational) Partner B, identify what type of number is on the right of Question 1. (Rational) Have partners discuss and then explain what we can do to compare these two numbers? (Find a rational approximation for the number on the left and compare it to the number on the right.) Partner A, what is the square root and perfect square that is closest to but less than 62? ( 49 = 7) Partner B, what is the square root and perfect square that is closest to but greater than 62? ( 64 = 8) Partner A, explain how to find the rational approximation of 62. (We can create a fraction that represents the difference between the number and the lower perfect square which is the numerator over the difference between the perfect squares which is the denominator.) Write your fraction Partner B, What is the rational approximation in the form of a mixed number? Partner A, explain how we can write the approximation as a decimal. (Divide the numerator by the denominator in the fraction, then add the whole number.) Partner B, what place value do we need to look at to compare the two values? (the ones place) Partner A, if we compare the place value of the ones, which one is greater? (8.12) Record the less than sign inside of the circle.

14 Mathematics Success Grade 8 T105 Step 2: Direct students attention to Question 2 on S51. Partner B, what type of number is the number on the left of Question 2? (Irrational) Partner A, what type of number is the number on the right of Question 2? (Rational) Explain what we can do to compare these two numbers. (Find a rational approximation for the number on the left and compare it to the number on the right.) Partner B, what is the square root and perfect square that is closest to but less than 29? ( 25 = 5) Partner A, what is the square root and perfect square that is closest to but greater than 29? ( 36 = 6) Partner B, explain how to find the rational approximation of 29 using the information from the chart. (We can create a fraction that represents the difference between the number and the lower perfect square - which is the numerator - over the difference between the perfect squares - which is the denominator.) Write your fraction Partner A, What is the rational approximation in the form of a mixed number? Partner B, explain how we can write the approximation as a decimal. (Divide the numerator by the denominator in the fraction, then add the whole number.) Partner A, what is the decimal form? (5.36) Partner B, what is the rational number on the right written as a decimal? (5.33) Partner A, if we line up the decimals of 5.36 and 5.33, which one is greater? (5.36) Record the greater than sign inside of the circle. IP, CP, WG: Have students complete Questions 3 6. After student pairs have completed the activity, review the answers as a whole group. Give students the opportunity to explain and justify their answers to review the process of approximating irrational numbers with rational values. {Verbal Description, Graphic Organizer}

15 T106 Mathematics Success Grade 8 Ordering Numbers Group Activity (M, GP, CP, WG) T123, T124, T125 for Cards M, GP, IP, CP, WG: Students will complete this activity to practice all of the skills that they have learned in the lesson. This is a group activity that requires teacher modeling and guiding. MODELING Ordering Numbers Group Activity *Teacher Note: Cutting out the cards ahead of time will make the activity run smoother. Should you have more than 36 students, you may create more cards by simply continuing up the number line with more irrational numbers. You also have the option to have students work in cooperative pairs. Be sure to mix cards so that they are not already in order for students. You will need to create a large classroom number line or use a number line that is available. Students will each be placing their sticky notes above a speci fic number on the line, so be sure they have enough room to complete the activity. The number line only needs to be labeled from 1 to 7 according to the cards provided. If you add cards that have larger numbers, you may need to adjust your number line. Step 1: At this time, distribute one card to each student and two sticky notes to each student. The first portion of the activity requires students to identify the rational approximation of the irrational square root they are given. Once students have seen what square root they received, ask them to place a sticky note over that square root, as to hide it from other students seeing. On this sticky note, students will find the rational approximation of their square root. If necessary, students can look back at S47 as a reference. If students need to, they may use the pictorial representation of the counters to help them. Step 2: At this time, direct students to the second sticky note. Now, students will need to convert their mixed number into a decimal. As a class review division of the numerator and denominator to find the decimal. On the second sticky note, have students write the original mixed number and the decimal value. Have students stick this to the back of their card so that there is now one sticky note on each side.

16 Mathematics Success Grade 8 T107 Step 3: Students will now begin the ordering activity. At this time, students will begin to form a human number line. Have students with decimals that are between 1 and 3.5 move to one side of the room and the students with decimals from 3.5 to 7 move to the other side of the room. Have students begin ordering themselves by looking at the decimals that they have calculated on the second sticky note. Once students are sure that they have their decimals in order from least to greatest, have students discuss with their neighbors the fraction that resulted from the rational approximation. Students standing next to each other should verify each other s work. Finally, start at 1 and have students reveal their original square root. If ordered correctly, the numbers should go in order from least to greatest. Students will be able to check this with their neighbors as they go. Step 4: Students will now complete the activity with the number line. Now that students have ordered their numbers and have worked together to verify the correct approximation for the irrational square roots, we want to see them place their numbers on the number line. On the second sticky note, ask students to now write their original irrational square root. Using their second sticky note, call groups of students to come to the number line and post the note that shows the original number, the mixed number approximation and the decimal. After students have posted, it s great to ask questions about why certain numbers are closer to whole numbers while others are not. The goal of this activity is that students have shown their understanding of approximation, they work with other students to verify solutions and they are able to plot their values on a number line after translating to decimal form. SOLVE Problem (WG, CP, IP) S52 (Answers on T120.) Remind students that the SOLVE problem is the same one from the beginning of the lesson. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. (Students have worked with finding the rational approximation for irrational values and compared and ordered them.) {SOLVE, Verbal Description, Graphic Organizer, Graph}

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