ࡱ>  oUGo] bjbj Wxx: kkk8k$4g&"KIaIaIaI[4]4]4]4]4]4]4$v8;4k 4aIaI4===aIkaI[4=[4==)Xh-aIܵT"+(G4404C+h;P;P--\;k.@=44=4; : Grade 4 Mathematics Unit 6: Fractions and Decimals Time Frame: Approximately five weeks Unit Description This unit develops an understanding of fractions with denominators through twelfths and decimals through hundredths. The unit focuses on fraction and decimal equivalents and adding, subtracting, and multiplying fractions. Student Understandings Students develop a strong understanding of fractions with denominators through twelfths. They use this understanding to compare and order fractions and to find equivalent fractions. Students are able to read, write and relate decimals through hundredths and connect them to fractions. Students are able to decompose fractions from mixed numbers and improper fractions into fractional units to add, subtract and multiply. Guiding Questions Can students model, read, write, compare, order and represent fractions with denominators through twelfths using region and set models? Can students estimate fractional amounts from pictures, models, and diagrams? Can students read, write and represent decimals through hundredths? Can students connect decimals with decimal fractions and find equivalent decimals for , ,? Can students generate equivalent fractions? Can students add, subtract fractions and mixed numbers with like denominators and add fractions with denominators of 10 and 100? Can students multiply a fraction by a whole number? Can studnts solve word problems involving fractions? Unit 6 Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS) Grade-Level ExpectationsGLE #GLE Text and BenchmarksNumber and Number Relations5.Read, write, and relate decimals through hundredths and connect them with corresponding decimal fractions (N-1-E) 6.Model, read, write, compare, order, and represent fractions with denominators through twelfths using region and set models (N-1-E) (A-1-E)7.Give decimal equivalents of halves, fourths, and tenths (N-2-E) (N-1-E)9.Estimate fractional amounts through twelfths, using pictures, models, and diagrams (N-2-E)CCSS for Mathematical ContentCCSS #CCSS TextNumber and Operations Fractions (NF)4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n a) / (n b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples:  EMBED Equation.3 = EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 ;  EMBED Equation.3 = EMBED Equation.3 + EMBED Equation.3 ; 2 EMBED Equation.3 =1 + 1+ EMBED Equation.3 = EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 . Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. 4.NF.4Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent  EMBED Equation.3 as the product 5(( EMBED Equation.3 ), recording the conclusion by the equation  EMBED Equation.3 = 5(( EMBED Equation.3 ). Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3(( EMBED Equation.3 ) as 6(( EMBED Equation.3 ), recognizing this product as  EMBED Equation.3 . (In general, n(( EMBED Equation.3 )= EMBED Equation.3 . Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? 4.NF.5Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express  EMBED Equation.3 as EMBED Equation.3 and add  EMBED Equation.3 + EMBED Equation.3 =  EMBED Equation.3 Measurement and Data4.MD.4Make a line plot to display a data set of measurements in fractions of a unit ( EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 ). Solve problems involving addition and subtraction of fractions by using information presented in line plots. ELA CCSSCCSS #CCSS TextWriting StandardsW.4.2dWrite informative/explanatory texts to examine a topic and convey ideas and information clearly. Use precise language and domain-specific vocabulary to inform about or explain the topic.Speaking and Listening StandardsSL.4.1 Engage effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grade 5 topics and texts, building on others ideas and expressing their own clearly. Follow agreed-upon rules for discussions and carry out assigned roles. Pose and respond to specific questions by making comments that contribute to the discussion and elaborate on the remarks of others. Review the key ideas expressed and draw conclusions in light of information and knowledge gained from the discussions. Sample Activities Activity 1: Fraction and Decimal Vocabulary Cards (CCSS: W.4.2d) Materials List: paper, pencil, index cards, zip-top bag or envelope for the vocabulary cards. Have students create fraction and decimal vocabulary cards, ( HYPERLINK "http://www.louisianaschools.net/lde/uploads/11056.doc" view literacy strategy descriptions) for the following terms: numerator, denominator, unit fraction, equivalent fractions, decimals, tenths, and hundredths. Vocabulary knowledge is one of the most essential pieces of understanding mathematics. Vocabulary cards will help students learn the content-specific terminology necessary for higher order understanding. Each vocabulary card has four parts: the definition, the characteristics, an example, and an illustration. These vocabulary cards will be used throughout the unit to review the key terms for fractions and decimals and to serve as future reference cards to deepen the understanding of fractions and decimals. Have students store the vocabulary cards in a zip-loc bag or in an envelope. Vocabulary card example: Definition Characteristics the number of parts of the number above the line the whole that are being in a fraction considered numerator Illustration Example  numerator 1  2 numerator Activity 2: Fractions as Regions and Sets (GLEs: 6) Materials List: Fractions as Regions and Sets BLM Review with students the definitions of numerator and denominator. Tell students that the denominator is the number of parts in a whole and that the numerator is the number of parts of the whole that are being considered. Ask students to draw a model of  EMBED Equation.3 . Some students may draw a square divided into 4 equal parts, with 1 part shaded. Others might draw a circle or a rectangle divided into 4 equal parts with 1 part shaded. These students have drawn a region model of the fraction  EMBED Equation.3 . Some students might draw 4 circles with 1 shaded in. Others might draw 4 stars or flowers or other objects, with 1 of the objects shaded in. These students have drawn a set model of the fraction  EMBED Equation.3 . Both models are very important to the understanding of fractions. Provide the students with the Fractions as Regions and Sets BLM. Ask students to tell what they notice about the models on the BLM. Explain to the students that the parts that are being considered in these pictures are the shaded parts of the whole. Ask students to identify the numerator and the denominator for each of the examples. Discuss the similarities and differences among the four examples. Guide the discussion to have students realize that all four examples show  EMBED Equation.3 . The fourth example could also be interpreted as  EMBED Equation.3 . Discuss the difference between the region model and the set model of fractions. A region model shows the whole divided into congruent parts. A set model uses a group of congruent objects as a whole. One way to illustrate the set model is to use students themselves. Call 4 students to come to the front of the class. Tell the other students to name a fraction that shows the number of students wearing tennis shoes to the whole group (possibly  EMBED Equation.3 ). Ask other questions about fractional parts of the group of 4 students, such as: What fraction of the group is wearing glasses? (possibly  EMBED Equation.3 ) What fraction of the group has blue eyes? (possibly  EMBED Equation.3 ) What fraction of the group is under 12 years old? (possibly  EMBED Equation.3 ) Provide students with multiple opportunities to identify the fractions in regions and sets as well as have students draw regions and sets for a given fraction. Activity 3: Exploring Fractions and Decimals (GLEs: 5, 7; CCSS: 4.NF.1) Materials List: base-10 blocks, Hundredths Grid Paper BLM, Place Value Chart BLM, pencils Students will use their knowledge of place value to explore how fractions and decimals are related using base-10 blocks. Provide students with the Hundredths Grid Paper BLM and base-10 blocks. Explain to the students that one flat will represent 1 whole unit. Have them place 1 unit cube on the flat or on the grid paper. Ask the number of cubes it would take to fill the entire grid. (100) Ask them how much 1 cube would represent (1 square out of 100 squares or EMBED Equation.3 ). Show them that  EMBED Equation.3 can be written as the decimal, 0.01. Place 10 cubes in a row on the flat or grid paper. Ask students to name this fraction ( EMBED Equation.3 ). Tell them that this can be written as the decimals 0.10. Ask them how many columns are covered (1) and how many columns are shown on the flat or grid paper (10). Have them replace the 10 cubes with 1 rod. Ask them to name the fraction covered now (1 column out of 10 columns or  EMBED Equation.3 ). Tell students that fraction  EMBED Equation.3 is written as the decimal 0.1. Tell students that  EMBED Equation.3 =  EMBED Equation.3 = 0.10 and 0.1. Introduce the word equivalent. Have students place 3 rods on top of the hundredths grid paper or shade in 3 columns on the paper. Display a place-value chart similar to the one below and distribute the Place Value Chart BLM to students. Have them state all of the fractions and decimals that name this amount ( EMBED Equation.3 , EMBED Equation.3 , 0.30, 0.3). Call out the fraction  EMBED Equation.3 and have students record it in the place value chart. Discuss how to write the decimal in word form and how to read the decimal. Have students record 0.3 and state the decimals in words. Tell students that decimals less than 1 whole are written with a zero in front of the decimal point to show that there are no whole numbers. Tell students that 0.30 should be read as three hundredths, not as zero point three zero. Example: HundredsTensOnes.Tenths Hundredths0.300.30.050.43 Continue modeling by showing that 0.05 is 5 hundredths  EMBED Equation.3 . Discuss how there is an 0 in the tenths place, so there must be an 0 between the decimal and the hundredths digit. Record this number in the place-value chart. Model 0.43 to show that it is the same as 4 tenths and 3 hundredths,  EMBED Equation.3 , or  EMBED Equation.3 and  EMBED Equation.3 . Discuss how the number can be shown using expanded form (0.4 + 0.03). Record this number in the place-value chart. Call out additional numbers for students to record on the Place Value Chart BLM. Have students model  EMBED Equation.3  on the grid paper. Ask how many cubes would be needed to cover  EMBED Equation.3 of the grid paper (50). Ask students to write an equivalent fraction and decimal for  EMBED Equation.3 . ( EMBED Equation.3 and 0.50) Ask how many rods would be needed to cover  EMBED Equation.3 of the grid paper (5). Ask students to write another equivalent fraction and decimal for  EMBED Equation.3 . ( EMBED Equation.3 and 0.5) Continue showing the fractions EMBED Equation.3 ,  EMBED Equation.3 and  EMBED Equation.3 , relating them to their fractional and decimal equivalents. Provide students with multiple examples for practice, stopping to discuss how each fraction is related to its decimal equivalent as needed. Observe how the students read, write, and model each fraction and decimal. Extend the activity by incorporating whole numbers with the decimals. Model how to read, write, and model each of the numbers. Gradually have students work only with pencil and paper and take away the base-ten blocks so they can rely on their knowledge of place value to explain how they found each digits value. Activity 4: Fractions and Decimals on Grids (GLEs: 5, 7; CCSS: 4.NF.1) Materials List: Fractions and Decimals on Grids BLM (2 pages), crayons, math learning log, pencil Use the Fractions and Decimals on Grids BLM to demonstrate fractions as parts of a whole. Help students understand the connection between fractions and decimals by relating them through money. Discuss the denominators that will be used to show varying amounts of money (the denominator for pennies in a dollar is 100, for dimes in a dollar is 10, for quarters in a dollar is 4, and for a 50-cent piece is 2.) Since students do not have to know denominators of 20, do not include nickels in this activity. Demonstrate for students how to color in 50 on the first grid of the Fraction and Grids BLM. Discuss the equivalent decimals and fractions for 50 ( EMBED Equation.3 , 0.50,  EMBED Equation.3 , 0.5,  EMBED Equation.3 ). Have the students complete the other grids on their own, stopping for class discussions when needed. (For example, 1 quarter is 0.25 or  EMBED Equation.3 , or  EMBED Equation.3  of a dollar. Show that  EMBED Equation.3 is 25 cents out of 1 dollar and since there are 4 quarters in 1 dollar, 25 cents can also be written as  EMBED Equation.3  of a dollar.) Use other examples such as 6 dimes is .60 or  EMBED Equation.3 , or EMBED Equation.3  of a dollar and 3 quarters is .75 or  EMBED Equation.3 , or  EMBED Equation.3  of a dollar, etc. Have the students create an Equivalence Table in their math learning logs ( HYPERLINK "http://www.louisianaschools.net/lde/uploads/11056.doc" view literacy strategy descriptions). This table will be used as a reference for future activities as well as a study guide for assessments. Have students add other money amounts and their corresponding fractions and any other equivalences as the unit continues. Example: Money amountEquivalent decimalEquivalent fraction1 penny = $0.010.01 EMBED Equation.3 1 dime = $0.100.10 = 0.1 EMBED Equation.3 =  EMBED Equation.3 25 pennies = $0.25.025 EMBED Equation.3 =  EMBED Equation.3 2 quarters = $0.500.50 = 0.5 EMBED Equation.3 =  EMBED Equation.3 =  EMBED Equation.3  Activity 5: Comparing Fraction Benchmarks (GLE: 6) Materials List: construction paper cut into 12 inch strips in a variety of colors, scissors, paper, pencils, math learning log Using 12 strips of construction paper, have students create individual fraction strips. Have students begin with one color strip and fold it in half. Ask the students questions such as the following: How many pieces or parts do you have? (2) What could you call each part? ( EMBED Equation.3 ) Have them mark each part as  EMBED Equation.3 . Using another color strip, have students fold it in half, and then in half again. This will show fourths. Ask questions similar to the above questions. Have them mark each part as  EMBED Equation.3 . Continue with eighths, thirds, sixths, twelfths, etc. Have students cut each fraction strip at the creases. Display the following fractions:  EMBED Equation.3 and  EMBED Equation.3 . Ask students what these fractions have in common. (They have the same denominator.) Explain that both fractions have the same size whole, so students can compare the two fractions based on the number of parts being considered. Have students use their fraction strips to make  EMBED Equation.3 and  EMBED Equation.3 . Have the students compare  EMBED Equation.3 and  EMBED Equation.3 .) Continue having students model fractions with the same denominators. Students should see that if the denominators are the same, the larger numerator will show the larger fraction. Tell the students to think of eating a pizza. If two pizzas are cut into the same size pieces (the denominator), they get more pizza if they get more pieces (the numerator). Just use 2 fractions in this activity. Continue the activity with fractions in which the numerators are the same, but the denominators are different. Display the following fractions:  EMBED Equation.3 ,  EMBED Equation.3 . Ask students what these fractions have in common. (They have the same numerators.) Explain that the fractions have the same size whole, so they can compare the fractions accurately. Have students use their fraction strips to make  EMBED Equation.3 and  EMBED Equation.3 . Have students compare the two fractions. Continue having students model fractions with the same numerator. Tell students to think of eating a pizza again. This time the pizzas will be cut into different numbers of slices. A pizza cut into 8 pieces will have smaller slices than a pizza cut into 6 pieces. Since the numerator is the same, the students will get the same number of slices from each pizza. Ask them from which pizza will they get the most to eat. Students should see that if the numerators are the same, the larger denominator means that the whole is split into more parts. The more parts, the smaller the parts will be. Ask students how they know whether a fraction is equal to  EMBED Equation.3 , greater than  EMBED Equation.3 , or less than  EMBED Equation.3 ? Explain to students that if the numerator is exactly half as large as the denominator, the fraction is equal to  EMBED Equation.3  (Ex:  EMBED Equation.3 or  EMBED Equation.3 ). If the numerator is less than half as large as the denominator, the fraction is less than  EMBED Equation.3  (Ex:  EMBED Equation.3 or  EMBED Equation.3 ). If the numerator is greater than half as large as the denominator, the fraction is greater than  EMBED Equation.3  (Ex:  EMBED Equation.3 or  EMBED Equation.3 ). Give students the following fractions:  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . Have the students determine in which fraction the numerator is half as large as the denominator ( EMBED Equation.3 ). Have the students write all the fractions with denominators of 8 that are less than a whole but that are greater than  EMBED Equation.3  ( EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 ). Have the students write all the fractions with denominators of 8 that are less than  EMBED Equation.3 but that are greater than 0 ( EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 ). Give students other fractions to compare to  EMBED Equation.3 . Have students draw 3 columns on a sheet of paper. Label one column as <  EMBED Equation.3 , one column as = EMBED Equation.3 , and the third column as >  EMBED Equation.3 . Call out different fractions and have students place them in the correct column. Teacher Note: The reason that determining if a fraction is <, >, or =  EMBED Equation.3 is so important is that the concept can be used to help students compare fractions. For the fractions  EMBED Equation.3 and  EMBED Equation.3 , all students have to think that  EMBED Equation.3 is less than  EMBED Equation.3  and  EMBED Equation.3 is greater than  EMBED Equation.3 , so  EMBED Equation.3 is less than  EMBED Equation.3 . Continue the activity with fractions in which they are all one piece away from being a whole. Give students the following fractions:  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 . Ask students what the fractions have in common. (They are all missing exactly one piece.) Reiterate that all of these fractions have the same size whole, so they can be accurately compared. Have students use their fraction strips to construct  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . Have students order the fractions from least to greatest ( EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 ). Continue having students model fractions that are missing one piece. Students should see that if the fractions are missing one piece, the number of pieces being considered is almost the same as the whole. The greater the number of pieces in the whole, the smaller the number of pieces being considered.  EMBED Equation.3 is  EMBED Equation.3 away from one whole.  EMBED Equation.3 is less than  EMBED Equation.3 or  EMBED Equation.3  because the whole is divided into smaller pieces. Give students plenty of fractions to compare using these strategies. Have students explain how they ordered each of the fraction sets. Teacher Note: Students should be taught to use number sense and these strategies to compare fractions before they learn about common denominators or use calculators to change the fractions to decimals to compare. Give students fractions for which they can use the above strategies. Activity 6: Comparing Fractions Using Fraction Strips (GLEs: 6; CCSS: 4.NF.1) Materials List: fraction strips from Activity 5, paper, pencils Have students use the fraction strips from Activity 5 to make comparisons. Have students compare  EMBED Equation.3  to  EMBED Equation.3  to determine which is greater. Remind them that they are comparing parts of the same whole. Continue with other fractions. Students need to realize that the more parts that a fraction is cut into, the smaller the parts will be. Have them order the fractions,  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . Having students compare fraction pieces with other fraction pieces helps them develop fraction sense by exploring the size relationship between two fractions, for example,  EMBED Equation.3 and  EMBED Equation.3 . Have students compare the  EMBED Equation.3  strip with the  EMBED Equation.3 strip to help them understand that it takes two  EMBED Equation.3 s to make a  EMBED Equation.3 . This idea can help students understand that  EMBED Equation.3 names the same region as  EMBED Equation.3 . Have them record their findings in their math learning logs ( HYPERLINK "http://www.louisianaschools.net/lde/uploads/11056.doc" view literacy strategy descriptions), as well as any observations they make about the equivalent fractions. Demonstrate on the board how to construct the following table. FractionEquivalent fractionEquivalent fractionEquivalent fractionEquivalent fractionEquivalent fraction EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  Observation: The denominators of equivalent fractions are multiples of the first denominator. Activity 7: Comparing and Ordering Fractions using Models (GLEs: 6) Materials List: fraction models, paper, pencils Have students draw models for different fractions (fourths and halves, thirds and sixths, fourths and eights, etc.) emphasizing that the wholes need to be the same size so that students can compare them later. For example:   Have students look at the models and compare which fraction is larger. ( EMBED Equation.3 >  EMBED Equation.3 ) Give students multiple opportunities to practice comparing fractions with denominators up to twelve. Give students three fractions to order from least to greatest. Have students draw and label each fraction model. For example, have students order  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 from least to greatest ( EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 ). Have the students draw three congruent wholes. Have the students draw and label each model and determine the order of fractions. Provide students with multiple opportunities to compare three fractions at a time. Make sure to include fractions with denominators through twelfths. Activity 8: Using Fraction Models to Demonstrate Equivalent Fractions (GLE: 6; CCSS: 4.NF.1, SL.4.1c) Materials List: Fraction Models BLM, paper, pencils Provide each student with the Fraction Models BLM. Ask students what fraction is modeled by the rectangles in row 1 ( EMBED Equation.3 ). Tell students to leave model A as it is showing  EMBED Equation.3 . In model B, have students draw a horizontal line through the center of the model to show that the number of equal parts doubles and the size of the parts is halved. Model this by multiplying  EMBED Equation.3  by  EMBED Equation.3  ( EMBED Equation.3 ). Tell students that they can do this because they are increasing the number of shaded parts and the number of parts in the whole by the same number. The comparison of shaded parts to number of parts in the whole remains the same. Discuss how the two fractions ( EMBED Equation.3  and  EMBED Equation.3 ) are still equal because their areas are equivalent regardless of the number of lines or parts the fraction has. In model C, have students draw two horizontal lines to divide the model into thirds. Tell students that this would give them 3/6, because they are multiplying  EMBED Equation.3 by  EMBED Equation.3 . In model D, have students draw three horizontal lines to divide the model into fourths. Tell students that this would give them 4/8, because they are multiplying  EMBED Equation.3 by  EMBED Equation.3 . Explain to students that all of the fractions are equivalent; that they name the same amount. Have students make connections among the similarities of the comparison of the number of parts in the whole to the number of parts being considered in each model. Have students begin to generate a rule for writing equivalent fractions. Example:      Continue this same process with the models E H to show equivalency. Activity 9: Estimating Fractions (GLEs: 6, 9) Materials List: paper, pencils, Estimating Fractions BLM Provide the students Estimating Fractions BLM that includes regional models of fractions that do not show the number of congruent parts. For example:  Have students look at the model to estimate the fraction. Engage students in questioning the content ( HYPERLINK "http://www.louisianaschools.net/lde/uploads/11056.doc" view literacy strategy descriptions) to help them estimate the fraction. Initiate the discussion by asking the students what the content is about (estimating fractions). Focus on the contents message by asking students what the word estimating means (to make an informed or educated guess). Link the information in the model to previous fraction activities.. Ask students how many parts the rectangle A has. (2) Ask if this means that the part being considered is  EMBED Equation.3 . (No, the fraction has to have equal parts in order for it to be  EMBED Equation.3 .) Tell students that they are estimating the fraction of the whole that is being considered without knowing the number of parts in the whole. Tell students that since they do not know the number of parts, they have to use their knowledge of fractions to be able to estimate what the fraction is. Link this to previous activities by asking which ways they compared fractions (closer to  EMBED Equation.3 , missing piece, same numerator, same denominator). Ask students which strategy(ies) would be most helpful here (closer to  EMBED Equation.3 , missing piece). Have students use these strategies to compare the model to  EMBED Equation.3 . (It is greater than  EMBED Equation.3  but less than 1 and is closer to EMBED Equation.3 than to 1.) Have students brainstorm different fractions that are closer to  EMBED Equation.3  than to 1 ( EMBED Equation.3 , EMBED Equation.3 , etc.). Have students divide the model into thirds or fifths to see if either of those fractions equals the part being considered in the model ( EMBED Equation.3 ). Identify any problems with understanding by asking if the process makes sense. Have students summarize the process by asking them what they would need to do to figure out the fraction for the other models. Give students multiple opportunities to practice. Encourage them to engage in the same questioning the content process to estimate the unknown fractional amount. Have them continue to reference the strategies used in previous units. Activity 10: Using Fraction Strips to Add and Subtract Fractions: Part I (GLEs: 6; CCSS: 4.NF.3a, 4.NF.3b) Materials List: fraction strips from activity 5, math learning log, pencil Have students use the fraction strips created in Activity 5 to group the fraction parts by the number of parts in a whole (all the halves, thirds, fourths, etc). Have students put together all of the third parts to make  EMBED Equation.3 or one whole. Have the students separate the parts to see that 3/3 =  EMBED Equation.3 +  EMBED Equation.3 +  EMBED Equation.3 . This is called fraction decomposition. Ask students if they see any similarities between fraction decomposition and multiplication. Discuss with students that multiplication is repeated addition. Remind them that 2 ( 4 means 2 groups of 4 or 2 fours or 4 + 4. Therefore,  EMBED Equation.3 means 3 groups of  EMBED Equation.3 or 3 one-thirds or  EMBED Equation.3  +  EMBED Equation.3 +  EMBED Equation.3 . If  EMBED Equation.3 is  EMBED Equation.3  +  EMBED Equation.3 +  EMBED Equation.3 , then  EMBED Equation.3 is the same as 3   EMBED Equation.3 . Give students other fractions such as 3/8 and have them create the fraction using their fraction strips and then separate the unit fractions to show that  EMBED Equation.3 =  EMBED Equation.3 +  EMBED Equation.3 +  EMBED Equation.3 . A unit fraction is a fraction with 1 as the numerator. Have students write these equations down as they go, so that they see the connection between the fraction strips and addition of fractions. Breaking fractions into unit fractions helps students make distinctions in the roles of the numerator and the denominator in addition and subtraction of fractions. Students need to understand that the denominator simply tells the number of parts in the whole and the numerator tells the number of those parts. A common misconception when students add and subtract fractions is to add and subtract the denominator as well as the numerator. For example, students adding  EMBED Equation.3 +  EMBED Equation.3 would get  EMBED Equation.3 instead of  EMBED Equation.3 . Have students use fraction strips to add  EMBED Equation.3  +  EMBED Equation.3 . When the students add the parts, they should see that there are 5 parts being considered and 5 parts in the whole. This process will help them realize they are adding a number of fifths rather than adding 5. Students need to think that 2 fifths and 3 fifths, simply gives 5 fifths. Give students other addition problems with like denominators to practice. Have students solve fraction subtraction problems like  EMBED Equation.3   EMBED Equation.3 =  EMBED Equation.3 . Have them make  EMBED Equation.3 with their fraction strips and subtract  EMBED Equation.3 to determine the difference of  EMBED Equation.3 . Give them multiple opportunities to do this making sure they write down each of the equations as they solve the problems. Activity 11: Using Fraction Strips to Add and Subtract Fractions: Part II (CCSS: 4.NF.3a, 4.NF.3b) Materials List: fraction strips from Activity 5, paper, pencil Have students work in a group of 4. Have each student display EMBED Equation.3 using one of the fraction strip pieces from Activity 5. Have one student find other strips that could be used to show the same amount. This student might find two  EMBED Equation.3  pieces. Tell the students that EMBED Equation.3 can be decomposed into  EMBED Equation.3 +  EMBED Equation.3 . Help students realize that  EMBED Equation.3 =  EMBED Equation.3  +  EMBED Equation.3  =  EMBED Equation.3 and that  EMBED Equation.3 is another way to represent the amount  EMBED Equation.3 . Have the other students in the group find different ways of representing using other fraction pieces. One student might use three  EMBED Equation.3  pieces. That student decomposed into  EMBED Equation.3  +  EMBED Equation.3 +  EMBED Equation.3 . Discuss that  EMBED Equation.3 =  EMBED Equation.3 +  EMBED Equation.3 +  EMBED Equation.3 =  EMBED Equation.3 and that  EMBED Equation.3 is another way to show the same amount as  EMBED Equation.3 . Another student might decompose  EMBED Equation.3  into four  EMBED Equation.3  pieces illustrating that  EMBED Equation.3 =  EMBED Equation.3  or into six  EMBED Equation.3 pieces illustrating that  EMBED Equation.3 =  EMBED Equation.3 . More advanced students might decompose  EMBED Equation.3  into one  EMBED Equation.3 piece and two  EMBED Equation.3 pieces illustrating that  EMBED Equation.3  =  EMBED Equation.3 +  EMBED Equation.3 +  EMBED Equation.3 . Give groups other fractions such as  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 and have them decompose these fractions into other fractions. Ask students how they could represent  EMBED Equation.3 into other fraction pieces. Students may see that  EMBED Equation.3 could be decomposed into two  EMBED Equation.3  pieces and that each piece could be decomposed into two  EMBED Equation.3 pieces, thus giving  EMBED Equation.3 =  EMBED Equation.3 . Lead students to see that what they are doing is showing the same amount with a different numerator and denominator. Teacher Note: Students will use this idea of decomposing one fraction to help them add and subtract two fractions where one denominator is a multiple of the other denominator. Give students the problem  EMBED Equation.3 +  EMBED Equation.3 . Have students show these amounts using fraction strips. Remind students that the denominators are not added together and that the denominators must be the same. Ask students which fraction they could decompose to make both fractions have the same denominator ( EMBED Equation.3 ). Have the students use the fraction strips to show that  EMBED Equation.3  =  EMBED Equation.3  +  EMBED Equation.3 , so the problem would now be  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3  or  EMBED Equation.3 . Give students the following problems:  EMBED Equation.3  +  EMBED Equation.3  ( EMBED Equation.3 ),  EMBED Equation.3 +  EMBED Equation.3  ( EMBED Equation.3 )  EMBED Equation.3  +  EMBED Equation.3  ( EMBED Equation.3 ),  EMBED Equation.3 +  EMBED Equation.3  ( EMBED Equation.3 ),  EMBED Equation.3  +  EMBED Equation.3  ( EMBED Equation.3 ),  EMBED Equation.3 +  EMBED Equation.3  ( EMBED Equation.3 ) When students become proficient, have them add fractions such as these:  EMBED Equation.3 +  EMBED Equation.3  ( EMBED Equation.3 ),  EMBED Equation.3 +  EMBED Equation.3  ( EMBED Equation.3 ), etc. Use this process to show subtraction of two fractions where one denominator is a multiple of the other denominator such as  EMBED Equation.3   EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3 , etc. Activity 12: Decomposing Mixed Number Fractions (GLEs: 6; CCSS: 4.NF.3a, 4.NF.3b, 4.NF.3c, 4.NF.3d, SL.4.1d) Materials List: fraction strips from Activity 5, paper, pencil Have students make predictions using directed learning-thinking activity ( HYPERLINK "http://www.louisianaschools.net/lde/uploads/11056.doc" view literacy strategy descriptions) or DL-TA. In this scaffolded activity, have students make predictions and then check their predictions before and after learning about fractions. This process teaches students to self-monitor their learning, which helps increase their attention and their curiosity. Activate their background knowledge by discussing how fractions are made up of individual parts. For example  EMBED Equation.3 is really  EMBED Equation.3 and  EMBED Equation.3 put together. Have the students use their fraction strips from Activity 6 to help them. Have students create mixed numbers by asking them to add  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3 . Students should realize they have more parts being considered than parts in a whole. Ask students how many wholes are represented in that fraction (1). Ask them what fraction of a whole is left over ( EMBED Equation.3 ). Tell them that this is a mixed number because it is a combination of whole numbers and fractions. Ask the students how many parts are being considered (5). Ask students where they usually write the number of parts being considered (as the numerator). Ask students how many parts are in a whole (4). Tell them that for an improper fraction, they should write the number of parts being considered over the number of parts in a whole regardless of whether the numerator is larger than the denominator. The numerator having the same number of or more parts being considered makes the fraction improper. In other words, the numerator is the same number or is greater than the denominator. A proper fraction has fewer parts being considered than the total number of parts in a whole. In other words, the numerator is smaller than the denominator. Tell the students that they will be making predictions about how to add and subtract fractions, improper fractions, and mixed numbers. Give students the problem: 1 EMBED Equation.3   EMBED Equation.3 = n. Have them make a prediction about how they will be able to find their answer. Guide the students through setting up the problem using fraction strips or another visual model of the fractions. For example:   Discuss with students how fractions can be decomposed into their unit fractions, a fraction with a numerator of one. Using the example above, 1 EMBED Equation.3 is the same as  EMBED Equation.3 +  EMBED Equation.3 +  EMBED Equation.3 +  EMBED Equation.3 +  EMBED Equation.3 and  EMBED Equation.3 is  EMBED Equation.3 +  EMBED Equation.3 +  EMBED Equation.3 . Discuss the different parts of a fraction (whole number, numerator, denominator) and how they are related. Have students stop and make any changes to their prediction. Some students may see the connection that they can subtract three  EMBED Equation.3 s and end up with two  EMBED Equation.3 s or  EMBED Equation.3 or  EMBED Equation.3 . Other students may decompose the mixed number into  EMBED Equation.3 +  EMBED Equation.3  and then combine the  EMBED Equation.3 and  EMBED Equation.3  to get  EMBED Equation.3 . They then would subtract from the  EMBED Equation.3 to get  EMBED Equation.3 or  EMBED Equation.3 . Have students check their predictions. Give students other examples such as  EMBED Equation.3 +  EMBED Equation.3 = n or  EMBED Equation.3   EMBED Equation.3 = n. Have students decompose the fraction into parts to be able to add or subtract the fractional parts to arrive at their answer. Have students make predictions before, during, and after they solve the problems. Extend the activity by giving students word problems. For example, Jeff and Kelly bought three buckets of crabs. Jeff ate 1 EMBED Equation.3  buckets and Kelly ate  EMBED Equation.3  of a bucket. How many of the buckets of crabs did Jeff and Kelly eat together? (Jeff ate  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3  and Kelly ate  EMBED Equation.3 .  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3 +  EMBED Equation.3 = 2, so Jeff and Kelly ate 2 buckets of crabs.) Have the students make predictions about what the answer will be before, during, and after they solve the problem. Use the predictions from the class as a discussion topic. Ask students what they expected to learn and what they actually learned. Have them continue this prediction process as they become exposed to new content. 2013 - 2014 Activity 13: Fraction Equivalency with Denominators of 10 and 100 (CCSS: 4.NF.5) Materials List: base-10 blocks, Hundredths Grid Paper BLM, Place Value Chart BLM, pencils Provide students with base-10 blocks, Hundredths Grid Paper BLM, and the Place Value Chart BLM. Tell students that each grid represents 1 whole. Tell the students that they are going to add  EMBED Equation.3 and  EMBED Equation.3 . Tell students to make  EMBED Equation.3 using the base-10 blocks on the Hundredths Grid Paper BLM. Have them write  EMBED Equation.3 in the Place Value Chart BLM. Ask students how many hundredths are equal to  EMBED Equation.3  ( EMBED Equation.3 ). Have students write  EMBED Equation.3 in the Place Value Chart BLM. Tell students to make  EMBED Equation.3 using the base-10 blocks on the Hundredths Grid Paper BLM. Have them write  EMBED Equation.3 in the Place Value Chart BLM. Have students add  EMBED Equation.3 or  EMBED Equation.3 and  EMBED Equation.3 to get  EMBED Equation.3 . Give students multiple opportunities to add fractions with denominators of 10 and 100. Adding fractions with unlike denominators such as 10 and 100 will set the stage for adding fractions with other unlike denominators in 5th grade. Activity 14: Creating a Line Plot to Add and Subtract Fractions (CCSS: 4.NF.3a, 4.NF.3c, 4.MD.4) Materials List: paper, pencils, ruler, pencils of different lengths that are all within 1 inch of each other Separate students into groups of four. Give each group of students several pencils (more than 4) to measure using the ruler to the nearest eighth of an inch. Tell students that they will be charting their measurements on a line plot. Explain that a line plot shows data on a number line with an x to show frequency. Tell students that since they were measuring to the nearest eighth of an inch, the number line will be marked using eighths instead of only using whole numbers. Discuss equivalent fractions for  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . Explain that rulers are labeled in simplest form but that this line plot will be labeled by eighths. Tell students if they get a measurement of 4 EMBED Equation.3 , they will list it as 4 EMBED Equation.3 . Have students draw a line plot using the straight edge of the ruler. Ask the students what whole numbers the lengths of the pencils were closest to. Explain that the whole numbers on their line plot would not be between 0 and 1 because the pencils were not smaller than one inch long. For example, if some of the pencils were 6 EMBED Equation.3 in, 6  EMBED Equation.3 in, and 6  EMBED Equation.3 in, the line plot would be labeled 6 in to 7 in. Have the students label the points on the number line based on the discussion and their measurements. Reiterate that a line plot shows frequency, so the students need to mark an x for each pencil that is that length. For example, if there are three pencils that are 6 EMBED Equation.3 inches, then there should be three xs above 6 EMBED Equation.3 inches. An example of a line plot with the following data is below: 4 EMBED Equation.3 , 4 EMBED Equation.3 , 4 EMBED Equation.3 , 4 EMBED Equation.3 , 4 EMBED Equation.3 , 4 EMBED Equation.3 . Ask students how this data would be listed on the line plot since only 8ths are listed on the plot. Students should respond that they have to find equivalent mixed numbers for the ones given or 4 EMBED Equation.3 , 4 EMBED Equation.3 , 4 EMBED Equation.3 , 4 EMBED Equation.3 , 4 EMBED Equation.3 , and 4 EMBED Equation.3 . X X X X X X  4 4 EMBED Equation.3  4 EMBED Equation.3  4 EMBED Equation.3  4 EMBED Equation.3  4 EMBED Equation.3  4 EMBED Equation.3  4 EMBED Equation.3  5 Ask students questions about the line plot such as: How many objects measured 4 EMBED Equation.3  inches? 4 EMBED Equation.3 inches? Which length was the most frequent? If the students put all the objects together end to end, what would be the total length of all the objects. What is the difference between the longest pencil and the shortest pencil? Activity 15: Line Plot Fractions (CCSS: 4.NF.3a, 4.MD.4) Materials List: Line Plot Fractions BLM, fraction strips, paper, pencil Show students the following problem and ask them the following questions. Cristal was comparing the amount of salt used in different recipes. Below is a line plot of all of the different amounts of salt in the recipes that she read. xxxxxxxxxxxxxxxxxxxxxxx EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 Amount of Salt in Recipes (in teaspoons) Ask questions such as these: How much more salt is the greatest amount of salt in a recipe compared to the least amount of salt? ( EMBED Equation.3 tsp) If she made all of the recipes that called for  EMBED Equation.3 tsp, how much salt would she use? ( EMBED Equation.3 tsp) How much more salt is in a recipe that asks for 5/8 tsp than one that asks for  EMBED Equation.3  tsp? ( EMBED Equation.3 tsp) How much more salt is in a recipe that asks for 7/8 tsp than one that asks for tsp? ( EMBED Equation.3 tsp) Have students use fraction strips to help them solve the questions. Ask them to explain how they determined that their fraction strips fit what the problem asked them. Have students write number sentences to represent their work. Provide students with the Line Plot Fractions BLM. Have students complete the questions on the Line Plot Fractions BLM. Activity 16: Addition and Subtraction of Fractions Word Problems (CCSS: 4.NF.3c 4.NF.3d, SL.4.1b) Materials List: Addition and Subtraction of Fractions Word Problems BLM, fraction strips, pencil, paper Provide students with the Addition and Subtraction of Fractions Word Problems BLM. Begin by having students solve addition and subtraction fraction word problems using models as guides if necessary. Have students decompose each of the fractions in the problems as they solve them. For example, the Jacobs family ate  EMBED Equation.3  ( EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 ) of a loaf of bread to make toast for breakfast. They ate  EMBED Equation.3  ( EMBED Equation.3 + EMBED Equation.3 ) of a loaf of bread for lunch to make a sandwich. How much bread did they eat? ( EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 = EMBED Equation.3 ) After reading each word problem, pick a student to play professor know-it-all ( HYPERLINK "http://www.louisianaschools.net/lde/uploads/11056.doc" view literacy strategy descriptions). The professor will read the problem aloud and write the number sentence on the board. The other students in the class will complete the problem while the professor works the problem himself. Have the students question the professor about how he/she solved the fraction problem. Repeat the activity again with other students and other problems so that students become exposed to many types of fraction word problems. Have students move on to more complex word problems involving mixed numbers. Have them decompose each of the fractions as they solve them. For example, Jack and Stephen need 5 EMBED Equation.3 feet of beans to decorate their Mardi Gras float. Jack has 3 EMBED Equation.3 feet of beads and Stephen has 1 EMBED Equation.3 feet of beads. How many feet of beads do they have altogether? (4 EMBED Equation.3 ) Will it be enough to complete the project? (No) Explain why or why not. (They need 5 EMBED Equation.3 feet and only have 4 EMBED Equation.3 feet. They need  EMBED Equation.3 more feet of beads to complete the project.) 2013-14 Activity 17: Multiplying Fractions (CCSS: 4.NF.4a, 4.NF.4b, W.4.2d) Materials List: fraction strips, pencil, paper Review with students how some fractions are multiples of a unit fraction. For example,  EMBED Equation.3 is a multiple of  EMBED Equation.3  because  EMBED Equation.3 =  EMBED Equation.3  +  EMBED Equation.3 +  EMBED Equation.3 or 3   EMBED Equation.3 . Ask the students to draw a model of  EMBED Equation.3 . Then ask them to draw a model of 3  EMBED Equation.3 . Discuss how  EMBED Equation.3 is really  EMBED Equation.3 +  EMBED Equation.3 1/5 and how 3  EMBED Equation.3 can be written as  EMBED Equation.3 +  EMBED Equation.3 +  EMBED Equation.3 . The model should look like this:  Ask the students how many  EMBED Equation.3 s are shown. (3) Have the students draw a model of  EMBED Equation.3 where each of the parts are next to each other. The model should look like this:  Ask the students how many  EMBED Equation.3 s are shown. (6) Discuss how 3  EMBED Equation.3 is the same as 3 2  EMBED Equation.3 or is the same as 6  EMBED Equation.3 which equals  EMBED Equation.3 or 1 EMBED Equation.3 . So 3  EMBED Equation.3 =  EMBED Equation.3 or 1 EMBED Equation.3 . Have students work through other examples of multiplying a fraction times a whole number with models. Have students participate in RAFT Writing ( HYPERLINK "http://www.louisianaschools.net/lde/uploads/11056.doc" view literacy strategy descriptions). For this assignment the RAFT will be: R Role of the Writer - The fraction that is being multiplied A Audience - The fractions that are being added together F Form - Letter T Topic - How fractions that are being multiplied (e.g., 8  EMBED Equation.3 ) are the same as fractions that are being added together (e.g.,  EMBED Equation.3 +  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3 ). The students will explain how multiplication of fractions is the same as repeated addition of fractions. Have students work with partners to write their RAFT. When they finish, have students share their RAFTs with a group or the whole class. Have students listen for accuracy and logic. Listen to your students RAFTs to assess the students understandings of addition and multiplication of fractions. 2013 - 2014 Activity 18: Multiplication of Fractions Word Problems (CCSS: 4.NF.4a, 4.NF.4b, 4.NF.4c) Materials List: Multiplication of Fractions Word Problems BLM, paper, pencils Provide students with the Multiplication of Fractions Word Problems BLM. Give students the following problem: In a track race, each runner runs  EMBED Equation.3 of a lap. If there are 6 team members, how many laps long is the race? Have students draw a model of  EMBED Equation.3 . Ask students how many models of  EMBED Equation.3  there would be if there are 6 team members (6). Ask students how they would find the total number of laps in the race (look to see how many wholes there are). Ask how many whole laps there are (3). Ask students to write their work in a number sentence ( EMBED Equation.3  6 = 3).  Have students draw models to help them understand what the word problem is asking. Ask them to explain how they determined that their model fit what the problem asked them. Have students write number sentences to represent their work. 2013 2014 Activity 19: Multiplying Fractions by Whole Numbers (CCSS: 4.NF.4c) Materials List: paper, pencils Give students a number sentence containing a whole number and a fraction such as 4  EMBED Equation.3 = n. Have students create text chains (HYPERLINK "http://www.louisianaschools.net/lde/uploads/11056.doc"view literacy strategy descriptions) using the number sentence. Model a sample text chain and then have students create their own. Sample Text Chain: Student 1: Ms. Diaz distributed markers to her class. Student 2: Each group got  EMBED Equation.3 of a box of markers. Student 3: There are 4 groups in her classroom. Student 4: How many boxes of markers would she need so that every group would get  EMBED Equation.3 of a box of the markers? Answer: 2 boxes because 4  EMBED Equation.3 is  EMBED Equation.3 or 1 EMBED Equation.3 . She would need two boxes of markers so that the last group would be able to get their 1/3 box of markers. When text chains are completed, make sure groups check for accuracy. Have groups exchange their text chains to solve, discuss, and clarify. Sample Assessments General Assessments Maintain portfolios containing student reports and samples of student work that includes representations of fractions and decimals using regions or set models. Show the student cards with a picture, a model, or a diagram on them. Have the student estimate the fraction that is represented. Give prompts such as the ones that follow and have the student record his/her thoughts in a personal math journal. Give real-life examples of events that use fractions and decimals. How would you use benchmarks to help you compare fractions if you could not use models or fraction strips? Activity-Specific Assessments Activities 3, 4: Provide shaded grids. Have the students describe the shaded area using a fraction and a decimal. Activity 7: Provide students with a list of fractions. Have the students compare and order fractions in order from least to greatest or greatest to least and explain their rationale for their order. Activities 16, 18: Give the students a variety of word problems to solve. 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qM͕DϤl֏+̍@}+ r`cls)Cia8yg9Iv4 9˲tXsZһR\žt*Ʒ%Hc۟FR3IG@PK-!2<>[Content_Types].xmlPK-!]  ,_rels/.relsPK-!3/A9(drs/shapexml.xmlPK-!Vdrs/downrev.xmlPK)7H  ^ YY?Rectangle 3333"@ÔPK!2<>[Content_Types].xmlAN0EH%,BM #{X$ccB{{&m TҞy?٫vՌ}Z_VH68O}6ōV;dnV]DVBz9v I&]Hd9DУkce\%C7;Dp{KU!["jK8 p&î8By/ O4;Tϐ#La\bDR7q[,į W pov+ fCPK!]  _rels/.relsj1 @h_B֐BWaLΖ5yB/dˠA 0xM e`J FG2p%}iRdITD10Z)4(I9`mtB{ƁmtπnTg 險,ܗr>j1W*@ŀӚZ7O#KOiPK!3/A9drs/shapexml.xmlQ(K-*ϳU23PRHKOKU qӵPR(.IKIKUL-VPK!xdrs/downrev.xmlDAk@Ս]R*>lhmȮI]030Sjf qM͕DϤFLJ5|H9j10L}K$Jw|ɲtXsZқ: zvC9SZOHcFR g_PK-!2<>[Content_Types].xmlPK-!]  ,_rels/.relsPK-!3/A9(drs/shapexml.xmlPK-!xdrs/downrev.xmlPK7`EH   ^ YY?Rectangle 3343"@ÔPK!2<>[Content_Types].xmlAN0EH%,BM #{X$ccB{{&m TҞy?٫vՌ}Z_VH68O}6ōV;dnV]DVBz9v I&]Hd9DУkce\%C7;Dp{KU!["jK8 p&î8By/ O4;Tϐ#La\bDR7q[,į W pov+ fCPK!]  _rels/.relsj1 @h_B֐BWaLΖ5yB/dˠA 0xM e`J FG2p%}iRdITD10Z)4(I9`mtB{ƁmtπnTg 險,ܗr>j1W*@ŀӚZ7O#KOiPK!3/A9drs/shapexml.xmlQ(K-*ϳU23PRHKOKU qӵPR(.IKIKUL-VPK!w_k drs/downrev.xmlDAk@B FWDW) ^g6}kn030Sj qM͕a "DdgMvۇs>Q_J$5\PZrf%Nwc]%MCF.l%֜,jwaQ}?\Je "7A%IG@nPK-!2<>[Content_Types].xmlPK-!]  ,_rels/.relsPK-!3/A9(drs/shapexml.xmlPK-!w_k drs/downrev.xmlPK`E@S. ! c > Text Box 335#"ÔPK!2<>[Content_Types].xmlAN0EH%,BM #{X$ccB{{&m TҞy?٫vՌ}Z_VH68O}6ōV;dnV]DVBz9v I&]Hd9DУkce\%C7;Dp{KU!["jK8 p&î8By/ O4;Tϐ#La\bDR7q[,į W pov+ fCPK!]  _rels/.relsj1 @h_B֐BWaLΖ5yB/dˠA 0xM e`J FG2p%}iRdITD10Z)4(I9`mtB{ƁmtπnTg 險,ܗr>j1W*@ŀӚZ7O#KOiPK!3/A9drs/shapexml.xmlQ(K-*ϳU23PRHKOKU qӵPR(.IKIKUL-VPK!$drs/downrev.xmlDj0@oCH()XZK I7mą:_;4U gj.5|~>AlqLaG)&|KJ!P&R"~Z]g1DٕtGmLXs\]EOk5﯅z/vnPa2l_@=~3%G@f7PK-!2<>[Content_Types].xmlPK-!]  ,_rels/.relsPK-!3/A9(drs/shapexml.xmlPK-!$drs/downrev.xmlPK|؎v  - " c > Text Box 336#"ÓPK!2<>[Content_Types].xmlAN0EH%,BM #{X$ccB{{&m TҞy?٫vՌ}Z_VH68O}6ōV;dnV]DVBz9v I&]Hd9DУkce\%C7;Dp{KU!["jK8 p&î8By/ O4;Tϐ#La\bDR7q[,į W pov+ fCPK!]  _rels/.relsj1 @h_B֐BWaLΖ5yB/dˠA 0xM e`J FG2p%}iRdITD10Z)4(I9`mtB{ƁmtπnTg 險,ܗr>j1W*@ŀӚZ7O#KOiPK!3/A9drs/shapexml.xmlQ(K-*ϳU23PRHKOKU 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