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1 Grade 6 Mathematics Unit 3: Fractions, Decimals, and Parts Time Frame: Approximately four weeks Unit Description The focus of this unit is on concepts and basic relationships of fractions and decimals. There is an emphasis on estimating outcomes prior to developing the computation algorithms that give the exact answers. Focus is also given to writing fractions in lowest terms. The development of the concept of rate, ratio, proportion, and percent continues by representing and working with miles/hour, dollar/pound, miles/gallon, and other derived rates and percents. Student Understandings Students understand that fractions, decimals, and integers can be compared by placement on a number line and/or by the use of symbols. They can solve ratio, proportion, and percent problems with models and pictures. Students understand place value to the ten- thousandths place. They can use rates to solve real-life problems. Guiding Questions 1. How can students represent and interpret values for decimals through ten- thousandths? 2. In what ways can students generate equivalent forms of fractions and decimals? Students use a drawing or paint program (Kid Pix or Windows Paint work fine) to create blocks (or other shapes) which represent fractions, the simpler the better such as halves, quarters, eighths, sixteenths, etc. Simple conversions to decimal equivalents can be printed on the blocks and the pages printed. Students can even challenge one another if they spot an error. 3. How can students predict reasonable outcomes for the addition and subtraction of fractions and decimals? Differentiated instruction can be accomplished using dual media and technology connections both at home and at school using the newspaper and television while bridging Guiding Questions. The link at http://nie.honoluluadvertiser.com/00-New-Resource- 2005-06/Using_NewspaperStandardsK-5.pdf contains a lengthy set of activities on using the local newspaper, one of which suggests watching the weather portion of the local news broadcast. Students keep track of daily weather facts like temperatures and barometric pressure on a hard copy log sheet. Using the computer, students can work individually or in small groups Grade 6 MathematicsUnit 3Fractions, Decimals, and Parts 1 2 to enter their data into a simple spreadsheet or even a table (with formulae or not as the teacher sees fit) in their word processor to average the facts as decimals and then convert the decimals to fractions. Student teams can even challenge each other to do the conversions correctly. 4. What strategies do students utilize to work with rates and ratios such as mph, mpg, and dollar/pound? Students can use the classroom, library, or lab computer with an Internet connection to research the value of the dollar and compare it to other currencies, such as the British pound or the Japanese yen. Currency conversion rates can then be determined and the facts utilized as a part of a larger activity, say a summer vacation to predict probable needs and outcomes. Unit 3 Grade-Level Expectations (GLEs) GLE # GLE Text and Benchmarks Number and Number Relations 4. Recognize and compute equivalent representations of fractions and decimals (i.e., halves, thirds, fourths, fifths, eighths, tenths, hundredths) (N-1-M) (N-3- M) GLE # GLE Text and Benchmarks 5. Decide which representation (i.e., fraction or decimal) of a positive number is appropriate in a real-life situation (N-1-M) (N-5-M) 6. Compare positive fractions, decimals, and positive and negative integers using symbols (i.e., <, =, >) and number lines (N-2-M) 7. Read and write numerals and words for decimals through ten-thousandths (N- 3-M) 10. Use and explain estimation strategies to predict computational results with positive fractions and decimals (N-6-M) 13. Use models and pictures to explain concepts or solve problems involving ratio, proportion, and percent with whole numbers (N-8-M) Measurement 18. Measure length and read linear measurements to the nearest sixteenth-inch and millimeter (mm) (M-1-M) 20. Calculate, interpret, and compare rates such as $/lb., mpg, and mph (M-1-M) (A-5-M) Data Analysis, Probability, and Discrete Math 31. Demonstrate an understanding of precision, accuracy, and error in measurement (D-2-M) Sample Activities Activity 1: Daily warm-up activity - First 100 days (GLEs: 4, 6) Grade 6 MathematicsUnit 3Fractions, Decimals, and Parts 2 3 As students continue marking each day of the first 100 days of school, review and pay close attention to days number 10, 20, and 25. Explore fraction/decimal/percent equivalence: 10 100 50 10 .10 10% 10 5 1 20 100 10 1 .20 20% 20 2 5 25 100 20 1 .25 25% 25 5 4 Have students make comparisons using symbols =, <, and > followed by class sharing. Activity 2: A Measuring We Go… (GLEs: 5, 18, 31) Have students work in groups of two or three. Provide students with measuring instruments that measure in both English and metric systems. Ask students in each group to find an item in the classroom to measure. The measurement should be recorded using both systems. English system measure should be recorded in fractional increments while the metric should be written in decimal numbers. Instruct students to find the most precise measure offered by their instruments (sixteenths of an inch/ millimeters). Once the measures are taken and recorded, focus discussion on the measures, how they are represented, what measurement tool was used, and how accurate each measure would be. Discuss precision of instruments, accuracy of the measurer, and other situations that may cause the measure to not be exact. Offer different types of tools to vary the outcomes and in turn promote further discussion- plastic rulers vs. wooden rulers; flexible meter sticks vs. rigid wooden ones; measuring tapes. Activity 3: Reading and Writing Decimals (GLE: 7) In the classroom, display a place value chart that students can refer to for this activity and throughout the year. The chart should range from billions to hundred-thousandths. Provide students with decimal values in a worksheet or individually on cards. Be sure to include decimals that have whole number parts, tenths, hundredths, thousandths and ten- thousandths (e.g., 2.67, .0235, 0.5, 0.43, 6. 324). Have students read and/or write the values. Then give students decimal values that are written in words and have students rewrite them in numerical form. Activity 4: Fraction Strips (GLEs: 4, 6, 18, 31) Have students work in groups and give each member of the group a sheet of different colored cardstock. Instruct each student to make fraction strips by cutting the cardstock into 6 one-inch wide strips. Remind students to use a ruler to make appropriate measurement. Ask students to fold one strip each into the following fractions: halves, fourths, eighths, thirds, sixths, and tenths and then write the fraction on each piece (each of the thirds will have 1 on it). Direct students to use a ruler to check their work to 3 Grade 6 MathematicsUnit 3Fractions, Decimals, and Parts 3 4 determine how accurate the lengths of the strips are to the nearest half, fourth, eighth, and sixteenth of an inch. During discussion, address the accuracy of the measurement. What could cause the measurement not to be precise? (Tools used, quality of measurement materials, student error, etc.) In the groups, use the strips to find equivalent fractions, 1 = 4 = 6 = 4 = 10 , etc. Ask 2 2 3 8 5 students to compare their fraction strips and write down comparisons using symbols of =, <, and >. Write fractions in lowest terms. Check students’ work in groups. Activity 5: Same or Different? (GLEs: 4, 6) Divide the class into two groups. One group will be the “fraction” group and the other the “decimal” group. Ask the “fraction” group to provide a fraction expressed in halves, thirds, fourths, fifths, eighths, tenths, or hundredths to the “decimal” group. Give the “decimal” group a specified time limit to provide an equivalent decimal representation of 8 25 the fraction. Note that the fraction group could give 16 instead of 1 or 100 instead of 1 , 2 4 etc. Have the two groups decide if the decimal expression provided is equivalent to the original fraction. If the decimal is equivalent, then the decimal group earns 1 point. If not, the fraction group earns 1 point. Next, have the “decimal” group provide a decimal representation of a fraction to the “fraction” group. Allow the “fraction” group a specified time limit to provide an equivalent fraction in lowest terms. Again, ask the two groups to decide if the fraction is equivalent. If correct, the “fraction” group earns a point; if not, then the “decimal” group earns a point. If an answer is determined to be correct, ask students to record each answer on an index card. Once complete, shuffle the index cards and then pass them out to individual students. Have students then form a “human number line” across the classroom. Order the fractions and decimals with the fraction/decimal equivalents standing one in front of the other. As a whole group, discuss the placement of students. Note that students will be paired across the number line. Place the cards on the wall so that students can view them and their placement at the end of the activity. Keep the cards for future use. Activity 6: Box Scores (GLE: 6) Depending on the season, use box scores from the newspaper’s sports section to get data to order decimals. For example, show the average yardage for rushes by different football players, rebounds for basketball players, on-base percentage or batting averages for baseball, times for track stats—dashes and pole vaults, etc. Write each statistic or number on an index card. As a class, order the stats on a number line. An example of batting averages would be .222, .234, .245, .255, .266, .289, etc. Help students understand how these data can be interpreted as rates. For example, a baseball player batting .250 gets a hit once every four times he bats, on average. After exploring the decimals, have students mix these index cards with those from the previous activity. Repeat the “human number line” with the additional cards. Because the number of cards may now surpass the Grade 6 MathematicsUnit 3Fractions, Decimals, and Parts 4 5 number of students, discuss with students how to solve this problem, i.e. pair the equivalent fractions and decimals- eliminate one-half of them. Activity 7: Rolling for Decimals (GLE: 7) Have students work in small groups of no more than 4 for this activity. Provide 3 number cubes per group. Ask students to take turns rolling the cubes and examining the numbers showing. Using those numbers as digits to create a decimal number, instruct groups to find the largest number possible and the smallest number. Have students record the number on paper and read the numbers correctly. After repeating this five or six times, have students write the numbers on note cards, turn the cards over and shuffle the cards. Instruct groups to turn the cards over and place them in order - largest to smallest. Repeat the process, this time from smallest to largest. To check, have the students read the correct word name for each number. Activity 8: Grocery Math (GLEs: 10, 13, 20) Provide students with grocery ads from a current newspaper. Have students work in pairs with a specified task: Given $20 to go to the grocery store, buy a variety of fruits and vegetables. Direct students to buy a combination of at least four different fruits and/or vegetables, compute tax at the rate of 10%, and report the findings to the class. The findings should include the estimated cost and the final cost. Sketch a rectangle to represent the $20. Divide the rectangle into approximate parts to show prices (i.e. if $5.00 is spent on apples, 1 of the rectangle should be marked and labeled “apples”). Use the rectangle as a 4 visual representation of the information. Considering the purchases, have students respond to the following: Do you have enough money to buy 3 lbs. of each? 4 lbs. of each? 5 lbs. of each? Use a calculator to determine the cost of each fruit/vegetable at 3, 4, and 5 lbs. If one item can be bought in three- pound bags for $2.99, select one item and decide if it is cheaper to buy by the bag or by the pound? Instruct students to record answers as a rate: 1lb. for $.997. Teacher Note: When computing 10% tax, mental math should be encouraged. Students should be able to explain how they arrived at the answer. Accept answers that show a clear understanding that “moving the decimal 1 places to the left” is because they are multiplying by .1 (one tenth or 10- hundredths). Activity 9: Tangram Ratio (GLE: 13) A tangram puzzle is made up of seven pieces: 2 large triangles, 1 medium triangle, 2 small triangles, 1 parallelogram, and 1 square. A large square can be formed using all 7 tangram pieces. Have each student make his/her own set of tangrams so that he/she can Grade 6 MathematicsUnit 3Fractions, Decimals, and Parts 5 6 have a direct understanding of the relationships between the parts and whole and among the pieces to one another. Simple directions for creating the tangrams are given below. Directions with visual representations can be easily found by putting the term tangram directions into an Internet search engine. After a quick review of the terms area and ratio, have students determine the ratio of the area of each piece to that of the other pieces by comparing the sizes of the pieces. For example, students should determine the ratio of a small triangle’s area compared to the medium triangle. Next, ask students to write the ratios of each of the tangram pieces to the whole (the completed puzzle). As an example, students should find that the ratio of a large triangle to the large square (the completed puzzle) to be 2 to 4, which reduces to a ratio of 1 to 2. This ratio compares the area of a large triangle to the area of the square (the completed puzzle). Have the students use the ratios to write proportions. When students complete the activity, have them rewrite the ratios of area of single pieces to the whole as fractions and then as percents. Fold and cut a square sheet of paper by following these instructions: 1. Fold the square in half diagonally, unfold, and cut along the crease into two congruent triangles. 2. Take one of these triangles. Fold in half, unfold, and cut along the crease. Set both of these triangles aside. 3. Take the other large triangle. Lightly crease to find the midpoint of the longest side. Fold so that the vertex of the right angle touches that midpoint, unfold and cut along the crease. You will have formed a middle-sized triangle and a trapezoid. Set the middle-sized triangle aside with the two large-size triangles. 4. Take the trapezoid, fold it in half, unfold, and cut. 5. To create a parallelogram and a small-sized triangle, take one of the trapezoid halves. Fold the right base angle to the opposite obtuse angle, crease, unfold, and cut. Place these two shapes aside. 6. To create a square and a small-sized triangle from the other trapezoid halve, fold the acute base angle to the adjacent right base angle and cut on the crease. 7. You should have the 7 tangram pieces: 2 large congruent triangles 1 middle-sized triangle 2 small congruent triangles 1 parallelogram 1 square 8. The pieces may now be arranged in many shapes. Try recreating the original square. Activity 10: Estimating with Positive Fractions and Decimals (GLE: 10) Provide students with a sheet of several “target ranges” of answers expressed as positive fractions or decimals. For example, “between 1 and 8 ” could be a target range. Also, 2 5 provide a sheet of addition and subtraction problems involving positive fractions and Grade 6 MathematicsUnit 3Fractions, Decimals, and Parts 6 7 decimals that would produce answers that would land in one of the target ranges provided. The fractions and decimals should be real-life examples, such as “1.43 lb of ground meat.” Have the students estimate solutions to the addition and subtraction problems by choosing a target range for the answer. Have each student discuss his/her estimation strategy in a small group and share strategies with the entire class. Activity 11: Vacation Math (GLE: 20) We’re going on vacation! Have the students predict how long it will take to drive at the posted speed limit. Allow students to make use of the Internet, maps, or atlases to locate the distance from home to a designated place. This distance with a variety of speeds will be used to determine trip length. Class discussion should focus on the distance formula with students discovering the formula instead of having it given to them. Questions student should explore include: If we are going to drive to visit our location, how long will it take to get there if we drive 60 mph? If the car we’re using gets 30 miles to the gallon, how much gas will we use to get there and back? If the price of gas is $1.50 per gallon, how much will it cost to go on our trip? Make a presentation to the class sharing information. Use drawing features on word processing program to sketch a model to show how time was calculated. Sample Assessments General Assessments The teacher will observe individual and group work throughout the unit. The student will create portfolios containing samples of experiments and activities. The teacher will facilitate small group discussions to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask might be: o How did you get your answer? o What are the key points or big ideas in this lesson? o How would you prove that? o What do you think about what ___ said? o Do you agree with your group’s answer? Why or why not? o How would you convince the rest of us that your answer makes sense? The student will create journal writings using such topics as: o The most important thing I learned in math this week was… o If I were the math teacher … o Explain today’s lesson to a student who was absent today. The student will submit a written reflection to the following two questions as a Performance Task Assessment of the unit: o How can you decide whether a fraction is closest to 0, ½, or 1? o When comparing two decimals such as 0.36 and 0.349, how can you decide which decimal represents the larger number? Grade 6 MathematicsUnit 3Fractions, Decimals, and Parts 7 8 Activity-Specific Assessments Activity 3: The teacher will create an assessment where the student matches decimal numbers to the written equivalent. Activity 4: The students will write a story using fractions and decimals appropriately, making at least five comparisons about the numbers by using the symbols <, >, or =. Activity 5: The student will bring a copy of a recipe from home. He/she will write the amount of each ingredient in fraction, whole number, and or mixed number amounts and convert the values to decimal numbers. Activity 9: Have the student create a poster with original definitions and/or pictures to convey the meanings in order to demonstrate understanding of the math terms covered- ratio, area, triangle, trapezoid, etc. Grade 6 MathematicsUnit 3Fractions, Decimals, and Parts 8