This lesson contains several opportunities and activities ...



left476250This lesson contains several opportunities and activities to explore and make sense of the unit fraction- a key concept for 3rd graders to understand. 00This lesson contains several opportunities and activities to explore and make sense of the unit fraction- a key concept for 3rd graders to understand. Understanding Unit FractionsNC Mathematics Standard(s):Understand fractions as numbers. NC.3.NF.1 Interpret unit fractions with denominators of 2, 3, 4, 6, and 8 as quantities formed when a whole is partitioned into equal parts; Explain that a unit fraction is one of those parts. Represent and identify unit fractions using area and length models.Standards for Mathematical Practice:Make sense of problems and persevere in solving them.Reason abstractly and quantitatively.Construct viable arguments and critique the reasoning of others.Model with mathematics.Use appropriately tools strategically.Attend to precision.Look for and make use of structure.Look for and express regularity in repeated reasoning.Student Outcomes:I can identify a unit fraction with 1 as the numerator, such as 1/2 or 1/4.I can recognize a unit fraction and I understand how to build fractions from unit fractions by adding unit fractions. Examples: 1/3+1/3=2/31/4 + 1/4 + 1/4 + 1/4 = 4/4 or 1 whole.I can understand that unit fractions are the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of whole numbers’I can divide the whole into equal-sized portions or fair shares (halves, thirds, fourths, sixths, and eighths).I can draw, fold and cut out equal shares and prove they do not have to be the same shape. (I know congruent squares can be folded in half to make different shapes.)I can understand that a fraction only tells about the relationship between the parts and the whole.I can understand that without a regional/area model, the fraction does not tell anything about the size of the whole or the size of the parts.I can understand that a unit fraction is the smallest of any equal parts of a whole.I can understand that the largest unit fraction is 1/2, then 1/3, then 1/4 etc.I can understand that the larger the denominator, the smaller the parts. (eighths are smaller than thirds.MaterialsColored pencils or thin markersScissorsPattern Blocks per studentHexagon Grid per studentClear tape or glueHandoutAdvance Preparation:Students partition rectangles and circles into two, three, or four equal shares.Students are able to describe the shares using the words halves, thirds, fourths, half of, a third of, and a fourth of.Students describe the whole as two halves, three thirds, and four fourths.Students recognize that equal shares of identical wholes need not have the same shape.Directions:The Focus of this lesson is Unit Fractions. Understand fractions are a part of a whole.Students should understand that unit fractions are the basic building blocks of fractions in the same sense that number 1 is the basic building block of whole numbers.Provide opportunities for students to make fraction strips made of paper or just drawing fractions (same length. Handout: Making Fraction Strips. The whole must be divided into equal-sized portions or fair shares.Provide models to show the part-whole concept of fractions and the meaning of the relative size to a part of the whole. (pattern blocks, geoboards, folding paper squares etc.Equal shares do not have to be the same shape but must have the same area.The fraction does not say anything about the size of the whole or the size of the parts. A fraction only tells about the relationship between the parts and the whole.The more fractional parts used to make a whole, the smaller the parts.The numerator tells how many of the equal parts you are counting. The numerator is the counting number.The denominator tells how many equal parts the whole is divided into – what is being counted.The larger the denominator, the smaller the part. For example, eighths are smaller than fifths.Model fraction language: halves, thirds, fourths, sixths, and eights for grade 3.Ask probing questions and provide learning activities to engage students, using various representations.Provide opportunities including models, communication with others and students’ explanations and experiences writing problems.The important anchors are 0, 1/2 and 1 whole. To know to which of these benchmarks a fraction is closest or if it is more or less than 1 or 1/2 are very valuable ideas for students.Provide multiple opportunities to communicate about their thinking and reasoning in order to build understanding.Listen and record students’ ideas and encourage flexibility in their thinking.Students need time to explore the effect that the whole has on how a fraction is named and its relationship to the whole. Students use pattern blocks to change the size of the whole. As the whole is changed, students adjust their thinking to make sense of the name of the fractional piece relative to the whole.Questions to Pose:Before:What can you tell or show me about folding a paper square into halves? fourths?Where have you used or heard the fraction 1/2?During:Where have you folded or cut 1/2 of something?What is meant by 1/2 of an hour? a brownie?What is a unit fraction? Give examples.Can you count to 1 using unit fractions?After:Can you tell or show an example where equal shares are not the same shape?411162529718000Using 1/6 as the unit fractions, show how you might count to 1 whole. (Record your strategy,)Show 3 different ways to cut the brownie.Possible Misconceptions/Suggestions:Possible MisconceptionsSuggestionsStudents may think of numerators and denominators as separate whole numbers. They are confused that 1/4 is larger than 1/8.Students have difficulty understanding the relationship of the numerator and the denominator.Present meaningful problems. Example:If you have 1/4 of an apple and your friend has 1/2 of an apple, who has the largest piece?Students need to see the 1//4 and 1/2 of two apples, same size.Students need many opportunities to find fractional parts of concrete objects, drawings and other visual representations. Students need opportunities to solve problems in a context based on their interests. Ex. What fraction of the cake was eaten?Special Notes: Students need many opportunities and time to explore the effect that the whole has on how a fraction is named and its relationship to the whole. Students use pattern blocks to change the size of the whole. As the whole is changed, participants adjust their thinking to make sense of the name of the fractional piece relative to the whole.Solutions: Students use pattern blocks, paper models, fraction strips, word problems and other models to make sense of the unit fraction, relative to the whole.Every fraction is a sum of unit fractions. Write the following as a sum of unit fractions.Example:3/8 = 1/8 + 1/8 + 1/82/4 =7/8 =7/6 =5/2 =Sharing BrowniesName Date Materials:Drawing of a Rectangular Brownies all the same size on different colored paper, scissors, rulers, blank paper, glue or tapeEach student cuts out 5 or 6 rectangles in several different colors.Pretend each rectangle you cut is one brownie.Exchange brownies with other students so each student can have brownies of different colors.Using only straight lines or straight cuts, cut a brownie in two equal parts.Find a different way to make straight cuts to cut a brownie in two equal parts. Provide time for students to share ways they cut a brownie.Students should be able to prove equal shares.Did anyone cut a brownie in half by cutting a diagonal line?Students prove their shares are equal by cutting, measuring, and folding.What fraction of the brownie does one person get?Adapted From: Dale Seymour Publications, White Plains, New YorkSharing BrowniesName Date Materials: Drawings of Rectangular Brownies. 5 large brownies, all the exact same size (can be same color); scissors, glue or transparent tapeDirections: Students cut out 5 brownies. Pretend each rectangle you cut out is one brownie. How can you cut your brownie to make equal shares? Use only straight lines or straight cuts to make equal shares.Tape or paste brownie pieces below to show how much brownie each person gets. After each problem, record the fraction that shows the amount of brownie that is each person’s share.Two people share one brownie. Each person gets Tape or paste brownie shares below.of the brownie.Four people share one brownie. Each person getsof the brownie. Tape or paste brownie shares below.Eight people share a brownie. Each person getsof the brownie. Tape or paste brownie shares below.Three people share a brownie. Each person getsof the brownie. Tape brownie or paste shares below.Six people share a brownie. Each person gets Tape brownie or paste shares below.Adapted From: Dale Seymour Publications, White Plains, New Yorkof the brownie.Brownies521335128270004008120128270005213351104900040754301104900052133523812500407543021971000What is a Unit Fraction?Name Date In math, fractions are a way to represent parts of a whole number. Imagine you have a large candy bar. The candy bar can be shared equally for 3 friends and yourself.Below is a picture to show the candy bar and the equal parts for 3 friends and yourself. The candy bar has been shared in equal pieces or fair shares.9144001657351/4441/41/41/4001/4441/41/41/4The fraction 1/4 is a unit fraction. The numerator (top number) tells how many of the equal parts you are talking about and the denominator (bottom number) tells what is being counted.Taylor wanted to share a large candy bar with 5 friends and himself. Draw and label a new candy bar below. Be sure the candy bar is divided equally.How many equal parts has the candy bar been partitioned into? Each slice of the candy bar is an equal part of the whole candy bar. What part is one piece of the candy?Examples of other unit fractions are 1/3, 1/6, 1/10. Write 5 different unit fractions.What are two or three examples of unit fractions we might use or hear? Example: 1/2 of an hour.What is a Unit Fraction?Name Date What do the fractions below have in common?1/2, 1/3, 1/4, 1/6, 1/8, 1/10Record your thoughts.What is different about all of the fractions in question #1?Draw a picture below of a brownie cut into 2 equal pieces. Draw another brownie (same size). Cut this brownie into 4 equal pieces. If you can have one part of one of the brownies, what part would you choose. Why?The fractions in Question #1 are defined as unit fractions.Other unit fractions are:1/5, 1/10, 1/12, 1/14, 1/25, 1/50, 1/100, 1/150 etc.In your own words, “What is a unit fraction?” Write 2 or more unit fractions below.Write 2 fractions that are NOT a unit fraction. What is the largest unit fraction? How do you know? Explain or draw pictures to prove your answer.Making Fraction SetsMaterials: 5 sheets of the same color paper for each group of 2 students, scissors, pencils, markers, rulersDirections: Work with 1 other student to create fractional parts of a whole. Each pair of students should have 5 sheets of paper, the same color.1st sheet – Cut into 2 equal piecesFold one sheet of paper in half. (horizontally)Unfold and mark the fold line with a pencil, if needed.Cut the paper into two equal pieces. Write 1/2 on each section, using a pencil.2nd sheet - Cut into 4 equal piecesright1358901/4001/4Fold one sheet of paper (horizontally) into 2 equal parts. Fold the paper again to make rectangles. (Fold vertically) Unfold and write 1/4 on each section in pencil.Mark the fold lines with a pencil if needed. Cut the paper into 4 equal parts.3rd sheet --Cut into 8 equal piecesFold one piece of paper into 8 equal parts.First fold paper into halves, then fourths (as above). Next fold the fourths, horizontally.Unfold and record 1/8 on each section with a pencil. Cut the paper into 8 equal pieces.4th sheet -- Cut into 3 equal piecesFold one piece of paper into 3 equal pieces, horizontally. Mark the fold lines with a pencil, if needed.Write 1/3 on each section in pencil. Cut the paper into 3 equal pieces .5th sheet --Cut into 6 equal piecesFold one piece as thirds and then fold again. Mark the fold line with a pencil, if needed.Keeping the fold above, fold the paper again to have 6 pieces.Write 1/6 on each section in pencil. (Some students may create sixths horizontally or vertically).Fraction StripsMaterials: 12 by 18 inch construction paper in 6 different colors. Cut each piece of construction paper into 4 x 18 inch strips. Each student will need 6 strips, each strip a different color. Students will need a thin black marker. Students may share markers.Directions:Each student should have 6 strips, each of a different color.Use the darkest strip as one whole.Have all students select a yellow strip (can be any color). Ask students to fold the strip to create 2 halves.Students should label one part as 1/2 and also one-half using words.1/2One-Half1/2One-HalfStudents select another color (all students select same color). Teacher shows the new strip and folds the strip to make one half. Students follow the same procedure.Teacher asks: If we fold to make a half one more time, how many sections will the strip have? Students predict and then fold the strip in half again.Students discuss why the strip has been divided into 4 equal parts. Students should label 1/4 of the strip and record one-fourth.1/4One-FourthStudents take another color (all students select the same color).Ask student to fold their new strip into halves and fold again to make fourths. Ask students to predict what will happen if they fold the strip again.Allow time for students to discuss how they folded the strip into eighths. Students should label 1/8 of the strip and underneath record One-Eighth.Student select a strip of a different color (all students select the same color). Folding the paper strips into thirds is difficult. Students might use rulers or other tools to fold the parts as equal as possible.1/3One ThirdStudents select their last strip. Ask students, “What strip could help us fold to get sixths? “(Students will again create thirds and then fold each third to get sixths.)Ask students (table groups) to record what they notice about the fraction bars. Allow students to share ideas.Ask students to discuss what they know about fractions and where they have seen fractions or where they have used fractions. Guide the discussion to include real-world situations (many may discuss sports), things that are typically divided into fractional parts.Making Sense of FractionsName Date Complete the chart below.Partition different models of wholes into equal parts.Shaded parts do NOT need to touch.Total Number of Equal PartsTotal Number of Shaded Equal PartsUnit FractionFraction Shaded821/82/8Making Sense of FractionsName Date Students draw models of wholes and divide the whole into equal parts. Shade a fraction of the model. Complete the chart. Shaded parts do NOT need to touch.Partition different models of wholes into equal parts.Total Number of Equal PartsTotal Number of Shaded Equal PartsUnit FractionFraction Shaded\Using Pattern Blocks as Unit FractionsName Part #1Date Materials Needed: 6 of each pattern blocks (yellow hexagon, green triangle, red trapezoid and blue rhombus); Hexagon grid paper for each student, pencils, colored pencils and color markers3162617165493Part #1 Students work with a partner. Each student will solve problems on his/her paper.If the yellow hexagon represents one whole, how might you partition the whole into equal parts using unit fractions?Using yellow pattern blocks, how many trapezoids can fit on one hexagon? What part of the whole hexagon is one trapezoid?Build and trace 2 trapezoids on your hexagon grid paper. Label each trapezoid using unit fractions. Example: Each trapezoid is 1/2 of the hexagon.A unit fraction always has 1 as the numerator. The denominator (bottom number) of the fraction labels the number of parts needed to cover the whole hexagon. Add the unit fractions to equal 1 whole.1/2 + 1/2 = 1 hexagonHow many rhombi (blue parallelograms) will equal one hexagon? Build a model using pattern blocks. Trace each blue rhombus on hexagon grid paper. Label each rhombus with a unit fraction.Add the unit fractions to equal 1 whole.How many green triangles will equal one hexagon? Build and trace each triangle on hexagon grid paper. Label each triangle with unit fractions.Add the unit fractions to equal 1 whole.Part #29398007801322743200673100Add unit fractions used to equal one pattern block.Record each solution on Hexagon grid paper. 1/2 + 1/6 + 1/3 = 1 whole00Add unit fractions used to equal one pattern block.Record each solution on Hexagon grid paper. 1/2 + 1/6 + 1/3 = 1 wholeBuild different hexagons made from 2 or more shapes. Trace and color the combination of pattern blocks to equal one whole. Label each pattern block using unit fractions.Continue building and finding different solutions for one hexagon. (You may notcount a different combination of the same pattern blocks as a different way.)Use different combinations of pattern blocks to build hexagons that arethe same size and shape as the yellow hexagon.Record and color each solution on your Hexagon Grid paper.Label each fraction.Add fractions to equal 1 whole. (See example above.)How many different solutions did you find?Ask students to show and explain solution strategies using a document camera or sharing in small groups.Part #3Use two hexagons to solve problems. Record on hexagon grid paper.Two yellow hexagons = 1 whole27311359652000Outline 2 hexagons on your hexagon grid paper.Name the unit fraction for each hexagon. Answer is 1/2How many unit fractions equal 1 whole? Answer is 2.Explain this solution to a partner or a small group.Outline 2 hexagons on your Hexagon Grid paper.273113515621000Using trapezoids as unit fractions, cover the whole.How many trapezoids equal the whole? Name the unit fraction for one trapezoid. Outline 2 hexagons on your Hexagon Grid paper.273113511557000How many blue rhombuses do you need to cover the whole? Record on hexagon grid paper.Name the unit fraction for one rhombus. How many green triangles do you need to cover the whole? Name the unit fraction for one triangle. Record on hexagon grid paper.Using the two hexagons as one whole, create a new design Using as many different pattern blocks that can fit on the whole.250253511493500Determine the unit fraction for each pattern block. Record your design on hexagon grid paper. Record unit fractions to equal the whole. Share and explain your design with partners or with the whole class. ................
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