Supplemental
Fraction Unit for Grade Three
based on the Common Core Standards
SCLME South Carolina Leaders of Mathematics Education
2011 SCLME recommends that district mathematics curriculum leaders support teachers with the implementation of this unit by providing the necessary content knowledge so that students gain a strong conceptual foundation of fractions.
Revised 3/30/12 Fractions A Supplemental Unit of Study based on Common Core Standards Time: 10-12 days (60 minute class periods)
Unit Overview Fractions are very difficult for students to learn and teachers to teach. This supplemental unit seeks to bridge a gap for students as we transition to Common Core standards. The learning activities provided herein should engage students in both hands-on and minds-on experiences. Students should have multiple opportunities to communicate about their thinking and reasoning in order to build understanding. Teachers should listen carefully to students’ ideas and encourage flexibility in their thinking.
In order for students to have a deep conceptual understanding of fractions, the focus on teaching should be with concrete materials and pictorial representations. In using best practices, virtual manipulatives should not take the place of concrete materials.
Anchor charts should be created and used throughout the unit. An anchor chart is a visual recording of students’ ideas and thinking about a certain concept. It serves to connect past teaching and learning to future teaching and learning. For example, on a piece of chart paper, the teacher will record students’ ideas about what means to them. This should include illustrations and labels of different representations of. (See Appendix A) Models for Fractions Area or Region Models – Fractions are based on parts of an area or region. Examples include: circular pie pieces, pattern blocks, regular/square tiles, folded paper strips (any shape), drawings on grids and partitioning shapes on geoboards.
Linear or Length Models – With length models the whole is partitioned and lengths are compared instead of area. Materials are compared on the basis of length. Examples include: fraction strips, Cuisenaire rods, number lines, rulers, and folded paper strips.
Set Models – In set models, the whole is understood to be a set of objects or group of objects, and subsets of the whole make up fractional parts. Examples; in a set of 6 marbles, is 3 marbles. This concept should be taught with concrete materials so that there would be 2 groups of 3 marbles so that each group is of 6. Big Ideas Understanding Unit Fractions Building on Unit Fractions Understanding Equivalent Fractions
Common Core Standards (Grade 3) Number and Operations-Fractions (3.NF) Develop understanding of fractions as numbers. 1. Understand a fraction as the quantity formed by 1 part when a whole is portioned into b equal parts; understand a fraction as the quantity formed by a parts of size. 2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size and that the endpoint of the part based at 0 locates the number on the number line. 2 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards b. Represent a fraction on a number line diagram by marking off a lengths from 0. Recognize that the resulting interval has size and that its endpoint locates the number on the number line. 3. Explain equivalencies of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions, (e.g.,=, = ). Explain why the fractions are equivalent, e.g., by using a visual fraction model. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 =; recognize that = 6; locate and 1 at the same point on a number line diagram. d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. *Limit to fractions with denominators 2, 3, 4, 6, and 8.
Understanding Unit Fractions 3 Days
Objective: Students will be able to identify and represent unit fractions as a part of unit wholes, as parts of a collection, and as locations on number lines.
Learning Goals: Understand fractions as parts of a whole. Numerator and denominator are not separate values but represent one number. Understand the conceptual meaning of the numerator and denominator. The whole must be divided into equal-sized portions or fair shares. Equal shares do not have to be the same shape. 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. Express whole numbers as fractions: 3 = Use fraction language: halves, thirds, fourths, sixths, and eighths. Linear models show that fractions can be found between any two whole numbers such as 0 and 1. Regional models show the part-whole concept of fractions and the meaning of relative size to a part of the whole. Set models show how subsets of the whole make fractional parts. Recognize unit fractions: ,,,, and .
Instructional Background Unit Fractions – A fraction with 1 as the numerator, such asis called a unit fraction. Numerator – The numerator tells how many of the equal parts you are talking about (the counting number). Denominator – The denominator is how many equal parts the whole is divided into (what’s being counted). 3 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards Equal Parts – Fractional parts are equal shares or equal-sized portions of a whole or unit. A unit can be an object or a collection of things. More abstractly, the unit is counted as 1. On the number line, the distance from 0 to 1 is the unit. Parts of the Fraction – The more fractional parts used to make a whole, the smaller the parts (the larger the denominator, the smaller the part). For example, eighths are smaller than fifths.
Lesson 1 – Making Fraction Strips (Length Model)
Lesson Learning Goals: Understand fractions as parts of a whole. The whole must be divided into equal-sized portions or fair shares. Use fraction language: halves, thirds, fourths, sixths, and eighths. Recognize unit fractions: ,,,, and .
Materials: 6 congruent strips of paper/student (1 in x 8 in)
1. Ask students to discuss what they know about fractions. Guide discussion to include the following: real-world situations things that are typically divided into fractional parts Explain how fractions represent exactly equal shares and how sometimes fractions are used in non-literal ways such as when a child says he is.
2. Give each student 6 pieces of equal length pre-cut strips of paper to use. Have students write 1 whole in the middle of one of their strips. Ask students to fold another strip in half. 1 whole Facilitate discussion about the two strips emphasizing the following points; The starting strip was 1 whole strip. The strip folded in half has 2 equal parts Each part represents 1 part out of two or one-half of the whole. One half is written as. Create an anchor chart for. (See appendix A) 3. Have students make a vertical mark on their strip at the fold and write 1/2 in each piece. Ask students to place this strip below the 1 whole on their desks. + =
4 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards One-half + one-half equals two halves or one whole. 4. Facilitate discussion with students about how they can fold parts for fourths, and eighths. As they discuss each new fractional part, have students fold, mark, and label new strips of paper. Continue with thirds and sixths. 5. Facilitate discussion which encourages students to look for relationships among different fractions. For example, how many fourths does it take to equal? How many thirds equal one whole? Is one-third larger or smaller than one-half? 6. To provide students practice with pronouncing fractions, stress the ending “th” or “ths”. Write the numbers 3, 4, 6, and 8 on the board. Then write,, , and . Have students read the numbers and the fractions as teacher points to them. Explain the exception to the rule – half and halves. 7. Allow students to work in small groups. One student writes a word name for a unit fraction,,,, or. Another student writes the numeral, and another sketches a representation of the same fraction. Post as anchor charts in the classroom.
Lesson 2 – Parts of a Fraction
Lesson Learning Goals: Numerator and denominator are not separate values but represent one number. Understand the conceptual meaning of the numerator and denominator. Use fraction language: halves, thirds, fourths, sixths, and eighths. Recognize unit fractions: ,,,, or .
Materials: fraction strips (made in lesson 1), (8) counters/student, paper/pencil
1. Writeon the board. Ask students to look at the numbers that they wrote on the fractions strips and tell you what they noticed about the bottom numbers. Guide students to notice that the bottom number tells the total number of equal parts. Tell students that the d stands for denominator and that the denominator tells the total number of equal parts into which a whole has been divided. The denominator names the kind of fractional part that is under consideration. The bottom number, denominator, tells what is being counted. The denominator is a divisor. 2. Tell students that a single part of an equally divided whole is called a unit fraction. Relate this to other units such as a unit of time might be minutes, or a unit of measurement might be inches. 3. Have students take two counters and put one counter on any two parts of the strip which represents eighths. Make sure students understand that the counters do not have to be placed in adjacent parts in order to illustrate the fraction. Discuss what part of the strip has counters. Explain 2 parts out of 8 have counters. Explain that of the strip has a counter. Repeat with 3, 4, 5, 6, 7, and 8 counters. Each time discuss, read and write the fraction with symbols and words. (add to anchor chart)
5 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards
4. Write on the board. Tell students that the n stands for numerator and the numerator is the number of equal parts in the fraction which is being counted. The numerator counts or tells how many of the fractional parts are under consideration. Therefore, the numerator is a multiplier – it indicates a multiple of the given fractional part. 5. Using the counters with the strips of paper ask how they might use the counters to represent,,,, or. (add to anchor charts) 6. Set Model - Provide students with scenarios illustrating fractions of a set (i.e. John has a bag of 8 cookies. 1 is chocolate. What fraction of the cookies is chocolate?) Show with counters and write the fraction. (add to anchor charts) 7. Linear Model - Have students draw a line horizontally on a piece of paper labeled to make a number line. Model for students how to find and label,,,, or on the number line. (add to anchor charts) 8. Encourage students to share their observations. Facilitate discussion about the order of the numbers and comparison of the sizes of the unit fractions on the strips of paper. It is important that students understand this concept before moving forward with other fraction lessons.
Lesson 3 – What is the whole?
Lesson Learning Goals: Express whole numbers as fractions: 3 = 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. Equal shares do not have to be the same shape. Shares must have the same fractional value but they do not have to look the same. Use fraction language: halves, thirds, fourths, sixths, and eighths. Regional models show the part-whole concept of fractions and the meaning of relative size to a part of the whole. Recognize unit fractions: ,,,, or . Materials: (2) 1in x 4in strips of paper/person for half of the class, (2)1in x 8in strips of paper/person for half of the class, grid paper or geoboards Part 1: The magnitude of the fraction is determined by the size of the whole. 6 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards 1. Students will work with a partner. One person will have 2 short strips and the other person will have 2 long strips. (The length of the strips is irrelevant as long as students can identify that the “whole” is different in length.)Tell students to pretend that the strips of paper represent long pieces of taffy candy that they will have to share. 2. Have each student in the group fold his strip to show half. Ask students to observe each person’s half. Say: “If you really like this candy and want to share half with someone, which person would you rather share with? Guide students into discovering the size of the unit fraction depends on the size of the whole.
OR
3. Repeat discussion with folding additional strips to show,, . (add to anchor charts)
Part 2: The whole may be divided in various ways to represent the same fractional value. 4. Using a sheet of grid paper or geoboards, ask students to determine how one region could be divided into halves, with all parts not appearing to be the same, as long as the area is the same.
5. Give each student a sheet of grid paper or a geoboard. Provide opportunities for students to explore this concept with thirds, fourths, and sixths.
Building on Unit Fractions 3 Days
Learning Goals: Understand the conceptual meaning of the numerator and denominator. A fraction is the quantity formed by n parts (copies) of size. For example: is and. I am counting in thirds and I have 2 of them. Understand a fraction as a number on a number line in the context of unit fractions. Recognize and utilize different interpretations of fractions such as a) a point on the number line, b) a number that lies between two consecutive whole numbers, c) as the length of a segment of the real number line, d) and as a part of a whole. Counting fractions can extend beyond the whole (ex. + + + = +=).
Instructional Background: 7 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards A unit fraction is between 0 and 1 on a number line and can generate any other fractions as copies of the unit fraction. (+ =) Numerator – The numerator tells how many of the equal parts you are talking about (the counting number). Denominator – The denominator is how many equal parts the whole is divided into (what’s being counted). When used alone, a fraction such as is a number or length, but when used in contexts such as “of an apple” the fraction represents a part of a whole. The phrase “I’ll take 3 oranges” is not about taking the number 3, but about counting 3 oranges. Similarly, “of an orange” is not about the number or unit fraction, but is a reference to a part of the whole orange. Lesson 4 – Non unit Fractions
Lesson Learning Goals: A fraction is the quantity formed by n parts (copies) of size. For example: is and. I am counting in thirds and I have 2 of them. Understand a fraction as a number on a number line in the context of unit fractions.
Materials: chart paper, markers, circles cut into eighths, unifix cubes, two-color counters, tiles, Cheerios, Goldfish, circles (Dinah Zykes foldables), Cuisenaire Rods, number lines
1. Introduce the lesson by reviewing unit fractions (anchor charts from previous lessons). Write the fraction in the middle of an anchor chart. Ask: What does this fraction mean? How do we describe what a model of this fraction looks like? a. is the amount we get by taking a whole, dividing it into 8 parts, and indicating one of those parts (partitioning). b. is the amount such that 8 copies of that amount put together, make a whole (iterating). Note: Make sure students understand that you can continue iterating past one. For example, = ++++++++
On the anchor chart, represent examples of the above, using area, set, and number line (linear) models. Students create individual charts. Such as:
Length Model a. 1 whole
←
b.
8 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards 1 whole
→
Set Model a. b. Δ Δ Δ Δ Δ Δ Δ Δ
Δ Δ Δ Δ Δ Δ Δ
Δ Δ Δ Δ Δ Δ Δ a/b.
Area Model 1 whole =+++++++ 2. Help students realize that a non-unit fraction is created by copying a unit fraction. To create a non-unit fraction, represent a unit fraction. Then copy the unit fraction representation until the number of unit fractions matches the non-unit fraction numerator.
9 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards
For example:
(unit fraction)
+ = (two copies of the unit fraction)
+ + = (three copies)
3. Give each student of a circle without displaying the whole.
Ask students what fraction this represents. Since students can only guess without seeing the whole, ask how they might find out. Lead students to determine that they could work together to build a whole circle. Ask why this strategy would tell them which fraction the piece represents. (The number of equal units iterated to make a whole indicates the size of the unit.) Have them put their pieces together and raise their hands when they have a whole. Discuss that the unit represents 1/8 because they needed to use 8 units to make a whole: Count the units: , , , , , , , = 1 whole Add the units: +++++++= = 1 whole 4. Repeat this procedure with at least one more unit representation. (i.e. ) Students may use clues to help them guess the unit size before building it. (i.e., Is it greater or less than the unit?). 5. To review and assess, display a variety of fraction representations and ask students to name the fractions. Include improper fraction representations. 6. Discuss the fraction: denominator (fraction being counted), numerator (number counted). Build representations of in a variety of ways. (unifix cubes, two-color counters, tiles, Cheerios, Goldfish, circles, Cuisenaire Rods, number line, etc)
10 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards
Lesson 5 – Representing Fractions
Lesson Learning Goals: Recognize and utilize different interpretations of fractions such as a) a point on the number line, b) a number that lies between two consecutive whole numbers, c) the length of a segment of the real number line, d) and a part of a whole. Understand the conceptual meaning of the numerator and denominator. A fraction is the quantity formed by n parts (copies) of size. For example: is and. I am counting in thirds and I have 2 of them. Materials: literature selection, chart paper, markers, glue, scissors, variety of materials representing fractions such as: egg cartons, string, rectangular piece of paper, paper plate, paper clips, two-color counters, unifix cubes, square tiles, beads and laces, dot paper, geoboard, grid paper, pattern blocks, Cuisenaire rods, small matching post-its, etc
1. Share a fraction-themed piece of children’s literature. For example, Full House: An Invitation to Fractions by Dayle Ann Dodds. 2. As a class, create a pictorial anchor chart for or another fraction related to your story. Draw a large in the middle. Surround it with area, set, and linear model drawings to represent. 3. Divide students into small groups. Give each group a piece of chart paper, glue, scissors, markers, etc. 4. Using the materials on their tables, each group will create an anchor chart with area, set, and linear representations of. Materials might include egg cartons, string, rectangular piece of paper, paper plate, 16 paper clips, 20 two-color counters, 8 unifix cubes, square tiles, beads and laces, dot paper, geoboard, grid paper, pattern blocks, Cuisenaire rods, 12 small matching post-its, etc. Students may glue consumable representations and/or draw nonconsumable representations. *Remind students that they must show the whole in each representation. 5. Have groups share their anchor charts. For groups who finish early, have developmentally appropriate fraction activities available: computer software, fraction games, fraction dominoes, fraction-themed books to read, etc.
Lesson 6 – More Representations
Lesson Learning Goals: Understand a fraction as a number on a number line in the context of unit fractions. Students must understand that each unit fraction generates other fractions (i.e., , , … or , , , ) and be able to locate these fractions on a number line. Students should understand that is the point to the right of 0 that demarcates the first segment created when the unit interval is divided into d equal segments. Points marking the endpoints of the other segments are labeled in succession with the numbers, ,, …These points represent the numbers that are called fractions.
11 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards Materials: popsicle sticks, chart paper, markers,
1. For review and assessment of Lesson 5, students will create individual anchor charts based on a different fraction. On enough popsicle sticks for each student to have one, write a different fraction with a denominator of 2, 3, 4, 6, or 8 and varying numerators. Give each student a small piece of chart paper. Each student will pull a labeled popsicle stick and create a pictorial anchor chart for the fraction on the stick. The charts will be displayed in the room, and the creator can add to the poster as new examples are found by the class. 2. Students will create and identify fractions (with denominators of 2, 3, 4, 6 and 8) on a number line. Using the pre-labeled popsicle sticks, have each student pull a stick and represent the fraction in as many ways possible with his/her premade fraction strips. As a label, each student will lay the popsicle stick beside the representation. Students will go on a gallery walk to view each person’s representation. 3. Teacher will discuss the number line as an important fraction model. On the interactive white board, utilize the activities found at: http://www.mathsisfun.com/numbers/fractions- match-frac-line.html http://visualfractions.com/IdentifyLines/identifylines.html As students identify the fractions, have them justify their answers by counting the unit fractions. Assessment Given any two of these – fraction name, part, whole- the students will use their models to determine the third. Example: If this rectangle is , what would the whole look like? (whole missing)
What fraction of the large rectangle does the small rectangle represent? (fraction missing)
Whole
If the first rectangle is the whole, what rectangle is? (Answer: b)
Whole a.
b.
12 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards c.
Missing Parts
1. If represents , what part is missing? ______
2. What fraction of this set of blocks is the cube? ______
3. If represents one whole, label on the circle.
4. If represents one whole, circle .
13 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards
5. If represents, draw one whole.
Understanding Equivalent Fractions 4-6 Days
Objective: Compare fractions by reasoning about their size and by explaining their equivalence using only number lines and visual models.
Learning Goals: Understand two fractions as equivalent if they represent the same quantity and are the same point on the number line. Recognize and generate simple equivalent fractions including those for whole numbers. Explain why fractions are equivalent using models. Compare two fractions with the same numerator or same denominator by reasoning about their size. Students should be able to justify their comparison verbally or with diagrams and models. Recognize that comparisons are valid only when two fractions refer to the same whole. Record the results of the comparisons with the symbols <, >, and =.
Instructional Background Students can develop an understanding of equivalent fractions and also develop from that understanding a conceptually based algorithm. Concept: Two fractions are equivalent if they are representations for the same amount or quantity – if they are the same number. Algorithm: To get an equivalent fraction, multiply (or divide) the top and bottom numbers by the same nonzero number. Only use fractions with a denominator of 2, 3, 4, 6, and 8. Equivalencies and comparisons should be made using models to build conceptual understanding prior to using algorithms. Same size parts may be different shapes.
Lesson 7 - Exploring with Pattern Blocks
Lesson Learning Goals: Explain why fractions are equivalent using models. Recognize and generate simple equivalent fractions including those for whole numbers.
Materials: pattern blocks, fraction dice labeled, , , , , and (1/pr of students)
1. Give each group of students a set of pattern blocks. Review the names of the pieces and list them on the board: triangle, square, parallelogram or rhombus, trapezoid, hexagon. 14 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards
2. Pose the following problem: Use different combinations of pattern blocks to build hexagons that are the same size and shape as the yellow hexagon. Try to find all the possible combinations. You may not count a different combination of the same pattern blocks as a different way. Look for different combinations of blocks.
*There are eight ways to build a shape congruent to the hexagon, including the hexagon as one way, but don’t tell the students this in advance.
3. Allow students time to work. Circulate around the room. If a group has built duplicates, tell them that some of the shapes are the same, but challenge them to spot the duplicate. Having students try to figure out when they’ve found all the combinations adds a logical reasoning dimension to the problem.
4. Once several groups have solved the problem, have students describe their solutions and record them on the board. Groups should check to make sure they have all the combinations posted and, if not, build them now. 2 red trapezoids 6 green triangles 3 blue rhombuses 1 yellow hexagon 1 red trapezoid, 1 blue rhombus, 1 green triangle 1 red trapezoid, 3 green triangles 2 blue rhombuses, 2 green triangles 1 blue rhombus, 4 green triangles
5. Model how to describe one of the ways with fractions. Beside 2 red trapezoids write + = 1. Ask students to think about why this describes a hexagon made from two trapezoids and call on someone to explain. Point out that this representation assumes that the hexagon is worth one whole.
6. Individually, have students represent each of the ways they built the hexagon using fractions. Students should record the block combination, the fractions used to represent the blocks, and then combine fractions when possible.
7. Day two – Play Wipeout (pattern blocks and fraction die) a) Partners decide if each will start with one, two, or three hexagons. b) Take turns rolling the fraction die. c) On your turn you may take one of the following options: remove a block if it’s the fractional part of the hexagon indicated by the fraction die exchange any of your remaining blocks for equivalent blocks; do nothing. d) Check with your partner to be sure he/she agrees with what you did. e) When you are finished with your turn, pass the die to your partner. f) The first person to discard all of his/her blocks is the winner.
15 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards Lesson 8 - Only One
Lesson Learning Goals: Understand two fractions as equivalent if they represent the same quantity and are the same point on the number line. Recognize and generate simple equivalent fractions including those for whole numbers. Explain why fractions are equivalent using models.
Materials: fraction kit (made in an earlier lesson) or Cuisenaire rods, fraction dice labeled, , ,, , (1/pr of students)
1. In this activity, students will build “trains” and then figure out how to represent it using only one fraction. Model this by showing students the strip for one whole. Cover the strip with , , and . (Note: the whole strip does not have to be covered.
Record + +. Ask: “How can I shorten what I’ve recorded?” Allow students to respond and explain why.
Record + . Ask: “What color piece can I use to build a train that’s the same length but uses pieces of only one color?” Allow students to respond and explain why. Build the train students suggest next to the one you’ve already built. Explain that the fractions recorded below are worth the same and are therefore called equivalent fractions.
+ + =
Continue until you’ve used all the possibilities. ( + + + + + =)
2. Students repeat this process for another train, again building, recording, then building another train using pieces of only one color, and finally recording the length of the train with only one fraction. Circulate as students work providing help as needed.
Lesson 9 - Equivalent Fraction Activities
16 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards Mix and match from the following activities. Note: These activities can be adapted to meet the needs of your class (e.g., by changing the denominators). The standards for grade 3 are limited to fractions with denominators 2, 3, 4, 6, and 8.
Different Fillers Materials: fraction strips or fraction rods, circle fraction pieces Using an area model, prepare a worksheet with two or three outlines of different fractions. Do not limit yourself to unit fractions. For example, if the model is circular pie pieces, you might draw an outline for,and . Students use their own fraction pieces to find as many single- fraction names for the region as possible. After completing the examples, students write about their ideas or patterns they noticed in finding the names. Conclude with a class discussion. Questions might include: “What names could you find if we had sixteenths in our fraction kit? What names could you find if you could have any piece at all?” Try to push students beyond filling in the region is a pure trial-and-error approach. Length models can be used such as rods or strips. Students use smaller rods to find fraction names for the given part. To have larger wholes and, thus, more possible parts, use a train of two or three rods for the whole and the part. (See Only One lesson above) Folding paper strips is another method of creating fraction names. Dot Paper Equivalencies Materials: dot grid paper Using dot grid paper, draw the outline of a region on a portion of the paper and designate it as one whole. Shade lightly a part of the region within the whole. 1. The task is to use different parts of the whole determined by the grid to find names for the part.
2. Students should draw a picture of the unit fractional part that they use for each fraction name.
3. The larger the size of the whole, the more names the activity will generate.
Example
=
17 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards Group the Counters, Find the Names (Set model) Materials: two-color counters
1. Have students set out a specific number of counters in two colors – for example, 24 counters – 16 red and 8 yellow. The 24 make up the whole. 2. Students group the counters into different fractional parts of the whole and use the parts to create fraction names for the red and yellow counters. 3. You might want to suggest arrays or allow students to arrange them in any way they wish. 4. Students record their different groupings and explain how they found the fraction names. Example
and Questions for prompting students: “If we make groups of four, what part of the set is red?” “If we make groups of one-half counters, what would the yellow set be called?
Divide and Divide Again Materials: Gator Pie, circles or rectangles cut into halves and/or thirds
Read aloud the book Gator Pie. In this story Alvin and Alice find a pie in the woods. Before they can cut it, another gator appears and demands a share of the pie. As the story continues, more and more gators arrive until there are 100 gators who want a piece of pie. Finally, Alice cuts the pie into hundredths. An interesting twist is to change the story so that the pie is cut before more gators appear. The problem is how to share it among a larger number once it is already cut. To illustrate, cut a circle, or rectangle, into halves or thirds, and then ask students to decide how to share it among a larger number once it is already cut. You may start going from halves to sixths. This is reasonably easy but may surprise you. After students have shared their approaches, progress into more difficult divisions. What if the pie is cut in thirds, and we want to share it in tenths? Students should be expected to identify the fractional parts they used and explain how and why they used those particular fractional parts.
Missing-Number Equivalencies Give students an equation expressing equivalence between two fractions but with one of the numbers missing. Such as:
= = = =
18 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards The missing number can be either a numerator or denominator. It can also be either larger or smaller than the corresponding part of the equivalent fraction. Students should find the missing number and explain his/her solution. You may want to specify a particular model, such as sets or pie pieces or you can allow students to select whatever methods they wish to solve the problems. One or two equivalencies in conjunction with a class discussion is sufficient for a lesson.
The ability to tell which of two fractions is greater is an aspect of number sense that should be built around concepts of fractions, not on algorithmic skill or symbolic tricks. Some of these approaches are finding common denominators and using cross multiplication. These rules can be effective in getting correct answers but require no thought about the size of the fractions. Teaching children these rules before they have had the opportunity to think about the relative size of various fractions lessens the chance that they will develop any familiarity with or number sense about fraction size.
Lesson 10 - Put in Order Note: These activities can be adapted to meet the needs of your class (e.g., by changing the denominators). The standards for grade 3 are limited to fractions with denominators 2, 3, 4, 6, and 8.
Lesson Learning Goals: Compare two fractions with the same numerator or same denominator by reasoning about their size. Students should be able to justify their comparison verbally or with diagrams and models. Recognize that comparisons are valid only when two fractions refer to the same whole. Record the results of the comparison with the symbols <, >, and =.
Instructional Background - Comparing and Ordering Fractions The ability to tell which of two fractions is greater is an aspect of number sense that should be built around concepts of fractions, not on algorithmic skill or symbolic tricks. Some of these approaches are finding common denominators and using cross multiplication. These rules can be effective in getting correct answers but require no thought about the size of the fractions. Teaching children these rules before they have had the opportunity to think about the relative size of various fractions lessens the chance that they will develop any familiarity with or number sense about fraction size.
Materials: 4 x 6 index cards, each with a fraction written on it (see Sample Sets below) OR Create a Promethean flipchart with fractions for the students to manipulate. Set 1: , , , , , , , , , , , Set 2: ,, , , , , , , , , , Set 3: , , , , , , , , , , ,
19 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards 1. Show students the fraction card with. Since is a benchmark fraction, it’s helpful for students to use it when comparing fractions.
2. Show students the other fraction cards and explain that it will be their job to decide where to place the cards on the chalk tray so that the fractions are lined up in order from smallest to largest. Also, when a student places a fraction, they must give a convincing reason for the placement.
3. Show the card. Have one of the students tell where it goes and why. When a student explains where to place a fraction, ask if anyone has a different way to explain. This sends the message that there are different ways to think about comparing and ordering fractions. It’s also helpful for students to hear a variety of strategies.
4. If a student places a card incorrectly, give him/her time to rethink his/her reasoning; then give others a chance to explain their thinking. If a card is placed incorrectly and no one challenges the placement, continue with the activity. Usually the error will be noticed after several other fraction cards are placed. However, if all the cards are placed and no one recognizes the mistake, say “I don’t agree with the placement,” and give students the opportunity to talk with a partner to locate the error.
5. Continue until all but one fraction has been placed. Use this fraction as an individual assignment. Have students write about where the last fraction should be placed.
6. Extend the lesson by having students create a number line. Fractions from the cards should be recorded on the number line in the appropriate place.
Comparing Fractions Note: These activities can be adapted to meet the needs of your class (e.g., by changing the denominators). The standards for grade 3 are limited to fractions with denominators 2, 3, 4, 6, and 8. The purpose of this activity is to determine if students understand how to compare numerators and/or denominators.
Which fraction in each pair is greater? Give one or more reasons. You may use drawings or models in your explanation. Record your answer using >, <, or = symbols. A. or F. or
B. or G. or
C. or H. or
D. or I. or
E. or J. or
Assessments:
20 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards Fraction Action 15 (Taken from AIMS: Actions with Fractions) – Use after the lesson Put in Order. Show the fractions named on each hexagon. Order the eight fractions from greatest to least.
a. b. c. d. 1/
e. f. g. h.
> > > > > > >
Fraction Action 22 (Taken from AIMS: Actions with Fractions) – Use after the lesson Exploring with Pattern Blocks. Show the named fraction in the hexagon on the left. In the right hexagon, build an equivalent fraction. Name the second fraction. a. b.
= =
c. d.
= =
Formative assessment, such as student responses to teacher discussions, student interviews, oral and written activities, should be an ongoing process in the unit of study. Listed below is a bank of items from which a summative assessment may be developed. 1. Which fraction matches this chain of unit fractions?
2. Shannon collects paper for recycling. If she collects of a pound of paper each week, how much paper will she collect in 4 weeks? 21 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards
3. Use the fraction strips to answer the question.
Which is more one twelfth or one fourth? 4. The triangle is one third of a polygon. Which polygon shows this?
5. Sue rode her bike mile. Draw a picture to show this. Use the number line below.
6. Jim ate one sixth bag of cookies. Show the correct amount using the circles below.
7. One half of the pizza is pepperoni. Draw a picture to show this. Use the circle below.
8. Sarah cut her candy into pieces each the same size. She ate one eighth of her candy. Into how many pieces could she have cut her candy? Draw a picture.
9. Bob lives three miles from the library. He walked 1 mile alone then met a friend who walked with him. What fraction of the distance did he walk alone? 22 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards 10. Shameka and Kia both have the same size pizza. Shemaka cut her pizza into 8 pieces each the same size. Kia cut her pizza into four pieces each the same size. Each girl ate one piece of their pizza. Who ate a larger slice?
11. Carol said that she sharpened 1/4 of the box of pencils. How many pencils could have been in the box? A. 1, B. 2, C. 3, D. 4
12. Are these figures equivalent? Why or why not?
13.The picture below shows the amount of chocolate milk Lance drank. About what fraction of the container did he drink?
14. The picture below represents the whole. Use the picture below to answer the questions.
is ______piece(s) of the whole is ______piece(s) of the whole is ______piece(s) of the whole is ______piece(s) of the whole is ______piece(s) of the whole
15. Markon the number line. 23 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards
16.You and your friend have a Hershey bar. You both eat of your own Hershey Bar. Your friend eats more of her Hershey bar than you do. Draw a picture and explain how it is possible for your friend to have eaten more of the bar even though you both have eaten of your own Hershey bar.
17. What fraction of the shape below is shaded?
A. , B. , C. , D.
18. Mark is offered the choice of a third of a pizza or a half of a pizza. He is very hungry and chooses the third of a pizza because it is larger. Draw a picture to show the two pizzas. Explain why he knows the third is larger.
19. Four friends are running a race. The fractions tell how much of the distance each has run.
Sandy Dale Christie Rita Carla Place the friend on a line to show where they are between the start and finish.
20. Three children share four brownies. Each child will get. Draw a picture to show how the children should equally share the brownies.
21.Which figures(s) is correctly portioned in fourths?
24 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards Unit Resources: The resources listed below can be referenced for activities to supplement this unit of study.
Fabulous Fractions/AIMS Activities Grades 3-6 pg. 1-19 Fair Shares/Grade 3/Sharing Brownies pg. 2-37 Math Expressions/Grade 3/Unit 11/ lessons 1, 2, 3, Beyond Pizzas and Pies/Top or Bottom With one Matters/Pg. 15-22 Lessons for introducing Fractions by Marilyn Burns Fractions as Parts of Sets/ pg 1 Exploring Fractions with Pattern Blocks/pg 39 Only One/pg 122 Put in Order/ pg 105 Everything Coming Up Fractions with Cuisenaire Rods Pg. 1-10 John Bradford A Collection of Math Lessons From Grades 3 through 6 by Marilyn Burns Fractions with Cookies/Grade 3/pg. 37-43 Teaching Student-Centered Mathematics Grade K-3 by John Van de Walle Unit Fractions/pg 252-258 Models for Fractions/ pg 254-256 Correct Shares/ pg 257 Teaching Student-Centered Mathematics Grade 3-5 by John Van de Walle Different Fillers/pg 152 Dot Paper Equivalencies/pg 152 Group the Counters, Find the Names/pg 153 Divide and Divide Again/ pg 154 Missing Number Equivalencies/ pg 155 Correct Shares/ pg 136
Children’s Literature: Fraction Fun by David A. Adler Full House: An Invitation to Fractions by Dayle Ann Dodds and Abby Carter Apple Fractions by Jerry Pallotta and Rob Bolster Piece = Part = Portion by Scott Gifford and Shmuel Thaler Working With Fractions by David A. Adler and Edward Miller The Hershey's Milk Chocolate Bar Fractions Book by Jerry Pallotta and Robert C. Bolster If You Were a Fraction (Math Fun) by Speed Shaskan, Trisha, Carabelli, and Francesca Whole-y Cow: Fractions Are Fun by Taryn Souders and Tatjiana Mai-Wyss Fraction Action by Loreen Leedy Equal Shmequal (Math Adventures) by Virginia L. Kroll and Philomena O'Neill Gator Pie by Louise Matthews
Appendix A
25 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards Sample Anchor Chart
one-half
Appendix B
26 Developed by SC Leaders of Mathematics Education 2010-2011 Fractions A Supplemental Unit of Study based on Common Core Standards Sample Concept Chart As you work through the lessons in the unit, create and post a list or chart to highlight key concepts that students should learn. Fractions
A fraction is the quantity formed when a whole is portioned into equal parts.
A fraction with 1 as the numerator, such as is called a unit fraction.
Fractions can be represented on a number line.
Fractions are equivalent (equal) if they are the same size, or the same point on a number line.
Whole numbers can be expressed as fractions (3 = or = 1).
Fractions with the same numerator or same denominator can be compared by reasoning about their size. The two fractions must refer to the same whole.
Unit fractions can be copied to create other fractions. Such as + =
The numerator tells how many of the equal parts you are talking about (the counting number).
The denominator is how many equal parts the whole is divided into (what’s being counted).
Fractional parts are equal shares or equal-sized portions of a whole or unit. A unit can be an object or a collection of things. More abstractly, the unit is counted as 1. On the number line, the distance from 0 to 1 is the unit.
The more fractional parts used to make a whole, the smaller the parts (the larger the denominator, the smaller the part). For example, eighths are smaller than fifths.
27 Developed by SC Leaders of Mathematics Education 2010-2011