Patterns in Fractions

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Thomasina Williams
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$Comparing Fractions using Creature Capture Patterns in Fractions Lesson time: 25-45 Minutes Lesson Overview Students will explore the nature of fractions through playing the game: Creature Capture.$

1 Comparing Fractions using Creature Capture Patterns in Fractions Lesson time: Minutes Lesson Overview Students will explore the nature of fractions through playing the game: Creature Capture. They will need to analyze both the properties of the numerator and denominator when comparing to other fractions. In addition, they will need to create strategies related to the cards to be played and where to place them on the board. Lesson Objectives Students will: Determine if given fractions are relatively larger or smaller than other fractions Determine if given fractions are closer to the benchmark of ½ when compared to other fractions Compare fractions through modeling (drawing) Explain the effect of the numerator and the denominator on the size of the fraction Estimate the relative size of two fractions added together Estimate the relative size of two fractions multiplied together Convert fractions to have a common denominator [Challenge] Anchor Standard Common Core Math Standards 3.NF.3d 4.NF.2 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 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols, =, or Lesson Plan Summary Introduction (1-5 minutes)

2 Game Play (15-25 minutes) Reflection (10-15 minutes) Materials, Resources, and Prep For the Student At least one computer for every two students Any handouts desired for the reflection activity For the Teacher Ensure you can load the game from your classroom: Decide if, and how, you want to activate prior knowledge before the game Decide how the vocabulary will be incorporated into the lesson Decide if the class will use power cards and what characteristics of them you will explore (i.e. keep it simple. A fraction + fraction results in a bigger fraction OR make it harder. What is the exact result of ¼ + 2/3?) Prepare at least one reflective activity for the class Lesson Plan Details Vocabulary This lesson has a number of words that can be incorporated. These should not necessarily be introduced at the beginning but the teacher should try to use as many as appropriate and assess understanding of the chosen words upon completion: Fraction: A number which represents equal parts of a whole. Numerator: The top number in a fraction which tells how many parts of the whole. A larger numerator means more a larger number (more parts shaded). Denominator: The bottom number in a fraction which tells how many equal pieces the whole was broken into. A larger denominator means smaller pieces (the whole has been broken into more parts) Benchmark(optional): A number we use to help us compare other numbers. 1 is a good benchmark. We know a fraction is greater than one if the denominator is less than the numerator (such as 4/3). ½ is another benchmark we often use. Fractions whose numerator is less than half of its denominator are below ½ (such as 3/8 is less than ½ since 3 is less than half of 8). Introduction

$Students will come into this lesson with varying experience of fractions.$

3 Students will come into this lesson with varying experience of fractions. In general, less talk at the start is better but activating some prior knowledge (especially student-to-student) may be helpful. Here are a few questions that might be worth discussing before (and definitely after) some game play. How can I fairly cut this pie (circle) or cake (rectangle) to share among 3 people? 4 people? How can I share 2 candy bars (strips) among 3 people? Which is bigger 1/3 or ¼? Why? [a third is a bigger slice than a fourth] Which is bigger 3/5 or 4/5? Why?[four somethings is bigger than the 3 of the same somethings] Which is bigger 5/6 or 7/8? Why?[7/8 because taking away an eighth from a whole will leave you with more than if you took away a sixth from a whole] Which is bigger 1/3 or 3/4? Why? [3/4 because we see these fractions every day!] Which is bigger 3/7 or 4/5? Why? [Hard to say. Maybe I recognize that 3/7 is less than 1/2 and 4/5 is not. Maybe I know how to get common denominators. Maybe I draw them out and color them and see that 4/5 is bigger] Is there a general rule (or set of rules) for telling if a fraction is bigger? TIP: Interrupting the students after 10 minutes of game play to re-focus on these questions as well as allowing them to share discoveries and strategies is recommended. Activity: Creature Capture Online Fraction Game The standard version of Creature Capture that is posted on the Center for Game Science web site allows the student to begin at any level. Ideally they will start from Level Pack 1 level 1 and work their way through. A detailed breakdown of the levels and other considerations are at the end of this lesson plan. Suggested In-Game Activity: When confronting fractions of unknown size, draw the two fractions either as number lines or as pie charts (with the numerator part shaded). To help in this effort, fractions in the players hand will usually reveal either a number line or pie representation by hovering over the card. Even a student who knows the fractions might be encouraged to draw four sets of comparisons: Two sets as number line and two sets as pie representations. For thirds and sixths, some younger students may need help drawing these.

4 If students are sharing a computer, they should alternate who is controlling the mouse on each level. The person without the mouse should be the decider and select which cards to play and where to play them. A goal would be to have all students be able to correctly know if a particular match-up of cards would win AND to be able to explain their thinking. Reflection & Wrap-up Research has shown critical that some form of debriefing takes place after any game in the class room for the learning benefits to be realized. Reflection is where the students transfer the play into learning outcomes. Based on your adaptation of this lesson, please choose at least one to implement in the classroom. Come up with a set of SIMPLE rules (or examples) for adding with power cards (for instance, if both are less than ½ then the result will be less than 1). Display these using combined pie charts. Come up with a set of SIMPLE rules (or examples) for multiplying by power cards (for instance, if multiplying by a number more than 1 the result is bigger whereas if the multiplying by a number less than 1 the result is smaller). Display these using crosshatched grids. Have students play the candy cut game (similar to Create Capture). See ds%29.pdf. There is a lot of information here including many variations of this game. Note that the game includes coloring a lot of pre-made fraction cards. Revisit the discussion questions. Have the students verbalize (and write if appropriate) the answers to two of these questions. Give the students 3 comparisons using pie charts, 3 with number lines, and 3 with numbers only. Have them circle the one that would win a fire battle (larger). Have them star the one that would win a grass battle (closest to ½ ) Have them determine which fraction goes between the other two. See Visual/English/1.pdf To further explore relative size compared to the benchmark ½, discuss strategies for deciding if the same as ½ (or less than) and then attempt the following: (or easier version: ish/1.pdf ) END OF LESSON PLAN. MATERIAL BELOW THIS POINT FOR REFERENCE ONLY. NO NEED TO PRINT (OR READ UNLESS CURIOUS).

5 Additional Common Core Math Standards which are connected to this lesson 3.NF.1 3.NF.3d 4.NF.2 4.NF.3a 4.NF.3b 4.NF.3c 5.NF.1 5.NF.2 5.NF.4a 5.NF.5a 5.NF.5b Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 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 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols, =, or 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. 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. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. Interpret the product (a/b) q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a q b. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n a)/(n b) to the effect of multiplying a/b by 1.

GOALS STUDENTS SHOULD HAVE: 1) To be able to recognize the outcome of any battle before the card is played and be able to explain their thought process to someone.

6 GAME AND PROGRESSION NOTES: For the basic level of this lesson plan, only the first Level Pack (1-12) needs to be played. The levels after this all involve Power Cards and you will need to decide on how you wish the class to address these. See the Power Card notes at the end of this section. GOALS STUDENTS SHOULD HAVE: 1) To be able to recognize the outcome of any battle before the card is played and be able to explain their thought process to someone. 2) To watch the animations during the battles and to use the mouse over animations to explore their cards if they are unsure as to their size. This exploration will improve their understanding of fractions and the game. 3) IF YOU ARE USING POWER CARDS, encourage the students to try to use the Power Cards on each turn. It is possible to win without them but using them will expose additional patterns related to multiplication and addition with fractions. Encourage them to risk a mistake by using the Power Cards rather than going for the sure win without using them. A brief outline of the levels and their contents is below. Students are given the choice of where they start. Ideally they should start at Level 1. From the Level Menu, you can tell if a level has been finished by a star (see level 5 on right). Level 1: Provides a board where the student cannot lose. This is simply teaching them to place tiles and that water battles are looking for the smallest number.

Level 2: The real challenge begins. Fire Battles are looking for what is largest.

In the example provided, the only mistake a student can make is by placing one of the cards on a tile that is not adjacent to the ½.

$In addition to understanding the relative size of a fraction, the student must be aware of which type of battle is being fought.$ The advanced student may be looking ahead one move to see if their win will be short-lived (for instance, placing a 1 on a fire battle will likely win that

The advanced student may be looking ahead one move to see if their win will be short-lived (for instance, placing a 1 on a fire battle will likely win that

7 Level 2: The real challenge begins. Fire Battles are looking for what is largest. The simplest concept of winning at this level is flipping a card to your color. In the example provided, the only mistake a student can make is by placing one of the cards on a tile that is not adjacent to the ½. Level 3: A mix of fire and water battles. In addition to understanding the relative size of a fraction, the student must be aware of which type of battle is being fought. Here the student can make a mistake if they do not recognize the water battle next to the ¾. The ½ and 2/3 will win but the ¾ and 5/6 will not. The advanced student may be looking ahead one move to see if their win will be short-lived (for instance, placing a 1 on a fire battle will likely win that turn, but if a water battle is adjacent to this, the rival will almost certainly take this card back on the next turn. On this level, the cards you see are all you will receive. Some students may notice that the cards are in order from smallest to largest. Level 4: Only one correct first move. ¼ on water next to ½. Level 5: Grass Battles indicates closest to ½ wins. Rival also gets to go first. A strategic best move MIGHT BE to play the ½ next to the ¾ as it cannot be flipped back once placed.

$Level 6: Mix of all battles. Student fraction cards are ¼, ½, ½, and 1. Level 7: Mix of all battles. Student fraction cards are ¼, 1/3, ½, and 1. Level 8: Fractions no longer perfectly ordered.$ $Player only gets one card at start. A new fraction card is drawn after each turn. They appear in this order: ¼, 2/3, ¾. At this level, students may opt to skip animations that appear during battles.$

8 Level 6: Mix of all battles. Student fraction cards are ¼, ½, ½, and 1. Level 7: Mix of all battles. Student fraction cards are ¼, 1/3, ½, and 1. Level 8: Fractions no longer perfectly ordered. Student fraction cards are ¼, 1/3, 1/2, 1 and 2/3. Level 9: Fractions no longer perfectly ordered. Student fraction cards are ¼, 1/3, 1/2, 1 and 3/4. Level 10: Begin drawing fractions. Player only gets one card at start. A new fraction card is drawn after each turn. They appear in this order: ¼, 2/3, ¾. At this level, students may opt to skip animations that appear during battles. Level 11: Start with 5 cards but draw after each one. Initial cards: 1,2/3, ½, 1/3, ¼ (always in reverse order like this). First card drawn is 1/6. Varies after that. Level 12: Repeat of Level 11 except board is different shape and distribution of battles. This level may require slightly more strategic thinking to win the level. POWER CARDS Levels Power Cards are cards that can add or multiply an existing card OR a future card. In the example, shown (Level 13), I might place the (x ½) card on the existing 1 on the board so it is smaller. Then placing the 1 (or ¾) on the fire battle will win (because the newly established ½ loses to ¾ in a fire battle). Note that the power card disappears after the battle and the card returns to its previous value. Level 13. One turn. Cards: 1, ¾. Power Card: (x2) Should use both power card and a number card. Note the should. They are not required to use the power card but this particular level is impossible to win without its use.

$Note these are addition power cards. Level 17: 3 turns. 4 cards: Unit fractions: ½, ¼, 1/3, 1/6. Power Card: (+1/2),(+1/3). Level 18: 4 turns. 5 cards: ½,3/4,2/3,1/3,1/5.$

9 Level 14: 4 turns. Cards: 1,1/3,2/3,1/4. Power Card: (x2/3),(x1/3). At this point the students do not have to know the exact value of the multiplication but they should understand when multiplying by a number less than 1, the resulting number is smaller than the original. Thus 2/3(x1/3) will be less than 2/3. Level 15: 5 turns. 5 cards: Random. No 5/6 or 1. Power Card: (x2/3),(x1/2)(x3/4). Level 16: Perfect Puzzle Level. Take all 4 at once. Note these are addition power cards. Level 17: 3 turns. 4 cards: Unit fractions: ½, ¼, 1/3, 1/6. Power Card: (+1/2),(+1/3). Level 18: 4 turns. 5 cards: ½,3/4,2/3,1/3,1/5. 3 Random Power Cards selected from (+1/3), (+1/4), (+1/2)(+2/3)(+3/4)(+1),(+4/3) (x1/3) Level 19: 6 turns. 5 cards: ¾,5/6,2/3,1/6,1. Power Card: (x1/2)(+1/2). Draw additional number cards on each turn BUT NO ADDITIONAL POWER CARDS. Level 20: 3 turns. 3 cards: ½, 1/6, 2/3, 1/3. Power Cards: (+3/4)(+1/2)(+1/3) Level 21: 4 turns. 5 cards: Random All. 2 Power Cards: Random All. Draw additional number AND POWER cards after each turn. Level 22: 5 turns. 5 cards: Random All. 3 Power Cards: Random All. More intelligent rival. This level will be challenging to win. Level 23: 6 turns. 5 cards: Random All. 2 Power Cards: Random All. More intelligent rival. Level 24: 8 turns. 5 cards: Random All. 2 Power Cards: Random All. More intelligent rival.

10 From this point on, it is essentially more practice with various board configurations and number of turns. Levels 25 and 26 have a slightly easier rival again but from level the rival is similar to playing a fairly smart human opponent. MORE NOTES ABOUT THE PROGRESSION: All students should play levels 1-1 to 1-5 to familiarize themselves with the game and the three battle types. If you want students to play using power cards, you might have them jump to level 13 AFTER FAMILIARIZING THEMSELVES WITH THE GAME. MORE IDEAS ON CLASSROOM MONITORING: It is possible for students to skip levels or accidently repeat the same level over and over again. When monitoring students look for the following: For any given card placement, can they explain what will happen and who will win? If not, here are possible forms of remediation: o Q: Can they explain the differences in the three battle types? o Q: Do they know the objective of a single turn? o Q: Can they explain who would win a fire battle between ½ and ¼? Inability to answer the first two sub-questions above might require them to go back and repeat earlier levels. It may be that they have skipped levels. From the main menu only completed levels will have stars. Difficulty with the last sub-question above might be address through a mini-lesson or having them repeat levels but paying closer attention to the animation and the card animations (received when hovering mouse over unplayed card). If students seem very far behind, it may be they continue to repeat the same level. They may be hitting the Replay Button at the bottom of the screen instead of Next Level button at the bottom right. If the main menu makes it appear that they have solved far more levels than you think possible, it s likely a cache issue. The cache is the memory that the browser keeps for the game. If another player from a previous class has been playing on the same computer, it may be that the computer is remembering this first player s history. You can simply ignore it, open an anonymous version of the browser (which will not have a cache and all stars from completed levels will be removed), or clear the cache (this process will vary by browser). More On Using Power Cards: Power Cards do not have to be played. A student may completely ignore them. You must encourage their use. One way to do this, is for a student to get a witness when they think they are using one correctly. They should write down what the play is (for instance, rival 1 (x1/2) fire battle vs. my ¾. I win because ½ < ¾ ). The witness should initial the statement if it played out as described. If they lose the battle, they can still get a point if the player and the witness can describe what actually happened.

Grade: 3 Lesson Title: Equivalent Fractions

Targeted Content Standard(s): Grade: 3 Lesson Title: Equivalent Fractions 3.NF.3 Explain equivalence of fractions in special cases and compare fractions by reasoning about their size. a. Understand two