Fraction Mania Lane ESD February 12, 2016

Transcription

1 Fraction Mania Lane ESD February 12, 2016

2 Common Difficulties Next whole numbers are obvious not so with fractions Multiplication makes amounts bigger except with fractions Division makes amounts smaller except with fractions Viewing the numerator and denominator as separate entities, rather than viewing the fraction as a single number Difficulties grasping the less than 1 idea; then difficulties extending on a number line past one. Not recognizing the role of the whole From Uncomplicating Fractions by Marian Small (201) What do we expect students to learn? Kindergarten Focus Number bonds, part-part-whole mats 2 2 part whole part Making whole of 10 ( Ten Friends ) 8 and? 4 and? How does this relate to fractions later on? 1

3 First Grade Focus Kindergarten ideas, plus Whole, halves, fourths, quarters No symbolic notation, just words is smaller than Second Grade Focus Halves, fourths, quarters and thirds Equal shares don t have to have the same shape, but must have identical wholes Fractions come in pairs (see Ten Friends ) three fourths one fourth one third two thirds Fraction Standards ALL fraction clusters in Grades through 7 fall into the Priority Clusters! Look at your grade s fraction standards. What are the big ideas that students must master at your grade level? Be prepared to contribute to a bullet point list. 2

4 How will we teach the fraction standards so students can learn? Five Big Ideas for Understanding 1: THE WHOLE STORY 2: COHERENCE WITH UNITS : EQUIVALENT FRACTIONS 4: COMPARING AND ORDERING Big Idea: The Whole Story What fraction is represented by the shaded area? 1: THE WHOLE STORY

5 Using Pattern Blocks 1: THE WHOLE STORY If = 1, what is? If = 1, what is? If = 1, what is? If = 1, what is? If = 1, what is? If = 1, what is? If = 1, what is? Tantalizing Tangrams (p. 26) 1: THE WHOLE STORY If the large square is 1 square unit, what fraction does each piece represent? Pizza Task: Grade 4 A pizza store makes two sizes of pizza shaped as rectangles. The large pizza is twice as long and twice as wide as the small pizza. Fadia ate 4 of a small pizza and Lori ate of a large pizza. 16 Fadia says she ate more pizza because 4 is greater than 16. Is she correct? Explain your reasoning. 1: THE WHOLE STORY 4

6 Big Idea: Coherence with Units cats + cats = 8 cats pennies + pennies = 8 pennies tens + tens = 8 tens inches + inches = 8 inches ( 1 4 inches) + (1 4 inches) = 8 (1 4 inches) = 8 4 2: COHERENCE WITH UNITS Wholes and Then Some What amount does the shaded part show? 1 Can students see the denominator within the whole, and count from there? 2: COHERENCE WITH UNITS Book Buddies Central Middle School has a book buddies program where sixth graders read to kindergarten students. Each sixth grader who volunteers is paired with one kindergarten student who wants a book buddy. If 2 of the sixth graders and of the kindergarten students have book buddies, what fraction of the combined kindergarten and sixth grade classes have book buddies? 2: COHERENCE WITH UNITS

7 Big Idea: Equivalent Fractions Try to avoid thinking procedurally just use the strips Organize the strips Work with partners to determine all the equivalent fractions that you can. Pick a set of equivalent fractions and use student language to prove why they are equal. : EQUIVALENT FRACTIONS Equivalent Work Use linking cubes or two-color counters to show why 2 = 4 6 Create all the equivalent fractions you can for where the denominator is less than 20. : EQUIVALENT FRACTIONS Equivalence on the Number Line Select and organize any three rows of your fraction tiles. How is each row like a number line? What would the numbers be on either end of the line? How would the marks in between be labeled? What would the number line look like if you included all three rows you selected? Draw a quick sketch. eighths sixths fifths Every fraction has a place in relation to the same whole. : EQUIVALENT FRACTIONS 6

8 Equivalent A-HAs Equivalent is another name for the same amount, just like: 20 = 2 tens or 1 ten and 10 ones or 4 groups of Say simplest form instead of reducing to reinforce the idea that it s just another way to say that fraction, not really smaller. Equivalent fractions on the multiplication chart can you find them? : EQUIVALENT FRACTIONS Multiplication Chart : EQUIVALENT FRACTIONS Big Idea: Comparing & Ordering Use only your tiles to find which is larger: 8 or 6 4 or 7 8 4: COMPARING AND ORDERING 7

9 Comparing Fractions Wholes must be the same Math Practice #6 Attend to Precision. Encourage students to find ways other than freehand drawing to visually compare fractions. Have a drawing lesson Teach using benchmarks such as halves and thirds. If it s close, find another way Outlaw circles except for halves and fourths 4: COMPARING AND ORDERING Grade : Comparing Fractions Compare with same numerator or same denominator Grade 4: Compare with different numerators and different denominators by creating a common numerator or common denominator Compare to benchmark fractions, such as 1 2. Grade : Use comparison skills to assess the reasonableness of answers Recognize that = 7 is incorrect because 7 < 1 2 4: COMPARING AND ORDERING Big Idea: Fraction Operations Believe it or not, both of these awesome bands make a contribution to this section. 8

10 Adding and Subtracting Use the fraction tiles to find a) + 7 (4.NF..a) 8 8 b) 1 (.NF.1) = 12 Back to the Unit because grapes + 9 grapes = 12 grapes. Units must be consistent you can t add grapes and bananas. If they don t match, make a smoothie! Relate to measurement. You wouldn t add inches and feet, right? Or pounds and ounces. One of them has to change. Interesting math fact: Adding the numerators and denominators of two fractions always results in a fraction between the two original ones. Try it. 9

11 Rock Around the Fraction Clock 1, 2, o clock, 4 o clock rock, 6, 7 o clock, 8 o clock rock 9, 10, 11 o clock, 12 o clock rock We re gonna rock around the fraction clock We gotta take two fractions, add em up If the D s the same then we re in luck We re gonna rock around the fraction clock We re gonna add some knowledge to our block We re gonna rock, rock, rock around the fraction clock If the D s are different, then we gotta find A common D by skip-counting or times (Repeat chorus) More than an Algorithm More than an algorithm more than an algorithm to meeeeee Using Models to Add & Subtract Fraction Tiles - requires guess and check Pattern Blocks - with denominators of 2,, and 6 Grid Model - any denominator within reason: = 11 1 Try it: Try it: Try it:

12 Using Models to Add & Subtract Try it: = 7 20 Improper Fractions Changing an improper fraction to a mixed number is considered simplifying. Introduce the idea in 4 th grade, work to mastery in th grade. More than an algorithm can students see how to convert? 8 Find the Operation For each problem, find the answer using the fraction tiles and write an equation that matches the numbers in the problem. 1. How many 1 tiles are in 2 wholes? 4 2. How many 1 tiles does it take to make 4 wholes?. Set out two 1 tiles. What tiles make 4 equal parts of what you set out? 4. What tiles make 1 of a 1 2 tile?. What tiles make 1 2 of a 1 tile? 11

13 Find the Operation The answer is 10. What is the question? 1 The answer is 10. What is the question? For bonus nerd points (not a th grade requirement!) The answer is 8. What is the question? The answer is 2. What is the question? Multiplying Fractions Math Practice #7 Look for and make use of structure. Commutative Property 8 8 If students know a x b = b x a, then 4 x = x 4 8 Some problems make more sense if you ask What is of 4? 8 Some problems make more sense if you ask What is 4 groups of? Write a story situation for each. What is 8 of 4? What is 4 copies of 8? Brownie Units Mrs. Logan went to the school bake sale to buy some brownies. All the pans of brownies were square. A pan of brownies cost $12. Customers could buy any fractional part of a pan and pay that fraction of $12 (For example, ½ of a pan costs ½ of $12.). Mrs. Logan bought ¾ of a pan that was ⅖ full. How much did she pay? 2: COHERENCE WITH UNITS Without doing the math yet, what steps do you need to take to solve this problem? Source: Doing What Works,

14 Find the Unit: Area Model Draw horizontal sections; shade 2 of them ( 2 ). 2: COHERENCE WITH UNITS Find the Unit: Area Model Draw 4 vertical sections; shade of them ( 4 ). The overlap shows 4 of 2. 2: COHERENCE WITH UNITS What s the Unit? So 4 of 2 is 6 out of 20 pieces, or x 2 = x $12 = 6 x 12 = If the whole pan is $12, how much will six pieces cost? 20 pieces = $ pieces = $ piece = $ pieces x $0.60= $ = = 6 = : COHERENCE WITH UNITS 1

15 Reflect on the Big Ideas Of the FIVE BIG IDEAS, which is one that you feel your students need more time to understand and practice? What are some steps you will take so they have that opportunity? Which of the FIVE BIG IDEAS helped you see something you hadn t seen before? 1: THE WHOLE STORY 2: COHERENCE WITH UNITS : EQUIVALENT FRACTIONS 4: COMPARING AND ORDERING Your Turn! Find the value of each expression using as many different methods as you can (include models, algorithms, conceptual explanations, etc). Start with one model for each. Give one-get one -- compare work with someone else. Share the first unique item you each have. Continue. R.A.F.T. Role, Audience, Format, Topic Low-prep way to differentiate Can be used on a choice day or as alternative homework If you have completed Hardest, choose one item from the RAFT Broadens kids responsibilities for quality work Role Audience Format Topic Fraction One Whole Letter Our relationship. 1/6 /6 Valentine You complete me. 2/ and /4 4 th Graders Instructions If you re going to compare us, at least do it fairly. 14

16 Two-Minute Speed Write What do I want to GET from today? 1

17 Use linking cubes or two-color counters to show why 2 = 4. Show your 6 work here. Create all the equivalent fractions you can for where the denominator is less than 20. Quick sketch of fraction number line with three denominators: 16

18 Multiplication Chart Use fraction tiles to solve. Draw a picture of how you solved

19 Grid Model for Adding and Subtracting (Use 1 Grid Paper) Try it: Answer: Notes: Try it: Answer: Notes: Try it: 1 4 Answer: Notes: 18

20 Find the operation. For each problem, find the answer using the fraction tiles and write an equation that matches the numbers in the problem. 1. How many 1 4 tiles are in 2 wholes? Answer: Equation: 2. How many 1 tiles does it take to make 4 wholes? Answer: Equation:. Set out two 1 tiles. What tiles make 4 equal parts of what you set out? Answer: Equation: 4. What tiles make 1 of a 1 2 tile? Answer: Equation:. What tiles make 1 2 of a 1 tile? Answer: Equation: 6. Write your own: Answer: Equation: 19

21 What s the Story? Write a story situation for each expression. What is of 4? 8 What is 4 copies of 8? 20

22 Reflection on the FIVE BIG IDEAS 1: THE WHOLE STORY 2: COHERENCE WITH UNITS : EQUIVALENT FRACTIONS 4: COMPARING AND ORDERING Of the FIVE BIG IDEAS, which is one that you feel your students need more time to understand and practice? What are some steps you will take so they have that opportunity? Which of the FIVE BIG IDEAS helped you see something you hadn t seen before? 21

23 Name Fractions and Decimals R.A.F.T. Challenge Options As you work, keep the following in mind: Include an accurate mathematical explanation Provide an example or solution within your piece Keep the format consistent Final work should be neat and easy to read and understand Bonus points for creativity, presentation and effort. Date Role Audience Format Topic Tenths & Hundredths 4th graders Instructions How to read me right! Fraction One whole Letter Our relationship 1/ 4/ Love Letter There s only one way we can be together Decimal Point Decimal Point Warning Straighten up or we re through adding! Cartoonist 4th graders Comic strip Fractions and Decimals: The Secret Twins of Math 1/6 /6 Valentine You complete me Mixed Number Improper Fraction Instructions How you can be like me. Disney Job application for one of Snow White s dwarves I m smaller than all those others 1/2, 1/, 1/4, 1/, etc. Speech (written) If you re going to compare us, at least do it fairly. 1/10 /4 and 2/ th 4 graders KEEP THIS PAGE IN CASE YOU NEED IT AGAIN DURING THIS UNIT! 22