Transitioning to the Common Core State Standards – Mathematics Pam Hutchison

Please fill in these 3 lines: First Name ________Last Name__________ Primary ______Alternate _______. School____________District______________

AGENDA Fractions and Fractions on a Number Line  Naming/Locating Fractions  Whole Numbers, Mixed Numbers and Fractions  Comparing Fractions  Equivalent Fractions/Simplifying  Adding and Subtracting Fractions  Multiplying Fractions  Dividing Fractions Stoplighting the CCSS

Spending Spree David spent of his money on a game. Then he spent of his remaining money on a book. If he has $20 left, how much money did he have at first?

Fractions

Fractions 3.NF.1. 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.

Fraction Concepts

So what is the definition of a fraction?

Definition of Fraction: Start with a unit, 1, and split it into ___ equal pieces. Each piece represent 1/___ of the unit. When we name the fraction __/__, we are talking about ___ of those 1/___ size pieces.

Fraction Concepts

Fractions on a Number Line

How many pieces are in the unit? Are all the pieces equal? So the denominator is And each piece represents ●

How far is the point from 0? So the numerator is And the name of the point is …… 0 1 ●

How many pieces are in the unit? Are all the pieces equal? So each piece represents 0 1 ●

How far is the point from 0?  How many pieces from 0? So the name of the point is …. 0 1 ●

Definition of Fraction: When we name the point, we’re talking about a distance from 0 of ___ of those ___ pieces. 4

The denominator is so each piece represents The numerator is and the fraction represented is ● 5

Academic Vocabulary What is the meaning of denominator? What about numerator? Definitions should be more than a location – the denominator is the bottom number They should be what the denominator is – the number of equal parts in one unit

Student Talk Strategy: Rally Coach Partner A: name the point and explain Partner B: verify and “coach” if needed  Tip, Tip, Teach Switch roles Partner B: name the point and explain Partner A: verify and “coach” if needed  Tip, Tip, Teach

Explains – Key Phrases Here is the unit. (SHOW) The unit is split in ___ equal pieces Each piece represents The distance from 0 to the point is ___ of those pieces The name of the point is.

Partner Activity 1

 Start with a unit, 1,  Split it into __ equal pieces.  Each piece represents of the unit  The point is __ of those pieces from 0  So this point represents 0 1 Definition of Fraction: ●

 Start with a unit, 1,  Split it into __ equal pieces.  Each piece represents of the unit  The point is __ of those pieces from 0  So this point represents 0 1 Definition of Fraction: ●

Partner Activity 1, cont. Partner B 5B. 6B. Partner A 5A. 6A.

||||||||||||||||||  The denominator is …….  The numerator is ………  Another way to name this point? Page 83

||||||||||||||||||  The denominator is ……..  The numerator is ………  Another way to name this point?

||||||||||||||||||  The denominator is ……  The numerator is ………  Another way to name this point?

||||||||||||||||||  The denominator is …..  The numerator is ………  Another way to name this point?

15 ||||||||||||||||||  Suppose the line was shaded to 5.  How many parts would be shaded?  So the numerator would be ……… 012 3

30 ||||||||||||||||||  Suppose the line was shaded to 10.  How many parts would be shaded?  So the numerator would be ……… 012 3

Rally Coach Partner A goes first  Name the point as a fraction and as a mixed number. Explain your thinking Partner B: coach SWITCH Partner B goes  Name the point as a fraction and as a mixed number. Explain your thinking Partner A: coach Page 93-94

Rally Coach Part 2 Partner B goes first  Locate the point on the number line  Rename the point in a 2 nd way (fraction or mixed number)  Explain your thinking Partner A: coach SWITCH ROLES

Partner A Rally Coach Partner B 6. 7.

Connect to traditional  Change to a fraction.  How could you have students develop a procedure for doing this without telling them “multiply the whole number by the denominator, then add the numerator”?

Connect to traditional  Change to a mixed number.  Again, how could you do this without just telling students to divide?

Student Thinking Video Clips 1 – David (5 th Grade) ● Two clips ● First clip – 3 weeks after a conceptual lesson on mixed numbers and improper fractions ● Second clip – 3.5 weeks after a procedural lesson on mixed numbers and improper fractions

Student Thinking Video Clips 2 – Background ● Exemplary teacher because of the way she normally engages her students in reasoning mathematically ● Asked to teach a lesson from a state-adopted textbook in which the focus is entirely procedural. ● Lesson was videotaped; then several students were interviewed and videotaped solving problems.

Student Thinking Video Clips 2 – Background, cont. ● Five weeks later, the teacher taught the content again, only this time approaching it her way, and again we assessed and videotaped children.

Student Thinking Video Clips 2 – Rachel ● First clip – After the procedural lesson on mixed numbers and improper fractions ● Second clip – 5 weeks later after a conceptual lesson on mixed numbers and improper fractions

Classroom Connections Looking back at the 2 students we saw interviewed, what are the implications for instruction?

Research Students who learn rules before they learn concepts tend to score significantly lower than do students who learn concepts first Initial rote learning of a concept can create interference to later meaningful learning

Discuss at Your Tables How is this different from the way your book currently teaches fractions? How does it support all students in deepening their understanding of fractions?

Compare Fractions Using Sense Making

Comparing Fractions A. B.

Comparing Fractions B. A. Common Numerator

Comparing Fractions A. B. Common Numerator

Comparing Fractions B. A. Hidden Common Numerator

Comparing Fractions A. B. Hidden Common Numerator

Benchmark Fractions |||||| 0 ½1 How can you tell if a fraction is:  Close to 0?  Close to but less than ½?  Close to but more than ½?  Close to 1?

Comparing Fractions B. A. A. B.

Comparing Fractions A. B.

Equivalent Fractions

Equivalent fractions can be constructed by partitioning equal fractional parts of a whole into the same number of equal parts. The length of the whole does not change; it has only been partitioned into more equal sized pieces. Since the length being specified has not changed, the fractions that describe that length are equal.

CaCCSS Fractions are equivalent (equal) if they are the same size or they name the same point on the number line.

Locate on the top number line. ● 0 1 Page 95

Copy onto the bottom number line. 0 1 ● ●

Are the lengths equal? 0 1 ● ●

0 1 ● ●

0 1 ● ●

● ● So 0 1

Order Matters! Locate 1 st fraction on number line Duplicate on 2 nd number line “Are they equal?” Split 2 nd number line “Are they equal?” Name point on 2 nd number line So Fraction 1 = Fraction 2

0 1 ● ●

Equivalent Fractions Let’s try a couple more 1. 2.

Fraction Families

SimplifyingFractions

Factors and GCF Well before we want to introduce simplifying fractions, student should learn or review factors and greatest common factors.

Which is simpler? 20 $1 bills OR 1 $20 bill 3 gallons of milk OR 12 quarts of milk 4 $1 bills OR 16 quarters 15 dimes or 6 quarters

Simplify Locate on the top number line. ● 0 1 Can we regroup the pieces to make larger groups with an EQUAL number of pieces? What size groups can we make?

Simplify Questions for students: Can we make equal groups of 2 pieces (with both the shaded part and the whole unit)? ● STOP STOP! We missed 9! 0 1

Simplify Can we make groups of 3 pieces? ● We can group both 9 and 12 evenly into groups with 3 pieces each. 0 1

Simplify Now, duplicate just the larger parts onto the bottom number line. Then name the point. ● ● 0 1 Mark the larger groups on the number line

Simplify ● ● 0 1 So the new name is Therefore:

One More Simplify

Simplify Fractions Practice with your partner using rows

Back to ● ● 0 1 So

Simplify Why couldn’t we break 9 into groups of 2 pieces? Because 2 is not a factor of 9.

Simplify We were able to make groups of 3 pieces. Why? So groups of 3 works because 3 is a factor of both 9 and 12. Let’s look at 2 important questions:  Is 3 a factor of 9?  Is 3 a factor of 12?

| | | | | | | | | | | | | | | | | 0 1 A Variation on the Traditional  Simplify

Simplify: 1)What is the greatest common factor of 16 and 24? 2) SO 2 3 1

Simplify using the alternative method: 1. 2.

Adding Fractions

Use equivalent fractions as a strategy to add and subtract fractions. 1. 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. For example, 2/3 + 5/4 = 8/ /12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) 2. 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. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

So what about adding fractions? What do we already know about adding?   How do we add whole numbers on a number line?

|||||||||||||||||| Add  Move a distance of 3  From that point, move a distance of 4  You end at 7 So =

What would you need to do to add? 5 inches + 9 inches 2 dimes + 3 dimes 4 hours + 3 hours 2 sevenths + 3 sevenths

Adding Fractions Remember our initial understanding of fractions means we have 5 pieces and each piece is in size

Add 0 1

0 1

0 1

What would you need to do to add?  5 inches + 1 foot  2 dimes + 3 nickels  1 hours + 40 minutes  1 fourth + 1 half How many of you thought “1 fourth + 2 fourth = 3 fourths”

WANT:  1/3 + 1/6  2/3 + 5/6

WANT: 01 01

ADD: 01 01

What would you need to do to add?  2 dimes + 3 quarters  1 third + 1 half

ADD: | | | | | | |

ADD: = | | | | | | | | | | | | |

Using Fraction Ruler

Subtracting Fractions

What do we know about subtracting so far?  579 – 34  5.79 – 3.4  Just like adding, we need likes to likes

0 1 / 3 2 / 3 3 / / 6 2 / 6 3 / 6 4 / 6 5 / 6 6 / 6 7 / 6 8 / 6

0 1 / 3 2 / 3 3 / / 6 2 / 6 3 / 6 4 / 6 5 / 6 6 / 6 7 / 6 8 / 6

2 dollars – 75 cents = 6 dollars - 3 dollars and 2 quarters 6 hours - 1 hour and 20 minutes 3 units – 1 and 1 fourth units

Subtract: | | | 4 5 6

Subtract: | | | | | | | | | | | | | | | | | | | | | | | | | 4 5 6

Subtract: | | | | | | | | | | | | | | | | | | | | | | | | | 4 5 6

Subtracting Use the number line to solve the following by adding on. Use the number line to solve the following by shifting the fractions.

Multiplying Fractions

4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. 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. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

x 2 Three groups of two

Multiplying Fractions What do we normally tell students to be when they multiply a fraction by a whole number?

0 1 / 2 2 / 2 3 / 2 4 / 2 3 x ½ Three groups of one-half 1( 1 / 2 ) 2( 1 / 2 ) 3( 1 / 2 )

Multiplying Fractions Remember our initial understanding of fractions of fractions Another way to write this is

Multiplying Fractions a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

Multiplying Fractions b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 ×(1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

Multiplying

Multiplying

Area Model 2 x 3 = 3 2 6

Area Model 1 1

Multiplying

1 1

Multiplying Fractions Solve each of the following problems using an area model

Multiplying Fractions

p r pr q s qs

Dividing Fractions

7.Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.

So, what about dividing on a number line?

= The question might be, “How many 2’s are there in 6?”

0 1/4 2/4 3/4 4/4 5/4 6/4 7/4 8/4 9/4 10/4 11/4 12/4 Draw a number line and partition it into ¼’s

0 1/4 2/4 3/4 4/4 5/4 6/4 7/4 8/4 9/4 10/4 11/4 12/ “How many ¼’s are there in 1?”

0 1/4 2/4 3/4 4/4 5/4 6/4 7/4 8/4 9/4 10/4 11/4 12/ “How many ¼’s are there in 2?”

0 1/4 2/4 3/4 4/4 5/4 6/4 7/4 8/4 9/4 10/4 11/4 12/ “How many ¼’s are there in 3?”

0 1/2 1

Dividing Fractions 3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Fraction Concepts Four children share seven brownies so that each child receives a fair share. How many brownies will each child receive?

Fraction Concepts Four children share three brownies so that each child receives a fair share. What portion of each brownie will each child receive?