Key strategies for interventions: Fractions

Today’s agenda Concepts and procedures Assessments Games, activities and simulation

General Strategies Use visual representations to explain concepts. Provide tasks that engage students’ thinking. Teach procedures explicitly, verbalizing your thinking. Require substantial practice where students are expected to explain their thinking. Provide corrective feedback. Connect types of fraction problems to types of whole number problems (putting together, taking from, repeated addition, measurement division, etc.) Provide guided practice with corrective feedback and frequent cumulative review.

Too many procedures 𝟒 𝟓 × 𝟏 𝟐 =___ Do I need common denominators? Do I cross multiply? A cake recipe makes 36 cupcakes. We only need 24. The original recipe requires 3½ cups of flour. How many cups of flour do we need if we want to reduce the recipe to make only 24 cupcakes?

What is a fraction? Several definitions – parts of a whole parts of a set a number on the number line a way to write division Introduce fraction circles and fraction bars, then relationship rods and pattern blocks. Build on 1st/2nd grade cutting and coloring circles, squares, etc. Illustrate each one with a drawing and symbols. Fluency involves identifying the fractional part of a whole (or part of a set), and drawing an illustration of a fraction as part of a whole or part of a set.

Comparing fractions same denominator same numerator relative to benchmark fractions (close to 0, close to 1, greater or less than ½) This is a conceptual task, not one that calls for procedural fluency.

Same denominator, same numerator Which is larger? Explain your reasoning. Use manipulatives or drawings. 𝟑 𝟖 𝟓 𝟖 𝟑 𝟒 𝟑 𝟓

Benchmark fractions Using fraction pieces, find a fraction that is close to 1 2 but smaller. Find a fraction that is close to 1 2 but larger. Find a fraction that is closer to 0 than 1 2 . Find a fraction that is closer to 1 than 1 2 . Find a fraction that is close to 1 but larger.

Ordering fractions Order this set of fractions from smallest to largest: 3 4 , 1 10 , 5 12 , 1 4 , 3 5 , 14 15 Explain how you figured this out. Test your understanding: http://phet.colorado.edu/en/simulation/fraction-matcher

More reasoning with fractions Compare 3 4 and 2 3 . Which is closer to 1? Why?

Using reason to think about addition Is this reasonable? Use manipulatives to check or show.

Number lines – a useful tool What’s hard about this for many students?

One approach Draw a line across a page. Using a fraction bar piece for 1 4 , measure lengths of the bar starting from the left end of the line. Mark the end of each length as 1 4 , 2 4 , 3 4 , 4 4 etc. (Where should you put zero?) Locate and label 5 4 , 6 4 , 7 4 on the number line. Locate and label 15 4 on the number line. How much is this as a mixed number? This is the introduction to mixed numbers.

Number line activities Almost! I almost got it exactly. I’m going to turn it over and try again to see if I can get the paperclip to land right on the ½ mark. ¼? I just moved my clip what I thought was half-way down the line and then cut that in half. I got pretty close.

From Bridges, 3rd grade CCSS supplement

Equivalent fractions 1 4 = 1×2 4×2 = 2 8 Use manipulatives to find multiple equivalent fractions Develop a procedure (scaling up both the total number of parts and the number we have, developing proportional reasoning). Fluency involves using this procedure to find equivalent fractions. 1 4 = 1×2 4×2 = 2 8

Comparing fractions (pt. 2) Which is larger, which is smaller? 3 4 5 8 3 4 6 10 First practice drawing the two fractions to see which is larger/smaller. Then find the equivalent fractions with common numerators or denominators. 4.NF.2 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.

Comparing fractions (pt. 2) Which is larger, which is smaller? 2 6 5 8 Compare each fraction to 1 2 . See the problem set in the handouts.

Is it reasonable? from Operations with Fractions & Decimals Packet

Fraction addition – common denominators 3 4 = 1 4 + 1 4 + 1 4 = 2 4 + 1 4 2 1 3 +1 2 3 = 7 3 + 5 3 = 12 3 =4 Adding thirds or fifths or fourths is like adding apples or oranges or pears. 2 thirds plus 2 thirds is like 2 pears plus 2 pears (= 4 pears, or 4 thirds). Model this with fraction manipulatives. This is a joining (putting together or adding to) problem. In this case the “pears” are 1/3-size pieces. This is the KEY learning.

Fraction addition – common denominators After a class party, 3 8 of one pan of brownies is left over and 2 8 of another pan of brownies is left over. How much is left over altogether? (Use fractions pieces or drawings, then explain the procedure.) Britney walks 3 4 of a mile to school each day. How far does Britney walk to school in one week (5 days)? Put a dot on the appropriate place on this number line to show your answer. Be as accurate as you can be.

Fraction addition – “friendly” denominators The procedure of finding a common denominator has to arise out of extensive work with fraction manipulatives, asking: How can we add two different things? Use fraction pieces to find this sum: 𝟑 𝟖 + 𝟏 𝟒 What procedure arises from work with the manipulatives?

C-R-A Try this addition on a number line: 𝟑 𝟖 + 𝟏 𝟒 Concrete (Objects) Representational (Picture) Abstract (Symbols) 𝟑 𝟖 + 𝟐 𝟖

Fraction addition – not “friendly” denominators The extension to any denominator is straightforward: Scale them both up until we find a common denominator. 𝟐 𝟑 + 𝟏 𝟒 Try it. 𝟐 𝟑 𝟒 𝟔 𝟖 𝟏𝟐 𝟏 𝟒 𝟐 𝟖 𝟑 𝟏𝟐 𝟖 𝟏𝟐 + 𝟑 𝟏𝟐 = 𝟏𝟏 𝟏𝟐

A great deal of work needs to be done to connect these kinds of fraction problems to joining and separating problems with whole numbers to develop the sense of proportionality in scaling up the fractions, and in eventually moving away from using manipulatives to using the multiplication procedure. Fluency comes from practice, where thinking about the trades gets replaced with the scaling up procedure.

Recommendations for fraction instruction Base early understanding on fair shares Use representations of different kinds Develop estimation skills for comparing fractions by basing comparisons on benchmark fractions such as 1/2. Develop the meaning of a fraction as a number (a place on the number line); connect a point on the number line to a fraction of a whole through the meaning of denominator and numerator Use real-world examples including measurement Develop the reasons behind procedures and expect students to explain their thinking

Practice vs. drill Practice is used to establish a procedure. Drill is used to get fast with it. What procedures have we looked at for adding fractions? Same denominator One denominator a multiple Family or no: common denominator

Practice games What do students learn playing this? Fraction Tracks, also called Fraction Game What do students learn playing this?

Practice problems 1/10 of the M&M’s in a bag are red and 1/5 are blue. What fraction of all the M&M’s are red and blue? What fraction of the M&M’s are NOT red or blue? You give 1/3 of a pan of brownies to Susan and 1/6 of the pan of brownies to Patrick. How much of the pan of brownies did you give away? How much do you have left? You go out for a long walk. You walk 3/4 mile and then sit down to take a rest. Then you walk 3/8 of a mile. How far did you walk altogether? Pam walks 7/8 of a mile to school. Paul walks 1/2 of a mile to school. How much farther does Pam walk than Paul? A school wants to make a new playground by cleaning up an abandoned lot that is shaped like a rectangle. They give the job of planning the playground to a group of students. The students decide to use 1/4 of the playground for a basketball court and 3/8 of the playground for a soccer field. How much is left for the swings and play equipment? Draw a picture to show this.

Drill problems Once students understand the type of problem and the procedure they’ll use, then they can do drill problems that are often just “naked number” problems.

Diagnostic assessments IISD Mathematics wiki inghamisd.org – WikiSpaces – Mathematics Fraction resources IISD Elementary Math Resources wiki, 4th-5th Grade inghamisd.org – WikiSpaces – Elementary Math Resources

Fraction multiplication The learning progression outlined in the CCSS needs to be followed systematically and explicitly. A whole number times a fraction (generalizing multiplication as repeated addition) A fraction of a whole number (generalizing work with geometric figures in 1st and 2nd grades) A fraction times a fraction (generalizing the area model)

Fraction multiplication A whole number times a fraction (generalizing multiplication as repeated addition) 5× 2 3 5 hops of 2 3 The key to fraction multiplication is knowing how to estimate the size of the answer.

Whole number times a fraction A game idea: Roll two dice, one with 1-6, one with fractions. Write the multiplication shown by the dice, e.g. 3× 3 4 , say the product out loud, 9 4 , then write 9 4 at the correct spot on a number line. It becomes obvious after playing the game awhile that a simple procedure is to multiply the whole number times the numerator. Dominoes can be used instead of fraction dice. Choose which fraction it can represent.

Fraction of a whole number A whole number times a fraction (generalizing addition of fractions) A fraction of a whole number (generalizing work with geometric figures in 1st and 2nd grades) 2 3 𝑜𝑓 6

You have 6 donuts. You want to give 1/3 of them to your friend Suzi, 1/3 of them to your friend Sam, and keep 1/3 of them for yourself. 2 3 𝑜𝑓 6 Why is this multiplication? How is it related to division? 1 3 2 3 3 3

C-R-A R: a visual representations - find 1 third of 6, then take 2 of that amount: A: an abstract method – divide the whole number by the denominator (6 ÷ 3 = 2) then multiply by the numerator (2 x 2 = 4). So 2 3 ×6=4.

Practice Create six practice problems, three contextual and three non- contextual, to practice this procedure. The procedure is: Divide the whole number by the denominator (to find one group), then multiply by the numerator (to find the total). Does it also work to multiply by the numerator first, then divide by the denominator? 2 3 ×6=4 (yes, but no meaning) What if the denominator is not a factor of the whole number? 3 4 ×10

Fraction multiplication A whole number times a fraction (generalizing addition of fractions) A fraction of a whole number (generalizing work with geometric figures in 1st and 2nd grades) A fraction times a fraction (generalizing the area model)

Type 1: Solving visually A track is 3 4 of a mile in length. If you run 2 3 of the track, how much of a mile have you run? https://www.teachingchannel.org/videos/multiplying-fractions-lesson

Simple visual problems 3/4 of a pan of brownies was sitting on the counter. You decided to eat 1/3 of the brownies in the pan. How much of the whole pan of brownies did you eat? 1 3 𝑜𝑓 3 4 , which is represented as 1 3 × 3 4 2 3 × 3 4 = ? For these problems, we don’t have a procedure other than using a drawing.

Will this visual approach work with 1 2 𝑜𝑓 3 4 or 3 8 × 4 5

Another visual representation What if the denominator is not a factor of the whole number? Start with a simple problem of a fraction times a whole number. 3 4 ×10 3 4 ×10 is 30 fourths, 30 4 10 40 fourths

Let’s generalize the area model from the picture of 3 4 ×10 to a picture for 1 2 × 3 4

C R A

Simplifying fractions You have 2/3 of a pumpkin pie left over from Thanksgiving. You want to give 1/2 of it to your sister. How much of the whole pumpkin pie will this be? The algorithm results in 2 6 but the drawing method results in 1 3 . Are they equivalent? Does the form matter?

Fraction division Learning progression: Division of a fraction by a whole number (use manipulatives or drawings) Division of a whole number by a fraction (generalize the “measurement division” or repeated subtraction process using drawings) Division of a fraction by a fraction (continue using drawings to support conceptual development; provide alternative algorithms)

Generalizing from whole numbers Partitive (fair shares) We want to share 12 cookies equally among 4 kids. How many cookies does each kid get? How would you solve this with objects? The number of groups is known; the number in each group is unknown.

Measurement division (repeated subtraction) For our bake sale, we have 24 cookies and want to put 4 cookies in each bag. How many bags can we make? How would you solve this with a picture? The number in each group is known; the number of groups is unknown. How many bags of 4 are in 24? Put out 12 cookies. Take the 2-cookie measure, and hold it up to the 12 cookies: How many groups are there? Or, repeatedly subtract 2 from the 12 (to make the bags) until none are left.

Why is this important? A box of Cheerios contains 12 1 2 cups. Each serving is 3 4 cups. How many servings are in a box of Cheerios? How much is left over? Is this Partitive or Measurement division? Write 3 additional problems like these. (We’ll solve this later.)

Dividing a fraction by a whole number We have ½ of a pizza and want to share it equally among 4 people. How much pizza does each person get? 𝟏 𝟐 ÷𝟒 Try this with fraction manipulatives. Doing these kinds of problems is essential to build number sense about the size of the expected answer. Try this: 𝟖 𝟑 ÷𝟒 While the approach is different, it’s still “fair shares” or partitive division.

Dividing a whole number by a fraction We have a dozen large cookies and want to give ½ cookie to each child. How many children can we serve? 12÷ 1 2 (How many times does ½ go into 12?) Solve this by making a drawing. The procedure: Each cookie is divided in half, making two pieces. There are 12 cookies, so 12 x 2 pieces.

Dividing a fraction by a fraction A serving size is ¼ cup. How many servings are in 5/4 cup? Solve it. Write it as an equation. 5 4 ÷ 1 4 How many ¼’s are in 5/4? This is the same as asking “How many 1’s are in 5?” Procedure: Get common denominators.

Dividing a fraction by a fraction A box of Cheerios contains 12 1 2 cups. Each serving is 3 4 cups. How many servings are in a box of Cheerios? How much is left over? Translated to symbols 12 1 2 ÷ 3 4 Procedure? Find common denominators... 50 4 ÷ 3 4 then ask “how many 3’s in 50?” Write two more similar problems to solve with common denominators.

Why does “invert and multiply” work Why does “invert and multiply” work? 𝟕 𝟖 ÷ 𝟏 𝟖 ( 𝟕 𝟖 × 𝟖 𝟏 )÷( 𝟏 𝟖 × 𝟖 𝟏 ) See pp. 29-34 in Operations with Fractions for an instructional approach.

Decimals The development of decimal fractions has several steps: Place tenths on the number line up through 3 or 4. (Equate 5/10 to 1/2 and 10/10 to 1). Introduce the new notation for tenths: 1/10 = 0.1, 2/10 = 0.2, 1 4/10 = 1.4, 2 7/10 = 2.7. Be clear that this is just a new name for tenths, a new way of writing tenths, called decimal numbers. The “decimal point” separates the ones from the tenths. Introduce hundredths through the money system: $4.55 is read 4 dollars and 55 cents, where each cent is 1/100 of one dollar (one hundred are needed to make one dollar). Use the hundreds block as a visual representation. Expand the place value system to show: hundreds tens ones . tenths hundredths

Why do we “count decimal places?” 2.3 x 1.8 .2 4 1.6 1.3 2 5.1 4 Yellow 2 x 1 Orange 0.3 x 1 Blue 2 x 0.8 Green 0.3 x 0.8 2.3 1.8 Decimals Forever!