Whats Proportional about Proportional Reasoning Making Math Magic

What’s Proportional about Proportional Reasoning? Making Math Magic www. makingmathmagic. net Copyright Protected Making Math Magic, LLC 2017

Reflections…. • Now that you have had a chance to think about yesterday…any questions? thoughts? next steps? • Please make sure your name card is up. Copyright Protected Making Math Magic, LLC 2017 ©

Copyright Protected Making Math Magic, LLC 2017 Using Common Core Standards to Enhance Classroom Instruction and Assessment, Robert Marzano, et al ©

The Road to Responsibility: Many of our students struggle because they have no “tools” to help them retain information. Students who are taught and understand strategies aimed at increasing conceptual knowledge take ownership of their learning. Copyright Protected Making Math Magic, LLC 2017 ©

Did you know? ? Every strand of middle and high school mathematics involves multiplicative and/or proportional thinking Geom Algebra etry s c Probability i t s i t a Nu St mb Measurement er Copyright Protected Making Math Magic, LLC 2017 ©

Your challenge: Uncle Fred always answers a question with another question or riddle. He has a farm and raises chickens and cows. I asked him how many chickens and how many cows he has, and he replied: “I have 21 heads and 66 legs. ” How many chickens and cows does he have? Copyright Protected Making Math Magic, LLC 2017 ©

Understanding Proportional Relationships between (and among) Decimals and Fractions is an important foundational concept… ls a m i c De s n o i t c Fra Making Math Magic Copyright Protected Making Math Magic, LLC 2017 ©

We still have to scaffold decimal concepts… Describe the value of digits in a multi-digit number (for example, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left) (5. NBT. A. 1) Copyright Protected Making Math Magic, LLC 2017 ©

Decimals…why do students struggle? § Decimals are taught in isolation, therefore students struggle… § To understand the decimal point is a place holder for the units position. § To see decimals as another way of writing a fraction § The traditional algorithm taught for addition and subtraction is based on digit placement. Decimal computation is based on place value. § “When you multiply a number by ten, just add a zero to the end!” Copyright Protected Making Math Magic, LLC 2017 ©

The Role of the Decimal Point! It Names the Unit Copyright Protected Making Math Magic, LLC 2017 ©

Common Misconceptions with Decimals 1. Longer is Larger: 0. 3578 is greater than 0. 67 because it has more digits 2. Shorter is Larger: 0. 4 is larger than 0. 97 because a tenth is larger than a hundredth. 3. Internal Zero: 0. 58 is less than 0. 078 because the “zero” has no impact on the number. (Appears as an issue with a number line because zero has no impact to the left. 4. Less than Zero: 0. 46 is less than 0 because they are not sure a decimal can be greater than 0. 5. Equality: 0. 4 - 4 tenths is equal to 40 hundredths or 400 thousandths Copyright Protected Making Math Magic, LLC 2017 Desmet, Gregoire and Mussolin, 2010 ©

Non. Linguistic Model 134 1 whole + 3/10 and 4/100 Similarities and Differences HUNDREDTHS ONES Copyright Protected Making Math Magic, LLC 2017 TENTHS ©

Questioning – To Encourage Conjecturing, Ask: 223 2 wholes + 2/10 and 3/100 What would happen if you had 2 and 3 hundredths? ONES Copyright Protected Making Math Magic, LLC 2017 TENTHS HUNDREDTHS ©

~THE M 3 APPROACH~ MOVING STUDENTS TOWARD UNDERSTANDING Math Talk Throughout! 3 Build It! Making Math Magic © Making Math Magic 3 Draw It! 2 Write It!

Developing the Fraction- Decimal Connection What do ½ and ¼ look like as decimals? You will need two 4 inch squares. Shade ½ of one square and ¼ of the other square. Copyright Protected Making Math Magic, LLC 2017 ©

Non-Linguistic Model Fraction Equivalence Grid Model for Tenths on Transparencies Copyright Protected Making Math Magic, LLC 2017 ©

Friendly Fractions to Decimals 1. Place the transparency grid over the square you labeled ½. 2. How many tenths are covered? 3. Now place the other transparency over the grid in the opposite direction. 4. Count number of squares 5. Write the number in decimal form. =. 50 Copyright Protected Making Math Magic, LLC 2017 ©

Friendly Fractions to Decimals 1. Place the transparency grid over the square you labeled 1/4. Questioning - To Encourage Conjecturing, Ask: 2. How many tenths are covered? 3. Now other transparency In our last problem, ½ was 50%. ½ =place 2/4, the so 2/4 = 50% over the grid in the opposite direction. In this problem, ¼ = 25%. . . 4. Countwhat number of squares Do you see a pattern here? Predict ¾ would look like as a decimal. 5. Write the number in decimal form. = 0. 2½ Copyright Protected Making Math Magic, LLC 2017 =0. 25 ©

Homework and Practice Copyright Protected Making Math Magic, LLC 2017 ©

Reflections…. • Questions? • Comments? Copyright Protected Making Math Magic, LLC 2017 ©

So What are the Problems with Percents? § Not understanding percent is a synonym for “hundredths” and can be described as a value “out of 100” or “out of a whole” unit. § Failing to see the relationship between percent, fractions and decimals. § Trying to apply a rule that is only partially remembered. Which number do I divide or multiply? Making Math Magic Copyright Protected Making Math Magic, LLC 2017 ©

$Percents…another way to write a fraction or decimal. x x x x x 1.$

Percents…another way to write a fraction or decimal. x x x x x 1. Model the fraction ¼ on the grid by coloring one out of every four squares. 2. Count the number of colored squares 3. Write the number in fraction and decimal form. x x x Copyright Protected Making Math Magic, LLC 2017 x x 1 4 ____ out of every _____ is . 25 Or 25% ©

Problem: How could we find 1 out of 5 as a percent? • Questioning – To Help Build Confidence, Ask: • What does Percent mean? How could you model this problem? • Questioning – To Promote Problem Solving and Make Connections, Ask: • What ideas/strategies have we used that might help? • Questioning – To Encourage Reflection, Ask: • Does your answer seem reasonable? Is 1/4 greater than or less than 1/5 ? • Is your answer greater than or less than the percent value for 1/4 ? Copyright Protected Making Math Magic, LLC 2017 ©

To Reinforce the Fraction-Decimal-Percent Connection • Break students in to small groups and give each group a shooting/hitting percentage of a professional baseball or basketball player. • Example, J. Turner has a batting average of. 356. – for every ten times he is up to bat, how many times would we expect for him to get a hit? • De. Andre Jordan has a 0. 677 field goal percentage in the NBA. When taking a shot would you expect for him to miss or make more often? How do you know? Copyright Protected Making Math Magic, LLC 2017 ©

$What fraction is shaded? What percent is shaded? Copyright Protected Making Math Magic, LLC$

What fraction is shaded? What percent is shaded? Copyright Protected Making Math Magic, LLC 2017 ©

$What fraction is shaded? What percent is shaded? What questions might you ask? Copyright$

What fraction is shaded? What percent is shaded? What questions might you ask? Copyright Protected Making Math Magic, LLC 2017 ©

WHERE are they? Approximating Fractions, Decimals, and Percents In teams of 2, we can use the number strips and paper clips to approximate the location of common fractions, decimals, and percents on a number line… 0 0 Copyright Protected Making Math Magic, LLC 2017 1 1 ©

Three Basic Types of Percentage Problems MISSING PART: There are 60 books on the shelf and 20% are cookbooks. How many cookbooks? MISSING PERCENT: There are 60 books on the shelf and 12 are cookbooks. What percentage are cookbooks? MISSING WHOLE: There are some books on the shelf and 12 are cookbooks. If 20% are cookbooks, how many books are on the shelf? Copyright Protected Making Math Magic, LLC 2017 ©

Ten-frame 10% 10% 10% or 10% Copyright Protected Making Math Magic, LLC 2017 10% 10% 10% ©

50% of 30 is 15 percent whole part Find a percent of a quantity as a rate per 100, Solve problems involving find the whole, given a Making part Math and a Magic percent. (6. R. RP. 3) Copyright Protected Making Math Magic, LLC 2017 ©

10 50% of 20 is ___ 10% 10% 10% Questioning – To Help If Students Get Stuck, Ask: • What Do We Know? • What Else Do We Know? • What Do We Want to Know?

40 ___% of 30 is 12 10% 10% 10% What do we know? What else do we know? What do we want to know? Making Math Magic

70 ___% of 50 is 35 10% 10% 10% What do we know? What else do we know? What do we want to know?

20 is 6 30% of __ 10% 10% 10% What do we know? What else do we know? What do we want to know?

50 is 20 40% of ___ 10% 10% 10% What do we know? What else do we know? What do we want to know?

30% of 40 is ___ 12 10% 10% 10% What do we know? What else do we know? What do we want to know? Making Math Magic

60 ___% of 80 is 48 8 8 10% 10% 10% What do we know? What else do we know? What do we want to know? Making Math Magic

What About UGLY Numbers? 65% of 70 is ___ 7 7 10% 10% 10% 7 7 10%

More“ugly numbers”? 1. 30% of 45 is _____ 2. ___% of 62 is 18. 6 3. 40% of 75 is ____ 4. 35% of 60 is ____ 5. ___% of 80 is 12 Copyright Protected Making Math Magic, LLC 2017 ©

What kinds of question are used most often in the classroom? • 20% are procedural • 60% can be answered by less than three words and require a RIGHT answer. • Only 20% of questions require higher level thinking. Teachers ask up to two questions every minute, up to 400 in a day, around 70, 000 a year, or two to three million in the course of a career Copyright Protected Making Math Magic, LLC 2017 S. Hastings 2003 ©

Fat and Skinny Questions • Fat questions require more than remembering a fact or reproducing a skill. (Explain why…. ) • Students can learn by answering fat questions and the teacher learns about each student from the attempt. • Fat questions may have several acceptable answers. (Predict what you believe will be the answer…) Copyright Protected Making Math Magic, LLC 2017 ©

Should All questions be “Fat”? • “Skinny” questions are more effective when giving factual knowledge and you want students want to commit those facts to memory. • If using “skinny” questions, the level of difficulty should elicit correct responses. • In classes above primary a mix of “fat” and “skinny” questions is superior to exclusive use of one or the other. Copyright Protected Making Math Magic, LLC 2017 ©

Reflection… Copyright Protected Making Math Magic, LLC 2017 ©

The coupon… I have a Kohl’s coupon that says I can get 20% off of 1 item. Should I use my coupon on a pair of socks that cost $4. 99 or on a new mixer that cost $199? Making Math Magic Copyright Protected Making Math Magic, LLC 2017 ©

Why focus on proportional Proportional reasoning has been referred to as reasoning? the capstone of the elementary curriculum and the cornerstone of algebra and beyond. Ratios Rates Proportions Making Math Magic Van de Walle, J. (2009). Elementary and middle school teaching developmentally. Boston, MA: Pearson Education.

• Proportional thinking is not developed just because a student gets older. • In 1999, Lamon stated over 50 percent of the adult population in the United States were not able to reason proportionally. • In 2012, Lamon raised the estimate to 90 percent. Copyright Protected Making Math Magic, LLC 2017 Susan Lamon, Teaching Fractions and Ratios For Understanding: Essential Content for Knowledge and Instructional Strategies for Teachers, 2 nd edition

Instructional Prerequisites to Proportional Reasoning ØStudents must know the difference between (additive) change and (multiplicative) change. ØStudents must be able to recognize whether a ratio is an appropriate comparison. ØStudents must recognize that each quantity in a ratio varies in such a way that the relationship between them is unchanged. ØStudents must be able to see things in “groups”(e. g. groups of six instead of six individual items). Copyright Protected Making Math Magic, LLC 2017 ©

How my students think! Copyright Protected Making Math Magic, LLC 2017 ©

Three spiders on a web each have 8 legs. How many legs are there altogether? 8 + 8 = 24 3 x 8 = 24 Copyright Protected Making Math Magic, LLC 2017 ©

Proportional reasoning requires multiplicative rather than additive thinking. Additive Thinker • Sees problems as “joining” or “separating” • Develops understanding of part/part whole relationships and direct comparisons only in absolute terms. • Struggles with communicating models or strategies that involve regrouping and place value. Multiplicative Thinker • Can work flexibly and efficiently with a range of numbers. • Has the ability to recognize and solve a range of problems involving multiplication, division and proportions • Has the means to communicate using a variety of models and strategies. What are your students? Copyright Protected Making Math Magic, LLC 2017 ©

Which class has more girls? They both have the same number of girls. The Comets, because 50 percent of the class is girls, while 40 percent of the Stars are girls.

What is the relationship between 20 and 30 ? Additive Thinker Copyright Protected Making Math Magic, LLC 2017 Multiplicative Thinker ©

Mr. Jones wanted to show the players on his football team a new play. His originally drew the play on a 5 inch by 7 inch piece of paper, but he needed it enlarged. He placed the sheet of paper on the copier and used the 150% option. Which is more square, the original or the enlargement? • A. The original is more square • B. The enlargement is more square • C. They are both equally square • D. Not enough information Copyright Protected Making Math Magic, LLC 2017 ©

To asses “thinking” requires multiple methods- each question type will reveal different information about your students’ thinking. Copyright Protected Making Math Magic, LLC 2017 Assessing Proportional Thinking, Mathematics Teaching in the Middle School, 2003 ©

Proportional Thinking… Making Math Magic

Fractions and Ratios? §Are they the same? §Why or why not? Copyright Protected Making Math Magic, LLC 2017 ©

Fractions vs. Ratios • Fractions are a subset of ratios! • Fractions only compare part-whole • Fractions must use the same measurement • Fractions must have the “same name” to combine. Copyright Protected Making Math Magic, LLC 2017 ©

A ratio is a comparison of two quantities by division. Ratios Part to Part Making Math Magic Copyright Protected Making Math Magic, LLC 2017 Part to Whole to Part Rate What do ratios compare? ©

$Equivalent fractions 3 4 Copyright Protected Making Math Magic, LLC 2017 = 6 8$

Equivalent fractions 3 4 Copyright Protected Making Math Magic, LLC 2017 = 6 8 = 9 12 • Same whole amount • Same portion • More parts • Smaller parts ©

Equivalent ratios • More total amount • Same size parts • More parts Cups of Red paint Total cups of paint Copyright Protected Making Math Magic, LLC 2017 3 4 6 8 9 12 • More red pigment ©

• Brittany is practicing for the upcoming basketball season. She must shoot 50 foul shots each day. She usually makes 35 of those 50 baskets. Represent this as a ratio. Baskets made Baskets shot 35 : 50 Copyright Protected Making Math Magic, LLC 2017 35 to 50 35 out of 50 ©

Taylor has 15 pair of jeans and 25 shirts in her closet. What is the ratio of jeans to shirts? Jeans Shirts 15 : 25 15 to 25 15 out of 25 We can not say 15 out of 25 because we are comparing part to a part. It would not make sense to say I have 15 jeans out of 25 shirts. Copyright Protected Making Math Magic, LLC 2017 ©

Ways to Think about Ratio Multiplicative Comparison: The comparison between the two items can go either way. The height of my first jump was 3 feet, while the height of my second jump was 4 feet. The ratio of the two jumps is 3 to 4, but this doesn’t really tell the relationship between the two jumps. My first jump was three-fourths as high as my second jump or my second jump was four-thirds higher than my first jump. Copyright Protected Making Math Magic, LLC 2017 ©

Ways to Think about Ratio Composed Unit- Think of the ratio as one unit: Kroger’s is selling 4 ears of corn for $1. 00. This means 8 ears $2. 00, and so on. Each comparison is based off of the original unit. Copyright Protected Making Math Magic, LLC 2017 ©

Proportional and Non-proportional Situations A comparison can be additive, multiplicative or constant. Additive Situation: means you add the SAME number to any x-value to get the corresponding y-value Multiplicative Situation: mean you multiply any x-value times the SAME number to get the corresponding yvalue Constant Situation: means y-value is not influenced by x. Copyright Protected Making Math Magic, LLC 2017 ©

Proportional and Non-proportional Situations A. Rhonda and Vonda are running on the track, each at the same rate. Vonda started first? When Vonda has run 6 laps around the track, Rhonda has run 2 laps. How many laps will Rhonda have run when Vonda finishes 12 laps? Additive: Vonda will still be 4 laps ahead of Rhonda. So Rhonda will have run 8 laps. B. Alice and Michael are planting corn on their farm. Alice plants 4 rows of corn and Michael plants 8. If Alice’s corn is ready to pick in 10 weeks, how many weeks will it be before Michaels corn is ready? Constant: It will take 10 weeks for the corn to grow no matter the number of rows. C. . Kevin and Brandon are making cookies using the same recipe. Kevin is making 3 dozen cookies and Brandon is making 6 dozen of cookies. If Kevin uses 4 sticks of butter, how many sticks of butter will Brandon use? Multiplicative: Copyright Protected Making Math Magic, LLC 2017 Brandon will use 8 sticks of butter ©

Sam the Snake is 4 feet long. When he is fully grown, he will be 8 feet long. Sally the Snake is 5 feet long. When she is fully grown, she will be 9 feet long. Which snake is closer to being fully grown? Explain how you know. Copyright Protected Making Math Magic, LLC 2017 ©

Is this legal? 7 8 = + 10 10 15 20 On Quiz #1 you got 7 points out of 10. On the second quiz you got 8 out of 10. What is your current grade? Copyright Protected Making Math Magic, LLC 2017 ©

Is this legal? 3 + 4 9 6 = 12 8 For your punch recipe, you mixed 3 cups of grape juice and 4 cups of pineapple juice. You decided you needed more punch so you added 6 cups of grape juice and 8 cups of pineapple juice. What is the ratio of grape juice to pineapple juice? Copyright Protected Making Math Magic, LLC 2017 ©

The never ending question…. What instructional strategies can teachers use to help students develop proportional reasoning?

Ways to Develop Proportional Reasoning…. Van de Walle suggests five categories of informal activities ~ § Identifying multiplicative relationships § Equivalent ratios § Comparing ratios § Developing a variety of strategies § Proportional reasoning across the curriculum Copyright Protected Making Math Magic, LLC 2017 ©

REFLECTIONS… Copyright Protected Making Math Magic, LLC 2017 ©

For TODAY…You can NOT use cross multiplication as a strategy for solving proportions! Copyright Protected Making Math Magic, LLC 2017 ©

Cross multiplication: A cause for students’ difficulties When students were taught to use the cross multiplication strategy, they were less successful at solving proportion problems than students who were never taught it (Fleener et al. , 1993). Copyright Protected Making Math Magic, LLC 2017 ©

Ratio Table Strategy Used to organize Brandon and Max are trading baseball information and help and basketball cards. Brandon gives students by using a Max 2 baseball cards for every 3 Build-up strategy and basketball cards that Maxfor gives him. finding unit rate How many basketball cards should Max give Brandon if Brandon gives him 6 baseball cards? Copyright Protected Making Math Magic, LLC 2017 ©

Grapic Organizer Baseball cards Basketball Cards Copyright Protected Making Math Magic, LLC 2017 2

Solve for “y” x 2 4 6 8 y 3 6 9 12 20 ? 2 2 2 3 3 3 Drew notices that every time the X’s increase by 2, the Y’s increase by 3 so he continues to add to both columns to find the missing value. Additive Thinker Copyright Protected Making Math Magic, LLC 2017 ©

x 2 4 6 8 y 3 6 9 12 20 ? 3 4 10 Copyright Protected Making Math Magic, LLC 2017 3 4 10 Spencer notices that when you “jump” in the table both numbers gets multiplied by the same factor; so he multiplies each initial number by 10 Multiplicative Thinking ©

Making Muffins Copyright Protected Making Math Magic, LLC 2017 ©

This student added 2/3 repeatedly to find that this can be done nine times

This student determined that 9 recipes could be made on the basis that 3 quantities could be made from 2 cups of milk and then multiplied 9 by 12 to get 108 muffins. Copyright Protected Making Math Magic, LLC 2017 ©

Strategy Using Tape Diagrams and Double Number Lines • Tape diagrams are best used when the two quantities have the same units. • Double number line diagrams are best used when the quantities have different units (otherwise the two diagrams will use different length units to represent the same amount). Copyright Protected Making Math Magic, LLC 2017 ©

Using a Tape Diagram • Katie is making a punch where she mixes 2 cups of orange juice with 3 cups of pineapple juice. How many cups of each juice will she need to make 80 cups of punch? Some students may initially need for you to fill in each segment with the unit being measured. Orange Juice 1 CUP Pineapple Juice 1 CUP Copyright Protected Making Math Magic, LLC 2017 ©

Using a Tape Diagram Orange Juice Pineapple Juice 5 parts 80 cups 1 part 80 ÷ 5 = 16 cups 2 parts 2 x 16 = 32 cups 3 parts 3 x 16 = 48 cups Orange Juice Copyright Protected Making Math Magic, LLC 2017 Pineapple Juice ©

Use a tape diagram to solve: Drew wants to buy a computer for $2100. His mother has agreed to pay $4 for every $3 that Drew saves. How much will each contribute? Copyright Protected Making Math Magic, LLC 2017 ©

Using the Double Number Lines to solve Ratio Problems Best used when the quantities have different units (otherwise the two diagrams will use different length units to represent the same amount). Copyright Protected Making Math Magic, LLC 2017 ©

Using a Double Number Line Katie is traveling at a constant rate of speed. If she travels 200 miles in 4 hours, how many miles did she travel in 1 hour? 0 hours 4 hours 0 miles 200 miles Copyright Protected Making Math Magic, LLC 2017 ©

Using a Double Number Line 0 hours 1 2 0 miles 50 100 4

Use a double number line to solve If 6 pounds of peaches costs $12.

Use a double number line to solve If 6 pounds of peaches costs $4.

25 % 100% 0 0 320 ? ? What is 25% of 320? Copyright

40 % 100% 0 0 320 ? ? What is 40% of 320? Copyright

Modified number line/tape diagram After a 20% discount, the price of a Super. Slick skateboard is $120. What was the price before the discount? 100% Original price ? ? 20% 20% 20% 80% Copyright Protected Making Math Magic, LLC 2017 Discounted price $120 ©

A Super. Slick skateboard costs $150 now, but its price will go up by 20%. What will the new price be after the increase? 100% $150 20% 20% 20% 120% Copyright Protected Making Math Magic, LLC 2017 new price ? ? ©

Let’s Go Shopping! GAME Copyright Protected Making Math Magic, LLC 2017 Use proportional relationships to solve multistep ratio and percent problems (for example, simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error) (7. RP. A. 3) ©

A Game for 3 to 5 people… • Player draws card from pile and shows players item drawn. • Player spins spinner to designate discount. • Without the use of a calculator or pencil and paper, each player writes their “estimate” on a post it note and places in the middle of the table. (Estimate can be either amount of discount or total sale price. ) • Players, using a chip or marker places on post it that they believe is the closest to the actual amount. Copyright Protected Making Math Magic, LLC 2017 • Use calculator/cheat sheet to confirm. Players who selected correctly, get a point. ©

There were 4 crates with 16 cans of pop in each. How many cans

2 10 35 There were 2 crates with 10 cans of pop in each.

Solve without using crossmultiplication 20 30 = ? 15 30 15 18 ? Copyright Protected Making Math Magic, LLC 2017 = 12 20 = 40 ?

Use any strategy, besides cross multiplication to solve the following problems. Two red chips cost $3 while 3 blue chips cost $4. Which is the better buy? Copyright Protected Making Math Magic, LLC 2017 ©

Six red chips are worth the same as 4 blue chips. Spencer spends $24.

I can buy 16 ounces (1 pound) of candy for $2. 98 or 24

Fraction, Decimal, Percent Rummy 1. Deal out 7 cards to each player. 2. Turn over the next card and start a discard pile. Lay it down on the table so everyone can see the numerical representation 3. Place the remaining deck of cards face down in the center of the table. 4. Check to see if you have any sets. A set is 2 or 3 cards that have equivalent values. IF you have a set, you may lay it down when it is your turn. 5. Each player in turn can either draw a card from the remaining deck or pick up the top card from the discard pile. If you wish to pick up more than the top card from the discard pile, you must be able to make a set with the last card in the stack that you picked up. 6. The game ends when a player has no more cards or when there are no more cards left in the deck. 7. The winner is the player with the most card sets. Copyright Protected Making Math Magic, LLC 2017

Take Aways… Copyright Protected Making Math Magic, LLC 2017 ©

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What’s Proportional about Proportional Reasoning? Making Math Magic www. makingmathmagic. net Tammy Wall tammylwall 81@gmail. com 606 -465 -5907 Copyright Protected Making Math Magic, LLC 2017