Just a darn minute Yesterday you said X

“Just a darn minute! Yesterday you said X equals two? ” Bar Modeling Model Drawing Tape Diagramming 1 Bar Modeling for Pelham UFSD: Presented by Beth Finkelstein, Pia Hattar, Kate Alix and Joanne Lombardi

Agenda • • • Why bar model? How do you help your child at home? Bar Modeling Progression Part-Whole Bar Models and Examples Comparison Bar Models and Examples Questions 2

Mathematical Practice Standards 3

Math Practice Standards and Bar Models 4

What is Bar Modeling? • Bar modeling is a systematic method of representing word problems and number relationships. • Purpose of bar modeling is to understand the concepts and work out a strategy for finding the answer. 5

Pedagogy of Math in Focus and Math Modules file: ///H: /My%20 Documents/013 -014%20 Math%20 Staff%20 Development/2014 -2015%20 September%20 Info/Building%20 Mathematical%20 Understanding. pdf 6

Think about this problem… Ron spent 2/5 of his money in the first week and 1/2 of the remainder in the second week. He spent $210 altogether. How much did he have left? 7

Many complex problems are easier to solve with algebra…. if you understand the algebra! The bar model helps us to understand the algebra. “Just a darn minute! Yesterday you said X equals two? ” 8

Types of Bar Models 1, 213 Part-Whole apples green red 368 ? 845 Comparison Red apples Green apples ? 368 9

Early Stages of Bar Modeling K-1 10

Bar Modeling Steps • Step 1: Read to understand the problem. • Step 2: Think: “Can I draw something? ” • Step 3: Draw the unit bar(s). • Step 4: Label the bars and fill in the data you know. • Step 5: Label the part you are looking for with a question mark. • Step 6: Correctly compute and solve the problem. • Step 7: Write the answer in a statement being sure it answered the question being asked. 11

Think…. • Who is the problem about? • What do you know? • Do you know who has more or less? • What are you trying to find out? 12

Part-Whole Bar Models 13

Part-Whole: Addition and Subtraction A supermarket has 368 green apples. There are 845 red apples. How many apples altogether? ? apples 368 green Grade 3, ch. 5 845 red 368 + 845 = 1, 213 14

Part-Whole: Addition and Subtraction Eva had 317 gumballs. Then, she gave 132 gumballs to her brother. How many gumballs does Eva have now? 317 gumballs brother ? 132 317 - 132= 185 15

Part-Whole: Addition and Subtraction A bookshop has 4, 320 books and magazines. It has 2, 169 books. The rest are magazines. How many magazines does the bookshop have? 4, 320 books and magazines 2, 169 books ? magazines 4, 320 - 2, 169=2, 151 Grade 2, Ch. 4 16

Part-Whole Model: Multiplication and Division Peter earned $5 a week for six weeks. How much did he earn altogether? We know the 1 part and the number of parts. ? weeks $5 5 x 6 = 30 17

Part-Whole Model: Multiplication and Division Sara earned $30 in six weeks. If she is paid the same amount each week, how much did she earn each week? We know the whole and the number of parts. $30 weeks ? 30 ÷ 6 = 5 18

Part-Whole Model: Multiplication and Division Sara earns $5 each week. How many weeks will it take her to earn $30? We know the whole and value of one part. $30 $5 ? weeks 5 x ? = 30 5 x 6 = 30 or 30 ÷ 5 = 6 19

Part-Whole Model: Multiplication and Division: Two Step A baker makes 32 rolls in the morning. He makes 64 rolls in the afternoon. He packs all the rolls equally into 4 boxes. How many rolls does each box have? ? rolls 32 morning 64 afternoon 32 + 64 = 96 96 boxes ? Grade 3, Ch. 9 96÷ 4= 24 20

Part-Whole Model: Multiplication and Division: Two Step A car can travel 96 km. on 8 liters of gas. How far can the car travel on 15 liters of gas? 96 km 96÷ 8 = 12 Liters ? = 12 km ? km Liters 12 km 12 x 15= 180 21

Comparison Model: Addition and Subtraction Ryan has 5 toy cars. Josie has 3 more than Ryan. How many toy cars do Ryan and Josie have altogether? We are comparing the number of cars. We are looking for the total number of cars. 5 Ryan Josie 3 5 ? 5 + 3=13 22

Comparison Model: Addition and Subtraction Measurement Callie is 1. 5 meters tall. Bill is 0. 18 meter taller than her and 0. 2 meter shorter than Tom. What is Tom’s height in centimeters? 1. 5 m Callie 0. 18 m Bill Tom 0. 2 m 1. 68 1. 5 + 0. 18= 1. 68 + 0. 2= 1. 88 ? 1. 88 x 100= 188 cm Tom is 188 cm tall. Grade 4, Ch. 12 23

Comparison Model: Addition and Subtraction There are 1, 287 men at a baseball game. There are 879 fewer women than men at the game. How many women are at the game? 1, 287 Men 879 Women Grade 3, 5. 2 ? 1, 287 -879= 408 24

Comparison Model: Addition and Subtraction A pet store donates 3, 500 pounds of dog food to an animal shelter. A farm donates 2, 500 pounds of dog food at first. Later it donates another 2, 000 pounds of dog food to the animal shelter. How many more pounds of dog food does the farm donate than the pet store? 2, 500 + 2, 000= 4, 500 - 3, 500=1, 000 3, 500 lbs Pet Store ? Farm 2, 500 lbs Grade 3, 5. 2 2, 000 lbs 25

Comparison Model: Multiplication and Division There are 8 blue marbles. There are 3 times as many red marbles as blue marbles. How many red marbles are there? 8 blue 8 x 3 = 24 red ? 26

Comparison Model: Multiplication and Division Sabrina is 69 inches tall. She is 3 times as tall as her baby sister Beth. What is Beth’s height? 69 Sabrina 69 ÷ 3 = 23 Beth ? 27

Comparison Model: Multiplication and Division A museum has 75 carvings in its collection. It has 10 more pieces of pottery than carvings. It has 3 times as many paintings as pieces of pottery. How many paintings does the museum have? 75 carvings 75 + 10 = 85 85 x 3 = 255 10 pottery paintings Grade 3, Ch. 9 ? 28

Comparison Model: Multiplication and Division A father is four times as old as his son now. If the father was 46 years old two years ago, find the son’s age now. 46 +2 Father Son ? 48 ÷ 4= 12 29

Comparison Model: Multiplication and Division Hannah and Francine have $120. Hannah and Peter have $230. Peter has 6 times as much money as Francine. How much money does Hannah have? $120 Hannah and Francine ? F 230 – 120= 110 $120 Hannah and Peter F F F $230 Grade 5, 2. 8 110 ÷ 5 =22 120 – 22= 98 30 30

Fractions with Bar Models Jill’s monthly allowance is $32. She puts ½ of her allowance into savings and gives ¼ of the remaining money to a local charity. How much money does Jill have left for herself each month? Jill’s Money savings charity 16 4 $32 16 ÷ 4= 4 4 +4 +4=12 ? 4164 4 31

Fractions with Bar Models Lena has some eggs in the refrigerator. She takes out 3/5 of the eggs to make waffles and scrambled eggs. She uses 2/3 of the eggs she took out to make waffles. What fraction of the total number of eggs does Lena use to make waffles? waffles and scrambled eggs waffles Lena uses 2/5 eggs for waffles. Grade 5, 4. 2 32

Fractions with Bar Models A high school basketball team scored a total of 108 points in their final game. Joanne scored exactly 1/3 of all the points the team scored, and Renee scored 23 points. How many points were scored by the rest of the team? 108 36 points Joanne GRADE 4 – NYS Sample 23 Renee ? 108 ÷ 3= 36 36 +23= 59 108 – 59= 49 33

Fractions with Bar Models Five-sevenths of the class has more than 1 pet. If 12 students have 1 pet or none, how many students are in the whole class? 1 pet or none More than 1 pet students 6 ? 6 12 ÷ 2 = 6 6 x 7 = 42 34

Fractions with Bar Models Ron spent 2/5 of his money in the first week and 1/2 of the remainder in the second week. He spent $210 altogether. How much did he have left? First week Second week ? money $210 ÷ 7 = 30 30 + 30= 90 35

Percents with Bar Models There are 800 members in an astronomy club, and 320 of them are females. What percent of the members are male? 10% Members 800 80 320 female or 40% Grade 5, Ch. 10 ? males 800 ÷ 10 = 80 80 x ? = 320 80 x 4 = 320 100% - 40%= 60% are males 36

Percents with Bar Models A vendor sells three types of watches. Of the watches in stock 20% are men’s watches, 40% are ladies’ watches, and the rest are children’s watches. There are 250 watches altogether. How many children’s watches are there? 20 %Men’s 40% Ladies’ watches 25 250 Grade 5, Ch. 10 ? Children’s 25 250 ÷ 10 = 25 25 x 4 = 100 37

Comparison Model: Percents Len bought a new car for $22, 500. After a few years, he sold the car at a discount of 25%. What was the selling price of the car? $22, 500 ÷ 10 = 2, 250 New Car 2, 250 25% discount 1, 125 + 2, 250=5, 625 Discounted Car Grade 5, Ch. 10 ? Selling price 22, 500 - 5, 625=$16, 875 38

Comparison Model: Percents Len bought a new car for $22, 500. After a few years, he sold the car at a discount of 25%. What was the selling price of the car? $22, 500 New Car 22, 500 ÷ 4 = 5, 625 25% discount Discounted Car Grade 5, Ch. 10 ? Selling price 22, 500 - 5, 625=$16, 875 39

http: //www. mathplayground. com/thinkingblocks. html 40

Questions 41