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Let’s Get Started… Directions: Use the whole numbers 1 through 9 only one time each to find the largest (or smallest) possible values for x. www.openmiddle.com 1 Learning Targets • I can model algebraic expressions and equations using concrete materials. • I can solve equations using concrete materials. • I can solve equations and problems using representations. 2 Concrete-Representational-Abstract • The C-R-A model is instructional approach in mathematics. • It can enhance student performance • Promote student learning and retention of conceptual knowledge • Supports understanding of underlying concepts, before learning rules and procedures Concrete Models Studies have shown that “students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas, and better apply these ideas to life situations.” Hauser , Jane. Concrete-representational-abstract instructional approach. Retrieved April 9, 2009, from the Access Center: Improving Outcomes for all Students K-8. Website: http://www.k8accesscenter.org/training_resources/CRA_Instructional_Approach.asp 4 Solving Equations the Concrete Way Count off 1, 2, 3 • 1’s begin with Cindy • 2’s begin with Magdalena • 3’s begin with Stacy Egg Timer Using Bar Models to Solve Equations (Representational) 3x + 7 = 2x + 20 • Bar Model • Decompose the Equation • Algebraic Method 6 Using Bar Models to Solve Equations with Variables on Both Sides (Systems) What is the value of showing students different ways to solve equations? • Bar Model • Decompose the Equation • Algebraic Method 7 Representations for Solving Word Problems • “The goal for model drawing is to build a pictorial bridge to abstract thinking.” Forsten, Char . Step-by-Step Model Drawing Solving Problems the Singapore Way. Peterborough: Crystal Springs Books, 2010. Print. • “Model drawing is wonderful for kids who are struggling with abstract math, because it literally gives them a picture of what’s happening in the problem. Model drawing gives word problems a visual context.” Walker, Lorraine. Model Drawing for Challenging Word Problems Finding Solutions the Singapore Way. Peterborough: Crystal Springs Books, 2010. Print. 8 Model Drawing for Word Problems If 3/8 of a sum of money is $384, how much is 1/4 of the money? Model Drawing for Word Problems 2/5 of all the boys from Central School and 3/4 of all the girls in the school attended the baseball game. If the number of boys at the game was the same as the number of girls, what fraction of the school's students attended the game? Model Drawing for Word Problems Cheryl had 1/12 as many books as Brenda. Later Cheryl bought 18 more books. If Brenda has 72 books, what will be the ratio of Cheryl's books to Brenda's after Cheryl's purchase? Model Drawing for Word Problems The ratio of Mia's balloons to Kiran's was 3:5. After Mia was handed 21 more balloons she had twice as many balloons as Kiran. How many balloons did Mia have initially? Model Drawing for Word Problems I think of a number, double it, and add 12. My answer is 54. What number am I thinking of? Model Drawing for Word Problems A pencil costs $2 less than a notebook. A pen costs 3 times as much as a pencil. The pen costs $9. What is the total price of a pen, pencil and notebook? Model Drawing for Word Problems Jake takes a board that is 50 inches long and cuts it into two pieces, one of which is 16 inches longer than the other. How long is each piece? Model Drawing for Word Problems There were 24 boys and 20 girls in a chess club last year. This year the number of boys increased by 25% but the number of girls decreased by 10%. Was there an increase or decrease in overall membership? Find the overall percent change in membership of the club. Model Drawing for Word Problems Sandy was buying new notebooks and pens for her home office. At one store she bought 2 notebooks and 2 pens for $13.14. At another store she found exact same products and pricing; this time she paid $25.08 for 5 notebooks and 2 pens. How much did each notebook cost? Model Drawing for Word Problems There are 73 children at the playground. Onethird of the boys and 2/7 of the girls are swimming in the pool. If there are 22 kids in the pool in all, how many boys are there at the playground? Model Drawing for Word Problems The sum of two whole numbers is 28. When one of the numbers is tripled and added to the other number, the sum is then 50. What are the two numbers? Model Drawing for Word Problems • Complete the handout. • For your assigned problem, model with a drawing. • Work with your group. • Discuss your strategies and thoughts as you work. 20 Model Drawing for Word Problems • How might the bar model help students? • How might the bar model challenge students? • The bar model may be a new concept for many students, how might you introduce this strategy to students? • Think about how you would transition from the bar model to a numeric or an algebraic model (representation to abstract). 21 Model Drawing for Word Problems • Online Tool for Creating Bar Models • http://www.mathplayground.com/thinkingblo cks.html 22 Day 1 Learning Targets • I can model algebraic expressions and equations using concrete materials. • I can solve equations using concrete materials. • I can solve equations and problems using representations. 23