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TO INFINITY AND BEYOND . . . GOING BEYOND ANSWER GETTING CATHY SHIDE, CONSULTANT 1 OBJECTIVES • Integrate the math practices with word problems • teachers and students going beyond “answer getting” • Use different modes of representation to solve problems with a focus on Fractions, Ratios, and Percents 2 “TAPE DIAGRAM” “A drawing that looks like a segment of tape, used to illustrate number relationships. Also known as a strip diagram, bar model, fraction strip, or length model.” Also referenced in “Visual Fraction Model” definition. - CCSSM (Glossary) p. 87 3 WHY NOT KEY WORDS? • Takes away reason for reading • Does not develop process for problem solving • Takes away creating equations and finding all the information • Takes away from thinking “What else do I know to solve the problem? What else may I need?” 4 EXPLORE • Caroline has 46 dolls in her collection. She has 12 more than her cousin, Tammy. How many dolls does Tammy have? • Alicia had $6 more than Bobby. If Bobby had $10, how much did they have altogether? • Pedro and Gina started out with an equal number of coins. Pedro lost 12 coins, and Gina collected another 36 coins. How many more coins did Gina have than Pedro? PROBLEM #1 Cathy and Joan started out with the same number of coins. Cathy lost 12 coins and Joan gained 36. How many more coins does Joan have than Cathy? 6 1. After reading the problem, mark the who or what and unit bars. 2. Re-read each sentence and adjust the unit bars accordingly. Cathy’s loses 12 and Joan gains 36 3. Record the ?, what students need to find. Cathy’s coins Joan’s coins Cathy’s coins Joan’s coins 12 36 12 Cathy’s coins Joan’s coins 36 ? 4. Students solve and try to 12 + 36 = 48 record an equation if CL + JG = ? possible. What Cathy lost plus what Joan gained equals how many more coins Joan has. 7 Example from the ISAT ER problem from 5th grade. 6.RP.3 Students were creating spirit necklaces to sell for a fundraiser. A necklace takes twice as many purple beads as white beads and 4 times as many purple beads as black beads. One necklace takes 28 beads. What is the number of each color of beads? 9 7.RP.3 A class had 32 students and twenty-five percent were boys. When some new boys joined the class, the percentage of boys increased to 40%. How many new boys joined the class? 10 7.RP.3 Two students were running for school president. Student A received 65% of the votes and had 900 more votes than Student B. How many students voted? 11 5.NF.4 The fundraising committee made 400 pizzas. The students sold 5/8 of the pizzas and took 1/5 of the remainder for a party. How many pizzas did the committee have left to sell? 12 GROUP PROBLEM SOLVING Work with your colleagues to create: • A manipulative model with your color tiles • A tape diagram (bar model) of your problem • An equation • A verbal description of your thought process • What other questions can be answered about your situation/problem? 13 WHAT DO YOU KNOW? WHAT CAN YOU ANSWER? A cran-apple mixture is made up of 3 parts apple juice and 1 part cranberry juice. The company will use 5 gallon containers for the cran-apple mixture. 14 WHAT ARE THE MATH PRACTICES? •Look at your Math Practices •What practices have you been engaged in? 15 ANSWER GETTING VS. LEARNING MATH • USA: How can I teach my kids to get the answer to this problem? Use mathematics they already know. Easy, reliable, works with bottom half, good for classroom management. • Japanese: How can I use this problem to teach the mathematics of this unit? Phil Daro, Writer of CCSS in Mathematics, Slide 16, http://www.cmcmath.org/resources/downloads/Daro%20PS%20Conference.p 16 POSING THE PROBLEM • Whole class: pose problem, make sure students understand the language, no hints at solution • Focus students on the problem situation, not the question/answer game. Hide question and ask them to formulate questions that make situation into a word problem • Ask 3-6 questions about the same problem situation; ramp questions up toward key mathematics that transfers to other problems Phil Daro, Writer of CCSS in Mathematics, Slide 80, http://www.cmcmath.org/resources/downloads/Daro%20PS%20Conference.p 17 WHAT PROBLEM TO USE? • • • • Problems that draw thinking toward the mathematics you want to teach. NOT too routine, right after learning how to solve. Ask about a chapter: what is the most important mathematics students should take with them? Find a problem that draws attention to this mathematics Begin chapter with this problem (from lesson 5 thru 10, or chapter test). This has diagnostic power. Also shows you where time has to go. Also near end of chapter, while still time to Phil Daro, Writer of CCSS in Mathematics, Slide 81, respond http://www.cmcmath.org/resources/downloads/Daro%20PS%20Conference.p 18 WHICH IS BIGGER? • 1/3 or 1/2 3/5 or 6/10 3/4 or 7/8 10/16 or 5/8 2/3 or 6/12 • Katie collected 20 seashells. Diego collected 3 times as many seashells as Katie. What was the total number of seashells collected? • Corey paid $84 for a kite and a CD player. The CD player cost 6 times as much as the kite. What is the price difference between the two items? • Of the 60 students in the fifth grade, we know that 3/5 are girls. We also know that 1/2 of the girls have blue eyes and one-fourth of the boys have blue eyes. How many of the students in the fifth grade have blue eyes?