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**Comparison Situations and Tape Diagrams**

This material was developed for use by participants in the Common Core Leadership in Mathematics (CCLM^2) project through the University of Wisconsin-Milwaukee. Use by school district personnel to support learning of its teachers and staff is permitted provided appropriate acknowledgement of its source. Use by others is prohibited except by prior written permission. Tuesday September 11, 2012 Common Core Leadership in Mathematics (CCLM ^2)

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**Learning Intentions & Success Criteria**

We are learning to… Understand how to solve comparison problems and use tape (strip) diagrams to illustrate the solutions. We will be successful when…. Clearly explain the difference between additive and multiplicative comparison situations. Understand CCSSM expectations for compare situations in grades 1 & 2 and grades 3 &4. .

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**Addition Comparison Problem Situations**

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Revisiting 1.OA.1 1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

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**Skilled by the end of Grade 1**

Four kindergarten problem subtypes Skilled by the end of Grade 1 Experience at Grade 1 and skilled by the end of Grade 2

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**A Comparison Word Problem**

Blakely has 9 plates and 5 cookies. If a whole cookie is put on each plate, how many more plates are there than cookies? Model this problem with cubes. Then draw a tape diagram to illustrate the solution.

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As children move to Level 2 strategies, “they no longer need representations that show each quantity as a group of objects.” (OA Progressions, p. 16)

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**Concrete Representational 5 cookies ? 9 plates 5 cookies ? 9 plates**

Blakely has 9 plates and 5 cookies. If a whole cookie is put on each plate, how many plates won’t get a cookie? Representational 5 cookies ? Concrete 9 plates Use concrete objects to form two groups to compare quantities. 5 cookies ? 9 plates

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**The Packers played the Bears for the NFL Championship**

The Packers played the Bears for the NFL Championship. The Packers scored 52 points. The Bears had 39 points. How many more points did the Packers score than the Bears? Use tape diagrams to show the relationship among the quantities in this compare situation.

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**A General Comparison Model for Addition and Subtraction**

larger quantity smaller quantity difference

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**Multiplicative Comparison Problem Situations**

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Standard 4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 equals 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. This will give us a glimpse into their understanding of multiplicative comparison problems/. Focus on the different vocabulary “ 35 is 5 times as many as 7 and 7 times as many 5.” This completes the student’s understanding of x situations. 3OA1 is about groups of objects and 3OA3 is groups, arrays and measurement. May have to refer them back to reading p. 22 and 24.

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**Standard 4.OA.1 Read standard 4.OA.1**

Divide your slate in half. On one side, rephrase this standard and on the other side, provide an example. Share with your partner. This will give us a glimpse into their understanding of multiplicative comparison problems/. Focus on the different vocabulary “ 35 is 5 times as many as 7 and 7 times as many 5.” This completes the student’s understanding of x situations. 3OA1 is about groups of objects and 3OA3 is groups, arrays and measurement. May have to refer them back to reading p. 22 and 24.

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**Multiplicative Comparison Problems**

Read the handout about Multiplicative Comparison Problems. Highlight as you read, noting ideas that you understand and those that still confuse you.

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Standard 4.OA.2 Cluster: Use the four operations with whole numbers to solve problem. 4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

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**Standard 4.OA.2 Read standard 4.OA.2**

Divide your slate in half. On one side, rephrase this standard and on the other side, provide an example. Share with your partner.

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Sense Making…. Share with your shoulder partner a few ideas that struck you as critical to developing a sound understanding of multiplicative comparison problems. 5 minute partner share Whole group discussion. Give them a situation like: The second paragraph is difficult to understand. We have to help them make sense of this. Draw on a chart paper. Blue hat $10 Red hat $30 Larger quantity unknown: The red hat costs $10. The blue hat cost 3 times as much as the red hat. How much does the blue hat cost? Smaller quantity unknown: The blue hat cost $30. The cost of the blue hat is three times as much as the red hat. How much does the red hat cost? Compare quantity unknown: The red hat cost $10 and the blue hat costs $30. The blue hat is how many times as expensive as the red hat?

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Tape Diagram A drawing that looks like a segment of tape, used to illustrate number relationships. Also known as strip diagrams, bar model, fraction strip or length model. CCSSM Glossary p. 87

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Your Turn … Connie ran 50 meters. Melissa ran 200 meters. On your slate, write three problems using this information: Larger quantity unknown Smaller quantity unknown Compare quantity unknown Four times as much Four times farther Fractions don’t need to have as much as or as far as – check the OA progressions Connie ran 50 meters. Melissa ran four times as far as Melissa. How far did Melissa run?

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A Parking Lot Problem A fourth grade class counted the number of vehicles that went by the front entrance of the school between 9 o'clock and 10 o'clock. The total number of vehicles counted was156. There were 3 times as many passenger cars as trucks. How many passenger cars and how many trucks were counted? Use a tape diagram to illustrate and solve this problem. Ask teachers to draw the strip diagram before showing them the sample. Emphasize that the structure of the problem is the same. You still have 3 to 1 comparison Picture is the same

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Farmer problem A farmer has 7 ducks. He has 5 times as many chickens as ducks. How many more chickens than ducks does he have? On your slate, draw a diagram to help you solve this problem. Record your thinking. (Singapore Math Primary Mathematics volume 3A, page 46, problem 4) 1 unit is 7 ducks Chickens are 5 units so 5 x 7 ducks 35 chickens 35 – 7 = 28 more chickens

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**Let’s Look at Some Student Problems**

Solve the problems (from Singapore Mathematics 3A) with your partner. How did the tape diagram help you think multiplicatively? 3A Exercise 19 and 20 is probably grade 2 or 3. Kids move through as they are ready.

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**The Power of Strip Diagrams**

With the aid of simple strip diagrams, children can use straightforward reasoning to solve many challenging story problems conceptually. Beckmann 2004

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**Reflecting on the Practices**

with mc cullum graphic. Discussions about which practices we saw/did today. How does clustering the practices help you with this discussion? Be explicit.

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**Learning Intentions & Success Criteria**

We are learning to… Understand how to solve comparison problems and use tape (strip) diagrams to illustrate the solutions. We will be successful when…. Clearly explain the difference between additive and multiplicative comparison situations. Understand CCSSM expectations for compare situations in grades 1 & 2 and grades 3 &4. .

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