Chapter 9: Developing Meanings for the Operations By: Melissa Zeltwanger

Objectives Analyze the 4 problem structures for additive story problems. Identify best teaching practices for addition and subtraction. Analyze the 4 problem structures for multiplicative story problems. Identify best teaching practices for multiplication and division.

Should teachers teach students that addition is “put together” and subtraction is “take away” when solving problems? No Yes

Analyze Problem Structures for Additive Story Problems Choose one story problem to solve Draw a picture or use counters to show how a student would solve the problem Write either an addition or subtraction equation that you think best represents your work

Join Anna had 6 apples. Gary gave her 4 more. How many apples does Anna have altogether? Anna had 6 apples. Gary gave her some more. Now Anna has 10 apples. How many did Gary give her? Anna had some apples. Gary gave her 4 more. Now Anna has 10 apples. How many apples did Anna have to begin with?

Separate Kim had 7 cookies. She gave 3 cookies to Tom. How many cookies does Kim have now? Kim had 7 cookies. She gave some to Tom. Now she has 4 cookies. How many did she give to Tom? Kim had some cookies. She gave 3 to Tom. Now Kim has 4 cookies left. How many cookies did Kim have to begin with?

Part-Part-Whole: Whole unknown Becky has 6 roses and 7 daisies. How many flowers does she have? John has 6 roses and Becky has 7 daisies. They put their flowers into a vase. How many flowers did they put into the vase?

Part-Part-Whole: Part unknown Becky has 13 flowers. Six of the flowers are roses, and the rest are daisies. How many daisies does Becky have? John and Becky put 13 flowers into the vase. John put in 7 flowers. How many flowers did Becky put in?

Compare: Difference Unknown George has 12 pennies and Sandra has 8 pennies. How many more pennies does George have than Sandra? George has 12 pennies. Sandra has 8 pennies. How many fewer pennies does Sandra have than George?

Compare: Larger Unknown George has 4 more pennies than Sandra. Sandra has 8 pennies. How many pennies does George have? Sandra has 4 fewer pennies than George. Sandra has 8 pennies. How many pennies does George have?

Compare: Smaller unknown George has 4 more pennies than Sandra. George has 12 pennies. How many pennies does Sandra have? Sandra has 4 fewer pennies than George. George has 12 pennies. How many pennies does Sandra have?

Teaching Addition and Subtraction Provide physical models

Teaching Addition and Subtraction Context problems – relate to children’s lives and recent experiences Provide time to discuss and explain their work Introduce symbols +, -, = Subtraction as “think addition” instead of “take away” 8-5 = ? “5 and what makes 8?”

Differentiation I have ___ ___________. I got ___ more. How many _____ do I have altogether? I had ____ _________. I got some more. Now I have ____ __________. How many more did I get? I had some ___________. I got ____ more. Now I have ____ _________. How many ___________ did I have to begin with?

True (green check) or False (red x) There are 4 categories of problem structures for additive story problems. 2. Teachers should encourage that addition is “put together” and subtraction is “take away.” 3. Students should use whatever physical models they need to solve the problem.

Analyze Problem Structures for Multiplicative Story Problems Choose a problem Show your work with a drawing or equation

Equal Groups: Whole unknown Mark has 4 bags of apples. There are 6 apples in each bag. How many apples does Mark have altogether? If apples cost 7 cents each, how much did Jill have to pay for 5 apples? Peter walked for 3 hours at 4 miles per hour. How far did he walk?

Equal Groups: Size of groups unknown Mark has 24 apples. He wants to share them equally among his 4 friends. How many apples will each friend receive? Jill paid 35 cents for 5 apples. What was the cost of 1 apple? Peter walked 12 miles in 3 hours. How many miles per hour did he walk?

Equal Groups: Number of groups unknown Mark has 24 apples. He put them into bags containing 6 apples each. How many bags did Mark use? Jill bought apples at 7 cents apiece. The total cost of her apples was 35 cents. How many apples did Jill buy? Peter walked 12 miles at a rate of 4 miles per hour. How many hours did it take Peter to walk the 12 miles?

Comparison: Product unknown Jill picked 6 apples. Mark picked 4 times as many apples as Jill. How many apples did Mark pick? This month Mark saved 5 times as much money as last month. Last month he saved $7. How much money did Mark save this month?

Comparison: Set size unknown Mark picked 24 apples. He picked 4 times as many apples as Jill. How many apples did Jill pick? This month Mark saved 5 times as much money as he did last month. If he saved $35 this month, how much did he save last month?

Comparison: Multiplier unknown Mark picked 24 apples, and Jill picked only 6. How many times as many apples did Mark pick as Jill did? This month Mark saved $35. Last month he saved $7. How many times as much money did he save this month as last?

Combinations: Product Unknown Sam bought 4 pairs of pants and 3 jackets, and they all can be worn together. How many different outfits consisting of a pair of pants and a jacket does Sam have?

Area and other Product-of Measures Problems Bob wants to tile his kitchen floor. The length of the kitchen is 12 feet and the width is 8 feet. What is the area of the kitchen floor?

Teaching Multiplication and Division Provide physical models Symbols and terminology “4 sets of 2” “8 divided by 4”

Teaching Multiplication and Division Context problems 1. Think about the problem, not the numbers. 2. Ask questions: Who is in the story problem? What is happening in the problem? What will the answer tell us? Will the answer be a small number or a larger number? What would be a good estimate? Differentiation Have students write a problem for a multiplication fact: 3 x 7

True (green check) or False (red x) There are 4 categories of problem structures for multiplicative situations. Arrays, counters and cubes are not the best physical models for multiplication. When solving context problems, we should first think about the problem not the numbers.