Addition and Subtraction

1. Addition and Subtraction Unit 2

2. Addition and Subtraction • Addition and subtraction are connected. • Addition names the whole in terms of the parts. • Subtraction names a missing part. • Often addition is incorrectly presented to children as “put together” and subtraction as “take away.” • When students develop these limited views of addition and subtraction, they can have difficulty later when addition or subtraction is called for but the structure of the problem is not “put together” or “take away.”

3. Four Types of +/- Problems • To develop meaning for the operations, children should solve contextual (story) problems. • There are four different kinds of addition and subtraction problems, based on the kinds of relationships involved: • Join problems • Separate problems • Part-part-whole problems • Compare problems • We are going to look at these four types of problems, all involving one “family” of numbers: 4, 8, 12.

4. Join Problems • For the act of joining, there are 3 quantities involved: • An initial or starting amount • A change amount (the part being added or joined) • The resulting amount • Any one of these could be unknown in a problem: • Sandra had 8 pennies. George gave her 4 more. How many pennies does Sandra have altogether? (result unknown; 8 + 4 = [ ]) • Sandra had 8 pennies. George gave her some more. Now Sandra has 12 pennies. How many did George give her? (change unknown; 8 + [ ] = 12) • Sandra had some pennies. George gave her 4 more. Now Sandra has 12 pennies. How many pennies did Sandra have to begin with? (initial unknown; [ ] + 4 = 12) • Note: joining does not always mean addition!

5. Separate Problems • In separate problems, the initial amount is the whole or the largest amount. • In join problems, the result was the whole. • There can be different unknowns in separate problems: • Sandra had 12 pennies. She gave 4 pennies to George. How many pennies does Sandra have now? (result unknown; 12-4=[ ]) • Sandra had 12 pennies. She gave some to George. Now she has 8 pennies. How many did she give to George? (change unknown; 12 - [ ] = 8) • Sandra had some pennies. She gave 4 to George. Now Sandra has 8 pennies left. How many pennies did Sandra have to begin with? (initial unknown; [ ] - 4 = 8) • Note: separating (or removing) does not always mean subtraction!

6. Computational and Semantic Forms of Equations • When you write an equation or number sentence that follows the order and meaning of the story problem, the equation is called a semantic equation. • The equations on the previous two slides are semantic equations. • An equation that isolates the unknown alone on one side of the equal sign is called a computational equation. • This is the equation you would need to use to solve the equation with a calculator. • When the unknown is the result, the computational and semantic equations are the same. • This is why instruction has tended to emphasize these kinds of addition & subtraction problems! • It is also why teachers have tended to connect join with addition and separate with subtraction.

7. Part-Part-Whole Problems • These problems involve two parts being combined into one whole. The combining can be physical or mental. • Unlike join problems, there is no meaningful distinction (e.g., initial, change) between the two parts being combined. • Either a part or the whole can be unknown: • George has 4 pennies and 8 nickels. How many coins does he have? (whole unknown; mental; 4 + 8 = [ ]) • George has 4 pennies and Sandra has 8 pennies. They put their pennies into a piggy bank. How many pennies did they put into the bank? (whole unknown; physicalaction; 4 + 8 = [ ]) • George has 12 coins. Eight of his coins are pennies, and the rest are nickels. How many nickels does George have? (part unknown; mental; 12 = 8 + [ ]) • George and Sandra put 12 pennies into the piggy bank. George put in 4 pennies. How many pennies did Sandra put in? (part unknown; physicalaction; 12 = 4 + [ ])

8. Compare Problems • These problems involve the comparison of two quantities. • There are three types, depending on which quantity is unknown: • George has 12 pennies and Sandra has 8 pennies. How many more pennies does George have than Sandra? (Difference unknown; 12 - 8 = [ ]) • George has 12 pennies and Sandra has 8 pennies. How many fewer pennies does Sandra have than George? • George has 4 more pennies than Sandra. Sandra has 8 pennies. How many pennies does George have? (Larger unknown; 4 + 8 = [ ]) • George has 4 more pennies than Sandra. George has 12 pennies. How many pennies does Sandra have? (Smaller unknown; 4 + [ ] = 12)

9. Teaching Addition and Subtraction • Using Context or Story Problems • Children need to gain experience with multiple problem structures, not just those with the result unknown • Use problems generated by the students, from their own lives and events in the classroom • Rather than a sheet of story problems, explore and discuss just one or two problems in depth • Gradually introduce symbolism into the discussion: • + (say “plus” or “and”) • - (say “minus” or “subtract”, not “take away”) • = (say “equals” or “is the same as”; not “the answer is …”) • Activity 2.1 “Equations with Number Patterns”

10. Teaching Addition and Subtraction • Using Model-Based Problems • Use models to help children understand what is happening in a problem and keep track of the numbers and solve the problem • Problems can also be posed using models when there is no context involved • Activity 2.1 “Equations with Number Patterns” • Activity 2.2 “Missing Part Subtraction”

11. Mastering Basic Facts • Mastery of a basic fact means that a child can give a quick response (< 3 seconds) • Typically mastery of all addition and subtraction facts (where both addends are <10) is expected in 2nd or 3rd grade • All children can master basic facts, including children with learning disabilities • All children can construct efficient mental tools that will help them • Number relationships provide the foundation for strategies that help students remember the basic facts • E.g. knowing how numbers are related to 5 and 10 helps • “Think addition” is a powerful way to remember subtraction facts • Think of 13 - 6 as “6 and what make 13?” • All of the facts are conceptually related - You can figure out unknown facts from those you already know!

12. Strategies for Addition Facts • “One more than” and “two more than” facts • Activity 2.3 “One/Two More Than Dice” • Facts with zero • Seems easy, but children sometimes overgeneralize the idea that “addition makes bigger” • Doubles • “Calculator doubles”: Students use the calculator as a “double maker” to practice facts like 7+7 • “Near doubles” (e.g. 7+8) • Make-10 facts • Activity 2.4 “Say the Ten Fact”

13. Strategies for Subtraction Facts • Facts involving 0, 1, 2 • Some children may initially have trouble with subtracting zero because they overgeneralize “subtraction makes smaller” • Subtraction as “think addition” • Children can make use of known addition facts • Missing number cards and worksheets - used to emphasize particular sets of facts (doubles, near doubles) • Activity 2.5 “Missing Number Cards”