A Journey into Alternate Algorithms There’s more than one way to get at the correct answer! Cindie Donahue and Misty McClain Instructional Trainer/Coaches

A Journey into Alternate Algorithms

There’s more than one way to get at the correct answer!

Cindie Donahue and Misty McClainInstructional Trainer/Coaches

Pasco County, Florida

2010 FCTM Conference

Ground Rules

1.Keep an open mind and think outside of the box!

2.Ask questions.3. Respect the learning needs

of the group by limiting side conversations and focusing on the goals of the day.

Developed by the Curriculum and Instructional Services Department District School Board of Pasco County

Try something new . . .

•”Never be afraid to try something new. Remember, that a lone amateur built the ark. A large group of professionals built the Titanic.”

-Dave Barry


Kindergarten

• MA.K.A.1.3 Solve word problems involving simple joining and separating situations. (This is based in use of manipulatives to model story situations.)


1st Grade• MA.1.A.1.1 Model addition and subtraction

situations using the concepts of “part-whole,” “adding to,” “taking away from,” “comparing,” and “missing addend”.

• MA.1.A.1.2 Identify, describe, and apply addition and subtraction as inverse operations.

• MA.1.A.1.3 Create and use increasingly sophisticated strategies, and use properties such as Commutative, Associative, and Additive Identity, to add whole numbers.

• MA.1.A.1.4 Use counting strategies, number patterns, and models as a means for solving basic addition and subtraction fact problems.

• MA.1.A.6.1 Use mathematical reasoning and beginning of understanding of tens and ones, including the use of invented strategies, to solve two-digit addition and subtraction problems.


2nd Grade

• MA.2.A.2.1 Recall basic addition and related subtraction facts.

• MA.2.A.2.2 Add and subtract multi-digit whole numbers through three digits with fluency by using a variety of strategies, including invented and standard algorithms and explanations of those procedures.

• MA.2.A.2.3 Estimate solutions to multi-digit addition and subtraction problems, through three digits.


3rd Grade

• MA.3.A.6.1 Represent, compute, estimate, and solve problems using numbers through hundred thousands.


4th Grade

• MA.4.A.6.1 Use and represent numbers through millions in various contexts, including estimation of relative sizes of amounts or distances.


Math Myth

The traditional algorithm is the most efficient and best procedure for solving

computation problems.


Math Myth

Students must solve computational problems by

starting on the right (ones position) and working to the left.


Math Myth

Students must first learn to add and subtract without regrouping.


Types of Addition and Subtraction Problems

• Involve actions:– Join Problems

• Elements are added to a given set– Separate Problems

• Elements are removed from a given set

• Do not involve actions:– Part-Part-Whole Problems

• Relationship between a set and its two subsets

– Compare Problems• Comparison between two disjoint sets


Join Problem TypesUnknown Example

Result Unknown5 + 2 = ?

Robin had 5 toy cars. Her parents gave her 2 more toy cars for her birthday. How many toy cars did she have then?

Change Unknown5 + ? = 7

Robin had 5 toy cars. Her parents gave her some more toy cars for her birthday. Then she had 7 toy cars. How many toy cars did Robin’s parents give her for her birthday?

Start Unknown? + 2 = 7

Robin had some toy cars. Her parents gave her 2 more toy cars for her birthday. Then she had 7 toy cars. How many toy cars did Robin have before her birthday?


Separate Problem TypesUnknown Example

Result Unknown8 – 3 = ?

Colleen had 8 guppies. She gave 3 guppies to Roger. How many guppies does Colleen have left?

Change Unknown8 - ? = 5

Colleen had 8 guppies. She gave some guppies to Roger. Then she had 5 guppies left. How many guppies did Colleen give Roger?

Start Unknown? – 3 = 5

Colleen had some guppies. She gave 3 guppies to Roger. Then she had 5 guppies left. How many guppies did Colleen have to start with?


Part-Part-Whole Problem Types

Unknown Example

Whole Unknown 6 boys and 4 girls were playing soccer. How many children were playing soccer?

Part Unknown 10 children were playing soccer. 6 were boys and the rest were girls. How many girls were playing soccer?


Compare Problem TypesUnknown Example

Difference Unknown

Mark has 3 mice. Joy has 7 mice. Joy has how many more mice than Mark?

Compared Set Unknown

Mark has 3 mice. Joy has 4 more mice than Mark. How many mice does Joy have?

Referent Unknown

Joy has 7 mice. She has 4 more mice than Mark. How many mice does Mark have?


Word Problem

• The two Scout troops went on a field trip. There were 46 Girl Scouts and 38 Boy Scouts. How many Scouts went on the trip?

• What type of addition/subtraction word problem is described?

• Solve using place value disks.

• Be prepared to explain your technique.


Left to Right Addition

• Also referred to as – Add Tens, Add Ones, Then Combine

46 + 38

70 40 and 30 is 70. 14 6 and 8 is 14.

84 70 and 14 is 84.

*No need for a regrouping procedure.


Add On Tens, Then Add Ones

46 + 38 76 + 8 80, 84

46 and 30 more is 76.Then I added on the other 8.76 and 4 is 80. (Make a 10 strategy)80 and 4 is 84.

*This can be done on an open number line.


Move Some to Make Tens

move 2

46 + 3844 + 40 84

Take 2 from the 46 and put it with the 38 to make 40. Now you have 44 and 40 more is 84.

*No need for a regrouping procedure.


Use a Nice Number and Compensate

• Also referred to as – Use a Friendly Number.

46 + 3846 + 40 8686 – 2 84

I picked 40 as a friendly number for 38.

46 and 40 is 86. 40 was 2 extra than 38.

So, remove the 2 to get 84.Developed by the Curriculum and Instructional Services

Department District School Board of Pasco County

Word Problem

• There were 75 students on the playground. 37 students came out to the playground with Ms. Engel. How many students are on the playground in all?


• Solve using an alternate algorithm. You may use the place value disks if they scaffold your thinking.

• Be prepared to explain your technique.Developed by the Curriculum and Instructional Services

Department District School Board of Pasco County

Word Problem

• Sara and her brother Sam were counting the pennies in their piggy bank. Sara counted 156 pennies. Sam counted 266 pennies. How many pennies were there in their piggy bank?





Word Problem

• Sam had 46 baseball cards. He went to a card show and got some more cards for his collection. Now he has 73 cards. How many cards did Sam buy?


• Solve using the place value disks.



Add Tens to Get Close, Then Ones

73 – 4646 + 10 = 5656 + 10 = 6666 + 4 = 70 (Make a 10)70 + 3 = 7310 + 10 + 4 + 3 = 27

46 and 20 is 66. Then 4 more is 70 and 3 more is 73.

That’s 20 and 7 or 27. *This can be done on an open number line.


Add Tens to Overshoot, Then Come Back

73 – 4646 + 10 = 5656 + 10 = 6666 + 10 = 7676 – 3 = 7310 + 10 + 10 - 3= 27

46 and 30 is 76. That’s 3 too much, so 30 – 3 would

be 27. *This can be done on an open number line.


Add Ones to Make a Ten, then Tens and

Ones73 – 46

46 + 4 = 5050 + 20 = 7070 + 3 = 734 + 20 + 3 = 27

46 and 4 is 50. 50 and 20 is 70. Then add 3 more to

get 73. So, 20 and 4 and 3 is 27.

*This can be done on an open number line.


Word Problem

• Juanita counted all of her crayons. She had 73 crayons in all. She took 46 to school in her backpack. How many crayons were left at home?


• Solve using the place value disks.



Take Tens from the Tens, Then Subtract

Ones73 – 46

70 – 40 = 3030 – 6 = 2424 + 3 = 27

70 minus 40 is 30. Take away 6 more to get 24.

Remember to add 3 ones from the 73 to get 27.


Take Away Tens, Then Ones

73 – 4673 – 40 = 3333 – 3 = 30 (Make a 10)30 – 3 = 27

73 minus 40 is 33. Now I need to take away 6, but I

can chunk it up. 33 minus 3 is 30, and 30 minus 3 is

27. *This can be done on an open number line.


Make a Nice Number and Compensate

73 – 4676 – 46 = 3030 – 3 = 27

I picked 76 as a friendly beginning number. 73 minus

46 is 30. Since 76 was 3 more than 73, I need totake away those extra 3 ones. 30 minus 3 is 27.


Word Problem

• Ethan had $150. He spent $84 to buy a used iPod. How much money does he have left?





Word Problem

• Mary has 114 spaces in her photo album. So far she has 89 photos in the album. How many more photos can she put in before the album is full?





Word Problem

• Jesus had 125 Bakugan in his collection. His brother Juan has 311 Bakugan. How many more Bakugan does Juan have than Jesus?





Caution: Avoid Key Words!

• Key words are misleading. Often a key word or phrase in a problem suggests an operation that is incorrect.

• Many problems have NO key words. Especially when you get away from the overly simple problems found in textbooks.

• The key word strategy send a terribly wrong message about doing mathematics. It encourages students to ignore meaning and find an easy way out. It is far more important for students to make sense of the problem.


Alternate Strategies (sometimes called invented strategies)

• Alternate strategies are – number oriented rather than digit oriented

– left-handed rather than right handed

– flexible rather than rigid


Alternate Strategies (sometimes called invented strategies)

• Base-ten concepts (place value, number sense) are enhanced.

• Students make fewer errors.

• Less reteaching is required.

• These strategies provide the basis for mental computation and estimation.

• Alternate strategies are often faster than traditional algorithms.

• Alternate strategies serve students at least as well on standardized tests.


How to Develop Alternate Strategies

• Use story problems frequently. Choose numbers carefully.

• Allow students to work together in a small group to solve the story problems.

• Allow students plenty of time to solve one problem.• Require students to share the strategies they use

to solve the problem, but avoid interjecting your own. Help students write their explanations on the board. Encourage students to ask questions of their classmates.

• Challenge students to find another method, solve the problem without models, or create a written explanation to extend the learning.

• Allow students to use the strategies presented by another student that make sense to them.


Debrief

• Did the presented information expand your view of addition and subtraction algorithms? How?

• Do you have any questions or concerns about the information presented on alternate algorithms for addition/subtraction?

• What is your personal plan for implementation with students?

• Further information is in your packet.


Credits/Sources

• Teaching Student-Centered Mathematics, Grades K-3, John Van de Walle and LouAnn Lovin

• Children’s Mathematics: Cognitively Guided Instruction, Thomas Carpenter, Elizabeth Fennema, Megan Loef Franke, Linda Levi, Susan B. Empson


Documents

A Journey into Alternate Algorithms There’s more than one way to get at the correct answer! Cindie Donahue and Misty McClain Instructional Trainer/Coaches