KEY STRATEGIES FOR INTERVENTIONS: COMPUTATION

Two types of fluency

Single-digit addition and multiplication “facts”

Multi-digit procedures or algorithms

The Framework

Acquisition Fluency

Generalization

Acquisition

Word problems to develop the concepts Use of strategies (developmental) Fact families

Foundational concepts (place value; array and area models; distributive property)

Multi-digit procedures (algorithms)

Word Problems

Word problems are critical for developing the concepts of the operations. Students must be able to recognize which operation is required in problem-solving situations. Joining, separating, comparing, part-

whole Grouping (including rate, price,

combination), multiplicative comparison Partitive division, measurement divisionYellow

handout

Strategies for Addition

1) Direct modeling2) Counting (counting on from first, from

larger, counting back)3) Derived strategies (sums to 5, 10;

doubles plus or minus 1)

Most facts are learned over time by solving

real-world and mathematical problems.

CGI interviews Number Talks 8+6

Strategies for Multiplication1) Skip counting (number line, arrays)2) Known facts plus or minus (6x3)3) Double-doubles

Number Talks 7x7Box of Facts

The Idea of Fact Families

Subtraction and division are the inverses of addition and multiplication – they are related in fact families

“Missing addend” and “missing factor” problems highlight this.

Alesha has 4 pretty sea shells. She gets some more for her birthday. Now she has 12 sea shells. How many did she get for her birthday?

4 + __ = 12 This can be solved by counting up, or by taking away the original amount from the whole. 12 – 4 = __

5 bags of candy with the same amount in each bag. 35 pieces of candy altogether. How many in each bag?

This can be thought of as 5 x ? = 35.

Summarize

What concepts do students need to acquire for single-digit fluency?

Think, pair, share at tables.

The Framework

Acquisition Fluency

Generalization

Fluency – Practice and Drill “Practice” refers to lessons that are

problem-based and that encourage students to develop flexible and useful strategies that are personally meaningful.

“Drill” is repetitive non-problem-based activity to help children become facile with strategies they know already in order to internalize (remember) the fact combinations.

From Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally

Games for Practice

Sum Search Multiplication Table V.M. The Product Game The Factor Game Pan Balance Games in Math Facts Packet

Math Squares

“After introducing Math Squares as a whole class activity, have students work in pairs to fill the empty squares or find the sum if all squares are filled. Always follow the small group work with a whole class discussion where students explain their methods. You may wish to ask questions such as:

“Is there another way?”

“Which way do you like best?”

“Is there a method someone showed today that you might use?”

Games for Drill

Computer games that are timed multiplication.com coolmath-games.com

Flash card and dice games with a competitive edge Multiplication Call-out “For each strategy or related group of facts,

make several sets of flash cards using all of the facts that fit that strategy.”

Games for Drill

“It is critical that you do not introduce drill too soon. Suppose that a child does not know the 9+5 fact and has no way to deal with it other than to count fingers or use counters. These are inefficient methods. Premature drill introduces no new information and encourages no new connections. It is both a waste of time and a frustration to the child.”

-Van de Walle

Assesssing Fluency

Fluency means… no counting. While you don’t know whether a student

used a strategy quickly, you do know they didn’t count.

See the handouts.

Generalization = Math Power Simple sums generalizes into missing

addend problems: 3+5 = ___ 3+___ = 8

Missing addend problems can be used to solve subtraction situations: Alesha has 4 pretty sea shells. She gets some more for her birthday. Now she has 12 sea shells. How many did she get for her birthday?

Multiplication fluency generalizes into division fluency (fact families).

Acquisition – for multi-digit Word problems to develop the concepts Use of strategies (developmental) Fact families

Foundational concepts (place value; array and area models; distributive property)

Multi-digit procedures (algorithms)

Word Problems

There were 27 boys and 35 girls on the playground at recess. How many children were on the playground at recess?

Jenise is shopping for a school party. She buys 15 bags of cookies. Each bag has 24 cookies in it. How many cookies will she have altogether?

A school is taking a field trip to the zoo. They will ride in buses. There are 320 students and 20 adults going on the field trip. Each bus can hold 50 people. How many buses will they need?

Concepts for multi-digit operations Strong place value understanding (not

just “lining up” but knowing that each place to the left represents ten of the number in the previous place – base ten blocks are important representations)

Estimation skill, or recognizing whether the answer is a good approximation

Base Ten Concepts

Using objects grouped by ten:

“There are 10 popsicle sticks in each of these 5 bundles, and 3 loose popsicle sticks. How many popsicle sticks are there all together?” This encourages/enables thinking about tens and ones. 50 + 3 = 53

The extension: The teacher puts out one more bundle of ten popsicle sticks and asks students “Now how many popsicle sticks are there all together?” What strategies would students use to answer this?

53 + 10 = 50 + 10 + 3

Multi-digit Add/Subt. conceptsDecomposing, splitting, regrouping:53 =

50+340+1343+1048+5

Pan Balance

Children’s Strategies

There were 27 boys and 35 girls on the playground at recess. How many children were on the playground at recess? Todd: Let’s see. 20 and 30, that’s 50,

and 7 more is 57. Then the 5. 57 and 3 is 60, and the 2 more from the 5 is 62. There were 62. (recall and derived facts)

Kisha: 20 [pause], 30, 40, 50 [pause], 57, 58, 59, 60, 61, 62. There were 62. (counting on by tens and ones)

Try this with other problems

Development of Algorithms Concrete – Representational – Abstract Objects – Pictures – Symbols

Multi-digit multiplication concepts Multiplying x10, x100 The distributive property Advanced area models related to

the distributive property

Last handout

Development of Algorithms

Concrete-Representational-Abstract

Concrete: Multiply 16 x 12 using base 10 blocks.

Procedures… The C-R-A

Concrete-Representational-Abstract

Representational:

National Library of Virtual Manipulatives nlvm.usu.edu

Procedures… The C-R-A

Concrete-Representational-Abstract

Abstract:

Multi-digit division conceptsPartitive division (fair shares)

We want to share 12 cookies equally among 4 kids. How many cookies does each kid get?

How would you solve this with a objects or a picture?

The number of groups is known; the number in each group is unknown.

Measurement division (repeated subtraction)

For our bake sale, we have 12 cookies and want to make bags with 2 cookies in each bag. How many bags can we make?

How would you solve this with a picture?

The number in each group is known; the number of groups is unknown.

Partial quotient method

6 )234 -120 20 114 -60 10 54 -30 5 24 -24 4 0 39

Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value. 4.NBT.6

This type of division is called repeated subtraction

You try it

24)8280

Now the standard algorithm

Keep in mind that 8280 = 8000 + 200 + 80 + 0 or 8200 + 80 or 82 hundreds + 8 tens

324)8280 72 10

The standard algorithm:

1) How many equal groups of 24 can be made from 82? 3 groups, with 10 left over. 82 what? 10 what? Why do we put the 3 there?

24 24 24 10

3424)8280 72 1080 96 12

The standard algorithm:

1) How many equal groups of 24 can be made from 82? 3 groups, with 10 left over. 82 what? Why do we put the 3 there?

2) How many equal groups of 24 can be made from 108? 4 groups, with 12 left over.108 what?Why do we put the 4 there?

1224 24 24 24

34524)8280 72 1080 96 120 120 0

The standard algorithm:

1) How many equal groups of 24 can be made from 82? 3 groups, with 10 left over. 82 what? 10 what?Why do we put the 3 there?

2) How many equal groups of 24 can be made from 108? 4 groups, with 12 left over.108 what? 12 what?Why do we put the 4 there?

3) How many equal groups of 24 can be made from 120? 5 groups, with 0 left over.120 what? Why do we put the 5 there?24 24 24 24 24

34524)8280 72 1080 96 120 120 0

8280 = 8000 + 200 + 80 + 0 or

= 7200 + 960 + 120

= 24x300 + 24x40 + 24x5

What about remainders?

The remainder means an extra is needed 20 people are going to a movie. 6 people

can ride in each car. How many cars are needed to get all 20 people to the movie?

The remainder is simply left over and not taken into account (ignored) It takes 3 eggs to make a cake. How

many cakes can you make with 17 eggs?

What about remainders?

The remainder is the answer to the problem Ms. Baker has 17 cupcakes. She wants to

share them equally among her 3 children so that no one gets more than anyone else. If she gives each child as many cupcakes as possible, how many cupcakes will be left over for Ms. Baker to eat?

The answer includes a fractional part 9 cookies are being shared equally

among 4 people. How much does each person get?

Fluency – Mental Math Practice

Coming to Know Number, Wheatley and Reynolds

“Two Ways present opportunities for students to construct number relationships in an interesting setting that provides a self-check. As students attempt to fill all the empty spaces in the various arrangements of given numbers, they will come to relate addition and subtraction.”

Fluency – Algorithm Practice Math is everywhere. We use it every day. Show students the importance of

quantitative thinking across the curriculum. Plan to spend some time at least once a week using numbers to quantify what you’re studying.

Take time to set up the numbers as math problems. Pose interesting questions, then give students the numbers to work with. Make sure they estimate first, to develop number sense.

Social studies, for exampleHow did the population of various countries or cities change over time? How did the percent of people living in cities change over our history? What does the census tell us about people in Michigan? How did travel time change between cities in the 1800’s with the advent of new modes of transportation? How many people are represented by each Congressman (or Michigan Representative, or US Senator or Michigan Senator)? What fraction of our national debt is owed by each individual? What proportion of people living in the U.S. are under 20? How does this compare to other countries?

Typical Learning Problems

Often children who come to you as interventionists have been taught the standard algorithm (or some variation) but they make mistakes that need to be corrected.

What is this mistake? What does the child know well? What does he or she still need to learn?

What did this child internalize from instruction? What does he or she still need to learn?

Multi-digit Problems

There were 53 geese in the farmer’s field. 38 of the geese flew away. How many geese were left in the field?

There were 28 girls and 35 boys on the playground at recess. How many children were there on the playground at recess?

Misha has 34 dollars. How many dollars does she have to earn to have 47 dollars?

Strategies? Counting single units. Direct modeling with tens and ones. Invented algorithms: Incrementing by tens and then ones, Combining tens and ones, Compensating.

Development of Algorithms

The C-R-A approach is used to develop meaning for algorithms.

Without meaning, students can’t generalize the algorithm to more complex problems.