Happy Spring!1) Addition and subtraction (last month)

2) Multiplication and division (today)

3) Measuring length, working with time and money (Apr. 17)

Agenda

• Your experiences in the last few weeks• Establishing fluency with addition and subtraction• The different ways that multiplication is represented and

learned– A sequential approach– Number talks and other fluency approaches– Internet resources for practice and drill

• Connection of multiplication to area of rectangles• Extensions to division

2nd grade critical areas

(1) extend understanding of base-ten notation;

(2) build fluency with addition and subtraction;

(3) use standard units of measure; and

(4) describe and analyze shapes.

3rd grade critical areas

(1) develop understanding of multiplication and division and strategies for multiplication and division within 100;

(2) develop understanding of fractions, especially unit fractions (fractions with numerator 1);

(3) develop understanding of the structure of rectangular arrays and of area;

(4) describe and analyze two-dimensional shapes.

How do we get to fluency?

Addition and Subtraction fluency

2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.Teaching strategy:

1. Lots of real-world problems to make sure students know the concepts

2. Focus on combinations that aren’t known.3. Practice in various ways to develop useful and

efficient strategies4. Drill in ways that reward speed See wiki and

fluency packet

1. Real-world problems

Joining, separating, comparing, part-whole• Lucy has 8 fish. She wants to buy 5 more fish. How many fish

would Lucy have then?• TJ had 13 chocolate chip cookies. At lunch she ate 5 of those

cookies. How many cookies did TJ have left?• Mark has 3 mice. Joy has 7 mice. Joy has how many more

mice than Mark?• 10 children were playing soccer. 6 were boys and the rest

were girls. How many girls were playing soccer?

2. Focus on specific combinations

3. Develop useful and efficient strategies

sums to 5 If you know the sums to 5, like 3+2, you can find other sums like 3+4, because 3+4 = 3+2 plus 2 more

sums to 10 If you know that 6+4 = 10, then you can figure out 6+5, because it’s 1 more than 6+4

doubles plus one 6+7 = (6+6) + 1 more

doubles plus two 7+9 = (7+7) + 2 more

nines Add ten then subtract 1

3. Practice

Sum SearchMath Squares• Always follow the small group work with a whole class

discussion where students explain their methods.

Number Talks 8+6

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

4. Drill

http://www.fun4thebrain.com/addition.html

Print triangle flash cards

Addition and Subtraction fluency

2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.Teaching strategy:

1. C2. R3. A4. Practice with corrective feedback (I do it, we do it,

you do it – Explain your thinking)

“…using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”

What’s a problem for which this solution strategy would be an easy approach?

• Johanna has 12 fish in her aquarium. Over the summer, she gets some more fish from the pet store. Now she has 32 fish. How many more fish did she get over the summer?

• 12 + ___ = 32

Multiplication and Division

• A silent activity: Write all the things you know about multiplication.

• Discussion: Go around the table and discuss what you wrote, one item at a time.

2nd Grade

2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. 2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

3rd Grade

3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Multiplication – Equal Groups

• Skip counting

Acquisition-Fluency-Generalization

Acquisition 1) modeling word problems2) skip counting in arrays3) using visual representations4) developing the connection to area

Whole-class activities

Developing Number Concepts Book 3: Place Value, Multiplication and Divisionby Kathy RichardsonDale Seymour Publications, 1999$34.95 through Math Perspectives – Teacher Development Center

Act. 2-23 Lots of Rectangles

The child selects a card, fills the rectangular shape with cubes or tiles, and writes the multiplication equation that describes the array that is formed.

Looking for equal groups in the real world

I see six panes in the window by my desk. Are there any other windows in our room that have six panes? I see two more. Yes, we have three windows, each with six panes. Six panes and six panes and six panes.

6’s 8’s 4’s 2’spanes in

each window

Six chairs at each of our

tables

watercolors in each set

tires on a car

legs on a horse

eyeseyebrows

Acting out multiplication stories

1: Using real objects • 4 chairs around 3 tables, 2 pencils for 4

children, 4 stacks of 3 books, 5 boxes of crayons with 8 in each box

2: Using counters with word problemsThere are 4 houses on Letitia’s street. The family in each house has 2 cars. How many cars is that in all?• Use your counters to show only the cars. How many cars are

in front of the first house?• How many cars are in the front of the second house? Show

this with counters.• Have you shown the cars for all the houses? Show me the cars

in front of the next house.• You made four groups of two cars. How many cars is that in

all?

Acting out multiplication stories

Other story problems

• Tim had three dogs. He gave each dog two bones. How many bones did he give all his dogs?

• Five girls went to the library. They each checked out three books. How many books did they check out altogether?

• There are five children in Dale’s family. Each child gets to carve one pumpkin for Halloween. How many jack-o’-lanterns will they have?

• Robin’s mother went shopping for school clothes for her three children. She bought three shirts for each child. How many shirts did she buy?

Other types of multiplication problems

Rate problemsA baby elephant gains 4 pounds each day…

Price problemsHow much would 5 pieces of bubble gum cost if…?

Combination problemsThe Friendly Old Ice Cream Shop has 3 types of ice cream cones. They also have 4 flavors of ice cream.

Other types of multiplication problems

Array and Area problems (symmetric problems)For the second grade play, the chairs have been put into 4 rows with 6 chairs in each row…

A candy maker has a pan of fudge that measures 8 inches on one side and 9 inches on another side. If the fudge is cut into square pieces 1 inch on each side…?

Building models of multiplication problems

• Towers or stacks• Rows• Groups or pilesBuild 2 rows of 3. How many altogether?Build 3 piles of 5. How many?Build 5 piles of 3. How many?Build 4 stacks of 2. How many?Build 2 stacks of 4. How many?

Modeling the recording of multiplication experiences

Misty stacks books into two piles. She puts four in each pile.

How many stacks is Misty making? Two.

2 stacks of How many books in each pile? Four.

2 stacks of 4How many books altogether? Eight.

2 stacks of 4 = 8

Introducing the multiplication sign

2 stacks of 4 = 82 x 4 = 8

Read the sign as “stacks of” (not “times”) (or groups of, rows of, piles of)

Interpreting symbols

Write 4x3. Let’s make rows this time. How many stacks will you make? How many in each stack?

Acting out stories to go with multiplication problems

What story can you tell for this problem?

Learning to write the multiplication sign

• Copying equations• Writing equations and checking

Acquisition-Fluency-Generalization

Fluency1) independent and small group activities2) games3) developing strategies that build to fluency

3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of one-digit numbers.

3.OA.5 Apply properties of operations as strategies to multiply and divide. Examples:

If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.)

3 × 5 × 2 can be found by 3 × 5 = 15 then 15 × 2 = 30, or by 5 × 2 = 10 then 3 × 10 = 30. (Associative property of multiplication.)

Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

(Students need not use formal terms for these properties.)

Doubling and Halving

1 x 162 x 84 x 48 x 216 x 1

Number Talks

Independent activities

Developing Number Concepts Book 3: Place Value, Multiplication and Divisionby Kathy RichardsonDale Seymour Publications, 1999$34.95 through Math Perspectives – Teacher Development Center

How many cups

How many groups?

How many rows?

How many towers?

Interpreting multiplication problems

Copying and solving problemsMaking up their own

Problems for partners

Roll and Multiply (Patterns)

Discovering Patterns: Cupfuls

Fluency – two steps

1.Practice – to learn strategies • The Product Game; Number Talks• Word problems clustered on 5’s (ORIGO)

2. Drill – to habituate the combinations• Some element of competition, where quick

recall is important• Multiplication call-out; Computer games

Number Talks Multiplication String: 7x7

Practice to develop strategiesThe numbers in the problems should be built up gradually and deliberately. While many teachers already focus on one set of “facts” at a time, this focused approach to story problems helps students see patterns and begin to develop strategies. Here is a set of “times 5” problems that would be used together over the course of several days:• There are bags of 5 apples for sale. If you buy 3 bags, how many apples

will you have?• It takes 5 minutes to fill a wheelbarrow with soil. How long will it take to

fill 6 wheelbarrows?• There are 5 rows of 8 chairs. How many people can be seated?• Nine cats each had 5 kittens. How many kittens is that altogether?• When Ben places 4 shoes end to end, they measure 1 yard. How many of

these shoes would be lined up to measure 5 yards?• Jacob wants to plant 7 rows of 5 seeds. How many seeds is that?

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 to develop recall strategieshttp://nlvm.usu.edu

Multiplication Table V.M.multiplication.comcoolmath-games.com

Primary Krypto @ Illuminations

Recall strategies for multiplication

1) Skip counting (number line, arrays)2) Known facts plus or minus (6x3)3) Double-doubles Number Talks 7x7

Box of Facts

Drill

Most 3rd graders won’t be ready for extensive drill to habituate the multiplication and division combinations. They should continue to play practice games. By the end of third grade, most of the combinations should be available to all students through strategies or memory.

Some drill might be necessary with some combinations for some students. Multiplication call-out and timed games are appropriate.

http://www.helpingwithmath.com/printables/others/fac0201fact_triangle01.htm

Early work with multiplication should mainly be devoted to conceptual understanding that multiplication is a shorthand notation for denoting multiple addition. Multiplication situations should be presented to children, and they should then use materials (beans, counters, etc.) to demonstrate the problem given and to generate an answer (a product). Do not be concerned with habituating the facts too early. It is essential that children understand that the multiplication problems expresses a relationship between the numbers involved and that they own the meaning of the symbolism –that the first factor in the problem denotes the number of sets and the second factor denotes the number or objects contained in each set. The product is then the total number of objects when the sets are joined.

As children progress in conceptualizing the meaning of multiplication and can express what the multiplication situation means, it is time to begin habituating the facts. We do not like the word “memorizing” because it communicates that the fact is a paired association, of 2 with 4 in the above example, but the student could be devoid of an understanding of what the multiplication symbol means. Instead, after much work with material objects, the association of 2 with 4 is habituated so the child can develop speed and accuracy in recalling the fact. Habituation does not contribute to understanding; it presumes that understanding precedes the habituation, and the fact is habituated in order for the child to gain speed in working with multiplication situations.

Acquisition-Fluency-Generalization

Generalization 1) division (unknown factor problems)2) solving two-step problems3) developing the connection to area4) distributive property

Two-step problems

3.OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations)).

A shape puzzle

• Fill this shape with two different color tiles or counters, making two rectangles.

• Represent the total number of tiles or counters with a number sentence that uses multiplication.

Act. 2-24, with classroom vignette on pp. 142-143

Area standards

3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement.

a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

3.MD.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

Area standards

3.MD.7 Relate area to the operations of multiplication and addition.

a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

3

4

Area standards3.MD.7 Relate area to the operations of multiplication and addition. (cont’d)c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

Division

3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

Division by Partitioning

• There are 6 pieces of candy. If two children share the candy, how many pieces does each child get?

Also known as “fair shares” division.

Division by Measurement

• There are 6 pieces of candy. Each person is to get 2 pieces. How many people can get candy?

Also known as “repeated subtraction.”

Write Two Division Stories

• Measurement division – how many in a group is known

• Partitive division – how many groups is known

20 marbles

Multiplication and division are taught separately in most traditional programs, with multiplication preceding division. It is important, however, to combine multiplication and division soon after multiplication has been introduced in order to help students see how they are related.

In most curricula, these topics are a main focus of third grade with continued development in the fourth and fifth grades.

3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = __÷ 3, 6 × 6 = ?

3.OA.6 Understand division as an unknown-factor problem. For example, divide 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Division Activities

Developing Number Concepts Book 3: Place Value, Multiplication and Divisionby Kathy RichardsonDale Seymour Publications, 1999$34.95 through Math Perspectives – Teacher Development Center

Division Activities

• Given what you’ve seen about multiplication activities, what can you imagine for division activities?

Remainders

• What’s the story that this picture represents?

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?

The remainder means an extra is needed20 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 the answer to the problemMs. 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 part9 cookies are being shared equally among 4 people. How much does each person get?

So much more than “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.

We have 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.