PS/IS 276 Grade 3 Parent Math Workshop Models for Multiplication February 28, 2013

Welcome! While you’re getting settled, please help us figure something out: Rebecca has 26 students in her class. If they count by fives around the class, what number will the 26th student say? Have paper and pencil available if participants want to use it. Others may solve mentally. Circulate to see if anyone uses the algorithm, partial products, doubling and halving, etc. Elicit responses and record.

Here’s how Rebecca’s kids solved the same problem Ben said: “I know that 20 groups of 5 is 100, and 6 groups of 5 is 30, so I put the 100 and the 30 together to make 130.” (Ben is making sense of the distributive property. He may not know that he is doing this, but he just solved the problem like this: 5(20 + 6) = 130 5 x 20 = 100 5 x 6 = 30 100 + 30 = 130 This kind of foundation of deep numerical understanding, and the flexibility to apply mental math strategies is what we aim for. Kids will develop deep understanding first, then match an equation to their work as they learn the language and notation of multiplication and division.

Here’s how Rebecca’s kids solved the same problem Nate said that he knew that when we count by tens 26 times, we land on 260. Since 5 is half of 10, we can cut the 260 in half to get 130. Nate knows that skip-counting and multiplication are related. This is the equation he is actually solving: 10 x 26 ÷ 2 = 130 This kind of foundation of deep numerical understanding, and the flexibility to apply mental math strategies is what we aim for. Kids will develop deep understanding first, then match an equation to their work as they learn the language and notation of multiplication and division.

Here’s how Rebecca’s kids solved the same problem Tate knew that 26 x 10 =260, so to figure out a “x5 combo” instead of a “times 10 combo,” he halved the 260. First he halved 200 to make 100, and then he halved 60 to make thirty, then he added them together to make 130, so 26 x 5 =130. 26 x 10 =260 260 x ½ = 130 This kind of foundation of deep numerical understanding, and the flexibility to apply mental math strategies is what we aim for. Kids will develop deep understanding first, then match an equation to their work as they learn the language and notation of multiplication and division.

The Common Core Standards for grade 3 and 4 state: Students develop fluency with efficient procedures for multiplying and dividing whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems. The beginning activity is an activity to promote the grade 3 and 4 fluency of multiply and divide within 100 Grade 5 fluency is multi-digit multiplication

Fluency Having efficient and accurate methods for computing Flexibility in computational methods Understand and explain methods Produce accurate answers efficiently Understands base 10 number system Understands number relationships

The Common Core Standards for grade 3 state: Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors.

The Common Core Standards for grade 4 state: Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving multi-digit factors. Student find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, and/or the relationship between multiplication and division.

The Common Core Standards for grade 5 state: Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving decimals to hundredths. Student find quotients and remainders with two-digit divisors, using strategies based on place value, and/or the relationship between multiplication and division.

Task: Use the grocer’s display to figure out the quantities of some of the items in the store. Record your thinking. Facilitators elicit responses and record, making commutative, distributive and associative properties explicit. Goal is to get participants thinking flexibly about how to figure out the quantities.

Using the rectangular array as a model for multiplication Use multiplication when you want to combine groups of the same size. Rectangular arrays of objects can be used to model same-size groups. Establish the image of an actual array of items

Common Core Focus – Multiplication- grades 3-5 Apply properties of operations as strategies to multiply and divide. 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.) Language may be technical, but maybe using this slide after the grocery store work enables us to place a name on some of the strategies used.

Breaking some (arrays of) eggs Arrays can be broken up to help students use multiplicative relationships as they develop fluency What multiplication expression can go with this array?

Don’t know 7 x 6? You can use combinations you know to get there, such as 7x3 7x6 = (7x3)x2 Breaking up an array to illustrate use of the associative property. (Using the distributive property, you could record (7x3) + (7x3))

“Go with what you know!” How about 5x6 and 2x6? 7x6 =(5x6) + (2x6) Breaking up the array this way highlights the distributive property. (And showing these different ways of breaking up the array reveal the versatility of this model.

From arrays of objects to grids Let’s take the stuff out and leave the “boxes” A geometric model for area Very imp’t to stress connection to 2-D measurement (area) ***Activity- MitC String A9: 5x7, 5x70, 5x71,5x69 (Prepare Grid Paper Arrays for the wow factor and discuss associative property in 5x70= 5x(7x10)

Tricks of the Trade - Doubling Use your 2 times table to figure out your fours and your eights (Doubling) 4 x 7 = 2 x (2 x 7) 8 x 7 = 2 x (4 x 7) Use your threes to figure out your sixes 6 x 7 = 2 x (3 x 7)

Tricks of the Trade – Doubling (4th grade) Use your 2 times table to figure out your fours and your eights (Doubling) 40 x 7 = 2 x (20 x 7) 80 x 7 = 2 x (40 x 7) Use your threes to figure out your sixes 60 x 7 = 2 x (30 x 7)

Assessments include showing mathematical proficiency Standards for mathematical practices are integrated into daily tasks and assessments. Students must be able to: Illustrate and explain by using equations, rectangular arrays, and/or area models. Represent problem situations in multiple ways and explain the connections Use clear and precise language in their reasoning Notice repetitive actions in computations to make generalizations

Student created the correct number of groups and showed how they created them by circling the amount in each box. If you notice the answer to question number 1, for box 1 they wrote 36 divided by 4 = 9, yet the supporting subtraction problems show subtraction of 4 nine times. This subtraction problem does not relate to that division problem and the same with box 3. This would be a point to discuss with the student and ask when we are dividing by a number what does that mean for the group of objects? Goal is to make sense of division

This student shows the equal groups and also supports their models with equations of the multiplication problem. In their answer to number 1, you can see their subtraction and then they counted and labeled the number of times that they subtracted for in order to get 9 groups. Also in number 2 they showed all of their thinking for each box type and circled their answer.

Here the student used words to explain that yes they can, yet supported their answer with a model or representation of how they would combine. This is a complete explanation. If it only had one part, it would only get half the points. For number 4, this student showed their thinking with a diagram and supported with an equation. For number 6, this student used a division equation with a variable and repeated subtraction.

Shift in Perspective Seeing math as a study of relationships enables us to see structural logic rather than just a series of random facts and procedures.

What can parents do at home? Have conversations that relate to everyday life and incorporate math questions (eg. How tall is that tree?) Become familiar with the new math standards Practice counting objects and money Practice telling time everyday Reason through questions with your children. Play board games