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Making Connections Through the Grades in Mathematics

A glimpse into thinking flexibly about numbers in the elementary grades and how it supports later

work in algebra.

Math fact expectations by end-of-grade level

Grade K - Add and subtract within 5 with accuracy and speed

Grade 1 - Add and subtract within 20 with accuracy and speed

Grade 2 - Add and subtract within 20 to compute with multi-digit numbers

Grade 3 - Add and subtract within 20 and multiply and divide within 100 with accuracy and speed

Grade 4 - Add and subtract within 20 and multiply and divide within 100 to compute with multi-digit whole numbers using efficient strategies.

Grade 5 - Use knowledge of basic facts to compute with fractions and decimals using efficient strategies.

Children come to school with a wealth of mathematical knowledge

When we receive children in preschool and kindergarten, children already use patterns and

relationships as they develop their mathematical understanding of the world around them. We build

on this by strategically selecting problems and numbers that develop big ideas in mathematics

throughout the grade level continuum.

We begin by understanding number.

You will have two seconds before they disappear.

I will flash dots on the screen. Your task is to figure out how many there are.

How many did you see?

Let’s try another one!

You will have two seconds before they disappear.

I will flash dots on the screen. Your task is to figure out how many there are.

How many did you see?

How did you see it?Did you count?

Are we having fun yet?

One more time!

Did you count this time?Now how many did you see!How did you see it?

Early Number Development

It is important for young children to not only count in the counting sequence, 1, 2, 3, 4, 5,…, but also comprehend that a number (symbol) represents a quantity.

This is a huge developmental milestone that takes time to develop.

Building Automaticity

It is also a big developmental idea for children to understand that numbers also contain other numbers. Meaning, a “5” contains the quantity “4” and that five is actually composed of a 4 and a 1 or a 3 and a 2…

It is important to push students beyond counting strategies to recognize numbers as being

composed of other numbers and that numbers can be decomposed into other numbers.

We look for patterns and make use of structure to make sense of and

organize our thinking.

Concrete and pictorial models help us to visualize.

Building NumbersUsing the Five Structure

-seeing three in relation to five

Building Numbers Using the Ten-Frame?

You will see dots in the ten-frame for 2 seconds.

Can you tell how many there are?

How did you see it?(5 + 3)(10 – 2)

Building a foundational understanding and making connections

Example:When students understand the commutative property of addition, learning the basic facts is easier.If a child is working on addition with 8 knows that 5 + 3 = 3 + 5, she/he can generalize and cut the number of facts to memorization in half.

Composing and Decomposing Numbers

Composing and decomposing numbers is related but different than adding and subtracting.

Five is composed of:

but there is always the quantity “five.”

2 & 31 & 4

3 & 24 & 15 & 0

Addition is

4 + 1 = 5

Subtraction is

5 - 1 = 4

We build on big ideas that are foundational for grades 1 an 2

• Counting• Composing and Decomposing numbers• Equivalence• Cardinality – understanding that the last

number said tells, “How many?”• Conservation of Number – understanding that

no matter how the objects are arranged the quantity remains the same.

Using the five-structure to help think about addition

Solve.

5 + 815 + 825 + 8135 + 8135 + 18

Some children in 1st or 2nd grade might think 5 + 5 + 3 , then

( )

What strategy did you use?

10 + 3 = 13

The associative property of addition

How can this idea help you solve these problems?

Children use their knowledge of decomposing numbers to find equivalent representations to make problem solving easier.

Children in early grades may use strategies to organize and make

sense of numbers like…

Use knowledge of 6 + 6 to help solve related problems like 6 + 7.

• Doubles Plus 1 • 6 + 7 = 6 + 6 + 1 or

• Doubles Minus 1• 7 + 7 - 1 = 13

( )

( )

And…

• Using 5 structure 6 + 7 = 5 +1 + 5 + 2

• Making 10 3 + 9 + 7 =

+ 3 = 1310

)( ( )

10 + 9 = 19

Children in early grades may use other strategies like…

• Using compensation

6 + 8 =

• Using known facts ( landmarks, friendly, familiar numbers)

6 + 8 = 14 so 7 + 8 must be 14 + 1 = 15

7 + 7 = 14

Addition Procedure with Regrouping

26

811 group of 10+ 55

1 Put down the 1 and carry the 1

7 tens plus 1 ten is 8 tens or 80.

1

Another Way to Record the Same Thinking for Regrouping

26

11+ 55

Or decompose the numbers and start with the tens

+ 7081

(20 + 6)+ (50 + 5) (70 + 11)

Partial Sums

In the intermediate grades students continue to build their algebraic

reasoning as it applies to multiplication, division and fractions.

Solve.

8 x 15 = ?

Decompose 15 and use a pictorial model to visualize it.

8

1510 5

80 40

= 120

+

MULTIPLICATIONOr, use a numerical representation

to illustrate this partial product strategy

8

8 x 15(10+ 5)

80 40

80 + 40 = 120

Decompose 15 into

ten and ones

(8 x 10) + (8 x 5)

Also known as th

e distributive

property of m

ultiplication over

addition

• Physical or concrete models, i.e. using manipulatives, coins, students acting it out,…

• Drawing models to visualize, i.e. using open number lines, arrays, coin images,…

• Numeric models: i.e. using the standard algorithm, algebraic properties,…

Models to Represent Your Thinking

Try that strategy yourself

6 x 29 = ?

MULTIPLICATIONUse the Open Array Model

to Illustrate this Partial Product Strategy

6

6 x 2920+ 9

120 54

120 + 54 = 174

Decompose 29 into

tens and ones

Fairfield Public Schools 2011-2012

Division

How would you divide

174 ÷ 6 = ?

Solve it mentally and think about how you approached the problem.

Fairfield Public Schools 2011-2012

How would you record your thinking?

2

-12 5

4

54

96 goes into 17 twice.Put down the 12 and

subtract from 17 to

get 5.Bring down the 4 to

make 54. 6 goes into

54 nine times.Digit

Oriented

Fairfield Public Schools 2013-2014

Or you could record your thinking?

20

-120 54

+ 96 goes into 174

twenty times.Put down the 120 and

subtract from 174 to

get 54.6 goes into 54 nine

times.Number

Oriented

A pictorial model

+ 9

54120 1746

20

Or record your thinking?

20-120 54

+ 9

6 goes into 174 twenty times.Put down the 120 and

subtract from 174 to

get 54.6 goes into 54 nine

times.Number

Oriented 29

174 ÷ 6 =

This division problem was solved the same way, using place value

concepts each time. The difference is how it is recorded.

Try this…

There is a sale on gift wrap for $2.98 a roll with a limit of 3 rolls. How much would it cost for three rolls?

Turn and Talk

How did you find your answer?

• Did you solve it using paper & pencil?

• Did you solve it with a calculator (or on your phone?)

• Did you solve it mentally?

• Did anyone decompose $2.98 into ($2.00 + .98)?

• Did anyone think of an equivalent value ($3.00 - .02)?

What strategy did you use?

• Did you use the standard algorithm?

2.98x 3

4

2

9 08 08 9 4

2

0+

OR,

Did you use a different strategy?

Did anyone decompose 2.98 and use partial products?

$2.98 x 3 can be thought of as …

2 cents less than $3.00Or ($3.00 - .02)

3 rolls at $3.00 is $9.00Then subtract (3 x 2 cents = 6 cents)

$9.00 subtract 6 cents is $8.94

Recording that thinking mathematically

3 x (3.00 - .02)

(3 x 3.00) – (3 x .02) = 9.00 – .06 = $8.94

Distributive property of multiplication over subtraction

Why is it important to use different strategies? (Turn & Talk)

• There is more than one way to solve a problem.

• Some ways are more efficient than others.

• Children may think differently than you about how to solve a problem.

• It is important to validate that thinking. Knowing their thinking will also help you when providing support.

FractionsComparing unit fractions

Which is bigger or ?

A common misconception children have is to think that 4 is bigger than 2 therefore is bigger than

Fractions can be decomposed too. pictorial model

is composed of .

12

12

14

14

14

14

Solve.

4 x 1 = ?

How did you solve it?

Use Concrete/Pictorial Representation

1

Concrete/Pictorial Representation

1 1

1 1

Decompose 1 into 1 and

Or 4 x (1 + ) == 6+ 24

(4 x 1) + (4 x )

Making the Link to Algebra 12 x 12 = (10 + 2) (10 + 2)

(10 x 10) + (2 x 10) + (10 x 2) + (2 x 2) =

10

10 2

2

+

+

(10 x 10)

(2 x 2)

(10 x 2)

(2 x 10)

100 + 20 + 20 + 4

A pictorial model

• Teachers, Principal and Math Science Teacher

• FPS website – Curriculum – Math • Parent Letters• Basic Facts practice• Homework –Teachers differentiate

homework based on student needs

• http://fairfieldpublicschoolsk5math.wikispaces.com/home

 Parent Resources:

 Thank You