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. As discussed in previous presentation…. Research suggests 3 second recall is speed and accuracy. . Clearly specified and targeted expectations at each grade level

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! This is fun isn’t it?

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

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. When a child shifts from using numbers to describe something (adjective) to representing a quantity (noun) is a big idea.

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. Visual representations are used to represent the information in problems graphically, so as to organize and make sense of the information before translating it into a mathematical notation. - Gerstan and Gonchar 2011 This is supports visualizing and mentally manipulating numbers.

Building Numbers Using the Five Structure Five frame: Seeing that three is in five and that five is made up of a three and two more. OR that 3 is two less than five. Five becomes a benchmark number and a friendly referent. -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? Provides a structure to visualize numbers.

(10 – 2) (5 + 3) How did you see it?

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. Children who understand the number relationships will free them up to do more complex problems rather than spending all their cognitive energy trying to remember facts.

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

but there is always the quantity “five.” Five is composed of: 1 & 4 2 & 3 3 & 2 4 & 1 5 & 0 but there is always the quantity “five.”

Addition is 4 + 1 = 5 You start with a quantity (4) and you add more to it (1) which give you a new quantity (5).

Subtraction is 5 - 1 = 4 Subtraction is when you start with a quantity (5) take some away (1) and you are left with a new quantity (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. Some children in 1st or 2nd grade might think 5 + 5 + 3 , then 5 + 8 15 + 8 How can this idea help you solve these problems? 25 + 8 ( ) 135 + 8 10 + 3 = 13 We help students to look at the numbers first before deciding on which strategy is most efficient given the problem. Our end-of-first grade standard is to know with accuracy and speed basic addition and subtraction facts through 10. 135 + 18 The associative property of addition What strategy did you use?

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 ( ) ( )

10 And… Using 5 structure 6 + 7 = 5 +1 + 5 + 2 Making 10 3 + 9 + 7 = ( ) ( ) 10 + 3 = 13 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 1 Put down the 1 and carry the 1 26 + 55 1 group of 10 81 1 7 tens plus 1 ten is 8 tens or 80.

Another Way to Record the Same Thinking for Regrouping 26 Or decompose the numbers and start with the tens Partial Sums + 55 11 (20 + 6) + 70 Decomposing 26 and 55 using place value + (50 + 5) 81 (70 + 11)

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. 10 15 + 5 8 80 40 = 120

MULTIPLICATION 80 + 40 = 120 Or, use a numerical representation to illustrate this partial product strategy 8 x 15 + 5) (10 8 Decompose 15 into ten and ones Also known as the distributive property of multiplication over addition (8 x 10) + (8 x 5) 80 40 Decompose 15 into (10 + 5) to make it easier to compute. 80 + 40 = 120

Models to Represent Your Thinking 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,… Although we encourage students to demonstrate their understanding with visual models, it is important to push them toward using numbers to represent their thinking. Important to point out that students will be asked to try each strategy but will use what is most efficient for them. Students will come to the algorithm with a stronger understanding of the various strategies for computational fluency.

Try that strategy yourself 6 x 29 = ?

MULTIPLICATION 120 + 54 = 174 Use the Open Array Model to Illustrate this Partial Product Strategy 6 x 29 + 9 20 Decompose 29 into tens and ones 120 54 6 120 + 54 = 174

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 9 6 goes into 17 twice. 4 Put down the 12 and subtract from 17 to get 5. -12 6 goes into 17 which is actually 17 tens, twice. That results in 12 tens. Subtract 12 tens from 17 tens which leaves you with 5 tens. Bring down the 4. 54. 54 5 Digit Oriented Bring down the 4 to make 54. 6 goes into 54 nine times. Fairfield Public Schools 2011-2012

Or you could record your thinking? 20 + 9 6 goes into 174 twenty times. Put down the 120 and subtract from 174 to get 54. -120 54 Number Oriented 6 goes into 54 nine times. Fairfield Public Schools 2013-2014

A pictorial model 20 + 9 6 120 174 54

Or record your thinking? 6 goes into 174 twenty times. Put down the 120 and subtract from 174 to get 54. -120 20 54 + 9 Number Oriented 6 goes into 54 nine times. 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 2 2.98 x 3 OR, Did you use a different strategy? 4 9 + 8 8 9 4

Did anyone decompose 2.98 and use partial products? $2.98 x 3 can be thought of as … 2 cents less than $3.00 Or ($3.00 - .02) 3 rolls at $3.00 is $9.00 Then 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.

Fractions Comparing unit fractions Which is bigger 1 2 or 1 4 ? A common misconception children have is to think that 4 is bigger than 2 therefore 1 4 is bigger than 1 2 .

Fractions can be decomposed too. pictorial model 1 4 1 4 1 2 1 2 1 4 1 4 1 2 is composed of 2 4 .

4 x 1 1 2 = ? Solve. How did you solve it? The same operations that occur with whole numbers also apply to fractions and decimals. How did you solve it?

Use Concrete/Pictorial Representation 1 1 2

Concrete/Pictorial Representation 1 1 2 1 1 2 A counting strategy can be used. I have 4 ones = 4 and 4 halves which = 2. 1 1 2 1 1 2 Decompose 1 1 2 into 1 and 1 2

4 + 2 = 6 Or 4 x (1 + 1 2 ) = (4 x 1) + (4 x 1 2 ) Algebraic properties that apply to whole numbers also apply to fractions and decimals. What does this look like on paper?

Making the Link to Algebra 12 x 12 = (10 + 2) (10 + 2) (10 x 10) + (2 x 10) + (10 x 2) + (2 x 2) = A pictorial model 100 + 20 + 20 + 4 10 2 + (10 x 10) (2 x 2) (10 x 2) (2 x 10) Do you remember FOIL? This is the work we are preparing our elementary students for. It is a sampling of what algebraic reasoning looks like in the early grades.

Parent Resources: 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

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