Making Connections Through the Grades in Mathematics

1. 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.

2. 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.

3. 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.

4. 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?

5. Let’s try another one!

6. 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?

7. How did you see it? Did you count?

8. Are we having fun yet? One more time!

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

11. 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.

12. 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.

13. 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.

14. Building NumbersUsing the Five Structure -seeing three in relation to five

15. Building Numbers Using the Ten-Frame? You will see dots in the ten-frame for 2 seconds. Can you tell how many there are?

16. (5 + 3) (10 – 2) How did you see it?

17. 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.

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

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

20. Addition is 4 + 1 = 5

21. Subtraction is 5 - 1 = 4

22. 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.

23. 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 10 + 3 = 13 135 + 8 135 + 18 The associative property of addition What strategy did you use?

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

25. 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 ( ) ( )

26. And… • Using 5 structure 6 + 7 = 5 +1 + 5 + 2 • Making 10 3 + 9 + 7 = ( ( ) ) 10 + 3 = 13 10 + 9 = 19

27. 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

28. Addition Procedure with Regrouping Put down the 1 and carry the 1 1 26 + 55 1 group of 10 81 1 7 tens plus 1 ten is 8 tens or 80.

29. 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 + (50 + 5) 81 (70 + 11)

30. In the intermediate grades students continue to build their algebraic reasoning as it applies to multiplication, division and fractions.

31. Solve. 8 x 15 = ?

32. Decompose 15 and use a pictorial model to visualize it. + 10 15 5 8 80 40 = 120

33. MULTIPLICATION 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 80 + 40 = 120

34. 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,…

35. Try that strategy yourself 6 x 29 = ?

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

37. Division How would you divide 174 ÷ 6 = ? Solve it mentally and think about how you approached the problem. Fairfield Public Schools 2011-2012

38. 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 • 54 5 Bring down the 4 to make 54. 6 goes into 54 nine times. Digit Oriented Fairfield Public Schools 2011-2012

39. Or you could record your thinking? 6 goes into 174 twenty times. + 9 20 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

40. A pictorial model + 9 20 174 120 54 6

41. 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

42. 174 ÷ 6 = This division problem was solved the same way, using place value concepts each time. The difference is how it is recorded.

43. 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

44. 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)?

45. 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 0 0 + 8 0 8 9 4

46. 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

47. 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

48. 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.

49. 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

50. Fractions can be decomposed too. pictorial model is composed of .