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August 7, 2013 Elementary Math Summit
August 7, 2013
Elementary Math Summit
Beth Edmonds
Math Intervention Teacher
Wake County Public School System
Participate actively.
Have an open-mind.
To become more familiar with Common Core State Standards.
To explore computational strategies.
To investigate resources and tools.
To help students achieve computational fluency.
Solve these math problems:
28 + 32 =
200 – 86 =
15 x 7 =
90 ÷ 6 =
Kindergarten: Add and subtract within 10, demonstrating fluency for addition and subtraction within 5.
1st Grade: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.
2nd Grade: Fluently add and subtract within 100, using place value and the properties of operations.
3rd Grade: Fluently add and subtract within 1,000; Fluently multiply and divide within 100.
4th Grade: Fluently add and subtract multi-digit whole numbers using the standard algorithm.
5th Grade: Fluently multiply multi-digit whole numbers using the standard algorithm.
Fluency includes three ideas: Efficiency means that the student can carry
out the strategy easily. Accuracy depends on precise recording,
knowledge of number relationships, and checking results.
Flexibility requires the knowledge of more than one approach to solve the problem and to check the results.
“Developing Computational Fluency with Whole Numbers in the Elementary Grades” – Susan Jo Russell
An understanding of the meaning of the operations and their relationships to each other.
The knowledge of a large “repertoire” of number relationships.
An understanding of the base ten system and the place value system.
“Developing Computational Fluency with Whole Numbers in the
Elementary Grades” – Susan Jo Russell
How many dots did you see?
How did you see that number?
How many dots did you see?
How did you see that number?
How many dots did you see?
How did you see that number?
“The most common and perhaps most important model” for number relationships is the ten-frame.
“Students must reflect on the two rows of five, the spaces remaining, and how a particular number is more or less than 5 and how far away from 10.”
(Van de Walle, 2006)
http://illuminations.nctm.org/activitydetail.aspx?id=75
“Every child’s journey in math begins with counting. But it is the next step - moving from counting to adding - that is critical. Unfortunately, many kids never make the transition from counting numbers one at a time to thinking abstractly and efficiently in groups. No wonder they find math difficult!” (Greg Tang)
Ask: Where are groups of counters that could help us count more efficiently?
Counting On is a difficult skill for many children.
A child will begin with the first number and count on or begin with the larger number and count on.
Once mastered some children do not progress beyond the Counting On strategy.
http://www.brainpopjr.com/math/additionandsubtraction/makingten/
“Knowing how to make ten is a fundamental building block for more complex mental math in our base-ten system.”
Games are an important way to practice making 10.
Make ten by subtracting the required amount from
one addend and adding it on to the other addend.
Example: 9 + 4 can become . . . .
Make ten by subtracting the required amount from one addend and adding it on to the other addend. Example: 9 + 4 can become 10 + 3.
Commutative Property of Addition:
If 8 + 3 = 11, then 3 + 8 = 11.
Associative Property of Addition:
To add 2 + 6 + 4, the second two numbers can be added to make a ten.
So, 2 + 6 + 4 = 2 + (6 + 4) = 2 + 10 = 12.
Common Core State Standards, Grade 1
57+34= Adding By Place: 50+30=80, 7+4=11, 80+11=91 Keeping One Number Whole and Adding On the
Other Number in Parts: 57+30=87, 87+4=91 Compensation Involves Adjusting Numbers to
Make Them Easier to Work With: (57+3)+34 = 60+34 = 94 94-3 = 91 (57+3)+(34-3) = 60+31 = 91
Solve: 48 + 25
48 + 25 40 + 20 = 60
8 + 5 = 13 60 + 13 = 73 Did you use important math knowledge
about: ◦ Place value (how to break two-digit numbers into
tens and ones)? ◦ Addition (how to take the addends apart and put
the parts back together in any order)?
Relearning to Teach Arithmetic: Addition and Subtraction (TERC)
48 68 73
48 + 25 =
+20 +5
Larger Numbers: 652 + 131?
Fractions: ¾ + ½ ?
Decimals: 1.2 + 0.9 ?
“Subtraction facts prove to be more difficult than addition. (Van de Walle, 2006)
How can we apply what we know about addition to subtraction?
◦ Place Value
◦ Properties of Operations
◦ The Relationship Between Addition and Subtraction
Join or Separate on a Mat
Join or Separate Bars of Cubes
Hops on a Number Line
0 1 2 3 4 5 6 7 8 9 10
Connecting Subtraction to Addition Knowledge
Start with 9 counters. Place them under a cover. Remove 6 from under the cover so they are visible. Ask: How many are covered?
(Van de Walle, 2006)
What goes with the part I see to make the whole? Six and what makes 9? What goes with 6 to make 9?
Children use known addition facts to determine the unknown quantity or part.
It is essential for addition facts to be mastered first if the think-addition strategy is to be used effectively.
(Van de Walle, 2006)
72 – 49 = Think: 49 + ____ = 72
49 + 1 = 50 (1)
50 + 20 = 70 (20)
70 + 2 = 72 (2) 1+20+2 = 23
49 50 70 72
+1 +20
+2
72 – 49 = 72 - (40+9) 72 – 40 = 32 72 – 9 = 23 72 – 40 = 32 32 – 10 = 22 (Made an easier problem – Subtracted 1 too many) 22 + 1 = 23 (Added 1 back to the answer)
(Partners for Mathematics Learning, Elementary K-2 Number Module 2008-2009)
Subtraction determines the space between two numbers. If both numbers are adjusted equally, then the answer to the problem stays the same.
Example: 81 +3 84
- 57 +3 - 60
24
92 - 35 Consider: “You can’t take a big number from
a little number.” This is not true because it will produce a negative number, but it can be done.
What should we say instead? “If we have only 2 ones, then we need to get access to more ones. We need to ungroup one of the tens to get access to more ones.”
(“Supporting the Struggling Student in Math: Emphasize the Mathematics, not the Tricks” Valerie Faulkner, 2008)
Designed by the teacher to help students develop an understanding of number relationships while working on computation strategies
Each number sentence is shown one at the time.
The students discuss how to solve each one.
The teacher encourages students to find a way to use the previous number sentence to solve the next one.
The students will be composing and decomposing numbers flexibly as they work.
Provide opportunities for students to use number lines and hundred boards in a meaningful way.
(Partners for Mathematics Learning, Grade 2, Module 6)
16 + 10 = ______
16 + 10 = ______
16 + 9 = ______
16 + 10 = ______
16 + 9 = ______
16 + 19 = ______
16 + 10 = ______
16 + 9 = ______
16 + 19 = ______
16 + 29 = ______
To explore the use of adding 10 and the understanding that adding 9 is like adding 10 and then subtracting 1.
“Because figuring in our heads is such an important life skill, it should have a regular role in your classroom math teaching.”
Choose problems that can be solved mentally or have students find an estimate rather than the exact answer.
Have students explain their ideas and listen to their classmates.
(“Marilyn Burns: Mental Math”, Instuctor Magazine, 2007)
26 + 57 =
If you have 65, how much more is needed to make 100?
How could you rename 578 as the sum of two smaller numbers?
Which answer is closest to the product?
148 x 21 (1,000; 2,000; 3,000; 4,000) (“Marilyn Burns: Mental Math”, Instuctor Magazine, 2007)
www.gregtangmath.com
Greg Tang’s goal is to revolutionize the way children learn math.
He uses rhymes, books, and games to show that there is much more to computation than memorization.
Children must move from additive understanding to multiplicative understanding (the ability to think about and operate with collections as a unit).
It is important that children see the relationship between addition and multiplication and the relationship between multiplication and division.
“Experiences with making and counting groups, especially in contextual situations, are extremely useful.” (Van de Walle, 2006)
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The Commutative Property:
The array is a powerful illustration of the order property. Turnaround facts should always be learned together. (Ex. 2 x 8 = 8 x 2)
The Zero Property:
Make up story problems to go with these problems.
(Ex. 5 x 0 = 5 hops of zero on the number line)
(Van de Walle, 2006)
The Identity Property:
“One is simple as can be, it’s known as the identity. The answer to identify? It’s the one to multiply!”
(Greg Tang, The Best of Times)
The Distributive Property:
It is very useful to relate one basic fact to another. The array model can be used to illustrate that a product can be broken up into two parts.
(Van de Walle, 2006)
7 x 5 = (5 x 5) + (2 x 5)
= 25 + 10 = 35
(Van de Walle, 2006)
5 x 5
2 x 5
Seven doesn’t take much time, even though it is prime. Here is all you have to do, first times 5 then add times 2! (Greg Tang, The Best of Times)
4 x 63
4 x 63
63 + 63 + 63 +63 (Repeated Addition)
2 x (2 x 63) (Double Twice)
(4 x 60) + (4 x 3) (Partition 63 into 60 + 3 and use the Distributive Property) (Partners for Mathematics Learning, 3-5 Number & Operations
Module 2008-2009)
How is 4 x 5 related to 4 x 50?
“Avoid saying ‘add a zero’ because when you actually add a zero, you are not changing a number’s value - Here you are not ‘adding’ a zero, you are multiplying by ten!”
(“Supporting the Struggling Student in Math: Emphasize the Mathematics, not the Tricks” Valerie Faulkner, 2008)
The biggest problems created by teaching the steps of “standard” algorithms are:
o The focus is on individual digits, not on whole quantities and relationships among the quantities.
o The central goal is mastering a series of steps rather than understanding the meaning and use of the mathematical operation.
Relearning to Teach Arithmetic: Addition and Subtraction (TERC)
Teachers should provide drill in the use and selection of computational strategies after they have been developed.
Practice: Problem-based activities in which students develop flexible and useful strategies.
Drill: Repetitive non-problem-based activities. Students have a strategy they understand and know how to use. Drill helps to make the strategy more automatic.
(Partners for Mathematics Learning, Elementary K-2 Number Module 2008-2009)
Different Strategies
Mental Math
Number Talks
Games
Computation Probes (Intervention Central)
