[PPT] - Addi,on & Subtrac,on Building a Founda,on for Later Grades PowerPoint Presentation

SLIDE 1

Building a Founda,on for Later Grades

Delise Andrews

Math Coordinator, Grades 3-5 Lincoln Public Schools dandrews@lps.org

Addi,on & Subtrac,on

SLIDE 2

Addi,on

On poster paper, together with several people

at your table…

Write a math story that could be solved by doing the addi:on problem: 8 + 7 = 15

Be sure to write with large, bold print so that

everyone in the room can read your problem.

2

SLIDE 3

Subtrac,on

On poster paper, together with several people

at your table…

Write a math story that could be solved by doing the subtrac:on problem: 15 – 8 = 7

Be sure to write with large, bold print so that

everyone in the room can read your problem.

3

SLIDE 4

Addi,on & Subtrac,on

Through numerous research studies, we know

that young students can solve contextualized mathema:cs problems through reasoning and making sense of the rela:onships in the story.

4

A., V. D., Lovin, L. H., Karp, K. S., & Bay-Williams, J. M. (2014). Teaching student-centered mathematics. Upper Saddle River, NJ: Pearson.

SLIDE 5

Problem Sort

Res Result lt U Unkn know

wn

Cha Change Unkno nge Unknown wn St Star art U Unknown nknown Add t Add to Take from Take from Total Unknown Total Unknown Addend U Addend Unknown nknown Bot Both Addends h Addends U Unknown nknown Pu Put Toget Together/ her/ Take Apart Take Apart Difference Unknown Difference Unknown Big Bigger er U Unknown nknown Smaller Smaller U Unknown nknown Co Compa mpare re

5

A. Lucy has two apples. Julie

has five apples. How many fewer apples does Lucy have than Julie? B . L u c y h a s 3 f e w e r a p p l e s t h a n J u l i e . J u l i e h a s f i v e a p p l e s . H

w

m a n y a p p l e s d

e

s L u c y h a v e ? C . S

m

e a p p l e s w e r e

n

t h e t a b l e . I a t e t w

a

p p l e s . T h e n t h e r e w e r e t h r e e a p p l e s . H

w

m a n y a p p l e s w e r e

n

t h e t a b l e b e f

r

e ?

E. Two bunnies sat on the
grass. Three more bunnies

hopped there. How many bunnies are on the grass now?

D. Five apples are on the table.

Three are red and the rest are green. How many apples are green?

I. Three red apples and two

green apples are on the

table. How many apples are
n the table?
J. Grandma has five flowers.

How many can she put in her red vase and how many in her blue vase?

G. Five apples were on the
table. I ate some apples.

Then there were three

apples. How many apples did

I eat?

K. Lucy has two apples. Julie

has five apples. How many more apples does Julie have than Lucy?

L. Two bunnies were sitting on

the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped

ver to the first two?
M. Julie has three more apples

than Lucy. Julie has five

apples. How many apples

does Lucy have? N . F i v e a p p l e s w e r e

n

t h e t a b l e . I a t e t w

a

p p l e s . H

w

m a n y a p p l e s a r e

n

t h e t a b l e n

w

?

O. Julie has three more apples

than Lucy. Lucy has two

apples. How many apples

does Julie have?

H. Lucy has 3 fewer apples

than Julie. Lucy has two

apples. How many apples

does Julie have? F . S

m

e b u n n i e s w e r e s i t t i n g

n

t h e g r a s s . T h r e e m

r

e b u n n i e s h

p

p e d t h e r e . T h e n t h e r e w e r e f i v e b u n n i e s . H

w

m a n y b u n n i e s w e r e

n

t h e g r a s s b e f

r

e ?

SLIDE 6

No,ce & Wonder

What did you no:ce about the types of

addi:on and subtrac:on problems?

What do you wonder?

6

SLIDE 7

Classify Table Problems

Determine a classifica:on for the addi:on and

subtrac:on problems you wrote earlier.

Clearly record the classifica:on on your chart

paper.

7

SLIDE 8

No,ce & Wonder

What did you no:ce about the types of

addi:on and subtrac:on problems we wrote?

What do you wonder?

8

SLIDE 9

Wri,ng Equa,ons

A situa:on equa:on represents a literal

transla:on of the math story context.

(represen:ng informa:on as it comes in the story)

A solu:on equa:on represents the

mathema:cs required to find the solu:on to the problem. (the unknown quan:ty is isolated)

9

Fuson, Karen C., Carroll, William M. and Landis, Judith(1996) 'Levels in Conceptualizing and Solving Addition and Subtraction Compare Word Problems', Cognition and Instruction, 14: 3, 345 — 371

SLIDE 10

Wri,ng Equa,ons

Joanna had some cookies. She gave 3 cookies

to Eric. Now she has 5 cookies. How many cookies did Joanna have to begin with?

Situa:on equa:on: ☐ – 3 = 5
Solu:on equa:on: 5 + 3 = ☐

10

SLIDE 11

Wri,ng Equa,ons

Your turn! Write a situa:on equa:on and a

solu:on equa:on for each story card.

– A situa:on equa:on represents a literal transla:on of the math story context. (represent

informa:on as it comes in the story)

– A solu:on equa:on represents the mathema:cs required to find the solu:on to the problem.

(isolate the unknown quan:ty)

11

SLIDE 12

No,ce & Wonder

Look at the equa:ons for all of the different

problem types. What do you no:ce?

What do you wonder?

12

SLIDE 13

Modeling Addi,on & Subtrac,on

Use the place value blocks at your table to

model these problems:

– Sandra had 8 pennies. George gave her 4 more. How many pennies does Sandra have altogether? – Sandra has 8 pennies and 4 nickels. How many coins does she have?

13

A., V. D., Lovin, L. H., Karp, K. S., & Bay-Williams, J. M. (2014). Teaching student-centered mathematics. Upper Saddle River, NJ: Pearson.

SLIDE 14

Modeling Addi,on & Subtrac,on

Compare the way you modeled the problems.

– How were your models similar? – How were they different?

14

SLIDE 15

Modeling Addi,on & Subtrac,on

Use the place value blocks at your table to model

these problems:

– Sandra had 12 pennies. She gave 4 pennies to George. How many pennies does Sandra have now? – George has 12 coins. Four of his coins are nickels, and the rest are pennies. How many pennies does George have? – George has 12 pennies and Sandra has 8 pennies. How many more pennies does George have than Sandra?

15

A., V. D., Lovin, L. H., Karp, K. S., & Bay-Williams, J. M. (2014). Teaching student-centered mathematics. Upper Saddle River, NJ: Pearson.

SLIDE 16

Modeling Addi,on & Subtrac,on

Compare the way you modeled the problems.

– How were your models similar? – How were they different?

16

SLIDE 17

Structure & Problem Difficulty

Consider the following set of problems…

– Maggie had 7 bracelets. She bought 8 more

bracelets. How many bracelets does Maggie have

now? – Maggie had 7 bracelets. She bought some more

bracelets. She now has 15 bracelets. How many

bracelets did she buy? – Maggie had some bracelets. She bought 8 more

bracelets. She now has 15 bracelets. How many

bracelets did Maggie start with?

17

A., V. D., Lovin, L. H., Karp, K. S., & Bay-Williams, J. M. (2014). Teaching student-centered mathematics. Upper Saddle River, NJ: Pearson.

SLIDE 18

SeTng the Stage for Mul,plica,on & Division

Operations and Algebraic Thinking 2.OA Represent and solve problems involving addition and subtraction.

1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

Add and subtract within 20.

2. Fluently add and subtract within 20 using mental strategies.2 By end of

Grade 2, know from memory all sums of two one-digit numbers.

Work with equal groups of objects to gain foundations for multiplication.

3. Determine whether a group of objects (up to 20) has an odd or even

number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

4. Use addition to find the total number of objects arranged in

rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

a. b.

18

COMMON CORE STATE STANDARDS FOR Mathematics

Common core state standards for mathematics. (2010). Washington, D.C.: National Governors Association Center for Best Practices (NGA Center).

SLIDE 19

SeTng the Stage for Mul,plica,on & Division

19

Children engage in coun:ng equal groups and

fair sharing (even with remainders) from an early age. K-2 students are well equipped to reason about such situa:ons given appropriate quan::es.

We can leverage this informal understanding

to set the stage for the more complex mul:plica:on and division situa:ons students will encounter in later grades.

A., V. D., Lovin, L. H., Karp, K. S., & Bay-Williams, J. M. (2014). Teaching student-centered mathematics. Upper Saddle River, NJ: Pearson.

SLIDE 20

Problem Sort

Unkno Unknown P wn Pro roduct uct Gro Group up S Size ize Unkno Unknown wn (“How ma (“How many in ea ny in each group?” Division) ch group?” Division) Number o Number of Gro f Groups Unkno ups Unknown wn (“How ma (“How many groups?” Division) ny groups?” Division) Equa Equal Gro l Groups ups Arrays, Area Arrays, Area Co Compa mpare re

!

20

A. A rubber band is stretched

to be 18 cm long and that is 3 times as long as it was at

first. How long was the

rubber band at first?

B. A blue hat costs $6. A red

hat costs 3 times as much as the blue hat. How much does the red hat cost? C . A r e c t a n g l e h a s a r e a 1 8 s q u a r e c e n t i m e t e r s . I f

n

e s i d e i s 3 c m l

n

g , h

w

l

n

g i s a s i d e n e x t t

i

t ?

D. If 18 plums are shared

equally into 3 bags, then how many plums will be in each bag? E . A r e c t a n g l e h a s a r e a 1 8 s q u a r e c e n t i m e t e r s . I f

n

e s i d e i s 6 c m l

n

g , h

w

l

n

g i s a s i d e n e x t t

i

t ? F . A r e d h a t c

s

t s $ 1 8 a n d a b l u e h a t c

s

t s $ 6 . H

w

m a n y t i m e s a s m u c h d

e

s t h e r e d h a t c

s

t a s t h e b l u e h a t ? G . A r u b b e r b a n d i s 6 c m l

n

g . H

w

l

n

g w i l l t h e r u b b e r b a n d b e w h e n i t i s s t r e t c h e d t

b

e 3 t i m e s a s l

n

g ?

H. If 18 apples are arranged

into equal rows of 6 apples, how many rows will there be?

I. A rubber band was 6 cm

long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first? J . I f 1 8 a p p l e s a r e a r r a n g e d i n t

3

e q u a l r

w

s , h

w

m a n y a p p l e s w i l l b e i n e a c h r

w

? K . T h e r e a r e 3 b a g s w i t h 6 p l u m s i n e a c h b a g . H

w

m a n y p l u m s a r e t h e r e i n a l l ? L . W h a t i s t h e a r e a

f

a 3 c m b y 6 c m r e c t a n g l e ? M . Y

u

h a v e 1 8 i n c h e s

f

s t r i n g , w h i c h y

u

w i l l c u t i n t

3

e q u a l p i e c e s . H

w

l

n

g w i l l e a c h p i e c e

f

s t r i n g b e ? N . I f 1 8 p l u m s a r e t

b

e p a c k e d 6 t

a

b a g , t h e n h

w

m a n y b a g s a r e n e e d e d ? O . A r e d h a t c

s

t s $ 1 8 a n d t h a t i s 3 t i m e s a s m u c h a s a b l u e h a t c

s

t s . H

w

m u c h d

e

s a b l u e h a t c

s

t ?

P. You have 18 inches of string,

which you will cut into pieces that are 6 inches long. How many pieces of string will you have? Q . T h e r e a r e 3 r

w

s

f

a p p l e s w i t h 6 a p p l e s i n e a c h r

w

. H

w

m a n y a p p l e s a r e t h e r e ? R . Y

u

n e e d 3 l e n g t h s

f

s t r i n g , e a c h 6 i n c h e s l

n

g . H

w

m u c h s t r i n g w i l l y

u

n e e d a l t

g

e t h e r ?

SLIDE 21

No,ce & Wonder

What did you no:ce about the types of

mul:plica:on and division problems?

What do you wonder?

21

SLIDE 22

Wri,ng Equa,ons

A situa:on equa:on represents a literal

transla:on of the math story context.

(represen:ng informa:on as it comes in the story)

A solu:on equa:on represents the

mathema:cs required to find the solu:on to the problem. (the unknown quan:ty is isolated)

22

Fuson, Karen C., Carroll, William M. and Landis, Judith(1996) 'Levels in Conceptualizing and Solving Addition and Subtraction Compare Word Problems', Cognition and Instruction, 14: 3, 345 — 371

SLIDE 23

Wri,ng Equa,ons

Joanna, Doug, and Eric each have the same

number of cookies. Altogether, they have 15

cookies. How many cookies does Eric have?
Situa:on equa:on: 3 × ☐ = 15
Solu:on equa:on: 15 ÷ 3 = ☐

23

SLIDE 24

Wri,ng Equa,ons

Your turn! Write a situa:on equa:on and a

solu:on equa:on for each story card.

– A situa:on equa:on represents a literal transla:on of the math story context. (represent

informa:on as it comes in the story)

– A solu:on equa:on represents the mathema:cs required to find the solu:on to the problem.

(isolate the unknown quan:ty)

24

SLIDE 25

No,ce & Wonder

Look at the equa:ons for all of the different

problem types. What do you no:ce?

What do you wonder?

25

SLIDE 26

Rela,ng Addi,on & Mul,plica,on

Which of the problem types for mul:plica:on

and division is most directly connected to addi:on and subtrac:on?

How can facility with different addi:on and

subtrac:on problem types support students’ ability to interact with mul:plica:on and division situa:ons?

26

SLIDE 27

A note about key words…

Suppose a student has been taught to use key
words. What opera:on are they likely to

perform to solve this problem?

Joanna has 8 cookies. She has 5 fewer cookies

than Eric has. How many cookies does Eric have?

27

SLIDE 28

A note about key words…

“What is the fundamental message the kids

get when told to look for the key/cue word? Don’t read the problem. Don’t imagine the situa:on. Ignore that context. Abandon your prior knowledge. … You don’t have to read ; you don’t have to think. Just grab the numbers and compute.”

28

Hyde, A. A. (2006). Comprehending math: Adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

SLIDE 29

Summarize & Reflect

Think about your instruc,on around math story problems.

I want to stop (or change)…
I need to keep...
I plan to start...

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SLIDE 30

Disclaimer

The National Council of Teachers of Mathematics is a public voice

f mathematics education, providing vision, leadership, and

professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all

students. NCTM’s Institutes, an official professional development
ffering of the National Council of Teachers of Mathematics,

supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of

viewpoints. The views expressed or implied in the Institutes,

unless otherwise noted, should not be interpreted as official positions of the Council.

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SLIDE 31

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