using the relationship between addition and subtraction to build fluency

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012

Using the Relationship Between Addition and Subtraction to Build

Fluency

Common Core Leadership in Mathematics (CCLM)Thursday June 28, 2012

This material was developed for use by participants in the Common Core Leadership in Mathematics (CCLM^2) project through the University of Wisconsin-Milwaukee. Use by school district personnel to support learning of its teachers and staff is permitted provided appropriate acknowledgement of its source. Use by others is prohibited except by prior written permission.


Our Motto

“Learning is a messy business, and constructing understanding

is hard work.”

-Fosnot & Dolk, Young Mathematicians At Work p.38.


Learning Intention and Success Criteria

We are learning to…• Recognize the role unknown addend situations and equations

play in developing fluency with single-digit problems.• Understand “decompose a number leading to a ten” and

“think addition to subtract” and explore how to build that understanding in students.

We will be successful when we...• Understand how to use addition and subtraction word

problems to support students’ ability to reason fluently.


Fluency


“Fluency” in the CCSSM

Revisit the discussion of fluencypp.18-19 OA Progression document (Starts with the last paragraph.)

Table groups: Discuss and chart a group response to your designated prompt.


Fluency

• Summarize what it means to be fluent and what kinds of thinking fluency involves for single digit addition and subtraction.

• What are the CCSSM expectations for fluency in your assigned grade and what are some examples of what students will be expected to demonstrate?

• How does this work on fluency link to the Standards for Mathematical Practice?


Decompose a number leading to a ten


Thinking about instruction…By the end of the K-2 grade span, students have sufficient experience with addition and subtraction to know single-digit sums from memory . . . this is not a matter of instilling facts divorced from their meanings, but rather as an outcome of a multi-year process that heavily involves the interplay of practice and reasoning.

p. 19 OA Progressions document

How might this statement impact classroom instruction around “single-digit addition and

subtraction” at all levels?


Progression toward Fluency – Subtraction

Methods for solving single-digit subtraction

Level 1: Direct Modeling or Taking Away Count, Count, Count

Level 2: Counting onThink addition or find unknown

addend

Level 3: Convert to an Easier Problem Decompose a number leading to a 10


Building Understanding Through Context

How would a student at each level approach this problem?

Note: For Level 3, be sure to examine both “Build up through ten” & “Back down through ten”

There were 13 cookies on the plate. Sam ate 8 of the cookies on the plate. How many cookies are on the plate now?


– Place 15 counters on the double ten frame.

– Completely fill one frame, place 5 on the other frame.

Decompose to ten: 15 – 6


How can you remove 6 counters in parts by decomposing it in a way that gets you to or “leads to a ten”?


Tell a story for 15 – 6.


– Place 15 counters on the double ten frame.

– Completely fill one frame, place 5 on the other frame.




Remove 5 counters to get to ten.

Decompose 6 to remove it in parts.



Remove 5 counters to get to ten.

Remove 1 more.

6 = 5 + 1



Decomposed 6 to remove 5 and then remove 1.

15 – 5 – 1 = 9or

15 – 5 = 1010 – 1 = 9

Write an equation(s).


– Use ten frames and counters to reason through the “Decompose to Ten” strategy.

– Write an equation(s) to show the reasoning. – Share and discuss in your small group.

Try it: 13 – 5 16 – 7

Brainstorm: What other subtraction facts would lend themselves well to this strategy?

Make a list of facts and try them out.


Unknown Addend Problems


Unknown Addend Problems Read K.OA.2, K.OA.4, 1.OA.1, 1.OA.4, 2.OA.1

1. Highlight essential ideas in each.2. Discuss the progression from

Kindergarten to grade 2 in developing the idea of an unknown addend.


Food for thought….

Learning to think of and solve subtractions as unknown addend problems makes subtraction as easy as addition (or even easier), and it emphasizes the relationship between addition and subtraction.

– K-5 OA Progressions document, p. 15


Problem Situations

Scan Table 1 revised“Addition & Subtraction Problem Situations.”

1.Which are the easier problem situations to solve? Why?

2.Which problem situations prompt Level 2 and Level 3 reasoning?


Result Unknown Change Unknown Start Unknown

Add To(join)

Eight frogs sat on the grass. Three frogs jumped out of the pond and joined them on the grass. How many frogs are on the grass now?

8 + 3 = ☐

Eight frogs were sitting on the grass. Some other frogs jumped over there and joined them. Then there were eleven frogs on the grass. How many frogs jumped over to join them?

8 + ☐ = 11

Some frogs were sitting on the grass. Three other frogs jumped over there to join them. Then there were eleven frogs. How many frogs were on the grass before those three joined them?

☐ + 3 = 11

Take From(separate)

Eleven grapes were in a bowl. I ate eight of the grapes. How many grapes are in the bowl now?

11 – 8 = ☐

Eleven grapes were in a bowl. I ate some of the grapes. Then there were three grapes remaining in the bowl. How many grapes did I eat?

11 – ☐ = 3

A bowl held a bunch of grapes. I ate eight grapes. Then there were three grapes remaining in the bowl. How many grapes were in the bowl before I ate some?

☐ – 8 = 3

Addition and Subtraction Problem Situations

Total Unknown Addend Unknown1 Both Addends Unknown2

Put Together

or Take Apart

The vase held three red flowers and eight yellow flowers. How many flowers were in the vase?

3 + 8 = ☐

Eleven flowers are in the vase. Eight are red flowers and the rest are yellow. How many flowers are yellow?

8 + ☐ = 1111 – 8 = ☐

Grandma has eleven flowers in a vase. Some of the flowers are red and some are yellow. How many flowers of each color might she have?

11 = ☐ + Difference Unknown Bigger Unknown Smaller Unknown

Compare

“How many more?” version:Nakeia has eight apps on her iPhone. and Justin has eleven apps on his. How many more apps does Justin have than Nakeia?

“How many fewer?” version:Nakeia has eight apps on her iPhone. and Justin has eleven apps on his. How many fewer apps does Nakeia have than Justin?

8 + ☐ = 1111 – 8 = ☐

“More” version with “more”:Justin has three more apps on his iPhone than Nakeia has on hers. Nakeia has eight apps. How many apps does Justin have?

“Fewer” version:Nakeia has three fewer apps on her iPhone than Justin has on his iPhone. Nakeia has eight apps. How many apps does Justin have?

8 + 3 = ☐

“More” version:Justin has three more apps on his iPhone than Nakeia has on her iPhone. Justin has eleven apps. How many apps does Nakeia have?

“Fewer” version:Nakeia has 3 fewer apps on her iPhone than Justin has on his iPhone. Justin has eleven apps. How many apps does Nakeia have?

☐ + 3 = 1111 – 3 = ☐


Pathways…Show how these standards are connected and build across grades by drawing lines and arrows.

K.OA: 2, 3 & 41.OA: 1, 3, 4 & 62.OA: 1 & 2

Graphically demonstrate these pathways on chart paper.


Revisit the Practices

• Divide your slate in half.• On the left side, choose a SMP that

links to the work on Unknown Addend Problems.

• On the right side, cite a specific example that illustrates the chosen practice.


Learning Intention and Success Criteria

We are learning to…• Recognize the role unknown addend situations and

equations play in developing fluency with single-digit problems.

• Understand “decompose a number leading to a ten” and “think addition to subtract” and explore how to build that understanding in students.

We will be successful when we...• Understand how to use addition and subtraction word

problems to support students’ ability to reason fluently.

using the relationship between addition and subtraction to build fluency

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