also by dr. nicki newton · 2016. 7. 18. · -ii- story problems can be great! story problems are...
TRANSCRIPT
Also by Dr. Nicki Newton
Guided Math in Action: Building Math Proficiency
Problem Solving With Math Models: Grade 1
Problem Solving With Math Models: Grade 2
Problem Solving With Math Models: Grade 3
Problem Solving With Math Models: Grade 4
Problem Solving With Math Models: Grade 5
Gigglenook Publications
P.O. Box 110134
Trumbull CT 06611
Email: [email protected]
Website: www.drnicki123.com
Produced by the Gigglenook Production Team
Copyright © 2012 Gigglenook Publications
All rights reserved. No part of the book may be reproduced in any form,
stored in a retrieval system, by any mechanical, photocopying, recording,
scanning, electronic means, or otherwise under Section 107 or 108 of the
1976 United States Copyright Act, without prior written permission in
writing from the publisher, except by a reviewer, who may quote brief
passages in a review.
Permission is given to individual classroom teachers to reproduce the
pages for classroom instruction use only. Reproduction of these materials
for an entire school or district is strictly forbidden.
For additional copies of this publication or permission to reproduce this
work, please contact Gigglenook.
Chief Operating Officer: Dr. Nicki Newton
Publisher: Gigglenook Publication
Cover Design: This Way Up Productions
Text Design and Composition: Bonnie Harrison-Jones
Printed in the United States of America
ISBN-13: 9781491253526
ISBN-10: 1491253525
Volume 1: September, 2013
Dedicated to Mom and Pops, Always
-ii-
Story problems can be great! Story problems are the stuff life is made of. If
we can make connections for children between their daily lives and the
problems we pose and solve in school, we will have much more success.
We need to provide scaffolds into the process.
The New Math Common Core (2010) places a big emphasis on problem
solving. The first mathematical practice mentioned states that students
should “Make sense of problems and persevere in solving them (CCSSM).
” It goes on to describe this by stating that mathematically proficient
students should be able to explain a problem and find ways to enter into it.
According to the New Math Common Core students should be able to
solve problems with objects, drawings and equations. Let’s take a closer
look at kindergarten. Kindergarten students should work with 4 of the 12
types of problems with numbers within 10.
The new national math standards – known as the Math Common Core,
actually adopted the framework for story problems, created by Carpenter,
Fennema, Franke, Levi & Empson, (1999); Peterson, Fennema & Carpenter
(1989). The research says that the more teachers understand these types of
problems and teach them to their students, the better students understand
the problems and are able to solve them. Furthermore, the research makes
the case that the KEY WORD METHOD should be avoided!
Students should learn to understand the problem types and what they are
actually discussing rather than key word tricks. Key words only work with
really simplistic problems and so as students do more sophisticated work
with word problems, the key words do not serve them well. Using a key
word strategy can actually lead students in the wrong direction, often
encouraging the wrong operation. For example, Susie has 10 marbles. John
has 4 more than she does. How many do they have altogether? Sometimes,
students using a key word method, incorrectly add 10 and 4 instead of
adding 10 and 14. Another example, Carol has 18 marbles. She has 3 times as
many as John does. How many does John have? Sometimes, students using a
-iii-
key word method, incorrectly multiply 3 times 18 instead of dividing 18 by
3.
This book is about giving students a repertoire of tools, models and
strategies to help students think about, understand and solve word
problems. We want to scaffold reasoning opportunities from the concrete
(using objects) to the pictorial (pictures and drawings) and finally to the
abstract (writing equations).
-iv-
I would like to thank many people for their support, expertise, guidance,
and encouragement during this project. First of all I would like to thank
God, without him this would not be possible. Second, I would like to thank
my mom, pa, big mom, and granddaddy. Third, I would like to thank my
family for all their love and support, especially my Tia that calls me every
day and ask me “What have you accomplished today?” I would like to
thank all of my friends that support me all the time. Finally, this book
series would not have been possible without the continual support of The
Gigglenook Book Production Team. Thank you to everyone that has
helped with this series along the way.
Problem Solving with Math Models© 2012 1
Adding to Problems
“Adding to” problems are addition problems that involve action.
There are 3 types. Kindergarteners have to know the first type of problem
according to the CCSS (2010). The first type is adding to problems where the
result is unknown. For example: Jenny had 5 marbles. John gave her 3 more.
How many marbles does Jenny have altogether now? In this problem type the
result is unknown. Teachers tend to tell these types of problems. They are
basic and straightforward.
Taking From Problems
“Taking From” problems are all about subtracting. There are three
types. Kindergarteners have to know the first type of problem according to
the CCSS (2010). The first type is taking from problems where the result is
unknown. For example: Jenny had 5 marbles. She gave John 3. How many
marbles does Jenny have left? In this problem the result is unknown. Teachers
tend to tell these types of problems. They are basic and straightforward.
Put Together/Take Apart Problems
“A Put Together/Take Apart” problem discusses the parts and the
whole. There are three types of Put Together/Take Apart Problems.
Kindergarteners are expected to know 2 types of these problems. The first
is a problem where the Whole is Unknown. For example: Susie has some
marbles. Five are red and five are blue. How many marbles does she have
altogether? We know both parts and the task is to figure out the whole.
Kindergarteners are also expected to know the third type of problem
which is a Part Whole with Both Parts Unknown. For example, John has 10
vehicles. Some are trucks and some are cars. How many of each could he have?
There are many possible combinations to the answer. Students must learn
to think about and solve these types of problems with pictures, diagrams
and tables.
2 Problem Solving with Math Models© 2012
There are several great tools to use for solving number stories. In this
book students will use three different tools to think about the word
problems. They will use Number Frames, Pictures and The Number Line.
Number Frames
Number frames provide a visual scaffold for children to understand
word problems. There are 3 different types of number frames- The Five
Frame, the Ten Frame and the Double Ten Frame. Five is the first anchor
number so we should always start with Five Frames, although few math
programs do. Ten is foundational and many math programs do use this
frame. Finally, the Double Ten Frame helps to increase the range of
number in word problems. Number frames are visual scaffolds that help
students to see what they are doing and think about the numbers in
different ways. Kindergarteners focus on the Five Frame and the Ten
frame.
FIVE FRAME
TEN FRAME
Problem Solving with Math Models© 2012 3
Drawings/Pictures
Drawings are a great strategy. Drawings allow students to think
about the problem, visualize it and then draw it out as a “pathway” instead
of just “jumping straight to a solution (CCSS, 2010).” It is important to
teach children to read the problem, stop, and think about what is
happening in the problem and then to draw what they pictured.
+
The Number Line
Number Lines help to scaffold children’s work. Number lines are
more abstract and so you want to scaffold student thinking from concrete
through pictorial to abstract. When working with the number line students
are working at an abstract level. You can most certainly use them with
facts through 10 for kindergarteners.
1 2 3 4 5 6 7 8 9 10
4 Problem Solving with Math Models© 2012
When you have the students solve problems with the number frames,
actually give them something to put on the frames. When students draw a
picture it has to match the story. Finally, when they are using the number
line, make sure they know where to start and how to jump across the
number line. You also want to show students what the equation to match
the story looks like. In kindergarten, students should be exposed to
expressions and equations but they don’t have to be able to write the
equations.
Problem Solving with Math Models© 2012 5
:
When you introduce the problem, be sure to tell the students what
type of problem it is.
Remember that you can take the same problem and rework it in
different ways throughout the week.
Work on a problem type until the students are proficient at
recognizing and solving that problem type. Also give them
opportunities to write and tell that specific problem type.
Be sure to contextualize the problems in the students’ everyday lives.
Using the problems in the book as models, substitute the students’
names and their everyday things.
Be sure to provide tons of guided practice. Solve problems together
as a class, with partners and in groups. Individual practice should
come after the students have had plenty of opportunities to work
together and comprehend and understand what they are doing.
Emphasize that there is no one correct way to solve a problem but
that there is usually only one correct answer.
Encourage students to always show their work.
6 Problem Solving with Math Models© 2012
hese types of problems are the easiest types of addition problems.
In these problems students are looking for what happened at the
end of the story. We know what we started with and what we added to
that part. We are trying to find out how many we have altogether now.
PROBLEM John had 4 marbles. Henry gave him 3 more.
How many does he have now?
MODEL
EQUATION 4 + 3 = ? 4 + 3 = 7
T
Unit 1 – Add to Result Unknown
Problem Solving with Math Models© 2012 7
Way#2: Draw pictures.
+
Way#3: Solve on the number line.
1 2 3 4 5 6 7 8 9 10
Way#1: Put some counters on your number frame.
Unit 1 – Add to Result Unknown
8 Problem Solving with Math Models© 2012
1. Zack had 1 airplane. His friend Kenny gave him 4
more airplanes. How many airplanes does Zack have now?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
Problem Solving with Math Models© 2012 9
2. Tiffany had 2 dolls. Amber gave her 2 more dolls. How
many dolls does Tiffany have now?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
10 Problem Solving with Math Models© 2012
3. Kristin had 1 chocolate chip cookie. She bought 1
more from Ashley. How many cookies does Kristin have now?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
Problem Solving with Math Models© 2012 11
4. Alex had 2 toy trucks. He got 1 more toy truck for his
birthday. How many toy trucks does he have now?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
12 Problem Solving with Math Models© 2012
5. Andrea has 1 dog. She wants to buy 2 more. How
many dogs will Andrea have then?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
Problem Solving with Math Models© 2012 13
6. Sabrina had 3 dolls. Tara gave her 1 more doll. How
many dolls does Sabrina have now?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
14 Problem Solving with Math Models© 2012
7. Ariel had 3 stickers. Riley gave her 2 more stickers.
How many stickers does Ariel have now?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
Problem Solving with Math Models© 2012 15
8. Kendall finished reading 6 books for Mr. Smith’s class.
She has 4 more books left to read. How many books will Kendall read altogether for Mr. Smith’s class?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
16 Problem Solving with Math Models© 2012
9. Erica had 5 pieces of candy. Timothy gave her 2 pieces
more. How many pieces of candy does Erica have now?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
Problem Solving with Math Models© 2012 17
10. Kevin had 8 toy elephants. He got 2 more elephants for
his birthday. How many toy elephants does Kevin have now?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
18 Problem Solving with Math Models© 2012
11. Tom had 8 marbles. He got 1 more marble for his
birthday. How many marbles does he have now?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
Problem Solving with Math Models© 2012 19
12. Denise did not have any dolls. Her Aunt Sue bought
her 3 dolls. How many dolls does Denise have now?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
20 Problem Solving with Math Models© 2012
13. David has 5 baseball cards. He needs 1 more baseball
card to complete his collection. How many baseball cards will David have once his collection is complete
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
Problem Solving with Math Models© 2012 21
14. Quinn had 4 books. Calvin gave him 4 more books.
How many books does Quinn have now?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
22 Problem Solving with Math Models© 2012
15. Summer had 3 dresses. Her older sister April gave her
4 more dresses. How many dresses does Summer have now?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 1 – Add to Result Unknown
Problem Solving with Math Models© 2012 23
Solve with a model: Use a ten-frame, draw a picture or
make a number line.
1. Sue had 5 apples. Her brother gave her 2 more. How many does she have now?
2. David had 4 toy trucks. He got 6 more for his
birthday. How many does he have now?
Unit 1 – Add to Result Unknown
24 Problem Solving with Math Models© 2012
3. Jamal had 7 marbles. He got 2 more from his friend
John. How many does he have altogether now?
4. Maria had 5 rings. Her mother gave her 2 more. How
many does she have now?
5. Carlos had 5 action figures. He got 5 more for his birthday. How many does he have now?
Problem Solving with Math Models© 2012 25
n these problems students are looking for what happened in the
end of the story. In this type of story we know what happened at
the beginning and also what change occurred. We are trying to find out
how many things remained after some things were taken away.
PROBLEM John had 3 apples. He gave 2 away. How many
does he have left?
MODEL
EQUATION 3 – 2 = ? 3 - 2 = 1
I
Unit 2 – Take From Result Unknown
26 Problem Solving with Math Models© 2012
Way#1: Put counters on the number frame and then
take some off.
Way#2: Draw a picture and cross out things!
Way#3: Make jumps on the number line and then jump BACK.
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
Problem Solving with Math Models© 2012 27
1. Chuck had 4 gummy bears. He gave 2 to Mickey. How
many gummy bears did Chuck have left?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
28 Problem Solving with Math Models© 2012
2. Ethan had 5 pieces of gum. He gave 3 pieces of gum to
Emily. How many pieces of gum did Ethan have left?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
Problem Solving with Math Models© 2012 29
3. Kayla had 5 puppies. She gave 4 away. How many
puppies did Kayla have left?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
30 Problem Solving with Math Models© 2012
4. Denver had 3 blocks. He gave Allison 2 blocks. How
many blocks did Denver have left?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
Problem Solving with Math Models© 2012 31
5. Charlie had 5 brownies. He gave 5 brownies away.
How many brownies did Charlie have left?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
32 Problem Solving with Math Models© 2012
6. Dawn had 5 dolls. She gave Maria 2 dolls. How many
dolls did Dawn have left?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
Problem Solving with Math Models© 2012 33
7. Trevor had 2 slices of pizza. He ate 1 slice. How many
slices of pizza did Trevor have left?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
34 Problem Solving with Math Models© 2012
8. Maria had 9 lollipops. She gave Jean 3 lollipops. How
many lollipops did Maria have left?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
Problem Solving with Math Models© 2012 35
9. Claudette had 10 pencils. She gave Wendy 1 pencil.
How many pencils did Claudette have left?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
36 Problem Solving with Math Models© 2012
10. Anna had 7 peaches. She ate 2 peaches. How many
peaches did Anna have left?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
Problem Solving with Math Models© 2012 37
11. Danny had 8 cupcakes. He gave Kelly 1 cupcake. How
many cupcakes did Danny have left?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
38 Problem Solving with Math Models© 2012
12. Katie had 6 carrot sticks. She gave Aaron 3 carrot
sticks. How many carrot sticks did Katie have left?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
Problem Solving with Math Models© 2012 39
13. Maggie had 10 posters. She gave Carla 2 posters. How
many posters did Maggie have left?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
40 Problem Solving with Math Models© 2012
14. Chris had 10 pieces of bubble gum. He gave Ricky 5 of
them. How many did Chris have left?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
Problem Solving with Math Models© 2012 41
15. Wayne had 7 jellybeans. He ate 1 of them. How many
did Wayne have left?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 2 – Take From Result Unknown
42 Problem Solving with Math Models© 2012
Solve with a model: Use a ten-frame, draw a picture or
make a number line.
1. Sue had 5 apples. She gave 1 away. How many did she have left?
2. Lucas had 7 trucks. He gave 2 to his brother. How many did he have left?
Unit 2 – Take From Result Unknown
Problem Solving with Math Models© 2012 43
3. Maria had 10 rings. She gave 5 to her sister. How
many does she have left?
4. Maya had 8 marbles. She gave 4 to her brother. How many did she have left?
5. Darrel had 5 candies. He gave 3 to his friends. How many did he have left?
44 Problem Solving with Math Models© 2012
hese types of problems are about sets of things. In them we
know both parts and we are looking for the whole. What
distinguishes a Put Together/Take Apart Problem from an Add to Result
Unknown problem is action. In a Put Together/Take Apart Problem there
is no action only a set of something.
PROBLEM John had five red apples and five green ones.
How many apples did he have altogether?
MODEL
EQUATION 5 + 5 = ? 5 + 5 = 10
T
Unit 3 – Put Together/Take Apart - Whole Unknown
Problem Solving with Math Models© 2012 45
Way#1: Use the number frame to show one part and
then the other part.
Way#2: Try drawing pictures to help!
+
Way#3: Put together the 2 numbers on the number line. Start at the first number and then hop to the next.
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
46 Problem Solving with Math Models© 2012
1. Miguel has 3 cats and 1 dog. How many pets does
Miguel have?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
Problem Solving with Math Models© 2012 47
2. Audrey has 2 chocolate cookies and 3 peanut butter
cookies. How many cookies does Audrey have?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
48 Problem Solving with Math Models© 2012
3. Nicole has 2 yellow hats and 2 pink hats. How many
hats does Nicole have?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
Problem Solving with Math Models© 2012 49
4. There are 2 orange butterflies and 1 yellow butterfly.
How many butterflies are there?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
50 Problem Solving with Math Models© 2012
5. Bridget has 4 brown teddy bears and 1 white teddy
bear. How many teddy bears does Bridget have altogether?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
Problem Solving with Math Models© 2012 51
6. Marta has 1 green shirt and 1 pink shirt. How many
shirts does Marta have?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
52 Problem Solving with Math Models© 2012
7. Hong has 2 brothers and 1 sister. How many siblings
does Hong have?
Way#1: Solve on the five frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
Problem Solving with Math Models© 2012 53
8. Maddie 5 brown hamsters and 1 white one. How many
hamsters does Maddie have altogether?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
54 Problem Solving with Math Models© 2012
9. Will has 5 green trucks and 5 blue ones. How many
trucks does Will have altogether?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
Problem Solving with Math Models© 2012 55
10. Parker has 7 pieces of lemon candy and 2 pieces of
cherry candy. How many pieces of candy does Parker have altogether?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
56 Problem Solving with Math Models© 2012
11. Edgar has 3 cupcakes with pink frosting and five
cupcakes with vanilla frosting. How many cupcakes does Edgar have altogether?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
Problem Solving with Math Models© 2012 57
12. There are 8 boys and 2 girls on the playground. How
many children are there altogether?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
58 Problem Solving with Math Models© 2012
13. Ari has 3 orange circles and 3 yellow ones. How many
circles does Ari have altogether?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
Problem Solving with Math Models© 2012 59
14. There are 2 men acrobats and 3 women acrobats at
the circus. How many acrobats are there altogether?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
60 Problem Solving with Math Models© 2012
15. Brian has 5 red marbles and 2 blue ones. How many
marbles does Brian have altogether?
Way#1: Solve on the ten frame
Way#2: Draw pictures
Way#3: Solve on the number line
1 2 3 4 5 6 7 8 9 10
Unit 3 – Put Together/Take Apart - Whole Unknown
Problem Solving with Math Models© 2012 61
Solve with a model: Use a ten-frame, draw a picture or
make a number line.
1. John has 5 orange marbles and 5 purple ones. How many marbles does he have altogether?
2. Elena has 7 green candies and 1 blue one. How many candies does she have altogether?
Unit 3 – Put Together/Take Apart - Whole Unknown
62 Problem Solving with Math Models© 2012
3. Trevor has 3 dinosaur books and 4 space books. How
many books does he have altogether?
4. Grace has 5 green rings and 1 pink one. How many rings did she have altogether?
5. Treshawn had 3 toy trucks and 2 toy cars. How many toy vehicles did he have altogether?
Problem Solving with Math Models© 2012 63
hese types of problems are Put Together/Take Apart problems
with both addends unknown. In these problems we are talking
about sets of something. In this particular type of problem we are talking
about all the ways a set can be put together. For example, We have 4
markers. Some are green and some are yellow. How many could be green and how
many could be yellow, if there are some of each? We could have 3 green and 1
yellow, 2 green and 2 yellow or 1 green and 3 yellow. Students have to find
all the ways to do that problem.
With this type of problem we could model our thinking by using real
objects, pictures, diagrams, or tables. When you have your students use
concrete materials, they can see what is happening. When you have the
students solve it with pictures they learn how to organize information and
draw it out. Then, when they use a table, they are working at an abstract
level. It is important to emphasize that information should be neat and
organized.
T
Unit 4 – Put Together/Take Apart
64 Problem Solving with Math Models© 2012
Green Markers Yellow Markers
Way# 2: Using the Table technique.
MARKER COMBINATION
Green Yellow 3 1
2 2
1 3
0 5
Way#1: Try drawing pictures to help!
Unit 4 – Put Together/Take Apart
Problem Solving with Math Models© 2012 65
1. Betty has 4 hats. Some hats are yellow and some hats
are pink. How many hats could be yellow? How many hats could be pink? What are all the possible
combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
HATS
Yellow Pink
3
2
1
0 5
Unit 4 – Put Together/Take Apart
66 Problem Solving with Math Models© 2012
2. Byron has 5 pets. Some of his pets are dogs and some
of his pets are cats. How many of his pets could be dogs? How many of his pets could be cats? What are
all the possible combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
PETS
Dogs Cats 4
3
2
1
0 5
Unit 4 – Put Together/Take Apart
Problem Solving with Math Models© 2012 67
3. Audrey saw 6 butterflies on a tree. Some of the
butterflies were yellow and some of the butterflies were red. How many butterflies could be yellow? How many
butterflies could be red? What are all the combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
BUTTERFLIES
Yellow Red 5
4
3
2
1
0 5
Unit 4 – Put Together/Take Apart
68 Problem Solving with Math Models© 2012
4. Courtney has 8 teddy bears. Some teddy bears are
brown and some teddy bears are white. How many teddy bears could be brown? How many teddy bears
are could be white? What are all the combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
TEDDY BEARS
Brown White 7
6
5
4
3
2
1
0 5
Unit 4 – Put Together/Take Apart
Problem Solving with Math Models© 2012 69
5. Samantha has 9 shirts. Some of her shirts are green
and some of her shirts are pink. How many shirts could be green? How many shirts could be pink? What
are all the possible combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
SHIRTS
Green Pink 8
0 5
Unit 4 – Put Together/Take Apart
70 Problem Solving with Math Models© 2012
6. Samuel has 5 marbles. Some marbles are red and
some marbles are green. How many marbles could be red? How many marbles could be green? What are all
the possible combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
MARBLES
Red Green 5
0 5
Unit 4 – Put Together/Take Apart
Problem Solving with Math Models© 2012 71
7. There are 7 kids on the playground. Some are girls and
some are boys. How many could be girls on the playground? How many could be boys on the
playground? What are all the possible combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
Unit 4 – Put Together/Take Apart
72 Problem Solving with Math Models© 2012
8. Kimi has 5 pairs of socks. Some are green and some
are blue. How many pairs of socks could be green? How many pairs of socks could be blue? What are all
the possible combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
0 5
Unit 4 – Put Together/Take Apart
Problem Solving with Math Models© 2012 73
9. Carolyn has 9 shapes. Some are squares and some are
circles. How many shapes could be squares? How many shapes could be circles? What are all the
possible combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
0 5
Unit 4 – Put Together/Take Apart
74 Problem Solving with Math Models© 2012
10. Francis has 5 hamsters. Some hamsters are brown
and some hamsters are white. How many hamsters could be brown? How many hamsters could be white?
What are all the possible combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
0 5
Unit 4 – Put Together/Take Apart
Problem Solving with Math Models© 2012 75
11. Parker has 7 toy trucks. Some toy trucks are green
and some toy trucks are blue. How many toy trucks could be green? How many toy trucks could be blue?
What are all the possible combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
0 5
Unit 4 – Put Together/Take Apart
76 Problem Solving with Math Models© 2012
12. Beverly has 4 books. Some books are about bunnies
and some books are about turtles. How many books could be about bunnies? And how many books could
be about turtles? What are all the possible combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
Unit 4 – Put Together/Take Apart
Problem Solving with Math Models© 2012 77
13. Martha has 4 cupcakes. Some cupcakes are vanilla
and some cupcakes are chocolate. How many cupcakes could be vanilla? How many cupcakes could
be chocolate? What are all the possible combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
0 5
Unit 4 – Put Together/Take Apart
78 Problem Solving with Math Models© 2012
14. Mr. Ted has 5 new students in his class. Some of the
students are girls and some of the students are boys. How many of each could there be? What are all the
possible combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
0 5
Unit 4 – Put Together/Take Apart
Problem Solving with Math Models© 2012 79
15. Mia has 10 skirts. Some of the skirts have polka dots
and some of the skirts have bows. How many of each could she have? What are all the possible
combinations, if there are some of each?
Way#1: Draw pictures
Way#2: Solve with a table
0 5
Unit 4 – Put Together/Take Apart
80 Problem Solving with Math Models© 2012
Solve with a model: Draw a picture or make a table.
1. Sue had 5 rings. Some were purple and some were pink. How many of each could she have had?
2. Tom had 7 toy trucks. Some were black and some were green. How many of each could he have had?
Unit 4 – Put Together/Take Apart
Problem Solving with Math Models© 2012 81
3. Raul had 4 marbles. Some were blue and some were
brown. How many of each could he have had?
4. Lucy had 8 cupcakes. Some were vanilla and some were chocolate. How many of each could she have had?
Final Word Problems
82 Problem Solving with Math Models© 2012
NAME:
DATE:
Solve the Problems. Show your thinking. Draw a picture, use a number line or make a table
1. Tom had 5 apples. Sue gave him 2 more. How many does he have altogether now?
2. Lee had 7 trucks. He got 3 more for his birthday. How many does he have altogether now?
Problem Solving with Math Models© 2012 83
3. Raul had 10 pieces of gum. He gave 9 away. How
many does he have left?
4. Lucy had 7 dolls. She gave 3 away. How many does she have left?
5. Marvin had 1 blue marble and 1 green one. How many marbles did he have altogether?
Final Word Problems
84 Problem Solving with Math Models© 2012
6. Carlos has 2 blue balls and 6 purple ones. How many does he have altogether?
7. Lynn saw 5 butterflies. Some were pink and some were orange. How many of each could she have seen?
What are all the possible combinations?
8. Marcus had 6 marbles. Some were blue and some were yellow. How many of each could he have had?
What are all the possible combinations?
Problem Solving with Math Models© 2012 85
9. Larry had 5 books about tigers and 4 books about
lions. How many books did he have altogether?
10. Maria brought 3 lemon cupcakes and 3 strawberry
ones to the school party. How many cupcakes did she bring altogether to the party?
86 Problem Solving with Math Models© 2012
Answer Key
Add to Result Unknown Practice
Problems
1. 5 airplanes
2. 4 dolls
3. 2 chocolate chip cookies
4. 3 toy trucks
5. 3 dogs
6. 4 dollhouses
7. 5 stickers
8. 10 books
9. 7 pieces of candy
10. 10 toy elephants
11. 9 marbles
12. 3 dolls
13. 6 baseball cards
14. 8 books
15. 7 dresses
Unit 1 Test
1. 7 apples
2. 10 toy trucks
3. 9 marbles
4. 7 rings
5. 10 action figures
Take From Result Unknown
Practice Problems
1. 2 gummy bears
2. 2 pieces of gum
3. 1 puppy
4. 1 block
5. 0 brownies
6. 3 dolls
7. 1 slice
8. 6 lollipops
9. 9 pencils
10. 5 peaches
11. 7 cupcakes
12. 3 carrot sticks
13. 8 posters
14. 5 pieces of gum
15. 6 jelly beans
Unit 2 Test
1. 4 apples
2. 5 trucks
3. 5 rings
4. 4 marbles
5. 2 candies
Problem Solving with Math Models© 2012 87
Put Together/Take Apart—Whole
Unknown Practice Problems
1. 4 pets
2. 5 cookies
3. 4 hats
4. 3 butterflies
5. 5 teddy bears
6. 2 shirts
7. 3 siblings
8. 6 hamsters
9. 10 trucks
10. 9 pieces of candy
11. 8 cupcakes
12. 10 children
13. 6 circles
14. 5 acrobats
15. 7 marbles
Unit 3 Test
1. 10 marbles
2. 8 candies
3. 7 books
4. 6 rings
5. 5 toy vehicles
Put Together/Take Apart—Both
Addends Unknown Practice Problems
1. Hats
Yellow Pink
3 1
2 2
1 3
2. Pets
Dogs Cats
4 1
3 2
2 3
1 4
3. Butterflies
Yellow Red
5 1
4 2
3 3
2 4
1 5
4. Teddy Bears
Brown White
7 1
6 2
5 3
4 4
3 5
2 6
1 7
88 Problem Solving with Math Models© 2012
5. Shirts
Green Pink
8 1
7 2
6 3
5 4
4 5
3 6
2 7
1 8
6. Marbles
Red Green
4 1
3 2
2 3
1 4
7. Kids
Boys Girls
6 1
5 2
4 3
3 4
2 5
1 6
8. Socks
Green Blue
4 1
3 2
2 3
1 4
9. Shapes
Circles Squares
8 1
7 2
9. Shapes
6 3
5 4
4 5
3 6
2 7
1 8
10. Hamsters
Brown White
4 1
3 2
2 3
1 4
11. Trucks
Green Blue
6 1
5 2
4 3
3 4
2 5
1 6
12. Books
Bunnies Turtles
3 1
2 2
1 3
13. Cupcakes
Vanilla Chocolate
3 1
2 2
1 3
Problem Solving with Math Models© 2012 89
14. Students
Girls Boys
4 1
3 2
2 3
1 4
15. Skirts
Polka
Dots
Bows
9 1
8 2
7 3
6 4
5 5
4 6
3 7
2 8
1 9
Unit 4 Test
1. Rings
Purple Pink
4 1
3 2
2 3
1 4
2. Trucks
Black Green
6 1
5 2
4 3
3 4
2 5
1 6
3. Marbles
Blue Brown
3 1
2 2
1 3
4. Cupcakes
Vanilla Chocolate
7 1
6 2
5 3
4 4
3 5
2 6
1 7
90 Problem Solving with Math Models© 2012
1. 7 apples
2. 10 toy trucks
3. 1 piece of gum
4. 4 dolls
5. 2 marbles
6. 8 balls
7. Butterflies
Pink Orange
4 1
3 2
2 3
1 4
8. Marbles
Blue Yellow
5 1
4 2
3 3
2 4
1 5
9. 9 books
10. 6 cupcakes
Problem Solving with Math Models© 2012 91
REFERENCES
Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999).
Children’s Mathematics: Cognitively Guided Instruction. Portsmouth,
NH: Heinemann.
Common Core Standards Writing Team (Bill McCullum, lead author).
(2011, May 29). Progressions for the common core state standards in
mathematics: K, Counting and cardinality; K-5, operations and algebraic
thinking (draft). Retrieved from:
www.commoncoretools.wordpress.com.
Peterson, P. L., Carpenter, T. P., & Loef, M. (1989). Teachers' Pedagogical
Content Beliefs in Mathematics. Cognition and Instruction, Vol. 6, No. 1,
pp. 1-40.