Author: Nicole Flores and Angela Han
School/Organization:
S. Weir Mitchell Elementary School
Year: 2010
Seminar: The Art and Craft of Problem Solving
Grade Level: 1-4
Keywords: Critical thinking, Everyday Math, multiplication, subtraction
School Subject(s): Math
Download Unit: Flores-Han_unit.pdf
The purpose of this curriculum unit is to develop students’ critical thinking skills. This unit is designed to further students’ understanding of addition, subtraction, multiplication, and division number stories (word problems) and to provide students with problem solving strategies. It addresses the following strategies: Guess and Check, Find a Pattern, and Draw a Picture. Using these strategies, students explore:
This unit is intended to complement the School District of Philadelphia’s pre-existing curriculum (Everyday Math). The allotted time for this unit is three weeks in the fall. The focus is addition and subtraction word problems. Its audience is third grade students in a low to mid-income urban neighborhood.
Mathematics is the single most important discipline to be taught because of its pervasiveness in our society. It is used every day, in almost every capacity of daily existence. Problem solving is essential because it teaches within a context. It develops the skills that are necessary to find solutions, and it motivates the mind much more than skills that are taught in isolation. Problem solving is a pathway for skills already learned, gives it the real life framework needed by students to persevere through increasingly complex problems. (Taplin, no date) It gives students confidence, a starting point, and the ability to transfer skills when faced with new mathematical situations (Higgins, 1999). The National Council of Teachers of Mathematics recommends a focus on problem solving because it involves skills used in everyday life. It allows for students to create and build upon their own theories about mathematics. It allows for the transference of previously learned material into new situations, which allows for deeper learning and understanding and adaptation to new situations. Problem solving also allows for logical reasoning, vital in today’s society. (Taplin, no date)
The School District of Philadelphia (SDP) uses Everyday Math as its curriculum, and solving the four basic operation number stories is a major theme in the SDP’s Planning and Scheduling Timeline for third grade. These four operations are fundamental skills in mathematics, while number stories apply these skills into everyday life. These concepts show up repeatedly in Everyday Math. Yet, the curriculum fails to provide students with critical thinking skills that are necessary for the students to develop an instrumental understanding of this topic. Thus, teachers need to supplement the curriculum with concrete approaches that will help students develop these necessary skills.
Problem solving involves discovery rather than passive learning. Because of the NCTM’s recent focus, it has resurfaced in standardized testing, giving teachers an increased need to integrate it into math lessons. (Higgins, 1999) The pedagogical focus of this unit centers on Guess and Check, Find a Pattern, and Make a Picture. These strategies will help students master number stories. The lessons in this curriculum unit address the shortcomings in the curriculum and present students with strategies to be used with a variety of word problems that will address the state standards. With these problem solving skills, students will achieve in higher grades and high stakes testing where critical thinking skills are expected and required. Students with problem solving capabilities were found to be better communicators of what they did, had a higher understanding of the usefulness of math and had higher levels of mathematical achievement than their peers who did not possess the same skills (Higgins, 1999). This will greatly enrich students’ learning experiences because they will have more tools at their disposal to enhance their understanding of the four basic operations. They can then apply this understanding into everyday life through number stories. They will develop a “relational understanding” instead of “instrumental understanding” of this topic. In the end, the hope is the students will then use this understanding in other aspects of their lives, their education, and their future.
The three strategies that are the most conducive to a third grade classroom are: Guess and Check, Find a Pattern, and Make a Picture. These strategies will be a springboard for the critical thinking that is the emphasis of this unit. They can be explained as follows:
The focus will be on addition/subtraction number stories in the fall. The three strategies will be taught within a series of word problems. The word problems will be aligned with the student’s depth of knowledge at that time of the year. They will be limited to single and double-digit addition, and subtraction. These lessons will build from the students’ previous learning. Once each strategy has been mastered, the next will be taught. When they all have been mastered, students will show their understanding by being able to choose the best strategy (for them) to use to solve the problem and explain why and how that strategy assisted them.
The goal of these lessons is to prepare the students for more advanced mathematic concepts such as multiplication and division.
E.g. 1: Jack has $10. He wants to buy 5 notebooks. Each notebook cost $1.75. Does Jack have enough money to?
Guess #1: The notebook is about $2. $2 x 5 = $10, so Jack has enough.
Check #1: $1.75 x 5 = $8.75
E.g. 2: Extensions of Multiplication and Division Facts
6 x 7 = 42 12 / 4 = 2
6 x 70 = 420 120 / 4 = 20
6 x 700 = 4200 1200 / 4 = 200
E.g. 3: What’s My Rule?
E.g. 4: Mike bought 9 packets of candy. There were 8 pieces of candy in each pack. How many pieces of candy did Mike buy?
8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8= 8 x 9 = 72
E.g. 5:Janice is planting carrots in her garden. She has 132 carrots seeds and she wants to plant 12 carrots in a row. How many rows of carrots can Janice plant?
After a brief introduction of Guess and Check, Make a Picture, and Find a Pattern and their usefulness in solving mathematical word problems, this unit will focus on each of the three strategies in depth.
Guess and Check is the first approach taught. The students are most familiar with this strategy. The emphasis will be on using logic to adjust their guess. Students often do not pay attention to the outcome of their guess, continue in the wrong direction, get frustrated and then give up.
Eg. #1: Find Differences
Which is more, 154 or 131? How much more?
Guess #1: 154 is more.
Check #1: 154 is after 131 on the number grid.
Guess #2: 154 is 5 more than 131.
Check #2: 131 + 5 = 136. 136 is a lot less than 154, so the guess needs to be a lot bigger than 5.
Guess #3: 154 is 25 more than 131.
Check #3: 131 + 25 = 156. 156 is now bigger than 154, so the guess needs to be smaller than 25. Also, 156 is two bigger than 154.
Guess #3: 154 is 23 more than 131.
Check #3: 131 + 23 = 154.
E.g. #2: Solving Problems with Dollars and Cents
Amy has $2. She wants to buy a notebook that costs $0.99 and a box of pencils that costs $1.49. Does Amy have enough money to buy the notebook and the box of pencils?
Guess #1: The notebook is about $1 and the box of pencils is about $1.50. $1 + $1.50 = $2.50, so Amy does not have enough.
Check #1: $0.99 + $1.4 = $2.48
Find the Pattern is the last approach taught. Problems may involve any of the four processes and will eventually extend beyond numbers to shapes and other objects. This skill lays the groundwork for many more complicated math problems that our students will face. Mastering this skill will assist them the most in their future schooling.
E.g. #1: Number grids
541 | 543 | 26 | 27 | ||
552 | 36 | ||||
561 | 563 | 46 |
E.g. #2: Frames and Arrows
E.g. #3: Extensions of Addition and Subtraction Facts
12 – 5 = 7 11 – 7 = 4
120 – 50 = 70 21 – 7 = 14
1,200 – 500 = 700 81 – 7 = 74
E.g. #4: What’s my rule?
Draw a Picture is the second strategy taught. During the fall, this strategy solidifies students’ understanding of addition and subtraction. In the spring, students use this strategy to gain a conceptual understanding of multiplication and division. They can actually see how it is different from addition and subtraction too.
E.g. #1: Parts-and-Total
Marissa read her book for 25 minutes on Monday and 30 minutes on Tuesday. How many minutes in all did she read?
Total | |
? | |
Part | Part |
25 | 30 |
Monday Tuesday
Number Sentence: 25 + 30 = 55
Answer: 55 minutes
Check: 55 minutes is greater than both parts.
E.g #2: Change Number Stories
Marcus had $25 in his wallet. He spent $16 at the store. How much money was in Marcus’s wallet then?
Start | End | |
25 | -16 | ? |
Number Sentence: 25 – 16 = 9
Answer: $9
Check: 16 + 9 = 25
E.g. #3: Comparison Number Stories
Jenna has $42. Her brother has $13. How much more does Jenna have?
Quantity | |
42 | |
Quantity | |
13 | ? |
Difference |
Number Sentence: 42 – 13 = 29
Answer: $29
Check: 13 + 29 = 42
Each lesson, in order to ensure student engagement, will consist of direct teacher instruction as well as partner and independent activities.
30 + 10 = 40
40 + 10 = 50
50 + 8 = 58
58 – 1 = 57
$1.49 + $0.49 = $1.98
Number | Changes To… | How? | Continue the Pattern |
17 | 24 | ||
Number | Changes To… | How? | Continue the Pattern |
30 | 60 | ||
100 | 150 | ||
200 | 125 | ||
135 | 16 |
Number | Changes To… | How? | Continue the Pattern |
98 | 71 | ||
64 | 83 | ||
162 | 147 | ||
186 | 208 | ||
302 | 275 |
Number | Changes To… | How? | Continue the Pattern |
80 | 73 | ||
100 | 123 | ||
167 | 140 |
Total | |
? | |
Part | Part |
8 | 5 |
8 + 5 = 13
Total | |
? | |
Part | Part |
16 | 23 |
16 + 23 = 59
= Hatched Eggs
Total | |
30 | |
Part | Part |
14 | ? |
30 – 14 = 16
Total
? |
|||
Part
13 |
Part
9 |
Part
7 |
Part
11 |
13 + 9 + 7 + 11 = 40
= child #1
Total = = child #2
Total
19 |
||
Part
6 |
Part
9 |
Part
? |
19 – 6 – 9 = 4
Bell, Max, Jean Bell, John Bretzlauf, Mary Ellen Dairyko, Amy Dillard, Robert Hartfield, Andy Isaacs, James McBride, Kathleen Pitvorec, and Peter Saecker. Everyday Mathematics: The University of Chicago School Mathematics Project (Teacher’s Lesson Guide, Vol.1, Grade 3). Wright Group/McGraw-Hill, 2007.
Higgins, Karen. “The Effect of Year-Long Instruction in Mathematical Problem Solving on Middle School Student’s Attitudes and Beliefs .” Journal of Experimental Education. Heldref Publications, 1999.
Healy, Lulu, Hoyles. “Visual and Symbolic Reasons in Math: Making Connections with Computers.” Thinking and Learning, 1999.
Pennsylvania Department of Education. “Academic Standards for Mathematics.” 22 Pennsylvania Code, Chapter 4, Web. 30 Mar. 2010.
Name: Date:
Grade: / Room:
Parts-and-Total Worksheet #1
Directions: Solve each problem. Remember to use the strategies we talked about in class.
Strategy Reminder:
Problem #1
There were 4 boys and 5 girls absent from Mitchell Elementary today. How many students were absent in all?
Total | |
Part | Part |
Picture |
Number Sentence: Answer:
Check:
Problem #2
Jane has 9 apples and 2 bananas. How many pieces of fruit does Jane have?
Total | |
Part | Part |
Picture |
Number Sentence: Answer:
Check: