HOVERBOARDING  THROUGH  THE MAGELLANIC  CLOUD

Star Bright Ideas About Solving Math Word Problems

 

Lesson Overview:

In this set of lessons, students will take turns acting as "math coaches" who will assist other students in solving word problems by identifying key words that usually indicate specific mathematical operations.

Instructional Objectives:


Students will:

 

Look for the "key" words.

Students work in groups that devise own lists of key words.

They should strive to create a list similar to the following:


Addition Clues


Sum

How may in all?

Total

Combined

Altogether

 

Multiplication Clues


Product

At the rate of ( speed)

Times

Double

Subtraction Clues


Difference

How many remain?

Take away

Fewer

How much more?

Change

How much less?

Decreased by

How much farther?

 Remains

Division Clues


Quotient

Mean

Equal shares

Percent

How many will each get?

Half

Goes into

Out of

Average

 

Next students create flash cards to review the relationships between key words and operations

Modeling and Think-Alouds are Great Teaching Devices

Sarah works in Washington D.C., at the Library of Congress. The library has over 600 miles of bookshelves. While working at the library she arranged books on 6 shelves. She put 1 book on the top shelf, 3 books on the second shelf, and 5 books on the third shelf. If she continues this pattern, how many books will Sarah put on the 6th shelf?
"I'll reread the last question. I learned that I need to look for a pattern.
I'll make a table and look for a pattern.
I will create two rows, one for the shelves, and the other for the books.
Then I will return to the problem and dig out the information I need.

shelf

1

2

3

4

5

6

book

1

3

5

7

9

11

I see the pattern. There are two more books on each shelf. I’ve completed the table to shelf 6.
Sarah put 11 books on shelf 6!"
 _______________________________________________________________
The shoe, in sandal form, was first worn in 1600 B.C. in Mesopotamia.
To find a person’s shoe size we use the formula:
Shoe Size equals three times the length of the foot in inches, minus twenty-three.
The Problem: find Emily’s shoe size if her foot is 9 inches long.
"Rereading the problem, I discover that Emily’s foot is 9 inches long and that we need to find her shoe size.
In order to translate the English sentence, "shoe size equals three times the length of the foot in inches minus twenty-three" into an equation, I must first label the variables.
Let S= shoe size.
Let L= length of foot in inches.
Plugging in the corresponding values for the variables, and then evaluating the expression, we find:
S = 3 * L – 23
S = 3 * 9 – 23 Multiply first
S = 27 - 23 Subtract next
Answer: S = 4 "
 ___________________________________________________________
The Navajo and Hopi tribes of Native Americans believe they can tell the outside temperature in Fahrenheit degrees by counting the number of times a cricket chirps in one minute. They then divide this number by four and add thirty-seven. If a cricket chirps one-hundred times in a minute, what is the outside temperature?
"Rereading the last question, we are told the cricket chirps 100 times each minute and that we need to find the outside temperature.
Let’s start by labeling variables.
T = temperature
C = cricket chirps
Let’s translate the English sentences into an Algebraic equation.
T = C/4 + 37
Let’s plug in numbers for variables and solve.
T = 100/4 + 37 Divide first
T = 25 + 37 Add next


Answer: T = 62 degrees Fahrenheit."

Strategy Street

Introduce these problem solving strategies

1. Understand the Problem

Read the problem entirely.

Ask, “Did I understand all the words used in stating the problem?”
Identify what is required.
Reread and restate the problem in your own words.
Draw models or pictures.
Define what answers you need (by reading the final question) and what units you need.

2. Devise a plan to solve the problem.

Work in an organized manner.
Brainstorm key words that usually indicate specific mathematical operations.
Underline important words and numbers.
Cross out irrelevant information.
Ask yourself, “Have I seen a problem like this before?”

3. Implement the plan.

Use pictures or models to help claculation.
Estimate the answer.
Carefully calculate the numbers

4. Reflect on the process


Check calculations
Ask yourself, “Does the answer make sense?”

Solving Word Problems through Translation 

Begin with simple problems to demonstrate techniques.

Ask students to study a model word problem and think about its elements.  Then work in small groups to solve the problem using the Strategy Street and Operational Clue sheets.

“Vondra owns a hoe store in Los Angeles.  Whenever she order a new style of shoe, she orders four pairs of that style in each of the most popular shoe sizes. The most popular sizes are 6,7,8,9, and 10. She also orders 2 pairs in size 5 and size 11. For every new style how many shoes does she need to order in total?


Creating their own word problems

Explain to students that the word problems they will write should include:
A start time and an end time (a time when they wake up in the morning and a time when they leave for school.)
A particular number of minutes it takes to complete each chore (brushing teeth takes four minutes).
A question at the end (Can I make my bus in time? Or What time do I leave the house?)
An answer to the problem (hidden on the other side of the paper).

 

 

Have students work in groups to write their own word problem. Explain that each group should pick a writer, artist, speaker, and presenter. The writer will write the word problem, complete with math using addition and subtraction. The artist will create a picture that represents the word problem. The picture can be of one sentence or the entire word problem. The speaker will ask questions within the group and keep the group on task while they write and solve the problem. The presenter will read the word problem to the rest of the class and announce the group's answer to the problem. All students will be responsible for creating the word problem, from the beginning to the end (including the correct answer). Inform students that when they work together, they need to listen carefully to one another and speak clearly.
Go around the classroom and guide the groups. Make sure that all students keep to their individual roles. After a few minutes, bring the class together and call on each group to present their problems and answers.

 

During the student’s recitation, the teacher should be asking:


1. How did you solve the problem?
2. Why do you think it is correct?
3. Has anyone else used a different strategy for arriving at the answer?

Research-based Assumptions

Value of Group Work
“Students observe other students’ work, and critique, evaluate, explain, and suggest ways for improvement” (Yun Tansil, 2004). Also, “Collaborative learning affords enormous advantages not available from more traditional instruction because group–whether it be the whole class or a learning group within the group–can accomplish meaningful learning and solving problems better that any individual can alone” 

Teach Key Vocabulary
Children who learn commonly used vocabulary in math have an advantage when approaching word problems. The words "total, altogether, combined and additional" indicate the need for addition. "Difference, fewer, decreased and remains" mean that subtraction of numbers needs to occur. More difficult for children are phrases such as "at this rate" which shows that multiplication will need to be used, and "ratio of" which is a division clue. Having a child keep a list of key vocabulary for math problems can empower him or her in taking steps to really investigate what the problem is asking for and what mathematical operation will need to be engaged.


Make Sure the Child Understands the Problem
Particularly in multi-step mathematical situations, reading comprehension can hinder the child's skill in solving a word problem. Having him or her read the question aloud and talk about what process is needed to find the solution can be helpful, but if the child's reading ability is not on par with the reading level of the material, he or she will be stuck before anything else can happen. Make sure the child can correctly read the material. Help him clarify unfamiliar words and make sure he can comprehend what is needed to understand how to solve the problem. If necessary, rephrase the problem with similar but simpler language.


Break It Down
Some word problems contain irrelevant numbers or details. There may also be multiple steps involved in arriving at a solution. Children who can separate out the important information are at an advantage over those who get overwhelmed and give up quickly. Teach the child to read each sentence of the problem to see where the critical information is and help her determine what words and numbers might be unnecessary to arriving at a solution. Additionally, teach children to break apart the problem to see if multiple steps are required, such as adding amounts then multiplying by another number. Assessing whether or not the final result is reasonable is another important tool children should be taught. Estimating before actually proceeding with the mathematics is a good way to check after working out an answer.


Use a Diagram
Children learn through many modes, and visual information can be a powerful tool to assist with math. Drawing diagrams, especially with part-to-whole comparisons, can be of tremendous help when lots of words can confuse a child. A problem such as "There are 22 kittens in a pet shop and seven puppies. How many more kittens are there than puppies?" can be more easily approached with by drawing comparison bars or aligned dots to indicate pets. If children are asked how many cars would be needed to drive 14 kids to a birthday party when five children can fit in each car, a solution can be visualized by drawing simple circles within larger circles for a visual representation of the problem and the resulting solution.