From Words to
Symbols:
Zooming in on
Interpreting
Mathematical
Language
About Translating Words to Symbols
Caution: Attaching Key Words to Specific Operations Can Cause Problems
Research has found that while teachers try to help identify key words to support student
thinking, students use key words to abandon reasoning about the problem (Kenney,
Hancewicz, Heuer, Metsisto, Tuttle, 2005; Sowder, 1988).
The reasons why a “key-word approach” can be detrimental to comprehension include:
(1) key words often have multiple meanings;
(2) the list of key words grows and grows over the years making it difficult to remember;
(3) the key words might indicate an operation, but the question may be asking for more
than the response to a computation;
(4) key word search doesn’t help students understand the meaning of the problem;
(5) for problems that involve multiple steps, it doesn’t work and students are then illprepared to read for understanding. Instead, students need opportunities to find
meaning in the reading;
(6) phrases can shift the meaning of words in the phrase. For example how, many, and
how many signify different things in mathematics; and
(7) addition of (or removal of) a single letter changes meaning (e.g. percent of versus
percent off, ten versus tens)
Appropriate (and Expanded) Attention to Key Words
It is important to develop vocabulary that is specific to mathematics. This point may
seem to contradict the previous section, but there is an important distinction. In the
“key word” approach, students memorize that a word (e.g., difference) means to
“subtract” regardless of the content of the sentence; In a precision-of-vocabulary
approach, students become familiar with the common phrases (e.g., more than) and
relationships among words (e.g., if – then) and how that interacts with the rest of the
sentence.
Sullivan (1982) found that students dramatically improved their computation scores
when she spent time explicitly teaching the meaning of the following words:
a, are, can, the, is, who, find, one, ones, ten, tens, and, or,
number, numeral, how, many, how many, what, write, it,
each, which, do, all, same, exercises, here, there, has, and
have.
Help students know what these words mean in the context of the story - a very different
task than asking students to define. Example: Helping students to understand what “it”
is in a story and how to find out what “it” is more broadly, can help students answer the
question, “How much will it cost?”
Translating Words to Symbols:
Steps for Supporting Student Reasoning
(think of this like the four steps to problem solving –
help students turn this into an internal mechanism for
figuring out how to go from words to symbols).
1. Develop Qualitative Understanding
Ask questions that help understand the relationship
among the items in the story (e.g., “Are there more
____ than ____?”)
2. Develop Quantitative Understanding
Ask questions that help move towards general case, like
examples, substituting. (e.g., “Suppose there were 100
___, how many ___ would there be?”)
3. Develop Conceptual Understanding
Ask all students to write their equation. Discuss the
equations – if they don’t work, why not? If two
different ones work, are they equivalent?
Ask students to check their rule by substituting in some
values and see if it turns out the way that they think it
should (based on questions 1 and 2).
Lochhead, J. & Mestre, J. P. (1988). From words to algebra: mending misconceptions.
In The Ideas of Algebra, 1988 Yearbook of the National Council of Teachers of
Mathematics. Reston, VA: NCTM.
Helping Students with Translation:
Words to Symbols
Lesson Plan – The next page is NOT a student activity sheet, as these prompts get tricky
too fast. They are a sampling of the types of phrases to pose to students.
Option 1: Use problems like these as WARM UPS – Imagine if students did one a day, each
day getting a little trickier, how they could move forward in this critical area!!
Option 2: Full lesson - Develop 5-8 at the level of items 1 – 3. Discuss. Use the three step
strategy on previous page to work with students on how to translate. Day 2: Move on to
harder translation problems.
Option 3: The Game of Telephone. This is best after doing some of Option 1 and 2. This is
words to symbols and symbols to words.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+++
Translate each English sentence into mathematical language. Use letters that fit the
situation. For example, use F for “Fred’s income.”
1. The combined incomes of Fred and Harry equal $490.
Math Language:
2. A man worked for 20 hours at $3 an hour plus 10 hours at $5 an hour for a
total of $110.
Math Language:
3. Larry is four times as old as his son Bob.
Math Language:
Hint: Rewrite the sentence with the word “equals” in it.
4. Ten pounds less than 50 pounds equals 40 pounds.
Math Language:
(the answer is not 10-50 = 40)
5. Four more than five times a number equals one less than six times that
number.
Math Language:
(the answer is not 5n + 4 = 1 – 6n)
6. A man worked a certain number of hours at #3 an hour and the same
number of hours plus 20 more hours at $4 an hour earning a total of $150.
Math Language:
(the answer is not 3n + n = 20(4) = 150)
A sampling of problems from Developing Mathematical Skills by Whimbey and Lochhead, 1981.
Classic Translation Tasks
Note: These are all one-step equations, yet the equations seem
counterintuitive based on students prior experiences with muliplication. Even
adults can have trouble with the distinctions. Use the three-step process
provided above and see if your students can think through these strategically!
1. Write an equation using the variables S and P to represent the
following statement: “There are six times as many students as
professors at this university.” Use S for the number of students and P
for the number of professors.
2. Write an equation using the variables C and S to represent the
following statement: “At Mindy’s restaurant, for every four people who
order cheesecake, there are five people who order strudel.” Let C
represent the number of cheesecakes and S represent the number of
strudels ordered.
3. Write a sentence in English that captures A = 7S, where A is the
number of Assemblers in a factory and S is the number of Solderers in
a factory
4. A man takes a photograph of some of the cows and pigs in a large field full
of cows and pigs. He is sure that he has photographed a typical sample for
the animals in the field. Write an equation to describe the relationship
between C, the number of cows, and P, the number of pigs, in the field. The
equation should allow you to calculate the number of cows if given the
number of pigs.
5. Write an equation that can be used to calculate the number of feet in a
measurement given the number of yards. Use F for feet and Y for yards.
Game of Telephone (with an Algebra Twist)
1. Prepare one 3 by 5 card for each group. Label each card with a
different letter (A – F, for example). One each card write one
of the phrases like the ones listed on the next page (you decide
how difficult they should be).
2. Give each group a stack of blank 3 by 5 cards (if there are five
groups).
3. Give students their first prepared card. Ask them to take the
letter on that card and write it on one of their blank 3 by 5
cards and a 1 to indicate Round 1 (for example, A-1).
4. On that newly labeled card, they are to write the expression
that fits the statement. DO NOT SOLVE.
5. When groups are finished, ask them to pass only they card
they prepared (with the expression or equation) to the next
group (they must keep the original secret).
6. Round 2. Now each group has an equation. Ask each group to
take a blank card and give it the letter of the card they just
received and a 2 for round 2 (for example, A-2). Ask them to
write the words that could be use to get the expression or
equation they received. (they can be creative or just straight
forward).
7. When groups are finished, they pass their sentence(s) to the
next group.
8. Repeat this process until the A problem is back with the group
that first had A and each group has their original probem
back.
9. Each group studies the progression of their problem and see if
it maintained equivalence through all the groups. If not, what
went wrong.
10. Provide time for students to share where their equations went
off track and why and/or to show how they knew the
expressions were equivalent.
M & M Equations
Lesson Objectives:
Students will be able to write an equation from words.
Students will be able to solve one- and two- step equations for a variable.
Materials:
M&Ms one bag per group of students and one for the teacher
Recording sheet (one per group)
Note Card (one per group)
Activity Instructions:
1. Explain to students that they are going to be using algebra equations to
figure out how many of each color of M&Ms someone else has.
2. Open the teacher bag and record on the board how many of each color are in
the teacher bag.
3. Tell students that in a minute they are going to be counting their M&Ms and
recording it on their notecard – but they must keep it secret from the other
groups.
4. Allow students to open and count their M&Ms.
5. Distribute the Recording Sheet (part 1). Ask students to write their group’s
name at the top.
6. Model how to do the first one by pretending that some group has 10 red
candies, showing how they would fill in the first question.
For example, if the teacher bag had 8 red candies, the first line would read:
1. I would have to add (or eat) _2__ red candies to have the same number
of red candies as the teacher. How many red candies do I have?
7. Ask students to do this for the other stories on their paper, still keeping their
numbers a secret.
8. Once all groups are done, they exchange papers with another group.
9. Explain that students are now going to write an equation to model the story
and then solve the equation to figure out the number of red M&Ms, etc.
Model how to do the first one by continuing with the example you had
started before.
1. I would have to add (or eat) _2__ red candies to have the same number
of red candies as the teacher. How many red candies do I have? _r = 10___
Equation: r – 2 = 8 [r is the number the group has, they eat two, which is
subtracting 2, and get the teachers amount of 8].
10. Observe students as they are working, making sure they are using equations
with a variable and that they are solving it.
11. When students are finished, they will have a list of the number of each
M&M the other group has. Then they meet up with the group that they traded
with to see if they solved the problems correctly (you can have one person in
each group meet with one from the other or do one larger group).
12. In debriefing, have students volunteer ones that didn’t work out – and the
class can find the error. This really helps students sort out their own
misconceptions.
NOTE: Part II can be done the same way (these are harder), or can be done with
individuals, then exchanging with a partner.
Activity Source: Borlaug, V. (1997). Building equations using M&Ms. Mathematics
Teaching in the Middle School, Reston, VA: NCTM.
M & M Equations: Part 1
Group Name(s)_______________________________________________
1. I would have to add (or eat) _____ red candies to have the same number
of red candies as the teacher. How many red candies do I have?_______
Equation:
2. I would have to add (or eat) _____ orange candies to have the same
number of orange candies as the teacher. How many orange candies do
I have?______
Equation:
3. If I doubled the number of tan candies I have, then I would have _____
tan candies. How many tan candies do I have?______
Equation:
4. If I had half the number of brown candies that I have, I would have
____ brown candies. How many brown candies do I have?_____
Equation:
5. If I tripled the number of green candies I have, then I would have ____
more (or less) than the teacher. How many green candies do I
have?________
Equation:
6. If I added 15 yellow candies to my bag, the teacher would have to add
______ yellow candies to his or her bag for us to have the same number
of yellow candies. How many yellow candies do I have?______
Equation:
7. If I double my blue M&Ms, then I would have ______ more (or less)
than the teacher. How many blue M&Ms do I have?______
Equation:
M & M Equations: Part 2
Name(s)___________________________________________________
1. If I tripled the number of yellow candies I have, I would have _____
more yellow candies than the teacher. How many yellow candies do I
have?
Equation:
2. If I ate 3 of my orange candies, then put my orange candies together
with the teacher’s orange candies, we would have ______orange
candies. How many orange candies did I start with originally in my
bag?
Equation:
3. Suppose another student had a bag of M&Ms exactly like mine. So we
each started with the same number of each color candy. If we combined
our candy, then I ate 5 of our red candies, we would have ______ red
candies left. How many red candies did I start with originally in my
bag?
Equation:
4. My brown, yellow, and green candies total ______. I have _______ more
(or fewer) brown candies than yellow candies. I have _______fewer (or
more) green candies than yellow candies. How many brown candies do I
have? How many yellow? How many green?
Equation:
Borlaug, V. (1997). Building equations using M&Ms. Mathematics Teaching in the Middle
School, Reston, VA: NCTM.