ESL Language Discovery Camp 2013 Math Curriculum
Table of Contents
Overview
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Introduction ……………………………………………………………….. 2
Calendar ……………………………………………………………………… 4
Vocabulary and Warm-Ups
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ExC-ELL Vocabulary Instruction Framework ……………….. 5
Deductive Thinking Skills Warm-Ups ……………..…………… 6
Activities
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The Parts of a Word Problem ……………………………………… 16
Logic Puzzlers ….............................................................. 21
Whole Number Computations ……………………………………. 23
Magic Squares …............................................................. 25
How Much – How Many …………………………………………..… 30
King Shamba’s Game ………………………………………………….. 33
Question Stem Phrases ………………………………………………. 40
Fibonacci Series ………………………………………………………….. 44
Describing Relationships in Math ……………………………….. 48
Delivery Dilemma ……………………………………………………….. 54
Analogies ……………………………………………………………………. 55
Psychiatrist …………………………………………………………………. 60
Metaphorically Speaking ……………………………………….……. 62
Appendix
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SIFE Alternate Activities ……………………………………………… 67
Extra Activities …………………………………………………………... 87
Answer Keys ………………………………………………………………. 95
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ESL Language Discovery Camp 2013 Math Curriculum
Introduction
You might be amazed at the rate at which English language learners struggle with and fail highschool level math courses. Often Algebra I or Geometry is the one course keeping these
students from promotion and graduation, and delayed promotion and graduation are key
factors in CMS’s high ELL dropout rate. Hopefully, at this point in the game we’re all past the
“math is the universal language, so ELLs should do fine in it” stage of naïveté. Math has its own
instructional language and symbolic system that is quite unique from all other courses.
Additionally, many of our students, especially late-arriving newcomers, like the ones who should
be attending this summer’s session, often lack the educational foundation or have a foundation
that is so fundamentally different from the American system that even if math were a “universal
language,” they’re not very fluent in it.
But ELLs generally do better on third through eighth grade math EOGs. It’s only when they
embark on the journey that begins with Algebra I or lead-in courses such as Intro Math or
Fundamentals of Algebra that they stumble. This might be attributable to the fact that highschool math encompasses a major shift in thinking: from the concrete to the abstract. And,
more than anything else, ability to grasp the abstract is firmly rooted in facility with language.
So, the math element of our summer school curriculum is going to try to attack both of these
areas of weakness. The HELPMath tutorial program adapts to each student’s skill set and
addresses fundamental arithmetic and pre-algebra skills, as well as developing Tier III math
vocabulary.
For additional vocabulary support and further incorporation of the ExC-ELL vocabulary
instruction techniques, critical Tier II terms gleaned from the 8th grade math curriculum and
assessment are provided.
This binder section is meant to focus on the development of abstract thinking and the language
that goes along with it. It’s not as mathy as you might think, which will hopefully make many
math-shy ESL teachers more comfortable with it. Instead the elements of abstract thinking,
such as finding and describing patterns, understanding analogies, representational thinking,
problem-solving and logic are addressed through puzzles, games and brain-teasers. To begin
the session, some of the complex language patterns unique to math instruction and assessment
will be addressed, but in a way that leads toward abstract thinking and (perhaps?) fun.
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ESL Language Discovery Camp 2013 Math Curriculum
So, the 2013 Language Discovery Camp math curriculum objectives are:
 Students will practice and develop basic arithmetic and pre-algebra skills through
individualized programs on HELPMath.
 Students will be able to understand and use Tier II math terms common to 8th grade
level math .
 Students will explore and develop abstract thinking skills
 Students will explore and develop the spoken and written language used to
communicate abstract ideas.
 Students will become familiar with complex language patterns unique to math
instruction and assessment.
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ESL Language Discovery Camp 2013 Math Curriculum
Monday
June 17
Workday
June 24
Tuesday
June 18
Workday
Wednesday
June 19
Deductive Thinking Skills Warm-Up
Daily Vocabulary: distance, square,
which of the following…, (1,2,3)
Activity: Parts of a Word Problem
June 25
Deductive Thinking Skills Warm-Up
Daily Vocabulary: triangle, cost,
charging, how many, (4,5)
Activity: Logic Puzzlers
Whole Number Computation
July 1
June 26
July 2
July 8
Workday
July 15
July 22
Deductive Thinking Skills Warm-Up
Daily Vocabulary: approximate,
maximum, can fit in, (9,10)
Activity: Question Stem Phrases
Fibonacci Numbers
HELP Math
July 9
July 10
Deductive Thinking Skills Warm-Up
Daily Vocabulary: base, above, what
is…, (11,12)
Activity: Fibonacci Numbers Project
July 17
Deductive Thinking Skills Warm-Up
Daily Vocabulary: comparing,
variable, below, amount of, (14,15)
Activity: Delivery Dilemma
Analogies
July 29
Last Day for Students
July 24
July 31
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Deductive Thinking Skills Warm-Up
Daily Vocabulary: collected, data,
displayed, (13)
Activity: Describing Relationships in
Math
Deductive Thinking Skills Warm-Up
Daily Vocabulary: results,
conclusion, represents, (16, 17, 18)
Activity: Analogies
July 25
HELP Math
July 30
Workday
July 4
Holiday
July 18
HELP Math
Deductive Thinking Skills Warm-Up
Daily Vocabulary: graphed, price,
given, (19, 20)
Activity: Psychiatrist
HELP Math
July 11
HELP Math
July 23
HELP Math
June 27
July 3
July 16
HELP Math
HELP Math
Deductive Thinking Skills Warm-Up
Daily Vocabulary: rectangle, x times
as ___, tripled, (6,7)
Activity: Magic Squares
How Much – How Many
HELP Math
Deductive Thinking Skills Warm-Up
Daily Vocabulary: shows, diagram,
container, (8)
Activity: King Shamba’s Game
Thursday
June 20
Deductive Thinking Skills Warm-Up
Daily Vocabulary: passes through,
which of the following…, about,
(21,22,23)
Activity: Metaphorically Speaking
August 1
ESL Language Discovery Camp 2013 Math Curriculum
ExC-ELL Vocabulary Instruction Framework
Every day, some math-related vocabulary has been assigned. For example, on Wednesday, June 19,
you’ll see:
The terms to be addressed are “distance”, “square”, and “which of the following…”. After the terms, the
numbers in parentheses indicate sample EOG problems (available in the next section of your binder)
that use these terms in context.
Use the framework below to teach each vocabulary word assigned for the day. Each word should take
3-5 minutes. Less is fine, as long as the full framework is followed.
Example of Seven Steps
STEPS
1. Teacher states the word in context from a text.
EXAMPLE
1. “A surveyor determined that the
distance across a pond is √2 ,2 55 feet..”
2. Say distance three times.
2. Teacher asks students to repeat the word three
times.
3. Teacher provides the dictionary definition(s).
3. The definition is “An amount of space between
two things or people.”
4. Teacher explains the meaning with studentfriendly definitions.
5. Teacher highlights features of the word:
polysemous, cognate, tense, prefixes, etc.
6. Teacher engages students in activities to
develop word/concept knowledge.
4. When we measure the space between this
table and this chair, we call it the distance.
5. Notice how we spell distance. Spell it with me.
What is the cognate in Spanish?
6. With a partner, pick two objects in the room and
measure the distance between them in steps. Tell
the group your measurement, using the word
distance.
7. Remember to use this word in your Exit Pass
today as you summarize what you read.
7. Teacher reminds students how this will be used
during class.
-- Adapted from Preventing Long-Term ELs: Transforming Schools to Meet Core Standards by Margarita Espino Calderon and Liliana MinayaRowe; 2011; p. 57.
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ESL Language Discovery Camp 2013 Math Curriculum
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Deductive Thinking Skills Warm-Ups
Support Tips and Solutions
General Suggestions:
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Model thinking through the answers for the first day or for several days, as needed.
These warm-ups should take no more than three to five minutes each day.
If the tasks prove to be too simple for your students, feel free to use more challenging deductive
reasoning warm-ups, such as http://www.internet4classrooms.com/brain_teasers.htm or
http://www.squiglysplayhouse.com/BrainTeasers/ ; or, if you’re saddled with a roomful of kids
who are arguably smarter than you are, try these little brain-teasers:
http://www.west.net/~stewart/lsat/ql_reaso.htm .
Review the vocabulary and phrasing of each warm-up, especially the elements that are critical to
finding the answer, so that you are ready to support students. The “Support tips” below have
tried to identify the language needs in each item. If you choose to use your own warm-ups, use
the support tips below as a model for how to analyze the language of each item with student
support in mind.
When reviewing the solution to each day’s warm-up, make it a continuing practice to have the
students explain WHY the solution is correct, in their own words, in complete sentences. This
may require guidance and modeling, but, as our main goal is to help students develop the
language of abstract thinking, this element is critical.
To make things easier for you, these warm-ups have also been arranged in a PowerPoint presentation
(“Deductive Thinking Skills Warm-Ups”) than can be displayed on the projector.
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ESL Language Discovery Camp 2013 Math Curriculum
Daily Warm-Ups
6/19:
1. Mary and James each work. One is a bricklayer. One is a skycap. The man is not a bricklayer. Who
does what?
Support tips:
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Because this is the first warm-up, you should model the thinking that goes into solving it.
Consider modeling through the first week, depending on how quickly students pick up on it.
“Bricklayer” and “skycap”, while probably unfamiliar terms, are not particularly important to the
exercise. Define them briefly and emphasize that the important fact is that they are the names
of two completely different jobs, and our goal is to figure out which job belongs to whom.
For these warm-ups, always make sure that students know the commonly expected gender
attached to names.
Solution: If the man (James) is not a bricklayer, then Mary is the bricklayer, which leaves James to be the
skycap.
6/24:
1. Malinda’s report card showed a C in math. Malinda’s mother was angry. She said that if Malinda’s
next report card didn’t show a better grade in math, Malinda would be grounded. Malinda’s mother
never lies. Malinda’s mother saw Malinda’s next report card but Malinda did not get grounded. What
must have happened?
Support tips:
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Review the American A, B, C, D, F grading scale.
Explain “grounding”.
Solution: Malinda’s mother requires a grade better than C on the new report card or she will punish
Malinda. If Malinda is not to be punished – and she isn’t punished -- her new report card must show a
grade higher than C (I guess, C+ through A+).
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ESL Language Discovery Camp 2013 Math Curriculum
2. Mrs. Raynor and Mr. Sartin each went to the store. One was supposed to buy cheese. The other was
supposed to buy eggs. Mr. Sartin was not supposed to buy cheese. What was each person supposed to
buy?
Support tips:
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Define “supposed to.”
Define “The other” as it is used in the third sentence.
Solution: If Mr. Sartin was not supposed to buy cheese, then he was supposed to buy the other item,
which was eggs. So, Mr. Sartin is buying eggs, and Mrs. Raynor is buying cheese.
6/26:
1. Mandy believes that everyone should go to a dentist at least twice a year. Mandy knows her brother
has been to a dentist only once in the last two years. What must Mandy believe about her brother?
Support tips:
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Clarify, perhaps by drawing a timeline on the board or using a calendar, the difference between
“twice a year” and “once in the last two years.”
Students might need help with the phrase “What must Mandy believe…”
Solution: Mandy probably believes that her brother doesn’t go to the dentist enough.
2. I am thinking of three colors. I like the first better than the second. I like the third better than the
first. Of these three colors,
a. Which one do I like the best?
b. Which one do I like the least?
Support tips:
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Review ordinals (first, second, third, fourth, etc. …) if necessary.
Review the comparative “I like… better than…” if necessary.
If students are struggling model how – or ask them -- to pick three colors, then label them
“first”, “second” and “third”, and then try to order them by which one is liked more.
Solution: a. The “third” is the color like best.
b. The “second” is the color like least.
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ESL Language Discovery Camp 2013 Math Curriculum
7/1:
1. Amy’s hair is longer than Marlene’s. Tanya’s hair is shorter than Marlene’s.
a. Who has the shortest hair?
b. Who has the longest hair?
Support tips:
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If needed, define “shorter than”, “longer than”, “shortest” and “longest” (comparatives and
superlatives).
Solution: a. Tanya has the shortest hair.
b. Amy has the longest hair.
2. Rocky and Terrible are a bird and an elephant. Rocky weighs more than Terrible. Who is what?
Support tips:
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Does everybody know what a bird and an elephant are?
The names are not important.
Define “weighs more than”.
Make sure the students understand what the question “Who is what?” is asking.
Solution: Elephants generally weigh more than birds. Rocky weighs more than Terrible, therefore Rocky
is probably the elephant.
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ESL Language Discovery Camp 2013 Math Curriculum
7/3:
1. If Randy were 5cm taller, his height would be 89cm. How tall is he?
Support tips:
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Explain how “If… were…” presents a conditional situation, or a situation that is not currently true
but where the solution will be found by pretending that it is true. In other words, Randy is not
5cm taller. However, in order to find the solution to the problem, it is necessary to imagine that
he is, and then compare that height to how tall Randy really is.
If necessary, explain that “cm”=centimeters and define or illustrate centimeters.
“Taller” in this sentence is defined as “taller than Randy is now” (see convoluted explanation of
conditional, above).
If necessary, define “taller” (in and of itself) and “height” and point out their relationship to one
another.
Solution: 89cm is Randy’s new pretend height that he attains with the addition of 5 imaginary cm. To
find out how tall Randy really is, subtract 5 from 89, which leaves 84cm.
2. If Rod were 10cm shorter, he would be the same height as Stacy. Stacy is 67cm tall. How tall is he?
Support tips:
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Remind students of the 6/27 warm up about how tall Randy is, because the principle, especially
of the conditional (if… then…) element is essentially the same.
If necessary, define “shorter.”
Make sure that students realize that “How tall is he?” refers to Rod, not Stacy. It’s actually a
pretty common confusion.
Solution: Rod’s real height minus 10cm is 67cm, so to find Rod’s actual height, add 10cm back to 67cm,
finding that Rod is 77cm.
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ESL Language Discovery Camp 2013 Math Curriculum
7/9:
1. An airplane was flying at an altitude of 1500 meters. A second airplane was flying at an altitude of
500 meters higher than the first. What conclusion can you draw from this?
Support tips:
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Define “altitude.”
This one may require you to model finding the solution.
Solution: The second airplane was flying at an altitude of 2000 meters.
2. We all know that no 2-year-old can ride a bicycle. Danielle is 2 years old. So, what must be true?
Support tips:
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Students may not be familiar with the construction “no 2-year-old can ride a bicycle” as opposed
to “2-year-olds can’t ride bikes”.
If students seem a little stuck, amend the second sentence to “What must be true about
Danielle?”
Solution: Danielle can’t ride a bike.
7/11:
1. Anyone who is completely happy has no worries. Ms. Neuman has a few small worries. What must
be true?
Support tips:
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Discuss how “completely happy” and “no worries” suggest absolute states (as would using
words like “all”, “always” or “every”) and that for those states to be “true” there can’t be any
flaws or little tiny deviations.
If necessary, define “a few.”
Solution: “A few small worries” is a flaw, however tiny, in the state of “no worries.” So, “no worries” =
FALSE, therefore the dependent state, “completely happy” must also = FALSE, therefore, Ms. Neuman
can’t be completely happy.
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ESL Language Discovery Camp 2013 Math Curriculum
2. Barry spends all of his spare time reading books. Mr. Darton is a grade school teacher. He belongs to
a teacher’s bowling league. What must be true?
Support tips:
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If necessary, define “spends… time”, “spare time”, “grade school”, “belongs to” and “bowling
league.”
Understanding the difference between first names (“Barry”) and last names (“Mr. Darton”) is
necessary to finding the solution.
You may need to model thinking on this one (see below).
Solution: This one is moving into slightly new territory, in which the authors are expecting the kids to be
hip to their devious, devious ways. The solution to this question depends entirely on the students
making an assumptive leap which winds up being definitively not true, but has to be made anyway.
In order to figure out what the question is asking, the students have to understand that it is possible for
“Barry” and “Mr. Darton” to be the same person. Once they understand that possibility, then they
realize the question “What must be true?” is really asking “Are Barry and Mr. Darton the same person?”
That’s the hard part. The final answer is actually pretty easy: no, Barry can’t be Mr. Darton, because Mr.
Darton spends some of his spare time bowling, and Barry spends all of his spare time reading. Another
one of those “absolute state” questions, like the one about Ms. Neuman.
7/16:
1. Ezra likes milk better than lemonade. But he likes lemonade better than soda. Of these three drinks:
a. Which does he like best?
b. Which does he like least?
Support tips:
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If necessary, define “likes… better than…”, “like best” and “like least”.
Solution: Ezra likes milk best and he likes soda least.
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ESL Language Discovery Camp 2013 Math Curriculum
2. Marsha’s mother sent Marsha to the store three times today. One time Marsha bought bread.
Another time she bought hamburger. The other time she bought mustard. She bought the mustard
before she bought the bread. She didn’t buy the hamburger last, but she didn’t buy it first, either.
What did Marsha buy on each of her three trips to the store?
Support tips:
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Have the kids make a little time-line and label three points on it “first”, “next” and “last”, then
try to use the clues from the question to sort out which .
Solution: Marsha bought the mustard first, the hamburger second, and the bread last. Marsha’s mother
clearly needs to get organized.
7/18:
1. A SUPER-8 is more expensive than a TIGER. A LEOPARD is cheaper than a FASTCAR. A TIGER costs
more than a FASTCAR. List these four products in order, starting with the one which costs the least.
Support tips:
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The names (SUPER-8, TIGER, LEOPARD, FASTCAR) mean nothing and should not have time
wasted on being defined. They seem to be four different brands of cars, but it totally doesn’t
matter to solving the problem.
Define, if necessary, “more expensive”, “costs more” and “costs the least”.
This is the first warm-up where four, rather than three items are to be put in a specific order. If
three items is still challenging some students, you should model solving this item. (I had to use
sticky notes, so I could move them around.)
Solution: From least expensive to most: LEOPARD, FASTCAR, TIGER, SUPER 8.
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ESL Language Discovery Camp 2013 Math Curriculum
2. Celeste had an apple, a pear, and a banana. She put them in a row. She put the apple to the right of
the pear. She put the banana to the right of the apple. Name the way she set the fruits from left to
right.
Support tips:
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If your students are struggling with right, left, “to the right” and “from left to right”, try
modeling the following practice item with sticky notes: “Cedric had a carrot, a tomato and an
onion. He put them in a row. He put the onion to the left of the tomato. He put the carrot to
the left of the onion. Name the way he sets the vegetables from left to right.” (Solution for
practice item: carrot, onion, tomato.) Then have the students do the actual item on their own.
Solution: Pear, apple, banana.
7/23:
1. Amos had a grape, a lemon, and a watermelon. He put them in a row. He put the largest one in the
middle. He put the smallest one to the right of the largest one. Name the way he set the fruits from left
to right.
Support tips:
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Make sure everyone knows that the grape is the smallest and the watermelon is the largest.
In addition to “to the right” like we saw yesterday, we also have “in the middle” today, so
define that if necessary.
Solution: Lemon, watermelon, grape.
2. The sum of the ages of Darlene and her mother is 29. Darlene’s age is 4. How old is Darlene’s
mother?
Support tips:
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If necessary, define “sum.”
Solution: Darlene’s mother’s age is 29 minus Darlene’s age (4), so Darlene’s mother is 25.
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ESL Language Discovery Camp 2013 Math Curriculum
7/25:
1. Two people share a ride to work each morning. Their first names are Sue and Terry. Their last names
are Rawls and Peters. Sue is almost never ready on time. Terry drives a blue car. Rawls likes to watch
TV. Peters is almost always ready on time. What is each person’s full name?
Support tips:
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If necessary, define “share a ride”, “ready on time” and “full name”.
This is a fun one, in that the key to solving it lies in eliminating a lot of useless information. The
question at the end indicates that the solution involves correctly connecting “Sue” and “Terry”
to “Rawls” and “Peters”, which is accomplished through finding something that a specific first
and last name must have in common OR a reason why they can’t possibly be together. The only
facts that help in this way are that Sue is almost never ready on time and Peters is almost always
ready on time, which means that Sue can’t be Peters. That they share a ride, that Terry drives a
blue car, and that Rawls like to watch TV are all useless bits of information. Help the students
see how they can put the distractions aside to find the hints that actually help (literally crossing
things out, for instance).
Solution: If Sue can’t be Peters, then it’s Sue Rawls and Terry Peters.
2. Three years ago, Jefferson was a year older than his brother. Jefferson’s brother is now 6 years old.
How old is Jefferson now?
Support tips:
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Don’t think we need any tips here, although I know some kids are surprised by the fact that, if
their brother is a year older than them now, he will always be a year older.
Solution: Jefferson is still a year older than his now-6-year-old brother, so Jefferson is 7.
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ESL Language Discovery Camp 2013 Math Curriculum
The Parts of a Word Problem
This activity is intended to help ELLs understand better how word problems work by looking at their
structure. Students need to understand that the mathematical information they need to select and
generate the operations that will provide the solution is found in the middle and final section of the
problem (referred to here as the “conditions” and “question”).
Objectives: Students will understand that word problems provide three separate stages of information:
the setup, the conditions, and the question. Students will be able to identify each of these stages in
examples of word problems.
Procedure: Use the first two pages of the activity to explain the three stages of information in a word
problem and to model separating a word problem into these stages. The latter two pages (the list of
word problems and the graphic organizer) are the activity the students will do, in which they cut apart
the problems and paste the appropriate sections into the organizer. This would be a good activity for
completing in groups. The students are not expected to solve the word problems in this activity (unless
they want to). The main purpose is the reading of the problems.
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ESL Language Discovery Camp 2013 Math Curriculum
The Parts of a Word Problem
Word problems follow a general pattern:
SET UP, OR ENVIRONMENT IN WHICH THE PROBLEM TAKES PLACE -- This is the first part
of the word problem. It will tell you about the people and things in the problem. This
part helps you understand what is going on in the problem.
CONDITIONS SPECIFIC TO THE PROBLEM -- This is the middle part of the word problem.
It will tell you about changes being made to things. There are generally numbers and
words that tell you what is happening to those numbers. The information in this section
will become an important part of the number sentence you write to solve the problem.
THE QUESTION YOU NEED TO ANSWER – The question is the last part of the word
problem. It works with the conditions section to tell you exactly what kind of math
problem you are going to write. The question will almost always include one of the
following words: find, what, which or how. Often there is other important information in
the question, such as units of measure, or words like approximately, most likely, or best,
that tell you more about what your answer should look like.
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ESL Language Discovery Camp 2013 Math Curriculum
Take a look at this word problem:
The regular price of a refrigerator is $1100. It is going to be discounted by 20%.
What is the discounted (sale) price?
SET UP, OR ENVIRONMENT IN WHICH THE PROBLEM TAKES PLACE
The regular price of a refrigerator is $1100.
CONDITIONS SPECIFIC TO THE PROBLEM
It is going to be discounted by 20%.
THE QUESTION YOU NEED TO ANSWER
What is the discounted (sale) price?
Now, work in groups to cut up the following word problems and glue their segments in the appropriate
part of the graphic organizer.
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ESL Language Discovery Camp 2013 Math Curriculum
A baseball league has 192 players and 12 teams, with an equal number of players on each team.
The number of teams was reduced by four but the total number of players remained the same.
How many players are on the new teams?
Julie bought a card good for 35 visits to a health club and began a workout routine.
After y visits, she had y fewer than 35 visits remaining on her card.
After 18 visits, how many visits did she have left?
Ron bought two comic books on sale.
Each comic book was discounted $1 off the regular price r.
If each comic book was regularly $2.50, what was the total cost?
In basketball, players score 2 points for each field goal, 3 points for each three-point shot, and 1 point
for each free throw made.
The Bobcats scores 23 field goals, 6 three-point shots, and 11 free throws.
What was the total score for the Bobcats?
Gary needs to buy a suit to go to a formal dance.
Using a coupon, he can save $60, which is only one-fourth of the cost of the suit.
What is the original cost of the suit?
Einar has $18 to spend on his friend’s birthday presents. He buys one present that costs $12.35. How
much does he have left to spend?
A leak in a commercial water tank changes the amount of water in the tank each day by -6 gallons.
When the total change is -192 gallons, the pump will stop working. How many days will it take from the
time the tank is full until the pump fails?
Julie is balancing her checkbook. Her beginning balance is $325.46, her deposits add up to $285.38, and
her withdrawals add up to $683.27. What is her ending balance?
There were 7 legs of the BT Global Challenge 2000 yacht race. The crew of the winning boat, the LG
Flatron, sailed at a rate of at least 6 knots (6 nautical miles per hour) continually on the leg between
Cape Town, South Africa and La Rochelle, France, a distance of 5820 nautical miles. How many hours
did this leg take?
Elaine runs the same distance every day. On Mondays, Fridays and Saturdays, she runs 3 laps on the
track and then runs 5 more miles. On Tuesdays and Thursdays, she runs 4 laps on the track, and then
runs 2.5 more miles. On Wednesdays, she just runs laps. How many laps does she run on Wednesdays?
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SET UP, OR ENVIRONMENT IN WHICH THE PROBLEM TAKES PLACE
CONDITIONS SPECIFIC TO THE PROBLEM
THE QUESTION YOU NEED TO ANSWER
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ESL Language Discovery Camp 2013 Math Curriculum
Logic Puzzlers
This activity is intended to extend on the technique of breaking down word problems into three
separate stages of information. The problems here are not actually math word problems, but rather
logic problems, but they follow the exact same format as math word problems. This will hopefully a)
keep the focus on the reading aspect of working with the problems while reinforcing the technique from
the previous lesson, helping students improve the skills they’ll be using in the warm-ups, and
encouraging abstract thinking skills.
Procedure: Ask students to identify the set-up, conditions and question of each problem, either by
highlighting them in three different colors or by circling the conditions, underlining the question, and
leaving the set-up blank.
Model solving the first problem. You could say something like:
The question is which person is wearing yellow, so I have to figure out from the clues given in the problem
which person is wearing yellow. The set-up tells me there are three people, so I’m going to put three
circles on the board, one for each person. I see a bunch of colors mentioned in the conditions section, so I
bet that’s where I will find the clues that help me solve this problem. Here it says: “’I’m not wearing red
or blue,’ says the first.” So my first circle is the first person, and I’m going to write “red” and “blue “ under
that circle, then cross them out to show that they are not wearing red or blue. Here it says: “’But one of
us is wearing yellow,’ says the second.” All that tells me is that one person is wearing yellow, but it
doesn’t tell me anything about which person, so I’m not going to write anything under the second circle,
which is my second person, just yet. Now the problem has the third person saying “I don’t see any yellow
or red on either of you.” This doesn’t tell us anything exactly for the third person, but it does tell us that
both the first and second person aren’t wearing yellow and they aren’t wearing red. I already have “red”
written under the first person and crossed out, but I don’t have it written and crossed out for the second
person, so I’m going to do that now. Also he said that neither the first nor second person were wearing
yellow, so I’m going to write “yellow” under each and then cross it out.
1
2
Red
Blue
Yellow
Yellow
3
So I can see that both the first and second person can’t be wearing yellow, but somebody is wearing
yellow, because the second person said so. Look at the circles: who’s left? That’s right, the third person
must be the one wearing yellow, because we’ve already figured out the other two can’t be.
Have students work in pairs to complete the other problems. Ask them to volunteer to explain how
they worked out the solution. Getting them to talk about this may take some coaching/modeling.
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WNC
Whole Number Computations
This activity is intended to help students review the terms strictly associated with mathematical
operations. You can see in the activity that the numbers (0-100, 1000) are written out (review, if
necessary) and the operations, which are the following terms and should be reviewed:






















Plus
Times
Minus
Divided by
The remainder of
The largest
Prime number
Less than
The smallest
Factor
Other than itself
Quotient
Doubled
Prime factor
Difference
Product
Sum
Negative
Square root
Cube root
To the third power
To the second power
Model problems 1, 5, and 17. The process should be to first write out the problem in numbers and
symbols, then to solve the problem. The directions on the sheet specify “do all the operations in the
order in which they are given” so do not worry about following the Order of Operations here, just do
‘em as they come. So, problem 1 would look like this:
Four plus seven times three minus six divided by three
4+7x3–6÷3
Which, if we’re going in order, would go:
11 x 3 – 6 ÷ 3
33 – 6 ÷ 3
27 ÷ 3
9
Always, when going over the answers on activities such as this, give the students the opportunity to
come to the front of the room and use the board to demonstrate how they solved the problem.
Students typically enjoy working on the board, and need to be given lots of opportunities to talk about
their work.
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Magic Squares
This is a fun-with-math activity with some reading thrown in. Make student copies of the following four
pages. Have students read the first two pages either through round-robin reading or having partners
read to each other. Go over the key facts: magic squares are math crossword puzzles, they’ve been
around for thousands of years, and the goal is to use all of the single-digit numbers (1-9) and have the
horizontal, diagonal and vertical rows add up to 15. On the second page are some example puzzles.
Have the students present on the board how to solve them or model solving them yourself, whichever is
necessary.
8
1
6
3
5
7
4
9
2
2
9
4
7 6
5 1
3 8
Item C on this page is taking things a step further. Try to get the students to note that, if there are 16
squares, you have to use each number, 1 through 16, one time. Also, the third row is completed, so the
target sum for all rows, columns and diagonals should be 34.
7
2
16
9
12
13
3
6
1
8
10
15
14
11
5
4
The final two pages provide instructions on how to construct a play a game around the concept of the
Magic Squares, which students can work in partners or in groups to complete and play.
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How Much – How Many
The question section of word problems frequently center on either the term “how much” or “how
many”. Which phrase is used depends on the count/non-count nature of the nouns in question. Count
and non-count nouns are a common source of confusion for ELLs, so this activity is meant to approach
the problem from the math angle and help elucidate what is being asked for when “how much” or “how
many” appears in a mathematical task.
Objectives: Students will be able to categorize nouns as “count” or “non-count”. Students will
recognize that “how much” is used in questions about non-count nouns and “how many” is used in
questions about count nouns.
Procedure: Use the first page of the activity as your mini-lesson guide and talk through the distinctions
between and examples of count and non-count nouns.
Model the first two items, found at the bottom of the first page. Have the students complete the
remaining items.
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ESL Language Discovery Camp 2013 Math Curriculum
How Much – How Many
Some things can be COUNTED. Some things can be MEASURED.
Examples of things that can be COUNTED:










students
pennies
rocks
M&Ms
pencils
toes
cars
books
points
birds
But, you can’t count some things, like water. Water, like any liquid, is only limited in size or shape by its
container, so we often describe water as a glass of water, a bucket of water, a tub of water. When we
talk about water or other liquids in a math problem, we usually describe them through measurement: a
gallon of water, a pint of milk, a liter of liquid nitrogen.
Likewise, you can’t count time. Time is infinite, so we generally refer, when we talk, to a segment of
time: some time, a long time, lots of time. In math problems we again usually describe time in terms of
measurements: a second, a minute, an hour, a day.
Things that you can count are called count nouns. Things like water and time that you can’t count are
called non-count nouns. We treat them differently when we use them in sentences.
In math, the main difference you will see is the way we ask about them. You can count pencils, so a
math problem will ask, “How many pencils?” You can’t count juice, so a math problem will ask “How
much juice?”
1. Jack had seventeen jelly beans. He ate six of them. How ___________ jelly beans did Jack have left?
(much/many)
2. The recipe called for two cups of apple juice and one cup of pineapple juice. How ______________
(much/many)
juice did the recipe need?
3. Judy spent one studying math, two hours studying science, and a half hour finishing her art project.
How _______________ time did Judy spend working on her homework altogether?
(much/many)
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4. Shari’s bucket can hold a pint of sand. She used four buckets of sand to build her sandcastle. How
____________ sand did Shari put in her sandcastle?
(much/many)
5. Byron’s yard has two hummingbirds, four bluebirds, and one mockingbird. How ______________
birds live in Byron’s yard altogether?
(much/many)
6. Larry’s tub holds ten gallons of water. Larry filled the tub completely, then drained 25% of the water.
How ____________ water was left in the tub?
(much/many)
7. Mrs. West requires that students leave their cell phones in a box on her desk during class. Mrs. West
has 30 students, two-thirds of whom have cell phones. How ____________ cell phones are in Mrs.
West’s box?
(much/many)
8. Leanne wants to put down carpet in her bedroom. Leanne’s bedroom is 8 feet long and 10 feet wide.
How _______________ carpet, in square feet, does Leanne need to buy to cover her whole floor?
(much/many)
Note: Money is a little bit odd with the how much/how many rules. Although it is possible to count
dollars and cents, generally the typical phrasing when asked about a total amount of money, such as a
price, is phrased “How much money?” So use your brain and try to figure out these last two!
9. Nigel had $2.40 in his pocket, earned $5 from Mrs. Hart for mowing her lawn, then spent $1.70 on a
soda. How ___________ money does Nigel have now?
(much/many)
10. Bethany saves all the pennies she gets in a jar. When the jar gets full, she goes to the bank and
deposits the pennies into her account. The last time she took the jar to the bank, she deposited $27.83
into her account. If that total amount came out of her penny jar, how _____________ pennies did
Bethany have in the jar?
(much/many)
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ESL Language Discovery Camp 2013 Math Curriculum
King Shamba’s Game
Like the “Magic Squares” previously completed, King Shamba’s Game is another hot math-fun/reading
practice combo. Again, have students partner up to read (the first page and the top of the second
page), go over the key facts in the history of the game, then have them work together to follow the
directions to build and play the game.
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ESL Language Discovery Camp 2013 Math Curriculum
Question Stem Phrases
Secondary math state testing questions are often characterized by not having concrete, absolute
answers, but rather by asking students to select the best of a set of flawed options. This activity is
meant to help students recognize when a question is calling for an exact, incontrovertible answer, and
when it is asking them to use the information in the question to compare and judge the options based
on the information given.
Objective: Students will be able to recognize when a test question requires an exact answer, and when
the question is asking them to use information given to select the best option available.
Procedure: The following lecture notes are also available as the PowerPoint presentation titled
“Question Stem Phrases.” If possible, use the PowerPoint. The work done by the students here is
embedded in the presentation.
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ESL Language Discovery Camp 2013 Math Curriculum
Question Stem Phrases
How are these two test questions DIFFERENT?
1. What is 2+2?
A. 2
B. 4
C. 6
D. 8
2. What is Mary’s favorite color?
A. pink B. blue C. red D. purple
Question 1 has a clear answer. Question 2 does not. Let’s improve Question 2:
Mary always wears red shoes and a red hat. Her house is red brick, and she rides a red motorcycle.
What is Mary’s favorite color?
A. pink B. blue C. red D. purple
You are probably guessing “red,” because there are clues that suggest that Mary likes red. But you don’t
KNOW that. Maybe red stuff was on sale. Maybe the four things listed above are red, but everything
else Mary owns is blue.
So actually, the “fixed” Mary question is still written incorrectly. On a test, the question should look like
this:
Mary always wears red shoes and a red hat. Her house is red brick, and she rides a red motorcycle.
What is most likely to be Mary’s favorite color?
A. pink B. blue C. red D. purple
The phrase “most likely” recognizes that you do not have all the facts, but asks that you use the facts
you have been given to make your best guess.
When you see the words LIKELY, MOST, MORE or BEST in the question part of a math test stem, it
usually means:



there is not one single correct answer, but many answers that may be correct
clues that will help you make a good guess are in the stem before the part where the question is
asked
the choice that matches best with the clues given in the stem is the right choice
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ESL Language Discovery Camp 2013 Math Curriculum
Which question is written correctly?
Mark asked for pizza for his birthday. Mark has a
pizza oven in his kitchen and has learned how to
make his own pizza. Mark has Dominos, Pizza Hut
and Papa John’s on speed dial.
Mark asked for pizza for his birthday. Mark has a
pizza oven in his kitchen and has learned how to
make his own pizza. Mark has Dominos, Pizza Hut
and Papa John’s on speed dial.
What is Mark’s favorite food?
What is most likely to be Mark’s favorite food?
Robert builds a path out of bricks. Each brick is a
square, six inches long on all sides. He lays
twenty-four bricks end-to-end.
Robert builds a path out of bricks. Each brick is a
square, six inches long on all sides. He lays
twenty-four bricks end-to-end.
How long is Robert’s path?
How long is Robert’s path most likely to be?
In the first example, “What is most likely to be Mark’s favorite food?” is correct, because we don’t have
enough facts to prove definitively that any one single food is Mark’s favorite… although we do have a
good guess.
In the second example, “How long is Robert’s path?” is the correct way to phrase the question. There is
only one right answer and no need for guessing, so there’s no need to point to a “best” or “more likely”
answer, there is only the right answer.
More practice – which is the right way to phrase the question?
1. Jenny runs twice a week. On Tuesdays she runs two miles. On Thursdays she runs four miles.
A. How many miles does Jenny most likely run in a week?
B. How many miles does Jenny run in a week?
2. Tom went to Johnson’s Store and bought two pairs of pants, five pairs of socks, and three shirts.
A. What type of store is Johnson’s Store most likely to be?
B. What type of store is Johnson’s Store?
3. Tom went to Johnson’s Store and bought two pairs of pants, five pairs of socks, and three shirts.
A. How many items did Tom most likely buy?
B. How many items did Tom buy?
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ESL Language Discovery Camp 2013 Math Curriculum
“Most likely” is common, but there are other phrases that let you know the answer is a best guess, and
not an absolute right answer:






Which graph best describes how well Diggity Dog Brand Dog Food has sold over the past year?
What scatterplot best fits the data set?
Which poster best represents why Harry should be class president?
Which route is Laura more likely to take?
Which is the best name for Paula’s restaurant?
Which argument best supports Richard’s answer?
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ESL Language Discovery Camp 2013 Math Curriculum
Fibonacci Series
More fun with math and reading! Start this one off by putting some of the Fibonacci sequence on the
board (1, 1, 2, 3, 5, 8, 13, 21, 34, 55…) and challenging the students to discover the rule and/or find the
next number in the pattern.
The explanation is in the reading that follows.
You can challenge the students with some other Fibonacci-related puzzles, such as:
List the five 3-digit Fibonacci numbers.
(144, 233, 377, 600, 977)
Which of the following is a Fibonacci number: 4666, 1077, 6685, 3114
(6685)
Which Fibonacci numbers under 100 are prime numbers? (1, 2, 3, 5, 13, and 89)
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Fibonacci Project
An interesting thing about the Fibonacci series is that it is found everywhere in nature. The text shows
three different flowers that present three, five and thirteen petals. Have students work in groups to find
images online or, even better, samples from outdoors, of plants and even animals that have parts in
multiples from the series. If working online, students can put together a slide show. If working with
actual bits of nature, have students mount the items on a poster or otherwise display it.
Product: Create a display or slideshow presenting examples of Fibonacci series numbers (1, 1, 2, 3, 5, 8,
13, 21, 34, 55…) in natural items (five fingers on humans, thirteen petals on ragworts, eight legs on
spiders).
Content objective: Students will be able to identify and explain the pattern that generates the Fibonacci
series. Students will recognize an example of how mathematical rules operate in nature (other
examples of mathematical rules operating in nature involve symmetry and fractals).
Language objective: Students will be able to explain orally and/or in writing the pattern that generates
the Fibonacci series. Students will be able to present their displays and explain the relationship
between Fibonacci numbers and the nature examples they have placed on their displays.
Rubric: [Teachers, please feel free to adapt the elements of this rubric in whatever way you think
appropriate to how your particular students will best execute the project.]
Didn’t
Completed Excellent
Completed
Do It
Correctly Example
Project Element
Product has at least five different images or samples,
each showing the presence of a different Fibonacci
number.
Group/student took on the challenge of finding two or
more examples of two-digit or larger Fibonacci numbers.
Display/slide show is neat and attractively presented.
Group/student gives an oral presentation explaining
each sample and its evidence of a Fibonacci number.
OVERALL
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ESL Language Discovery Camp 2013 Math Curriculum
Describing Relationships in Math
Use chart paper or poster board to recreate the eight images that follow. Each image illustrates a
common relationship between numbers and/or figures used in math. These relationships can be
described by multiple terms.
For each image, you will have a list of terms. Write these out on index cards, sticky notes or segments of
sentence strips. If you think of more terms than the ones provided, fantastic! Use those as well. The
final product is going to be a word wall that the students can use as a tool, so make the lettering large
and neat.
Have the students affix the cards with the terms on them to the appropriate posters. If the students
think of more terms, that’s also fantastic and they should be added.
A. within, inside of, in, inside
B. beside, next to, on the side, to the side, to the left of (you could also make a poster illustrating “to the
right of”)
C. above, on top of, over, on
D. under, below, beneath, on the bottom of, underneath
E. together, grouped together, with, separate from
F. equals, equal to, equivalent to, the same as, as much as, as big as
G. less than, smaller than, not as much as
H. greater than, more than, larger than
Expansion activities:

Students could write complete sentences, using the terms you just sorted (“The small square is
above the large square.”) Or, to make it more amusing, they could draw their own figures, using
cartoon images or goofy symbols (or stickers, whatever) and write sentences about those (“The
cat is next to the unicorn.”) As long as they’re using the phrases or terms describing the
relationships.
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ESL Language Discovery Camp 2013 Math Curriculum


Students could draw a figure (say a red triangle on top of a blue circle) then cover it, describe it
to a partner, have the partner draw it, then see if they match.
Find different objects or ideas around the room that can be compared (books that are bigger,
smaller, and the same size; students who are older, younger and the same age) and orally or
using cards with <,>, and = signs line things up and compare them.
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Model Wall Charts for Describing Relationships
A
B
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C
D
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E
(
)
F
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G
H
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Delivery Dilemma
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Analogies
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Find more worksheets at http://englishforeveryone.org/Topics/Analogies.htm
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Analogies
Objective: Students will be able to identify then explain the rules that support analogies.
A
dog
B
puppy
C
cat
D
A
dog
B
puppy
C
cat
D
kitten
Explanation:
B is ________ to A, so D must be ________ to C.
rule
rule
B is the baby to A, so D must be the baby to C.
A puppy is the baby to a dog, so a kitten must be the baby to a cat.
A
yes
B
no
C
up
D
A
yes
B
no
C
up
D
down
C
light
D
Explanation:
B is opposite to A, so D must be opposite to C.
No is opposite to yes, so down must be opposite to up.
A
day
B
night
Explanation:
B is ________ to A, so D must be ________ to C.
rule
rule
_____________________________________________________________________________________
A
pine
B
tree
C
rose
D
Explanation:
A is ________ to B, so C must be ________ to D.
rule
rule
_____________________________________________________________________________________
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A
Washington, D.C.
B
United States
C
Paris
D
Explanation:
A is ________ to B, so C must be ________ to D.
rule
rule
_____________________________________________________________________________________
A
morning
B
breakfast
C
noon
D
Explanation:
B is ________ to A, so D must be ________ to C.
rule
rule
_____________________________________________________________________________________
A
head
B
hat
C
feet
D
Explanation:
B is ________ to A, so D must be ________ to C.
rule
rule
_____________________________________________________________________________________
A
pen
B
write
C
scissors
D
Explanation:
B is ________ to A, so D must be ________ to C.
rule
rule
__________________________________________________________________________________________________
A
summer
B
hot
C
winter
D
Explanation:
B is ________ to A, so D must be ________ to C.
rule
rule
__________________________________________________________________________________________________
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ESL Language Discovery Camp 2013 Math Curriculum
Psychiatrist
Objectives: Students will practice identifying patterns and rules. Students will describe these patterns or rules
orally.
In the game of Psychiatrist, “It”, or the Psychiatrist, leaves the room for a minute, while the rest of the group
decides on a weird behavior for the Psychiatrist to diagnose. The Psychiatrist then returns to the room, interacts
with the group, and attempts to determine what the rule of the selected weird behavior is.
The group needs to agree upon a simple rule that will guide their behavior when the Psychiatrist returns. For
example, everyone could decide to lie whenever the Psychiatrist asks them a question. Or they have to say a color
every time they answer. Or they could choose to pat their head every time the Psychiatrist looks at them.
Probably it would be good to model generating the rule for the students, until they get the idea and take over.
Encourage goofiness – this game can be hilarious. When the Psychiatrist correctly guesses the rule, let the
Psychiatrist pick his or her successor, or draw a name from a hat.
The challenge for newcomer ELL students – and what they have to do in order to succeed at the game – will be
communicating the diagnosed rule accurately and effectively. In the case of this game, the rule will be determined
by WHO is doing WHAT and WHEN.
WHO will probably be “everybody,” so a good sentence starter for the solution would be “Everybody is…” If it’s
not everybody, it will be an identifiable group, such as “all of the boys,” so then the sentence starter would be “All
of the ________ are…”
WHAT is the action, be it lying, head-patting, color-mentioning, or whatever. Given the way we’ve started the
solution sentence, this should be expressed as a present participle: “Everybody is lying…”, “Everybody is patting
their head…” Now a scaffolded framework for the solution sentence would read: “Everybody is ______ing…” or
“All of the _______ are _______ing…”
WHEN is the conditions under which the students perform the action of the rule. Everybody is lying when the
Psychiatrist asks them a question. Everybody is patting their head when the Psychiatrist looks at them. Now the
end of the solution sentence is, for the Psychiatrist: “… when I (do something).” So…
Everyone is ________ing when I __________.
or
All of the _________ are ___________ing when I _________.
I would have these frameworks up on the board for the Psychiatrist to reference (and for the rest of the students
to reference while developing a new rule), and I would demarcate the different elements of the rule within the
sentence:
All of the _________ are ___________ing when I _________.
WHEN
WHO
WHAT
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Metaphorically Speaking – Parsing Our Most Hackneyed Sayings
Sometimes people say, “a bird in the hand is worth more than two in the bush.” This means, “What you
have is better than what you think you can get. Because you have it.”
How can we guide students from the aphorism to the plain English translation?
The aphorism is a metaphor that paints a picture. Literally, you can imagine a person holding one bird,
and covetously eyeing two other birds sitting in a bush some ways off.
There’s a story in this picture. Get the students to tell it to you, and explain why the bird in the hand is
worth more than the two in the bush.
Now, point out that “A bird in the hand is worth more than two in the bush” is something we say all the
time. However, do people walk around valuing actual birds in their actual hands on a regular basis? No,
because pecking, and hysterical elimination, and also probably mites. “A bird in the hand…” is meant to
be useful advice. Discuss with your students how this concept can actually be helpful advice. This
should guide them to a good approximation of the definition already given above.
Two key points fall out here: 1) it’s not about birds, even though it says “birds” and 2) it’s supposed to
be helpful advice. Show the students “Don’t judge a book by its cover” and “The squeaky wheel gets the
grease.” If the first aphorism isn’t about birds, then what should each of these NOT be about? Books.
Wheels. Grease. The point is not wheel maintenance and book selection, the point is helping people
understand something about themselves or other people.
In this activity, cut out the aphorisms and separate them from their definitions. Post the definitions
around the room, give the students the aphorisms, and ask them to find the correct definition and
match them up.
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Take it further: After students find their definition, have them draw a cartoon that literally represents
the aphorism, then another that represents the advice or wisdom meant to be conveyed.
Challenge: Which one of the following aphorisms isn’t real, is actually something I made up five seconds
ago?



Don’t go punching hippos if the mud is not sticky enough.
If you can’t stand the heat, get out of the kitchen.
Least said, soonest mended.
Can the students write their own definitions of these aphorisms?
A stitch in time saves
nine.
Doing the job well now
saves doing a bigger job
to fix things later.
Hint 1: “in time” here means “at the right time”
Hint 2: “Nine” what? Nine stitches, later on.
A dictionary could help with: stitch
People in glass houses
People who aren’t
should not throw stones. perfect shouldn’t talk
about what others are
doing wrong.
Six of one, half a dozen
of the other.
Both choices are the
same.
A dictionary could help with: half a dozen
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Birds of a feather flock
together.
People who are the same
stay together.
Hint 1: “of a feather” means “that are the same”
A dictionary could help with: flock
Great minds think alike.
We had the same idea,
therefore we are both
very smart.
The early bird gets the
worm.
You can get what you
want if you get there
earlier than everyone
else.
Don’t judge a book by its You can’t decide who a
cover.
person is just by looking
at them.
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The squeaky wheel gets
the grease.
People who say what
they want are more likely
to get what they want
than people who stay
quiet.
A dictionary could help with: squeaky, grease
In for a penny, in for a
pound.
If I am going to do this
(probably bad) thing, I’m
not going to do it a little
bit, I’m going to do it all
the way.
Hint 1: this saying comes from England, where a “pound” is similar to a dollar.
Hint 2: “In for” here is like with poker: you have put in a penny so that you can play the game – you’re
“in for” a penny.
Idle hands are the Devil’s People with nothing to
workshop.
do are likely to get in
trouble.
Hint 1: if necessary, elucidate who the Devil is
A dictionary might help with: idle, workshop
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Too many cooks spoil the If many people try to
broth.
control a project, that
project will probably turn
out badly.
A dictionary might help with: spoil, broth
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SIFE-Appropriate Activities
Even though, believe it or not, the preceding curriculum was designed with Newcomers in mind, it still
obviously presumes a) a certain level of literacy, b) a certain level of mathematics ability and c) some
ability to speak, read and understand English. Some of your kids are not going to be at this point in one
or all areas.
We have provided you with Oxford Picture Dictionaries for the Content Areas, and you should have the
boxed set of reproducible as well. The math reproducible collection is very nice, although much of it
involves too much reading for SIFE students and absolute newcomers. However, there are illustrated
vocabulary cards that are handy.
In the main Oxford Picture Dictionary for the Content Areas and its attendant workbook Topics 70-75
focus on math and the workbook pages are pretty doable.
Here we’ve also included some non-verbal activities, tangrams, mandalas and Celtic knot work that
could also serve as alternative activities for these students.
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Tangrams
A tangram is a geometric puzzle of sorts, consisting of a set group of polygons that can be rearranged to
make an infinite number of shapes and patterns. Work with tangrams helps familiarize students with a
host of geometry concepts, especially about polygons and angles.
First, print out the tangram template below, and cut along the lines to create the tangram pieces (gluing
to a pasteboard or construction paper backing will make the pieces sturdier).
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1. Sort the tangram pieces using your own classification or rules.
2. Put two or more of the tangram pieces together to make others shapes.
3. Put two or more of the tangram pieces together to form shapes that are congruent.
4. Use all of the tangram pieces to make a square. DO NOT look at the existing pattern.
5. Use the seven tangram pieces to form a parallelogram.
6. Make a trapezoid with the seven tangram pieces.
7. Use two tangram pieces to make a triangle.
8. Use three tangram pieces to make a triangle.
9. Use four tangram pieces to make a triangle.
10. Use five tangram pieces to make a triangle.
11. Use six tangram pieces to make a triangle.
12. Take the five smallest tangram pieces and make a square. 13. Using the letters on the tangram
pieces, determine how many ways you can make:
- squares
- rectangles
- parallelograms
- trapezoids
(Be sure to list all the ways possible to make the above.)
14. Work with a partner to come up with as many mathematical terms or words related to tangrams
as you can.
15. Make a rhombus with the smallest three triangles, make a rhombus with the five smallest pieces
and make a rhombus with all seven pieces.
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Below are some patterns you can challenge students to reproduce – using all of the tangram pieces
with none left over. After that, what new pictures can they design themselves?
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Mandalas
A mandala, common to Indian meditation traditions, is a repetitive design within a circle. Encourage
students to develop color patterns when coloring the following designs. The final mandala page is
actually a blank mandala, where students can create their own mandala.
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Celtic Knots
Maybe it’s because, if you trace all the lines of my heritage (except the apocryphal Cherokee one) back
far enough, you hit a Celt tribe somewhere, but I think Celtic knots are nifty as all get-out. I’m including
some fun with them here because a) they’re beautiful and b) they are patterns with allowances for onthe-fly rule-making, which fits in nicely with some of our work on abstract thinking.
We’ve got some knot work from the Book of Kells to study and color, directions on how to build knots
using the traditional method, and directions on how to build easier knots that I made up myself during
Finite Math back during my freshman year in college (D+!).
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Making Celtic Knots (dots method)
This is a method of drawing Celtic knots that I found in a book of Celtic and Anglo-Saxon painting. It
called the knots 'interlace', and said "Interlace is not a motif that can be learned by simply looking at a
model. One must know the 'trick', and from unfinished interlace borders we can tell how it was usually
made up." This suggests that there was only one method, but looking at examples of Celtic knots, I
suspect that several methods was used. This method would only work for close-weave knots in a simple
border.
Start by drawing dots in a diamond lattice pattern like this. You would normally draw
this in black, but I'm making the dots red to contrast with the later lines.
The dots should be diamond-shaped themselves. When you start drawing lines, draw
them alongside the dotes rather than through the centre, otherwise this technique
doesn't work.
The patterns below are from the Durham Gospel.
Simple plait with four strands
Draw the top left diamond. Draw the top left and bottom right sides only. Keep
inside the dots. This is the first strand.
Draw a curved line at the top. This represents the strand bending round to go
downwards.
Draw the lower diamond the same, still keeping inside the dots. This will make the
long line look wonky. This is the second strand.
Draw the middle diamond. This time you draw the bottom left and top right sides.
Keep within the dots! This is the third strand.
Draw the top diamond and the top curve, as before. This continues the second
strand.
Draw a bottom curve and bottom diamond, to start the fourth curve.
The middle diamond continues the first strand.
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The top diamond and the top curve diamond continues the fourth strand.
Continue to complete the knot. I have changed the red dots to black so you can see
the finished effect. There is a suggestion of a black background as well, to heighten
the effect.
Twists with four strands
This design starts the same as the last one.
Continue the top curved line twice as far as last time. It's better to rub out the
surplus dot altogether.
Continue with the next two diamonds, the same as last time.
Make a second shorter curve, below the top one.
Make a long curve at the bottom, remembering to remove the surplus dot.
Make a short curve above the bottom curve.
Draw the second middle diamond.
Draw the second middle diamond.
Draw the two outer curves...
... then the middle two diamonds.
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Here is the final result.
Entangled loops
Draw a line straight down in the centre. This is the start of a new loop.
Draw in a normal under and over from bottom left heading up and right, and
curve it round.
Make a short curve, bending round the top of the loop.
Draw in another under and over next to the previous one, but this time bend
it round with a long curve. removing the middle dot.
Make a long curve at the bottom in the same way.
Draw a line straight down in the centre. This is the end of the old loop.
Repeat.
The final pattern.
I don't think this can have been a design tool for Celtic knots, since it's quite easy to get lost (which is
why I've broken it down into small steps). But if you designed a rough draft using a looser design
technique (see below), then this method could be useful for transferring your pattern to the final copy.
It would also be useful for bending patterns round curves, to fit inside letters, for example. It can be
hard to predict the angles of the lines, but you could mark a pleasing regular pattern of staggered dots,
then fit the pattern round it.
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Making Celtic Knots – Square Matrix Method
1. First, draw small squares around the perimeter of a square or rectangle:
There must be a minimum of 4 squares on a side – but anything more than 4 is fine. This is a square
with 5 small squares on each side.
2. The small squares are the holes in the knots. Now you need to draw the knots around the squares:
The knots are formed by sets of parallel lines on the sides of the small squares that cross over and under
each other.
3. The finished basic weaving:
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ESL Language Discovery Camp 2013 Math Curriculum
4. Now the ends of the strings are hanging loose, and they need to be connected. This is the fun part.
Use loops and arcs to connect one loose end to another:
5. When the ends are all connected, the final product will look like a very simple Celtic knot:
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Extra Activities
These final pages are simply spare activities that are left over from the gleaning done to flesh out the
curriculum. They fit in nicely with all the other things you have been doing, so feel free to pull these up
if you’ve gone and run out of things to do, or you want to take the day’s math lesson in a slightly
different direction.
Secret Codes
Falsehood Follies
Peculiar Patterns
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Secret Codes
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Falsehood Follies
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Peculiar Patterns
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Answer Keys
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How Much – How Many
1.
2.
3.
4.
Many
Much
Much
Much
5.
6.
7.
8.
Many
Much
Many
Much
9. Much
10. Many
Analogies
Sentence Analogies 1:
1. C
2. B
3. C
4. A
5. A
6. A
7. A
8. C
9. B
10. C
4. B
5. A
6. A
7. A
8. B
9. C
10. B
Sentence Analogies 2:
1. A
2. A
3. B
Analogies
Dark: Night is the opposite of day, so dark must be the opposite of light.
Flower: A pine is a type of tree, so a rose must be a type of flower.
France: Washington, D.C. is the capitol of the United States, so Paris must be the capitol of France.
Lunch: Breakfast is the meal you eat in the morning, so lunch must be the meal you eat at noon.
Shoes/Socks/Etc.: A hat is worn on the head, so shoes must be worn on the feet.
Cut: You write with a pen, so you must cut with scissors.
Cold: It is hot during the summer, so it must be cold during the winter.
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