Math Solutions Lesson from the Classroom
What’s the Secret Code?
A Lesson Plan for Grades 6–8
Linda Dacey and Karen Gartland
Featured in Math Solutions Online Newsletter, Issue 37
Overview of Lesson
Strand: Number
Focus: Fractions, decimals, and percents
Context: Solving secret number puzzles
The lesson is excerpted from Linda and Karen’s award-winning resource, Math for All:
Differentiating Instruction, Grades 6–8. The resource was selected as one of 2010’s most
outstanding teaching and learning products by the Association of Education Publishers
(AEP). It received three Distinguished Achievement Awards, deeming it the best of
Curriculum and Instruction: Math; Professional Development: Differentiated Instruction 6–8;
and Professional Development: Curriculum and Instruction 6–8. For support on how to
create your own tiered lessons, see page 110 of the resource.
Sometimes tasks need to be tiered in order to be successful with a wide range of
student readiness. Such activities allow students to focus on the same general concept or
skill, but to do so according to their levels of readiness. Finding the right combination of
accessibility and challenge is the goal of a tiered approach. This lesson is a tiered example
from a seventh-grade classroom. Students are given the task of solving a mathematical
mystery. Number puzzles that include clues to find a mystery number are used to reinforce
students’ recognition of classes of numbers, for example, square numbers, and familiarity
with the associated mathematical terms. They also provide opportunities for students to
strengthen their problem-solving abilities to reason deductively, make an organized list,
eliminate possibilities, and make inferences.
© 2010 Math Solutions, mathsolutions.com. Reproducible for one teacher’s classroom use only.
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What’s the Secret Code? Grade 6–8, continued
Materials
• copies of number puzzles (see Figure 4–4 at end of lesson)
• chart paper
• colored folders: 1 red, 1 blue, 1 green
Preparation
Create a chart listing students’ names in three groups according to their levels of readiness. Colorcode each group to correspond with a folder: red is associated with the first level of the task,
blue with the second, and green with the third—the most challenging level. Place copies of the
corresponding code problem (see Figure 4–4 at the end of lesson) in each folder.
Lesson Outline
Focus or Warm-Up
1. Begin the lesson by asking students, “Do you like to solve mysteries?” With most of the
students nodding “yes,” ask them what they need to solve a mystery. Students will likely
respond with ideas about how investigators collect and study evidence and clues. Draw
on their responses to connect their conversation to the idea that they will be solving a
mathematical mystery.
Introduction
2. Introduce students to a number puzzle by doing one as a whole class, sharing one clue at a
time with them in order to capture their attention and better assess their thinking. Here’s the
entire number puzzle:
What Could Be My Secret Code?
The number is between 400 and 410.
The tenths digit is one-third the hundredths digit.
All of the digits are different.
The tenths digit is one-half the ones digit.
There are five places in the number.
© 2010 Math Solutions, mathsolutions.com. Reproducible for one teacher’s classroom use only.
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What’s the Secret Code? Grade 6–8, continued
3. When you give the first clue, The number is between 400 and 410, ask students to think about
what numbers are possible. Follow similar questioning when unveiling the remaining clues.
Give students the opportunity to work in small groups and discuss their thinking.
Examples of Questions to Ask Students
• What numbers are possible?
• What does the word between mean?
• What might you learn from knowing that all the digits are different?
Exploration
4. When students are ready to investigate similar problems on their own, show them the chart
that lists the students’ names in three groups. Explain to students that within each group they
are free to form partnerships or to work alone. Hand the corresponding folder to each group
(see the “Preparation” section of lesson).
5. Ask students if they have any questions before turning the activity over to them. As students
solve their number puzzle, circulate and ask questions. See the vignette, “From Karen’s
Classroom: Explorations and the Red Group,” at the end of this lesson on page 4 for insights
into such interactions.
Summary
6. Although the students have worked on different code problems, it is important to have the
students come back together to discuss their thinking. Talking about the different tasks takes
the secrecy out of what each group was doing and reveals the similarity among the tasks.
Also, sharing ideas and experiences helps maintain their sense of community as a class.
See the vignette, “From Karen’s Classroom: Summarizing” on page 8 for insights into such
interactions.
© 2010 Math Solutions, mathsolutions.com. Reproducible for one teacher’s classroom use only.
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What’s the Secret Code? Grade 6–8, continued
7. Finally, ask students to identify problem-solving strategies they used to find the codes. As
they brainstorm, record the ideas for all to see. Examples may be:
Problem-Solving Strategies
• Read carefully to see if there is ths at the end.
• Make lists.
• Think about easier numbers to understand a clue.
• Talk together and share ideas.
• Make lines for the number to show where the digits can go.
• Decide the best clue first.
From Karen’s Classroom: Exploration and the Red Group
(For additional vignettes focused on the other groups, see pages 100–109 of
Math for All, Grades 6–8.)
The students who have been assigned to do the red (first tier) task have some difficulty figuring
out how to get started. As Karen observes them, she notices that several students are looking at
the written clues and several others are looking for pencils and a comfortable place to sit. Karen
knows that it will take this group a bit longer to get going, but that it will be worth it to have them
determine how best to work together in order to accomplish their task. She knows that if she
steps in too early, the students will not learn how to bring themselves to the table to work on their
own. So, she waits until they are settled before checking in on them. Their task is similar to the
one completed as a class, purposely repeating a clue that refers to one digit being one-third of
another.
Karen checks on the other two groups and once she observes that each group is engaged
in solving its number puzzle, she moves back to the red group, which has now settled in to work.
The second clue identifies the number of places in the code with hopes that it will help students
© 2010 Math Solutions, mathsolutions.com. Reproducible for one teacher’s classroom use only.
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What’s the Secret Code? Grade 6–8, continued
(From Karen’s Classroom: Exploration and the Red Group, continued)
organize the later clues. Mark immediately draws six blanks and writes the digits 0–9 above
each blank as possible choices for each place, including the thousands place not recognizing
immediately that the digit in the thousands place must be a 3. (See Figure 4–5.)
The task is a bit simpler than the whole-class puzzle in that there is only one possible code
number. Once they identify the code, students have to explain how two of the clues helped them
find the solution.
Several of these students seem to be stuck on the clue that tells them that the number
code is 0.41 less than a whole number. Tonya refers to the importance of thinking about zeros and
Mark doesn’t think there is any number that would fit all of the clues. Karen listens for a while and
then decides to ask the students how addition and subtraction might help them.
They seem a bit perplexed by this question until Mark says, “Maybe we could subtract from
one, but the numbers won’t line up.”
Tonya then connects to this idea by saying, “Oh, that must have been why I was thinking
about zeros; we need to do it like this,” and writing 1.00–0.41.
A couple of students complete the subtraction and once they do so, a few of the other
students pay closer attention
and appear to understand why
this clue identifies the last two
digits in the number. When a few
students in the group find the
secret code, there is growing
interest by the remaining students
to find it as well. When they do
so, they appear to believe that
their work is completed. Karen
knows that it can be a challenge
(continued)
© 2010 Math Solutions, mathsolutions.com. Reproducible for one teacher’s classroom use only.
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What’s the Secret Code? Grade 6–8, continued
(From Karen’s Classroom: Exploration and the Red Group, continued)
for some of these students to explain their thinking. She reminds them that they can work
together. Each person can talk about how one clue was used and then they will each have two
clues to explain. This idea seems to reenergize the students.
Ramie and Todd talk to each other about the first two clues and then write their responses
separately. Ramie recognizes that the first two clues allowed her to determine the first two digits
of the number, but does not explain how they do so. (See Figure 4–6.) Todd correctly recognizes
that knowing the digits are all odd allows him to determine the hundredths digit. (See Figure 4–7.)
(continued)
© 2010 Math Solutions, mathsolutions.com. Reproducible for one teacher’s classroom use only.
6
What’s the Secret Code? Grade 6–8, continued
(From Karen’s Classroom: Exploration and the Red Group, continued)
© 2010 Math Solutions, mathsolutions.com. Reproducible for one teacher’s classroom use only.
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What’s the Secret Code? Grade 6–8, continued
From Karen’s Classroom: Summarizing
The red group explains its work first. The teacher has displayed the clues to this task at the front
of the room so that presenters can point to the clues as needed and the rest of the class will have
access to the puzzle. As soon as the clues are revealed, several students who were in the other
groups begin to work on the puzzle. The teacher, wanting to maximize the time spent in wholegroup conversations, advises, “You may spend two minutes getting to know this puzzle, but be,
(From Karen’s Classroom: Summarizing,continued)
prepared to stop at that time and listen to the reporters. Your job is to ask questions as they
present their work.” Mark begins the report by saying, “We thought that this puzzle was going to
be really difficult to solve, but once we got going, it was easier than we thought.” Liza, a member
of the blue group, says, “That’s what happened to us, too.” When the blue group reports, the
teacher notes that members of the red group are staying engaged. Ramie asks, “How did you
know that four times one-fourth would get to a whole number?” Liza thinks for a few seconds
and then responds, “I thought of quarters. If you get four of them, you get a whole dollar.” Ramie
appears satisfied with this response. When Zak reports, “At first, we were not happy that our clues
didn’t get us to one answer,” Lucy, a student from the green group pipes in, “Yeah, us too!” The
teacher is particularly pleased that students remain engaged during the green group’s explanation.
She knows the other students can’t necessarily complete this version of the task, but being
familiar with the format allows them to feel comfortable to ask questions such as “How did you
use the third clue?”
© 2010 Math Solutions, mathsolutions.com. Reproducible for one teacher’s classroom use only.
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What’s the Secret Code? Grade 6–8, continued
© 2010 Math Solutions, mathsolutions.com. Reproducible for one teacher’s classroom use only.
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