Utilization Strategies

Lesson Plans

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by Karen Byers


TIME ALLOTMENT: Three 1 hour lessons

SUBJECT MATTER: Mathematics: algebraic thinking, proportions

Students need many early experiences with algebraic concepts before they formally begin to study the subject. Algebra can be defined as the study of patterns. Children learn to see relations and make predictions, generalizations, and connections through the study of patterns. In this lesson students will be assessed before the lesson to determine their level of understanding. Students will use colored beads in two problems and will explore the use of proportional reasoning as a problem solving method. Through careful questioning and modeling the teacher will encourage proportional reasoning. Working in groups, students will explain their reasoning strategies orally, and through drawings. Presenting their work validates students’ method s of thinking and increases confidence. They will observe another class demonstrating the lesson.


Students will:

  • Understand that patterns are things that repeat in an orderly way
  • Identify, replicate and create their own patterns
  • Understand and use proportional reasoning to solve problems
  • Be able to communicate understanding of the problem orally, with manipulatives, or with a drawing
  • Recognize patterns in the natural world and manmade world


National Council of Teachers of Mathematics Principles and Standards for School Mathematics, Grades 3-5, http://standards.nctm.org

Algebra Standard:

  • Understand patterns, relations, and functions
  • Use mathematical models to represent and understand quantitative relationships

    Problem Solving Standard:
  • Build new mathematical knowledge through problem solving
  • Apply and adapt a variety of appropriate strategies to solve problems
  • Monitor and reflect on the process of mathematical problem solving

    Reasoning and Proof Standard:
  • Make and investigate mathematical conjectures

    Communication Standard:
  • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others

    Connections Standard:
  • Recognize and apply mathematics in contexts outside of mathematics

    Representation Standard:
  • Use representations to model and interpret physical, social, and mathematical phenomena


Tell your own story of why you have a beaded section of a broken necklace. Use real objects and visual representations that model how the children will explain their work. Explain that we will watch a video showing some young mathematicians working together on a similar problem.


  • Pre-assessment worksheet
  • TV/VCR
  • Math journal
  • Markers
  • Pad of large paper
  • Large beads
  • String
  • Digital camera or Polaroid camera


Mathline Algebraic Thinking Math Project “Bead-dazzling”.


The complete Mathline website with ideas for many math topics.

National Council of Teachers of Mathematics website http://www.nctm.org

A site for female mathematicians http://www.agnesscott.edu/


This video is for teachers and students alike. As teachers view it they can focus on the pre-assessment and interview sections and the verbal prompts used by the teacher to elicit responses and analyze responses for understanding.

View the beginning of “Bead-dazzled”. Here the teacher demonstrates and talks about the interview process. (about 3 ½ minutes). Skip until you see the outside of a school. The next section shows how the teacher sets up the class with a personal story to entice the students. Students will see a model of other students’ classroom behavior, interest in the lesson, and methods of working.

Preview the video for your understanding and to know where your pause points will be for your students. Cue the video. Write the steps to the lesson on the board so the students will know what is expected of them during this lesson. (Introduction, The problem, Methods for solving and sharing) Prepare the materials.

PRIOR KNOWLEDGE This lesson should follow previous multiple experiences with many kinds of patterns.


Lesson 1
Step 1: Pre-assessment. This step will help you design instruction that helps the development of incomplete or missing concepts, pinpoint the error or misunderstanding about the nature of student thinking. Give each student a copy of the pre-assessment sheet to complete. Conduct one-on-one interviews to check their level of understanding of proportional thinking. Record information to keep track of for each student.

Step 2: View the video. Your questioning and explanations throughout the video will provide the FOCUS FOR MEDIA INTERACTION. Say, “Let’s see how this class works together to solve their problem.” PLAY showing how the students use drawing, discussing, numbers and writing to solve their problem.

STOP and ask what your students saw happening in the classroom. SKIP to the spot where a group of 3 girls and 1 boy work through the problem together.

PAUSE before the teacher talks about the 6 packs of beads. Ask what they saw happening in the group. Point out how they all wrote on their pads using numbers.

PLAY through the segment showing the problem-solvers using drawings. PAUSE and point that out. PLAY through the segment that shows the teacher explaining the next step until after the students explain their work at the board. STOP. Ask what your students saw on the group’s paper taped on the board. Did they notice how well the students explained their work?

Step 3: Use mathematicians’ names as you form groups. Some suggestions are; Fibonacci, Pythagoras, Euclid, Archimedes, Pascal, Descartes, Gauss, Boole. (We study historical figures in mathematics throughout the year. They come alive through their fascinating stories.) Don’t forget to include women who contributed to math.

Step 4: In your problem, the original necklace contained 20 beads (2 colors) and 5 are left from the broken necklace. Can you recreate the whole necklace from these 5 beads? Discuss the problem. Volunteers will explain their reasoning. Hand out pre-made packages of beads that contain either 2, 4, or 5 beads of 2 different colors. Each group gets one package so that different groups have different numbers of beads. Groups will answer the question: How many packs of beads will it take to make the original 20 bead necklace? Hand out chart paper and markers to each group. Check each group’s progress. One member of each group will present their solution at the board using their chart paper. Introduce the words, “ 1 package is to 2 beads as blank packages is to 20 beads, or 1 package is to 4 beads as blank is to 20, or 1 package is to 5 beads as blank is to 20.”

Lesson 2
Step 1: A new problem. Students will solve a similar problem. They will also assign values to the bead bags and determine the cost of the necklace. Give each group a pack of 4, 8, or 12 beads with 2 colors, chart paper and markers. Ask how many packs of beads they would need to make a necklace of 32 beads. They will create a chart that shows their reasoning and answer. Explain that there might be more than one way to get an answer. They can make any pattern they chose and must show their pattern on the chart. They will get the necessary packs and make the necklace they designed. One member of each group will present their solution at the board using their chart paper, their necklace, and the language of proportions.

Step 2: What is the cost of our necklace? Explain that a pack of 4 beads costs $4, a pack of 8 beads costs $5, and a pack of 12 beads costs $7 dollars. They will determine the cost of their necklace. Using the same procedures as the previous problem, each group will present their conclusions and reasoning.

Step 3: In the classroom, display all the work.


Lesson 3

Step 1: Students will look for patterns in nature and in the manmade world. Take a walk around your community and, using a digital camera or Polaroid, let students take photos of any patterns they see, (at least one for each student).

The student will decide if his pattern repeats (not all will) and will draw and describe the pattern in his math journal.

Student will apply their knowledge of: ____ is to _____ as _____ is to _____ in their descriptions.

Students will present their work at the board.

Step 2: Make a mural of the photos and students’ charts. Student Assessment: Read the math journals for improved understanding.


Look for patterns in the animal world and discuss why animals might need this characteristic. (camouflage, mate attraction, a signal to stay away.)

Language arts:
Daily use of a math journal.

Students can apply their new knowledge as they design and make a beaded bracelet on small looms. This will take a few days to accomplish. You’ll demonstrate the method in the first lesson. Students will design the bracelet using graph paper and colored pencils, The bracelet will be 3 beads wide.

Let the students work independently on this however you wish. They must make a repeating pattern and they must calculate how many beads they will need to complete a 5” beaded section. Make a class quilt with repeating patterns.

Students can use proportional reasoning to determine how many pieces of each color they will need.

Community Connections:
Weavers and quilters often use proportional reasoning. Invite one to the class to explain his/her craft.


For additional lesson plans and ideas relating to this topic and many others try TeacherSource at PBS Online! You will find activities, lesson plans, teacher guides and links to other great educational web sites! Search the database by keyword, grade level or subject area! Mathline and Scienceline are also great resources for teachers seeking teaching tips, lesson plans, assessment methods, professional development, and much more!

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