Rudy G Castro
Strategy Overview
Physical Representations are a way that you can use items around the house to solve a question. Items could be LEGO’s, coins, your cat, maybe a small cube such as a dice or box. Physical things are things you can touch, so if you can use your hands (without getting in trouble or hurt) then it works! A representation is a way to show your work, so think of whatever you may write down to add 2 apples plus two oranges. You can draw it, make a figure or show two people with 2 fruits.
For what types of problems is this strategy helpful?
- When the problems involve multiple items or certain circumstances.
- Guess and Check Questions!
When is it NOT very useful?
- When you are low in friends or have too many scenarios
- When you are in a fast-paced environment, it can take a lot of time
Common mistakes or misconceptions to watch out for:
- Probably that the questions are going to be fast or easy to answer…
How does this strategy help students become better mathematical thinkers? Give a sales pitch.
- Physical Representations allow you to literally visualize and have freedom to try and infinite amounts of possibilities!
- More or less, it allows you to see things from a new perspective.
Example Problems
Problem 1:
True Equations
Use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to make true equations. Use each digit once. (Johnson & Herr, 2001, p. 320)
Problem 2:
Baseball Seating
A family of five, consisting of Mom, Dad, and three kids (Alyse, Jeremy, and Kevin) went to a baseball game. They had a little trouble deciding who was to sit where. Alyse would not sit next to either of her brothers. Kevin had to sit next to Dad. Mom wanted to sit on the aisle but not next to any of the children, although she could sit next to her daughter as needed. How was the seating arranged? (Johnson & Herr, 2001, pg. 319):
Worked Examples + Think-Alouds
Problem 1:
Problem 2:
Student Self-Talk Questions
- Have I tried this way?
- What if I move ___ with ____?
Teacher Notes and Instructional Tips
Lesson Launch Ideas
- Ask the class to sort a list of 20 numbers (range 1-10000). Give one group a small cut-outs one for each number to move around. Give another group a list with the numbers, with no ability to use a writing utensil.
- While unorthodox, this will show the benefits of physical representations.
- To finish, ask groups what are the benefits of using manipulatives.
Teacher Questions & Anticipated Student Responses
- (T) Is there just one way to attack this question?
- (S) No
- (T) How could we act it out?
- (S) We all take one role
Differentiation Suggestions
-
Support:
- Have you tried ____ going there?
- Why not move _____ around?
-
Extend:
- Why can _____ not go there?
- Is there just one answer to this?
- Can you find another possible way?
Formative Assessment Tools
- A 15 minute quiz with 2 questions asking students to try and work out the problem with manipulatives
- Goal is to see their work, not accuracy.
- Students could record themselves working it out.
Student Reflection Prompts
- How did you start the problem?
- Is there possibly another problem, how does it start?
- What was the first step?
What prior knowledge do students need?
- They need to know about combinations, so that multiple things can be paired but it is not by force
What classroom challenges commonly arise?
- Time, the method while good at seeing the problem does take a significant amount of time to work out. Also limitations of resources could make it harder to create an accurate physical representation.
Media Attributions
- Using Digits to Make True Equations © Johnson & Herr is licensed under a All Rights Reserved license
- Solving the True Equations Problem © Rudy G Castro is licensed under a CC BY-NC (Attribution NonCommercial) license
- Solving the Baseball Seating Problem © Rudy G Castro is licensed under a CC BY-NC (Attribution NonCommercial) license
