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  • Three Heights

Three Heights Problem



In this class we will explore: (a) How the children deal with comparisons, (b) How they draw inferences from comparisons, and (c) How they represent comparisons between three unknown amounts.


Further work with addition (and subtraction) as functions, instead of simply operations on particular numbers.


Overheads, Handouts


Contextualized Situations

Describing Magnitudes

Full Class Discussion

Function Representations

Interpretation of Stories

Production of Algebraic Expressions

Production of Tables

Representing Variables

Small Group Work


Activity Plan:

Reasoning about Relative Heights

1. The Problem [Whole Class ]

Present the Three Heights Problem on the overhead (page 1) to whole class.

Enact the heights in the class by comparing the height of three volunteer children.

2. Drawing the Heights [Whole Class/Group Work ]

Ask children to represent the three heights on the handout (page 2). 

Choose two or three representations to discuss.

Attribute some possible values for one of the characters in the problem and ask the children to determine the height of the others.

3. Table of Possible Values [Whole Class/Group Work]

Show the overhead on page 3 and, together with the children, fill out the first row of the Table of Possible Answers.

Distribute the handouts (also page 3) and give special attention to how they fill out the last three lines where one of the heights is represented by letters.

Children present, discuss, and explain their answers to the whole class.

4. Homework (Pages 4 & 5)

Overhead: The Heights Problem            (Page 1)






Overhead and Handout: Showing the Heights (Page 2)


Name: _________________________________ Date: ____________


Tom is 4 inches taller than Maria.

Maria is 6 inches shorter than Leslie.

Draw: Tom’s height

Maria’s height

Leslie’s height

Show what the numbers 4 and 6 refer to in your drawing.





























Overhead and Handout: Possible Stories about Height                                                                                                                                             (Page 3)


Name: _________________________________ Date: ____________


Tom is 4 inches taller than Maria.

Maria is 6 inches shorter than Leslie.

Fill out the table for each story.


What if…

Tom’s height is:

Maria’s height is:

Leslie’s height is:

Story 1

34 inches



Story 2


37 inches


Story 3



37 inches


Describe Maria’s height, using the letter N .

[Hint: Is Maria taller, shorter, or the same height as Tom?]

Relations 1

N inches




Now describe Tom’s height, using the letter R .

[Hint: Is Tom taller, shorter, or the same height as Maria?]

Relations 2





Now try this one where Z describes Leslie’s height.

Relations 3





Overhead and Homework (Page 4)


Name: _________________________________ Date: ____________



Money Rule 1: “Add $4.00 to the input amount.”

Money Rule 2: “Take away $6.00 from the input amount.”



Complete the following table:

Input Amount

Apply Money Rule 1 to the Input Amount

Apply Money Rule 2 to the Input Amount













K dollars





Overhead and Homework (Page 5)


Name: _________________________________ Date: ____________



Here is the Heights problem you worked on today.  Complete the table:


Tom is 4 inches taller than Maria.

Maria is 6 inches shorter than Leslie.


Maria’s height

Tom’s Height

Leslie’s Height

45 inches



46 inches



47 inches



50 inches



P inches





Make a rule to get Tom’s Height from Maria’s height:





Make a rule to get Leslie’s Height from Maria’s height: