Tape Diagrams & Double Number Lines - Sixth Grade Mathematics

Visual models like tape diagrams and double number lines provide concrete spatial representations of proportional relationships, making complex multi-step ratio problems intuitive and transparent.

The Tape Diagram (Bar Model)

A tape diagram (also known as a bar model or strip diagram) is a visual drawing composed of rectangular blocks of equal size used to illustrate the parts of a ratio.

The Fundamental Tape Diagram Principle:
EVERY single block (unit) in a tape diagram represents the EXACT SAME value!

Modeling Ratio of Red Paint to Blue Paint (2 : 3):
Red:   [   ] [   ]
Blue:  [   ] [   ] [   ]

If the total paint mixture is 40 liters:
Count total blocks: 2 + 3 = 5 blocks.
Value of 1 block: 40 liters / 5 blocks = 8 liters per block.

Fill in each block:
Red:   [ 8 ] [ 8 ]               = 2 * 8 = 16 liters
Blue:  [ 8 ] [ 8 ] [ 8 ]         = 3 * 8 = 24 liters
Check: 16 + 24 = 40 liters.

Solving "Difference" Problems with Tape Diagrams

Tape diagrams are exceptionally powerful when word problems describe the difference between two quantities rather than their total.

Problem: The ratio of cats to dogs at a shelter is 3 : 5.
There are 14 more dogs than cats. How many total animals are at the shelter?

Step 1: Draw the tape diagram:
Cats: [   ] [   ] [   ]
Dogs: [   ] [   ] [   ] [   ] [   ]
                        |---------|
                      Extra 2 blocks = 14 dogs!

Step 2: Calculate the value of 1 block:
        2 blocks = 14 animals  -->  1 block = 14 / 2 = 7 animals.

Step 3: Calculate requested quantities:
        Cats = 3 blocks * 7 = 21 cats.
        Dogs = 5 blocks * 7 = 35 dogs.
        Total animals = 21 + 35 = 56 animals.

The Double Number Line Model

A double number line diagram uses two parallel, aligned number lines that share the origin zero, with tick marks spaced proportionately to track matching values between two different measurement units.

+-------------------------------------------------------------------------+
|                  DOUBLE NUMBER LINE: DISTANCE VS TIME                   |
|                                                                         |
| Distance (Miles)                                                        |
|   0        15        30        45        60        75        90         |
| --|---------|---------|---------|---------|---------|---------|-->      |
|                                                                         |
| Time (Hours)                                                            |
|   0         1         2         3         4         5         6         |
| --|---------|---------|---------|---------|---------|---------|-->      |
+-------------------------------------------------------------------------+

Navigating Double Number Lines

In the double number line above, the top line tracks miles while the bottom line tracks hours: Every tick mark aligns matching proportional values: 15 miles aligns with 1 hour; 30 miles aligns with 2 hours; 45 miles aligns with 3 hours. To find a fractional value, such as 2.5 hours, locate the exact midpoint between 2 and 3 hours on the bottom line. Move vertically up to the distance line: the midpoint between 30 and 45 is 37.5 miles.

Choosing the Best Visual Tool

Choose between a tape diagram and a double number line based on the units involved: Tape Diagrams: Best suited when comparing quantities with the same units (e.g., cups to cups, dollars to dollars, animals to animals) or when parts combine into a tangible sum or difference. Double Number Lines: Best suited when comparing two completely different measurement units (e.g., miles to hours, dollars to pounds, kilometers to liters) and tracking continuous rates.

Exercises

Exercise 1. The ratio of fiction to non-fiction books on a shelf is 4 to 1. There are 60 books total. Draw and describe a tape diagram to find how many fiction books are on the shelf.

Exercise 2. A baker uses a ratio of 5 cups of flour to 2 eggs. If the baker uses 18 more cups of flour than eggs, use a tape diagram to determine the number of eggs used.

Exercise 3. Draw a double number line showing that 3 movie tickets cost $36. Use the diagram to find the cost of 5 movie tickets.

Solutions

Solution to Exercise 1. Draw 4 blocks for fiction and 1 block for non-fiction, giving 4 + 1 = 5 total blocks. The total collection is 60 books. Divide 60 by 5 to find the value of each block: 60 / 5 = 12 books per block. Fiction has 4 blocks: 4 * 12 = 48 books. Non-fiction has 1 block: 1 * 12 = 12 books. There are 48 fiction books on the shelf.

Solution to Exercise 2. Draw 5 blocks for flour and 2 blocks for eggs. The difference between flour and eggs is 5 - 2 = 3 blocks. The problem states there are 18 more cups of flour than eggs, so the 3 extra blocks represent 18 units. Value of 1 block: 18 / 3 = 6. Eggs are represented by 2 blocks: 2 * 6 = 12 eggs. (Flour is 5 * 6 = 30 cups; check: 30 - 12 = 18).

Solution to Exercise 3. Set up two parallel lines starting at 0. Top line is Cost ($), bottom line is Tickets. Place a tick mark aligning 3 tickets with $36. Dividing by 3 establishes the unit rate: 1 ticket aligns with $36 / 3 = $12. Mark consecutive tickets: 1 ticket = $12; 2 tickets = $24; 3 tickets = $36; 4 tickets = $48; 5 tickets = $60. Five movie tickets cost $60.