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How to use a double number line for ratios

The idea: two lines, one pair of marks, then scale

A double number line is two parallel lines, one for each quantity, drawn so that matching amounts sit one above the other. The recipe pins one pair - say $2$ cups of milk under $6$ teaspoons of chocolate - and everything else follows from it. The move that solves almost every task of this type is finding the unit rate: how much of the second quantity goes with exactly $1$ of the first. Divide both numbers by the first one, and mark that pair on the lines:

\frac{6 tsp}{2 cups} = 3 tsp per cup

After that, any other amount is one multiplication away: for $n$ units, the second quantity is $n$ times the unit rate. Equal steps on the top line always match equal steps on the bottom one - that is the whole point of drawing them together.

Problem

Worksheet tasks about a chocolate milk recipe, all on a double number line. Example 1. The recipe uses $1$ cup of milk and $4$ teaspoons of chocolate powder. How much powder is needed for $1$ cup? Show it on the line. Example 2. Same recipe. How much powder is needed for $4$ cups of milk? Example 3. A different recipe uses $2$ cups of milk and $6$ teaspoons of powder. How many teaspoons go with every single cup?

Example 4. Orders of 5, 10, 15, and 20 textbooks weigh 16, 32, 48, and 64 kg. Write an equation for total weight $y$ in kg in terms of the number of textbooks $x$ .

The double number line for 4 teaspoons per cup: equal steps on the top line match equal steps on the bottom one.

Answer

1

cup

\to 4

teaspoons;

4

cups

\to 16

teaspoons; and a recipe of

2

cups to

6

teaspoons means

3

teaspoons per cup. For the textbook orders,

y = 3.2 x

The original three ratio examples and the added proportional table were checked by hand.

Step-by-step solution

Example 1: the amount for 1 cup

The recipe already states the pair: $1$ cup of milk goes with $4$ teaspoons of powder. Nothing has to be computed - the answer is read straight off the recipe:

1 cup \to 4 teaspoons

On the lines, put $4$ on the bottom line directly under the $1$ on the top line. That pair is the anchor for everything else.

Example 2: scaling up to 4 cups

With $4$ teaspoons per cup, four cups need four times as much:

4 cups \cdot 4 \frac{tsp}{cup} = 16 teaspoons

On the line the same thing is visible as counting in equal steps: $0, 4, 8, 12, 16$ under $0, 1, 2, 3, 4$ . Each step on the top line of one cup matches a step of four teaspoons below it.

Example 3: from 2 cups to the unit rate

This recipe pins a different pair: $2$ cups to $6$ teaspoons. To find what goes with a single cup, divide both quantities by $2$ :

\frac{6 tsp}{2 cups} = 3 \frac{tsp}{cup}

So the box under the $1$ holds $3$ . Once that mark is in place, the rest of the line is just repeated addition of $3$ : $0, 3, 6, 9, 12$ .

Example 4: from matching values to an equation

For the textbook table, divide the weight by the number of books: $16/5 = 32/10 = 48/15 = 64/20 = 3.2.$ The same ratio appears in each row: 3.2 kg per book. On a double number line, the mark for 1 book therefore matches 3.2 kg. For $x$ books, multiply by $x$ to get $y = 3.2 x .$ This proportional relationship passes through $(0, 0)$ : zero books have zero total book weight.

How to check yourself

Pick any pair of marks that sit one above the other and divide the bottom number by the top one. Every pair on a correct double number line gives the same result - the unit rate. If one pair disagrees, that mark is wrong.

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