A double number line is two lines drawn one above the other, marked so the pairs line up.
Step 1: Picture It
Start with something you already know.
A ruler with two edges
- Some rulers have inches along one edge and centimeters along the other.
- Both edges start at the same 0.
- The 1-inch mark sits beside about 2 and a half centimeters.
- The 2-inch mark sits beside twice that, about 5 centimeters.
A double number line is that ruler: two scales, one shared 0, moving together.
Find a mark on one line, and its partner sits directly opposite it.
Where the picture stops: a ruler's pair never changes; each problem sets its own.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A punch recipe uses 3 cups of juice for every 5 cups of water. Priya makes it with 12 cups of juice. How many cups of water does she need?
cups
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Equal steps on both
If the top line goes up by 3 each mark, the bottom goes up by 2 each mark. Every column is the same ratio.
What it is for
Find the column you know, then read the answer off the other line.
Why bother drawing it
A table hides the spacing. A double number line shows that halfway along the top is halfway along the bottom.
Step 3: Try It Yourself
Change the ratio and watch both lines re-mark themselves.
| Cups of juice | Cups of water |
|---|---|
| 3 | 4 |
| 6 | 8 |
| 9 | 12 |
| 12 | 16 |
Every row is the same ratio. 3 to 4 is the same comparison as 12 to 16.
Step 4: In Real Life
Miles and minutes
A runner does 3 miles in 30 minutes. Two lines: miles on top, minutes below. 6 miles lines up with 60 minutes. Coaches plan pace this way.
Step 5: Watch an Example
One step at a time.
Watch Ana Use a Double Number Line
Juice to water is 3 : 4. Ana needs 12 cups of juice.
- Step 1
Ana draws two lines, one above the other, with juice on top and water underneath. Both start at 0, in the same column.
Watch Sam Read the Other Way
Sam mixes paint. Red to yellow is 4 : 3. He has 12 cups of yellow, and you fill in the rest.
- Step 1
Sam puts red on the top line and yellow underneath. Both start at 0, and the first mark carries 4 cups of red above 3 cups of yellow.
Step 6: Your Turn
Practice makes it stick.
The Bake Sale
Problem 1 of 15
Muffins are sold 3 for $2. A stall needs 12 muffins. Reading across the double number line, what do the 12 muffins cost, in dollars?
dollars
The Car Wash
Problem 2 of 15
A car wash charges $5 for every 2 cars. A club paid $30. Reading across the double number line, how many cars were washed?
cars
Read Across the Lines
Find the column you know first. Count how many marks along it sits, then take the same number of steps on the other line.
Problem 3 of 15
Flour to water is 3 : 5. What does the water line show in the column where the flour line shows 9?
cups
Problem 4 of 15
Red to blue is 4 : 7. What does the red line show in the column where the blue line shows 21?
beads
Problem 5 of 15
Miles to minutes is 2 : 9. What does the minutes line show in the column where the miles line shows 10?
minutes
Problem 6 of 15
Sugar to flour is 6 : 10. A halfway mark has been added between 0 and that column. What does the flour line show at the halfway mark?
cups
Problem 7 of 15
On this double number line the third mark shows 15 on top and 24 below. What does the bottom line show at the first mark?
Whose Step?
One line in each page is a slip. Find it, then fix it.
Problem 8 of 15
Dev was asked what the water line shows where the juice line shows 12, on a double number line where juice to water is 3 : 8. One line is a slip. Which?
Dev's answer for: juice to water is 3 : 8, juice line shows 12
- 1juice to water = 3 : 8
- 2juice line: 3 × 4 = 12
- 3water line: 8 + 4 = 12
- Answerthe water line shows 12 cups
Problem 9 of 15
What should Dev's line 3 read?
Problem 10 of 15
Maya was asked what the sticker line shows where the card line shows 20, on a double number line where cards to stickers is 5 : 3. One line is a slip. Which?
Maya's answer for: cards to stickers is 5 : 3, card line shows 20
- 1cards to stickers = 5 : 3
- 2card line: 20 ÷ 5 = 4 marks along
- 3sticker line: 3 × 5 = 15
- Answerthe sticker line shows 15 stickers
Problem 11 of 15
What should Maya's line 3 read?
Pick the Move
Decide the first move before you make it. Look at which line carries the number you were given.
Problem 12 of 15
The first column of this double number line holds 4 cups above $6. You need the dollars in the column holding 10 cups. What is the first move?
Problem 13 of 15
The first column of this double number line holds 7 pages above 8 minutes. You need the pages in the column holding 48 minutes. What is the first move?
Same Column?
Decide what is true without working any answer out.
Problem 14 of 15
A double number line holds 3 above 4 at its first mark. Select every pair that sits in a later column of this same line.
Problem 15 of 15
Select every statement that is true of any double number line.
Step 7: Quick Check
Show what you know.
Question 1 of 1
Ratio 2 : 9. If the top shows 6, what does the bottom show?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A double number line holds 6 above 15 in one column and 10 above 25 in another. Every mark is a whole number on both lines. Which pair sits at its first mark?
Stretch 2 of 2
A stall sells shirts at one fixed price. On its double number line the column holding 3 shirts holds $21. Noor wants a column holding exactly $50. Can one be drawn with a whole number of shirts?
What You Learned
- A double number line marks two quantities so their pairs line up.
- Both lines take the same number of steps.
- Find the value you know, then read straight across.
For the grown-up
The misconception this lesson targets is additive scaling. A child who needs the top line to go from 3 to 12 adds 9, then adds 9 to the bottom line as well, and the mix has quietly changed — 3 : 8 becomes 12 : 17, which is a different drink. A double number line is the best place to catch this, because the columns visibly stop lining up. Two smaller slips ride alongside it. The first is multiplying by the wrong number: by the other line's step rather than by how many marks along the column sits. The second is uneven spacing — marks that are not equally spaced make the line lie about size, which is the one thing it is drawn to show. Asking "how many marks along is it, on both lines?" catches all three.
Two number lines, kept in step — A double number line puts one quantity on the top line and the other underneath, aligned so that matching values sit above each other. Move along the top by a factor and the bottom moves by the same factor. It is a ratio table drawn to scale.
Reading a value off it — If 3 kg costs $12, mark 3 above 12. Halving gives 1.5 above 6. Doubling gives 6 above 24. To find the cost of 4 kg, get to 1 kg first — 1 above 4 — then multiply up. The line makes the intermediate steps visible.
When to use it rather than a table — A table is quicker to write; a double number line shows the relative sizes, so it is better when you need to judge whether an answer is sensible. Many people sketch the line to think and use the table to record.
The spacing has to be honest — Equal steps along the line must represent equal amounts. A line where 1, 2 and 3 are close together and 10 is just after 3 will mislead you about size. If you are drawing it to think with, draw it roughly to scale.
Two lines, one scale — Top line: cups of flour 0, 3, 6, 9. Bottom line: cups of water 0, 2, 4, 6. The marks line up because both were multiplied by the same numbers. To find water for 12 flour: 12 is 4 lots of 3, so water is 4 lots of 2 = 8.