Bar Models — A Complete Journey

What a bar can do

A bar is a picture of an amount. Long bar means more, short bar means less. Two bars side by side compare two amounts.

Draw a bar for $40. Mark off the $15 she spent.

What's left is the part of the bar that isn't shaded red. That's $40 − $15 = $25.

Two bars side by side

Two people, two bars. Jerry's bar is some length we don't know. Tom's bar is the same length plus an extra $40.

Two equal bars together make $80. So one bar is $80 ÷ 2 = $40.

Which bar should be your "1 unit"?

So far the problems told you who had the smaller amount. But what happens when there are three people? Watch carefully — the choice of which one to call "1 unit" matters a lot.

Count up all the units: 1 + 2 + 4 = 7 units = $140. So 1 unit = $20.

When someone has "more than" or "less than"

The rule about picking the smallest changes a little when the comparison isn't purely multiplicative. Watch this one.

Bina is described two ways: "twice as much as Anya" (multiplicative) and "$20 less than Carl" (additive). Anya is described only one way — as the thing Bina is built from. Anya is the cleanest starting point.

Count units: Anya 1 + Bina 2 + Carl 2 = 5 units. Plus Carl's extra $20.

So 5 units + $20 = $200. Cover the $20 piece: 5 units = $180. So 1 unit = $36.

Fractions of a whole

Draw the whole jar as one bar. Split it into 5 equal parts (the denominator). Two of them are blue.

2 parts = 30 marbles. So 1 part = 15 marbles. Total = 5 parts = 75 marbles.

When the story goes forward but the unknown is at the start

Many problems describe what happened in order — first this, then that — but they ask about the very beginning. Your instinct might be to start drawing from the first sentence. Resist that instinct.

Step 1: Where the story ends

After everything, Tom has $15. Draw that.

Step 2: Undo lunch

Before lunch, Tom had "what was left after the toy." He spent half of it on lunch. So $15 is the other half. The full amount before lunch was twice $15 = $30.

Step 3: Undo the toy

Before the toy, Tom had $30 + $20 = $50. We just glue the $20 back on.

Backwards with fractions

The known number is 90, at the end. Start there and walk backwards.

Step 1: End of day

90 apples left after the afternoon sale.

Step 2: Undo the afternoon

In the afternoon, she sold 1/4 of what she had. The remaining 90 is the other 3/4. So 3 parts = 90 apples, which means 1 part = 30 apples. Before the afternoon, she had 4 parts = 120 apples.

Step 3: Undo the morning

In the morning, she sold 1/3 of her apples. The remaining 120 is the other 2/3. So 2 parts = 120, meaning 1 part = 60. She started with 3 parts = 180 apples.

The language of ratios

A ratio is what you get when you say "for every this many of one thing, there are that many of another." Bars handle ratios beautifully — each unit in the bar is a part of the ratio.

"3 : 5" means red is 3 parts, blue is 5 parts. Draw 3 red units, 5 blue units. Eight units in total.

8 units = 40 marbles. So 1 unit = 5 marbles. Red = 3 × 5 = 15. Blue = 5 × 5 = 25.

Age problems — the secret trick

Age problems sound tricky because time moves and so do all the ages at once. But there's a secret that makes them easy:

Right now: daughter = 1 unit, mother = 4 units. Draw two bars.

The mother is 3 units older than the daughter today. That gap of 3 units is the age difference — and it stays the same forever.

In 10 years, both have aged by 10. The mother is now twice the daughter, so daughter = 1 new unit, mother = 2 new units. The mother is 1 new unit older than the daughter.

From "now": daughter = 1 old unit. Mother is 3 old units older.
From "future": daughter is 1 new unit = 3 old units. So in 10 years, daughter is 3 old units. But she also gained 10 years.

So: 1 old unit + 10 = 3 old units → 2 old units = 10 → 1 old unit = 5.

The kind of problem that bends bars

Pick the smallest as 1 unit: Bina is 1 unit, Aiden is 3 units.

Now Aiden spends $48 and Bina spends $8. Try to mark those off on the bars.

But here's the trouble: we don't know what one unit is worth. We can't tell how much of a unit $48 takes up, or how much $8 takes up. Both the "starting" and "ending" moments contain unknowns. Drawing-from-the-end won't help this time — the end is just as unknown as the start.

The best we can do is express the leftover amounts in mixed form:

• After spending, Bina has (1 unit − $8)
• After spending, Aiden has (3 units − $48)
• Aiden's leftover is twice Bina's leftover: (3 units − $48) = 2 × (1 unit − $8)

The bar gets a name: x

Look at the Aiden-Bina problem. We said "Bina has 1 unit." Let's call that unit x. Same idea, shorter to write.

The two rules for working with x

You can treat x like a number you just don't know yet. There are two basic moves you'll use over and over.

Move 1: You can do the same thing to both sides of an equation.

If two things are equal, and you change both of them the same way, they're still equal. Add the same amount to both sides — still equal. Subtract the same amount — still equal. Multiply or divide both sides by the same number — still equal.

This is how you simplify an equation step by step until x is alone on one side.

Move 2: You can open up parentheses by multiplying.

2(x − 8) means "2 copies of (x − 8)." Which is 2 copies of x minus 2 copies of 8 — that is, 2x − 16.

This is called distributing. The 2 reaches inside the parentheses and multiplies everything in there.

That's it. Two rules. With these, you can solve our problem.

Solving the Aiden-Bina problem

We had:

3x − 48 = 2(x − 8)

Step 1. Open up the parentheses on the right side using Move 2:

• 2 × x = 2x
• 2 × 8 = 16
• So 2(x − 8) = 2x − 16

Now we have: 3x − 48 = 2x − 16

Step 2. Use Move 1 to get all the x on one side. Subtract 2x from both sides:

• Left side: 3x − 2x − 48 = x − 48
• Right side: 2x − 2x − 16 = −16

Now we have: x − 48 = −16

Step 3. Use Move 1 again. Add 48 to both sides:

• Left side: x − 48 + 48 = x
• Right side: −16 + 48 = 32

So x = 32.

That means Bina started with $32. Aiden started with 3x = 3 × 32 = $96.

When to use which

You now have two tools, and a few rules for choosing how to use them:

And a few rules for drawing well:

• Pick your unit small. The smallest amount is 1 unit. Build up from there. Avoid fractions of units.
• Find the moment you know. Start your drawing from a moment where a real number appears in the problem. Walk outward from there — backwards or forwards — by undoing or redoing the story's changes.
• When stuck, look for the new tool. If both ends of the story are unknown, the bars can only set up the relationship. The x finishes it.

Bars and algebra are not different math. They're the same thinking in different costumes. The bar is a picture you can see. The x is the same picture with a shorter name.

Strong mathematicians use both. They draw a bar to think. They write x to solve. They switch back and forth depending on which one makes the problem easier.

That's the whole secret.

How to debug your bar

When your answer doesn't check out, the bar is almost always trying to tell you something. Here are the four most common ways the bar goes wrong, and how to spot each one.

Mistake 1 — Picked the wrong amount as "1 unit"

Mistake 2 — Drew forward when the unknown is at the start

Mistake 3 — Forgot the extra piece is glued on, not part of the units

Mistake 4 — Bars not actually proportional

The check-yourself ritual

1. Substitute your answer back into the original story. Does every sentence in the problem come out true?
2. Add the parts and make sure they equal the stated total.
3. Look at the bar. Does the picture still look right with your answers written on it?

Practice — try them yourself

Twelve problems from across the lesson. Type your answer in the box, click CHECK. If you get stuck, click SHOW SOLUTION to see the worked walkthrough.

Tip: draw a bar first. Don't try to do these in your head.