Bar Models and Visual Ways Into Word Problems

By Laura Lurns · Reviewed by Philip Weaver · Last reviewed 2026-10-22

What the federal guide says about word problems

A US federal guide on extra math help came out in 2021 (Fuchs and colleagues). It covers children who struggle with math, from kindergarten to sixth grade. It is about help in small groups or one to one, at school.

It is not a guide to dyscalculia. Its studies were of help given in school, not at home. Nothing in the guide tests Learning Success.

The guide rates the evidence for teaching word-problem types as strong. Strong means its studies kept finding gains, across different groups of children. The grade is for the advice as a whole, not for each step in it.

Each study tested a whole package of help. So the guide cannot say what one step adds on its own.

Here are the steps the guide gives:

• Teach one type of problem at a time.
• Start with a story that gives every number.
• Teach a way to solve each type. Use a worked example and a card with the steps.
• Then change the problem. Move which number is missing. Add a step, a number that is not needed, or a chart.
• Teach the words that word problems use.
• Mix old types in with new ones.

The guide's examples include Change, Combine, Compare, Part-Part-Whole, Equal Groups and Ratio. In 12 of its 18 word-problem studies, children learned to sort problems by type. Then they used a way to solve that type.

Where bar models fit

Bar drawings appear across the guide's examples. The guide gives no separate grade for bar models. In its examples, drawings show how the amounts in a problem relate.

The guide rates the evidence for hands-on and drawn models, linked to the numbers, as strong. That grade is for the advice as a whole, not for any one kind of drawing. One of its steps is to show the drawing and the written numbers together. Which model suits which topic is the panel's advice.

The panel suggests quick sketches, like circles for objects, so drawing does not take up the time.

Key words can point the wrong way

The guide says not to teach key words that link a word to an operation. It shows problems where the key word points the wrong way.

Here is an example of our own. "Sam has 3 more cards than Ana. Sam has 8 cards. How many does Ana have?" The word "more" seems to say add. But the answer comes from taking away: 8 − 3 = 5. A bar drawing shows why. Sam's bar is 3 longer than Ana's. Take the extra 3 off Sam's 8, and Ana's bar is 5.

Bar drawings to try at home

The home routines on this page are ours. We hold no study that tests them at home. Do them alongside your child's math instruction, not in place of it.

Each one uses the guide's type names. The drawings are ours.

A Combine problem. "Mia has 5 red beads and 3 blue beads. How many beads does she have?" Draw a bar for the 5. Draw a bar for the 3 right after it. Draw a bracket over both, and put a question mark on it. Your child writes 5 + 3 = 8.

A Change problem. "Leo had 9 stickers. He gave away 4. How many are left?" Draw one long bar for the 9. Mark off a part for the 4 he gave away. The rest of the bar is the question. Your child writes 9 − 4 = 5.

A Compare problem. "Ava read 7 pages. Ben read 3 pages. How many more pages did Ava read?" Draw Ava's bar. Draw Ben's bar under it, starting at the same place. The extra bit at the end of Ava's bar is the answer. Your child writes 7 − 3 = 4.

Then move the question. Keep the story and hide a different number. "Mia has 5 red beads and some blue beads. She has 8 beads. How many are blue?" The drawing stays the same. Only the question mark moves.

A few things we suggest, from our own reasoning:

• Ask your child to name the type of story before drawing.
• Keep the bars rough. Letters or numbers on them are enough.
• Write the number sentence under the drawing.
• Let your child hold the pencil.

Is it the reading or the math?

Word problems ask for three things at once. The child reads the sentence. They work out which operation it describes. Then they do the arithmetic. A child who misses word problems may be missing any of the three.

Read the same problem aloud. If the answer is now right, the reading was at least part of the trouble. That is worth knowing before anyone changes the math.

Our position is that word problems need both reading and math thinking. Support both areas together.

When to ask for help

Each of these is a reason to ask for help on its own.

• Little change, even with steady practice and good teaching.
• Math well below your child's other schoolwork.
• A teacher who raises a concern about math.
• Extreme upset about math.
• Complete avoidance of math.
• Worry about math that spreads into the rest of learning.
• Sadness or worry that lasts.
• A skill that was there and has gone. Our position is that a lost skill goes to a doctor first, not a tutor.

If no skill has gone, start with the teacher. Say what you are seeing at home, and what you have tried. Beyond the teacher, the right person depends on what it turns out to be about. If the work is too hard, someone who works on that school skill. If worry or mood is the thing, someone who works with children's mental health. In our view, if you are worried enough to ask, a professional evaluation is worth asking about.

Any talk of hurting themselves is a same-day call to your doctor or a crisis line.