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Drawing a Problem

The bar model for grown-ups, in four lessons: parts and a whole, more and less, equal units, and how a bar becomes an equation. Then let the child at your table teach you.

Spark & Whetstone · sparkwhetstone.org/learn/drawing-a-problem · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. Parts and a whole

Children in many schools now draw a word problem before they solve it. The drawing is a bar: one long rectangle for the whole amount, cut into pieces for the parts.

The point of the bar is to show what you know and what you do not. If you know the whole and one part, the missing part is the rest of the bar, and you find it by taking the known part away.

Three questions do most of the work: what is the whole? What do we not know? What stays the same? Ask them before anyone reaches for a sum.

Question. A sofa and a chair cost $900 together. The sofa costs $640. Which drawing fits?

  1. One bar for $900, cut into a $640 piece and an unknown piece.
  2. Two separate bars, one for $900 and one for $640, set side by side.
  3. A bar for $640 with another $900 added on the end.

Say it back. In your own words: what are the three questions you ask before drawing the bar?

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Lesson 2. More and less

Comparison problems use two bars, one under the other, so you can see the difference. If Ana has more than Ben, Ana's bar is longer, and the extra length is the difference.

The trick is to cover up the extra. Once the difference is taken off, the two bars are the same length, and the rest is sharing equally.

For example: Ana has $15 more than Ben, and together they have $85. Take the $15 away and $70 is left in two equal bars, so each equal bar is $35. Ben has the short bar; Ana has $35 plus the extra.

Question. Ana has $15 more than Ben. Together they have $85. How much does Ana have?

  1. $50
  2. $35
  3. $42.50

Say it back. Say it back: why do you take the difference away before you share?

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Lesson 3. Equal units

When a problem is about a ratio, the bar is cut into equal units. A ratio of 3 to 2 means three units of one thing for every two of another.

Once you know what one unit is worth, every other amount follows. Find the unit first; everything else is multiplying.

A cook who uses flour and sugar in the ratio 3 : 2 draws three equal boxes for flour and two for sugar. If the flour is 450 g, three boxes are 450 g, so one box is 150 g.

Question. Flour and sugar go in the ratio 3 : 2. You use 450 g of flour. How much sugar?

  1. 300 g
  2. 675 g
  3. 225 g

Say it back. In your own words: why is finding one unit the first step in a ratio problem?

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Lesson 4. From a bar to x

Algebra is the bar model written in letters. Call the unknown bar x, write down what the drawing shows, and you have an equation.

In the Ana and Ben problem, Ben's bar is x. Ana's bar is x and an extra 15. Both bars together are 85. So x + x + 15 = 85, which is 2x + 15 = 85.

Solving the equation is the same as the drawing: take the 15 away from both sides, then share what is left by two. A child who can draw it can usually follow the letters.

Question. Ben's bar is x. Ana has 15 more than Ben, and together they have 85. Which equation matches the drawing?

  1. 2x + 15 = 85
  2. x + 15 = 85
  3. 2x = 85 + 15

Say it back. Say it back: how does a bar drawing turn into an equation?

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At the table: The paper-strip table: the child teaches

Players: a grown-up and a child, 7 or older. You'll need: strips of paper, a pencil, scissors and one word problem from homework.

  1. The child reads the problem aloud. The grown-up does not solve it.
  2. The grown-up asks the three questions, in this order: what is the whole? What do we not know? What stays the same?
  3. The child draws the bar on a strip and cuts it into its parts. The grown-up copies the drawing, and gets corrected.
  4. Only then does anyone write a sum. Swap roles with the next problem.

Answers

  1. One bar for $900, cut into a $640 piece and an unknown piece. — The $900 is the whole, so it is the bar. The sofa is one piece of it and the chair is the rest: $900 − $640 = $260.
  2. $50 — Take the $15 away: $70 is left in two equal bars, so one bar is $35. That is Ben. Ana has $35 + $15.
  3. 300 g — Three boxes are 450 g, so one box is 150 g. The sugar is two boxes: 2 × 150 g.
  4. 2x + 15 = 85 — Ben is x and Ana is x + 15, so together they are x + x + 15. That equals 85.