TEAS Math Percents and Proportions: Visual Ways to Understand Them

Short answer: three pictures cover most TEAS percent and proportion problems. Each one stands for an equation, so draw the picture, then write the equation it shows, then solve.

The picture is there to help you set the problem up. It only helps if you can say which number in the picture is which number in the equation (more on that below).

Why percent trips people up

Percent problems pack a lot into a few words. In a long review of how students learn percent, Parker and Leinhardt (1995) point to that compressed language, and to the small word “of,” which does several different jobs. Here are the same three numbers asked three ways:

• What is 35% of 240? 0.35 × 240 = 84
• 84 is what percent of 240? 84 ÷ 240 = 0.35 = 35%
• 84 is 35% of what number? 84 ÷ 0.35 = 240

On a bar, the difference is easy to see. The first question shades a piece of a bar you already have. The second asks how far along the bar a known piece reaches. The third gives you the piece and its percent and asks how long the whole bar is.

The percent bar

Draw one bar for the whole and mark 0% at one end and 100% at the other. In a design study of a long-term teaching sequence on percent, Van den Heuvel-Panhuizen (2003) used a bar like this as a tool for estimating and reasoning about percent. Students tend to start from friendly marks like 50% and 10% before they use the formal method (Lembke & Reys, 1994), and the bar lets you use both.

Percent of a number. A nursing program received 240 applicants and accepted 35%. Mark the bar: 10% is 24, so 30% is 72, and 5% is half of 24, which is 12. 72 + 12 = 84 applicants. 0.35 × 240 = 84. ✓ The bar also tells you 84 should be less than half of 240 (120).

A discount. Scrubs cost $80 and are 25% off. A quarter of the bar is $20, so you pay the other 75%: $80 − $20 = $60. ✓ If you answered $20, you found the discount, not the price.

Tax. $60 plus 8% tax. Add a piece past the 100% mark: 10% of $60 is $6, 1% is $0.60, so 8% is $6 − $1.20 = $4.80. $60 + $4.80 = $64.80. ✓

Percent change. A price goes from $50 to $65. The whole bar is the old price. The new piece is $15, and 10% of $50 is $5, so $15 is 30%. 15 ÷ 50 = 0.30 = a 30% increase. ✓ Dividing by the new price (15 ÷ 65, about 23%) is the most common slip.

Try TEAS percent problems with FurahaA fresh problem in place. Answer it, or tap “Walk me through it” for one small step at a time. Free, no account.Start

Tape models for word problems

A tape model draws the quantities in a word problem as strips, so you can see the structure before you write any algebra. Singapore’s “model method” uses it so children can solve word problems that would otherwise need letter algebra (Ng & Lee, 2009).

Example. A class of 40 students has men and women in a 3 to 5 ratio. How many women? Draw 3 equal boxes for men and 5 for women: 8 boxes hold 40 students, so each box is 5. Women: 5 × 5 = 25. Men: 3 × 5 = 15. ✓ (15 + 25 = 40, and 15 to 25 is 3 to 5.)

The strongest evidence for diagrams like this comes from large classroom trials of schema-based instruction, which teaches students to match each problem to a diagram of its structure. In a randomized study of 1,163 seventh graders, it beat the control classes on proportion problems, percents included, with an effect of g = 1.24 right after teaching and 1.27 six weeks later (Jitendra et al., 2013). Two cautions: those students learned a whole method built around the diagrams, not a picture alone, and the gains did not show up on a broader standardized test. They were also seventh graders, not adults preparing for nursing school.

Double number lines

A double number line puts two quantities on matched lines. The Common Core math standards name it, along with tape diagrams, as a tool for ratio problems and unit conversions (standard 6.RP.A.3). It’s a standard classroom tool, but we found no trial that tests it on its own, so treat it as a sensible way to see a proportion, not a proven one.

Example. You read 12 pages in 20 minutes. How many pages in 50 minutes? Put minutes on the top line (0, 20) and pages on the bottom (0, 12). Halve both: 10 minutes, 6 pages. Then count up to 50 minutes, five steps of 10: 5 × 6 = 30 pages. ✓ A ratio table is the same thing in rows: 20 → 12, 10 → 6, 50 → 30.

Cross-multiplication, with the reason

Write the same problem as two fractions: 12 over 20 equals x over 50. Cross-multiplying gives 20 × x = 12 × 50 = 600, so x = 600 ÷ 20 = 30. ✓ Why it works: you’re multiplying both sides of the equation by 20 and by 50, which clears both fractions.

That reason matters. Researchers have long warned that students use cross-multiplication without knowing why it works, and that it often replaces proportional thinking instead of supporting it (Lesh, Post & Behr, 1988). A quick sense check keeps you honest: 50 minutes is 2.5 times 20 minutes, and 12 × 2.5 is 30.

In the TEAS Math Coach, proportions are drawn one over the other, the way you’d write them on paper. On the cross-multiply step, an X of arrows shows which pair is being multiplied. That drawing is our design choice; no study compares it with other layouts.

Try ratios and proportions with FurahaA fresh problem in place. Answer it, or tap “Walk me through it” for one small step at a time. Free, no account.Start

When a picture hurts

More pictures are not automatically better. Multiple representations help only when the design, the job each one does and the task are right (Ainsworth, 2006). A paper on teaching fractions warns that unless students can read each representation and connect it to the others, extra ones “may actually confuse” (Rau & Matthews, 2017).

The practical rule: label your picture with the numbers from the problem, and say out loud which part of the picture is which part of the equation. If you can’t, go back to the equation. That’s why the coach’s pictures are optional, sit behind the chart button instead of on top of the problem, and show the answer as “?” until you reach the step that finds it.

For doing these in your head, see mental math for nurses. For how both coaches are built, see how the NurseSavvy math coach teaches. For the TEAS math section itself, see TEAS math practice questions.

Sources

• Parker, M., & Leinhardt, G. (1995). Percent: A privileged proportion. Review of Educational Research, 65(4), 421–481.
• Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9–35.
• Lembke, L. O., & Reys, B. J. (1994). The development of, and interaction between, intuitive and school-taught ideas about percent. Journal for Research in Mathematics Education, 25(3), 237–259.
• Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40(3), 282–313.
• Jitendra, A. K., Star, J. R., Dupuis, D. N., & Rodriguez, M. C. (2013). Effectiveness of schema-based instruction for improving seventh-grade students’ proportional reasoning: A randomized experiment. Journal of Research on Educational Effectiveness, 6(2), 114–136.
• National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 6, Ratios and Proportional Relationships (6.RP.A.3).
• Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93–118). Lawrence Erlbaum & NCTM.
• Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183–198.
• Rau, M. A., & Matthews, P. G. (2017). How to make ‘more’ better? Principles for effective use of multiple representations to enhance students’ learning about fractions. ZDM, 49(4), 531–544.