Strategies for Adding and Subtracting Decimals

By Cassandra Hatfield, RME Assessment Coordinator

Why is it that many kids struggle when adding and subtracting decimals? After working as a math specialist in elementary schools, I have some theories about why students that I taught in middle school often struggled with this concept.

When an elementary student is asked to solve 36 + 4, some strategies I have seen include:

Counting On

Making 10

The US Standard Algorithm

How would a child’s strategy change if we asked them to solve 3.6 + 0.4? The truth is, it should not:

• Counting on: If the student understands that tenths increase just as the ones place increases, they can still do this strategy.

• Making 1: Have you ever thought to relate making 10 with whole numbers to making 1 with tenths? Or making 100 with whole numbers and making 1 with hundredths? Consider teaching a lesson comparing the ways to make 10 with the ways to make 1 using tenths and the ways to make 100 using hundredths, or even tenths and hundredths. This will support students in solving with mental math instead of the standard algorithm.

• The standard algorithm: Let’s be truthful, when solving the whole number problem 36 + 4 with the standard algorithm, would you “line up the decimals?” Technically, yes. However, you weren't aware because the decimal was not visible. If students are taught to add whole numbers with the standard algorithm by “lining up the place values,” we can teach the same principal as it applies with decimals. The standard algorithm was invented to create an efficient uniform way of computing. The common theme in using the standard algorithm in addition and subtraction is that the place values are lined up. This ensures that the computation is accurate. This same principle applies when students add 3.6 + 0.4 - without a deep understanding of place value many students misplace the decimal point.
• In early elementary, students practice counting around the class or by multiples of whole numbers. Have you ever thought about counting around the class by increments of decimals?

Assessing Decimal Addition and Subtraction
In Teaching Elementary and Middle School Mathematics (2013), a suggested activity for formative assessment is to ask students to compute the sum of a problem involving different numbers of decimals places.
For Example:

75.35 + 4.7 + 0.671

For this assessment, interview students estimating the sum and then computing the exact answer. The goal of this assessment is to record “whether they are showing evidence of having an understanding of decimal concepts and the role of the decimal point. Note whether students get the correct sum by using a rule they learned in an earlier grade but have difficulty with their explanations. Rather than continue to focus on how to add or subtract decimals, struggling students should shift their attention to basic decimal concepts.”

Summing It All Up
If we place more value on mental math strategies, as well as lining up the place values when computing with the standard algorithm,  students may develop a deeper understanding for the skill.

Van de Walle, J. A., Karp, K. S., Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally. Upper Saddle River, NJ: Pearson Education, Inc.

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Comparing Fractions

By Cassandra Hatfield, RME Assessment Coordinator

4 × 5 is 20. 8 × 1 is 8.
So, ⁵⁄₈ is the greater fraction.

As a middle school math teacher, I found it interesting that when I would ask students to compare two fractions, many were quick to give me an answer, but when I asked them to order fractions it took those same quick students quite a bit more time and they often got the wrong answer. Intrigued by this I began to ask my students “how do you know?” I quickly found out that many students were comparing fractions using the “butterfly method” or cross multiplication. In this example, because the larger cross product is on the left, the larger fraction is on the left.

This “trick” will not take a student very far in their journey of mathematics. Although a student can arrive at the correct answer, the “trick” does not require any thought about the relative size of the fractions. If a student does not understand anything about the relative size of the fractions, how would the student order three or more fractions or think conceptually about this word problem: Max paid $12 for his portion of dinner. This was one-third of the total bill. How much was the total bill?

Instead of using the “trick” above, students should be given opportunities to compare fractions that have been intentionally selected by the teacher for investigation. This is not to say, that this should be a lesson titled “Many Ways to Compare Fractions.” Students should be given the opportunity to compare the fractions and share strategies with their classmates in a discussion. The teacher should listen for students who use specific strategies and poster those strategies in the classroom.

Below are some fractions I have intentionally selected for comparison. I’ve shared one strategy that could be used for comparison.

(Van De Walle, Karp, Bay-Williams, 2013, p. 310-311)

Summing it All Up
For students that are struggling with the abstractness of the above strategies, the use of number lines could be helpful. The IES Practice Guide, which is supported by research evidence, states, “conceptually, number lines and number paths show magnitude and allow for explicit instruction on magnitude comparisons" (IES Practice Guide, 2010).

First, see if the student understands how to partition a number line into equal parts and identify distances on the number line. If the student does, then models like the one shown below might be helpful for students to visually see the comparison strategy. 

Model of comparing ⁴⁄₅ and ⁹⁄₁₀ to a whole.

Now it’s your turn. Share with us a different strategy you would have used to compare the fractions above.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013) Elementary and middle school mathematics: Teaching developmentally. Upper Saddle River, New Jersey: Pearson.

Siegler, R., Carpenter, T., Fennel, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., Wray, J. (2010) Developing effective fractions instruction for kindergarten through 8th grade: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/.

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Converting Fractions to Percentages

By Beth Richardson, RME High School Mathematics Coordinator

As a high school math teacher, I taught a wide range of students from ESL Algebra 1 and regular Geometry to Pre-AP Algebra 2. The resounding similarity I saw between all of my students was that, for some reason, students cringe when they see rational numbers. They feel like rational numbers automatically make the problem “hard”. I was amazed that by high school, students were still struggling with something as simple as converting from a fraction to a percent. Perhaps this is because, as teachers, we sometimes teach our students shortcuts that leave out the logic behind the scenes of the procedures they learn.

The IES Practice Guide, which is supported by research evidence, recommends that teachers “'help students understand why procedures for computations with fractions make sense’ and ‘develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication…’ (Siegler et al., 2010).”

Some common shortcuts teachers use are changing the fraction to a decimal then multiplying by 100 or changing the fraction to a decimal then moving the decimal to the right twice.

Examples:

Neither method above leads students to a percentage as the final answer, unless the student “remembers” to tag it on at the end. Units are crucial when converting in any context. In order for students to understand why they must multiply by 100% rather than 100 when converting from fraction to percent, units must be used properly.

Instead, students should be taught to set up proportional relationships, including units, between the fraction and unknown out of 100%. It is important that students understand that when the units of the numerator and denominator are the same, they cancel and the fraction is unit-less.

Example:
25 students went on a field trip and 5 wore a hat. What percentage of the students wore a hat?

20% of the students wore a hat on the field trip.

Through the process above, students see why they are multiplying by 100% and why the units in their answer must be a percentage. Also, students can use number sense to reason that x must be a percentage between 5 and 100.

Summing It All Up
Fellow teachers: it’s not safe to assume that our students understand why they are doing a particular procedure, even if it is one they “should” have mastered several grade levels ago. If we take a little more time to illustrate examples with labeled units and explanation, we will hopefully catch any previous misconceptions our students have and steer them on the right path towards math success.

Now it’s your turn. Share with us common misconceptions, similar to what we described above, that you’ve found in your classroom!

Resources:
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Siegler, R., Carpenter, T., Fennel, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010) Developing effective fractions instruction for kindergarten through 8th grade: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/