Why can’t we just use a calculator? Can’t the computer do it? Do you ever hear this from students who don’t want to do math? They have a point. Calculators and computers can be very useful in figuring out complex problems or crunching large amounts of data, but, as the National Council of Teachers of Mathematics (NCTM) in the United States says:
“The use of calculators does not supplant the need for students to develop proficiency with efficient, accurate methods of mental and pencil-and-paper calculation.”
Kids need paper and pencil calculation—and they need mental math. Mental math helps students develop number sense. That is they learn how numbers work and they start to see patterns. Gaining control and command over numbers builds confidence as math gets harder or more complex. Greater confidence and understanding helps overcome math phobia. And scientists are even studying how mental math may be related to emotional health. There’s a lot to be said for mental math, even if your students don’t see it right away.
If you can’t convince them why they need to do it, maybe you can simply convince them that they want to do it by mixing it up, keeping it fun, and knowing where they might be getting stuck (and how to get around it).
Teaching 5 key mental math strategies 101
Let’s take a look at how to teach the mental math strategies of counting on, compatible numbers, near doubles, partitioning, and estimating.
You’ve seen students counting on, perhaps using fingers in the early stages. The key part of this strategy is to start with the larger number of the two numbers and “count on” the smaller number. For example, if you were adding 4 + 7, you would start with the larger number (7) and then count on 4: 8, 9, 10, 11.
Counting on from the larger numbers is much more efficient than beginning with the smaller number and counting on the larger number. Students may get stuck by starting on the first number they come to, so remind them to start with the larger number.
To practice, let students use their fingers, counters, or a number line/track to count on. Get students to first circle the larger number and then count on. Once they can visualize and have the hang of it, encourage them to try without any tools (including fingers).
If your students have mastered their doubles facts, they are ready for near doubles. We use this mental math strategy to add two consecutive numbers, such as 5 and 6. If children know that 5 + 5 = 10, getting to 5 + 6 = 11 isn’t too much of a jump.
Use a number line to show students that 5 + 6 is the same as 5 + 5 + 1. Then ask students to try with another set of numbers, 7 + 8 for example.
Compatible numbers produce a “tidy sum,” one that usually ends in zero, or multiples of 10, when they are added together. For example, 8 and 2 are compatible because they add up to 10 (end in zero). Also, 40 and 60; 12 and 8; 28 and 42 are also compatible numbers because they add up to a multiple of ten.
Once you’ve explained the concept, give students 10 counters. Ask them to show the different compatible number sets that add up to 10. You can repeat this with a larger number, such as 20 or 100. Have students chart their responses if you do several number sets that are multiples of 10. Ask students to notice patterns, such as a 3 in the ones place always pairs with a 7 in the ones place to get to a multiple of 10.
If you modeled near doubles as 5 + 6 is the same as 5 + 5 + 1, you’ve already introduced partitioning. When we use partitioning we break numbers into parts to make the math simpler. For example, take 5 + 18.
What happens if we break it down (often using compatible numbers) to make it easier to solve?
Write this example on the board:
5 + 18
Explain that 18 = 15 + 3. So you could write 5 + 18 as 5 + 15 + 3.
Then using compatible numbers, students can see that 5 + 15 = 20. And 20 + 3 is easy.
Note that there are other ways to partition. Challenge students to come up with other ways to partition this example and work through the steps. Some other examples might include:
3 + 2 + 18 = 3 + 20 = 23
5 + 5 + 13 = 10 + 13 = 23
Give students another example and ask them to use partitioning to solve it.
Estimation is important because it helps us judge whether an answer is reasonable. We use a number of skills to estimate including rounding, finding compatible numbers, compensating and using benchmarks.
Rounding is a way of simplifying numbers to make them easier to work with. Students need to know what they are rounding to (the nearest 10th? The nearest whole number? The nearest 100?). They also need to remember that rounding is an estimate, not an exact number. Rounding is useful, but may not be appropriate when exact numbers are needed.
We discussed compatible numbers above, and they can be used for estimating often with other tools. For example, compensation means readjusting math problems to make them easier to do in your head. So if you were using compensation to solve 94 + 27, you might change it to 94 + 6 + 21 (note that 94 and 6 are compatible numbers).
To use a benchmark, look at a quantity and apply to a different setting. For example, if you knew that in one section of a theater there are 200 people, you could use that number and the number of sections to estimate the total number of people at the show.
Make it a game to see how many different ways students can solve something like
84 + 26
Have them do the mental math … and then ask them to explain how they did it. Write down their steps and then point out the different strategies they used. Ask several students to explain their method to help kids realize that there are many ways to approach a problem to get to the solution.
Want some done for you help to teach mental math and keep it fun? Mental Math Strategies, Posters and Activities has you covered.
In this helpful pack you get:
- An explanation of each of these 5 strategies with examples
- Posters to assist in explaining each strategy
- Activity ideas to help your students practice the strategy
- Games to provide further consolidation of the strategy.
Get your mental math strategies bundle here >> Mental Math Strategies, Posters and Activities.