Intro to Number Sense

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Overview

A combination of rapidly recalling simple facts and applying methods to mentally calculate values can allow someone to start thinking flexibly about numbers. This will allow you to do seemingly daunting, but useful mental calculations such as mentally multiplying two and three digit values or mentally calculating percentages.

Doubling and Halving

Before useful methods are introduced, it is worth practicing how to rapidly recall doubles (adding a number to itself) and halves (finding a value that can be added to itself to get the number you want half of) for all numbers between 1-9.

A way to practice this would be to take a deck of cards, remove all the face cards, treat the A as a 1, and run through the deck while saying the doubles or halves aloud. If you are doubling, you might pull a 5-card and say “ten” aloud. If you are halving, you might pull a 9-card and say “four and a half” aloud. Once you get the hang of it, you should pull two cards and attempt to double or half that two digit number. For example, pulling a “4” and then a “2” gets you the number 42. You would then say “84” aloud. Constantly doing this practice will help develop quick recall when performing more complex calculations.

Decomposition

Decomposition is breaking a number into pieces by place value, that can be added back together to form that number. The pieces are used to perform easier calculations and then combined back together.

For example, 5 = 4 + 1.

A use for decomposition is taking half of odd numbers. You can take half of 4 to get 2, but it might be difficult to take half of 5. Decomposition lets you set 1 aside, take half of both parts, and then put the parts back together.

Half of 5 = Half of 4 + Half of 1 = 2 + ½ = 2½, so half of 5 is 2½.

Extending Decomposition

Beyond the digits 1-9, all numbers can be decomposed by place value. For example, the number 358 is made up of 3 hundreds, 5 tens, and 8 ones. You can extend this to decimals (0.358 is 3 tenths, 5 hundredths, and 8 thousandths) and larger numbers of any size.

The known values for doubling and halving digits 1-9 can apply to double-digits and beyond. The double of 3 (3 ones) is 6 (6 ones); therefore, the double of 30 (3 tens) is 60 (6 tens). This can also extend to halving.

Thinking about numbers in terms of their place values will allow you to double and half values beyond a single digit. Doubling the above, 358 can be broken down to 300 + 50 + 8. If you double each part, you have 600 + 100 + 16 = 716.

Be careful when halving odd hundreds or tens. You can use decomposition to set aside 1 place value to half 300 or half 50.

Half of 300 = half of 200 + half of 100 = 100 + 50 = 150.
Half of 50 = half of 40 + half of 10 = 20 + 5 = 25.

In conclusion, half of 358 = 150 + 25 + 4 (half of 8) = 179.

Combining our Three Strategies to Multiply

Decomposition, doubling, and halving can be useful to recall multiplication facts. You might already know some multiplication facts by memory, or know how to calculate using a traditional method, but understanding how you can use known facts to create new ones will build more flexible and confident thinking.

It is easy to multiply a whole number by 10 by adding a 0 on the right side of the number.
6 x 10 = 60 or 70 x 10 = 700. You can use 10s facts to help you calculate 5s facts. It is known that half of 10 is 5, so every 5s fact is half of a 10s fact.

If 6 x 10 = 60; then, 6 x 5 = 30 (half of 60).

Doubling can be used for recalling 2x, 4x, and 8x facts. You can get a 2s fact by doubling a number [2 x 4 = 4 + 4 = 8]. You can double a 2s fact to get a 4s fact, and double a 4s fact to an 8s facts [2 x 8 = double of 2 x 4 = 8 + 8 = 16]. A common multiplication fact that is typically forgotten is 7 x 8, but doubling 7 x 4 = 28 can get 7 x 8 = 28 + 28 = 56.

Decomposition can help with the rest of the single digit facts. 3s can be formed by adding a 2s fact with the same 1s fact [3 x 8 = (2 x 8) + (1 x 8) = 16 + 8 = 24]. Much like 3s facts, 6s facts can be formed by adding a 5s fact with the same 1s fact. 9s can be formed by subtracting a 10s fact from a 1s fact [8 x 9 = (8 x 10) – (8 x 1) = 80 – 8 = 72]. 7s facts can be formed by adding a 5s fact with the same 2s fact.

Putting it All Together

You can combine all the above methods to multiply double-digit values.

Consider 50 x 25:

1. Decomposition: 50 x 25 = (50 x 20) + (50 x 5) = 1250
2. Double and 10s fact: 50 x 20 = double of (50 x 10) = double of 500 = 1000
3. 5s fact: 50 x 5 = half of (50 x 10) = half of 500 (decompose if needed) = 250
4. Add all the pieces together: (50 x 20) + (50 x 5) = 1000 + 250 = 1250

One might be tempted to use a calculator when the calculations get this large; however, relating math facts together using these methods will train you to become more flexible with numbers and allow you to estimate and reconsider the reasonableness of calculations. It might not be easy at first, but consistent practice will yield more flexible thinking.

Poll: Where is Mental Math Used?

Investigate

Thinking flexibly will allow you to perform easier calculations and approach problems in unique ways.

1) Stephanie is making punch for a large party. The recipe calls for twice as much fruit juice as club soda. If she uses 57 cups of club soda, how much fruit juice should she use?

In this problem, you need the double of 57. Using decomposition, the problem becomes:

Double 57 = double 50 + double 7 = 100 + 14 = 114

So, Stephanie needs 114 cups of fruit juice.

2) Rey donated 15 twelve-packs of t-shirts to a homeless shelter. How many t-shirts did he donate?

Because there are 15 packs of 12 t-shirts, the problem is 15 x 12. You can decompose this in two useful ways:

1) Break the 12 down: (15 x 10) + (15 x 2) = (10s fact of 15) + (double 15) = 180

2) Break the 15 down: (10 x 12) + (5 x 12) = (10s fact of 12) + (half 10s of 12) =
=120 + 60 = 180

Notice that both ways will conclude that Rey will donate 180 t-shirts.

3) Javier owns 300 shares of stock in one company. On Tuesday, the stock price rose $12 per share. How much money did Javier’s portfolio gain?

In this problem, we have 300 x 12. We could do the usual decomposition, but we could also break the 300 into 100 x 3. This changes the problem to:

300 x 12 = 100 x 3 x 12 = 100 x 36 = 3600

We needed to use 100s facts on this problem, but it is similar to the 10s facts: you add two zeros instead of one. In conclusion, Javier’s portfolio gained $3600.

4) Carlton got a $250 raise in each paycheck. He gets paid 24 times a year. How much higher is his new annual salary?

So now we have 250 x 24. Let’s break this down again:

250 x 24 = (250 x 20) + (250 x 4) =

For the first part, 20 can be broken into 2 x 10 which means we can double the 10s fact. For the second part, we can use the method for 4s facts which is doubling twice.

Double (250 x 10) + (double double 250) =
Double 2500 + double 500 = 5000 + 1000 = 6000

Decomposing 24 into 20 and 4 allowed us to apply 10s facts, 4s facts, and doubling to quickly calculate that Carlton’s new salary is $6000 more than last year.

Let’s try approaching this by decomposing 250 instead:

250 x 24 = (200 x 24) + (50 x 24) =
Double (100 x 24) + Half (100 x 24) = Double 2400 + Half 2400 =
4800 + 1200 = 6000

With this approach, we took advantage of the relationship between 200, 100, and 50. 200 is a double of 100 and 50 is half of 100, so we could use that relationship to get the same answer as before.

The more practice you get, the easier it is to identify ways to break down numbers and apply multiplication facts to perform mental calculations.

1) 16 x 5 =

2) (500)(4) =

3) 600 x 15=

4) (25)(32) =

5) Kathryn bought 8 flats of impatiens for her flower bed. Each flat has 24 flowers. How many flowers did Kathryn buy?

Answer Key

1) 80

2) 2000

3) 9000

4) 800

5) 192 flowers