Middle School Number Sense Lesson 111: Converting Gallons to Cubic Inches

This concept appeared 2 times this year, with a median placement at question # 65.

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Number Dojo Level: 185

We have already established that a gallon is equivalent to 128 fluid ounces.  But we also need to know (because the number sense test writers said so) that a gallon is equivalent to 231 cubic inches.  We may be asked to convert either direction.
 
The number 231 happens to be 3 x 7 x 11, and I have seen questions using different combinations of those factors. 

• 3 x 77 = 231
• 7 x 33 = 231
• 11 x 21 = 231

 
Here are a few examples.
 
Example 1: 1/11 gallon = ___ cubic inch(es)

1. Multiply by 231
2. 1/11 x 231 = 231 ÷ 11 = 21

 
Example 2: 77 cubic inches = ___ gallon(s)

1. Divide by 231
2. 77 ÷ 231 = 1/3

 
Example 3: 1/7 gallon = ___ cubic inch(es)

1. Multiply by 231
2. 1/7 x 231 = 231 ÷ 7 = 33

 
Example 4: 693 cubic inches = ___ gallon(s)

1. Divide by 231
2. 693 ÷ 231 = 3

 
Example 5: 3/7 of a gallon = ___ cubic inches

1. Multiply by 231
2. 3/7 x 231 = 231 ÷ 7 x 3 = 33 x 3 = 99

 
Example 6: * 143 gallon = ___ cubic inches

1. Multiply by 231
2. 143 is exactly 1001 ÷ 7.  So divide 231 by 7 (which is 33) and multiply by 1001.
3. 33 x 1001 = 33033

High School Number Sense Lesson 108: Converting from Base 2 to Base 8

This concept requires a good understanding of BaseNtoBase10 and Base10toBaseN.  It appeared 3 times this year, with a median placement at question # 32.

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Number Dojo Level: 305
​
Since 8 is a power of 2, we can use a special method of converting from base 2 to base 8.

How to Solve:

1. Starting at the right, split the base 2 number into triples.
2. Convert each triple of base 2 numbers into a base 10 number, and write the result down.

Example 1: 1101 (base 2) = ___ (base 8)

1. We have 2 triples: 1 and 101
2. 1 (base 2) = one 1, which = 1 (the first digit in the answer)
3. 101 (base 2) = one 4 + zero 2's + one 1, which = 5 (the second digit in the answer)
4. Your answer is 15

Example 2: 11100 (base 2) = ___ (base 8)

1. We have 2 triples: 11 and 100
2. 11 (base 2) = one 2 + one 1, which = 3 (the first digit in the answer)
3. 100 (base 2) = one 4 + zero 2's + zero 1's, which = 4 (the second digit in the answer)
4. Your answer is 34

Example 3:

​​Example 4:

​​Example 5:

Up Next for High School: Mult%3

High School Number Sense Lesson 106: Converting Distance Units with Volume--3 Dimensions

This concept is an extension of ConvDistArea.  So far, I have only seen problems with English unit conversions (as opposed to metric unit conversions).  It appeared 3 times this year, only on high school tests, with a median placement at question # 15.
 
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Number Dojo Level: 187

How to Solve:

1. Decide which direction you’re converting.  If you are converting from a larger unit to a smaller one, multiply.  If going from a smaller unit to a larger one, divide. 
2. Multiply or divide (as determined above) by the conversion factor, 3 times—once for each dimension.

 
Example 1: 9 ft x 6 ft x 3 ft = ___ cubic yards

1. We are converting to a larger unit, so we will divide
2. There are 3 feet in a yard, so we will divide by 3, three times.
3. 9 ÷ 3 = 3.  6 ÷ 3 = 2.  3 ÷ 3 = 1.
4. Now multiply: 3 x 2 x 1 = 6.

 
Example 2: 2 cubic feet = ___ cubic inches

1. We are converting to a smaller unit, so we will multiply.
2. There are 12 inches in a foot, so we will multiply by 12, three times.
3. 123 = 1728.  2 x 1728  = 3456.

 
Example 3: 18” x 24” x 30” = ___ cubic feet

1. We are converting to a larger unit, so we will divide—in this case, by 12 (3 times).
2. 18 ÷ 12 = 3/2.  24 ÷ 12 = 2.  30 ÷ 12 = 5/2.
3. Now multiply: 3/2 x 2 x 5/2 = 3 x 5/2 = 15/2.

 
Example 4: 1/3 yard x 3 yards x 2 yards = ___ cu. ft.

1. We are converting to a smaller unit, so we will multiply—in this case, by 3 (3 times).
2. 1/3 x 3 = 1.  3 x 3 = 9.  2 x 3 = 6.
3. Now multiply: 1 x 9 x 6 = 54.

 
Example 5: 12 ft x 6 ft x 4 ft = ___ cubic yards

1. We are converting to a larger unit, so we will divide—in this case, by 3 (3 times).
2. 12 ÷ 3 = 4.  6 ÷ 3 = 2.  4 ÷ 3 = 4/3.
3. Now multiply: 4 x 2 x 4/3 = 32/3.

 
Example 6: 2 cubic yards = ___ cubic feet

1. We are converting to a smaller unit, so we will multiply—in this case, by 3 (3 times).
2. 2 x 3 x 3 x 3 = 54.

High School Number Sense Lesson 105: Converting Percents to Decimals

This concept is very similar to ConvDeci%.  It appeared 4 times this year, with a median placement at question # 6. 
 
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Number Dojo Level: 14

Most of these problems at a high school level have fractional percentages.  Since we are going from a percent to a decimal, we need to divide by 100. 
 
How to Solve:

1. Remove the % sign.
2. If necessary, convert the fraction (or mixed number) to a decimal.
3. Move the decimal 2 places to the left.

 
Example 1: 13.7% = ___ (decimal)

1. Remove the % sign: 13.7
2. Move the decimal 2 places to the left: .137

 
Example 2: 3 1/5% = ___ (decimal)

1. Remove the % sign: 3 1/5
2. 3 1/5 = 3.2
3. The answer is .032

 
Example 3: 3/8 % = ___ (decimal)

1. Remove the % sign: 3/8
2. 3/8 = .375
3. The answer is .00375

 
Example 4: 1 5/8% = ___ (decimal)

1. Remove the % sign: 1 5/8
2. 1 5/8 = 1.625
3. The answer is .01625

 
Example 5: 12 3/8% = ___ (decimal)

1. Remove the % sign: 12 3/8
2. 12 3/8 = 12.375
3. The answer is .12375​

Elementary School Number Sense Lesson 22: Converting Decimals to Percents

This concept is one of the easier conversions in number sense.  It appeared just 1 time this year—at question # 53.
 
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Number Dojo Level: 13

The word percent means per 100.  If we are moving from a decimal to a percent, we multiply the number by 100 to get our answer.  The easiest way to do this is to move the decimal 2 places to the right.
 
Example 1: .44 = ___%

1. Move the decimal 2 places to the right.
2. The answer is 44

 
Example 2: 1.87 = ___%

1. Move the decimal 2 places to the right.
2. The answer is 187

 
Example 3: 0.043 = ___%

1. Move the decimal 2 places to the right.
2. The answer is 4.3

 
Example 4: 0.004 = ___%

1. Move the decimal 2 places to the right.
2. The answer is .4

 
Example 5: 7 = ___%

1. Multiply by 100.
2. The answer is 700

 
Example 6: 115 = ___%

1. Multiply by 100.
2. The answer is 11500

Middle School Number Sense Lesson 99: Converting Money

This concept is one of the most practical number sense questions around.  It showed up 4 times this year, with a median placement at question # 28.
 
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Number Dojo Level: 46

In the past 10+ years, I have only seen questions about four denominations of United States currency:

• Quarter (25¢)
• Dime (10¢)
• Nickel (5¢)
• Penny (1¢)

 
I would love to see questions about half-dollars, dollar coins, and/or $2 bills…they’re my favorite since nobody seems to use them anymore.
 
Example 1: 64 nickels = $___

1. Multiply by .05 (or divide by 20)
2. 64 x .05 = 3.2; the answer is 3.20 (because we have two write 2 decimal places)

 
Example 2: $11.15 = ___ nickel(s)

1. Divide by .05 (or multiply by 20)
2. 11.15 x 20 = 22.3 x 10 = 223

 
Example 3: 14 quarters + 13 nickels = ___ ¢

1. 14 quarters = 14 x 25¢ = 350¢
2. 13 nickels = 13 x 5¢ = 65¢
3. 350¢ + 65¢ = 415

 
Example 4: 7 nickels, 3 quarters, and 29 pennies = ___¢

1. 7 nickels = 7 x 5¢ = 35¢
2. 3 quarters = 3 x 25¢ = 75¢
3. 29 pennies = 29 x 1¢ = 29¢
4. 35¢ + 75¢ + 29¢ = 110¢ + 29¢ = 139

 
Example 5: 17 quarters + 6 dimes = $___

1. 17 quarters = 17 x .25 = 4.25
2. 6 dimes = 6 x .10 = .60
3. 4.25 + .60 = 4.85

 
Example 6: 23 quarters – 5 dimes = $___

1. 23 quarters = 23 x .25 = 5.75
2. 5 dimes = 5 x .10 = .50
3. 5.75 - .50 = 5.25

 
Example 7: 14 dimes + 7 quarters = ___ nickels

1. 14 dimes (x 2) = 28 nickels
2. 7 quarters (x 5) = 35 nickels
3. 28 + 35 = 63

High School Number Sense Lesson 98: Converting between Metric Units--with Operations

A few weeks ago, we introduced metric unit conversions in ConvMetric.  This concept is an extension of that one.  It appeared 4 times this year on high school tests, with a median placement at question # 16.

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Number Dojo Level: 94

Take the time to review the metric conversions if you need to.  Then you can perform operations while converting from one unit to another.

How to Solve:

1. ​Look at the unit of measure required for the answer.
2. Try to convert each term in the operation to the unit of measure required for the answer.
3. Add or subtract as you go.

Example 1: 3 hectometers + 20 meters = ___ meter(s)

1. 3 hectometers = 300 meters
2. 300 + 20 = 320

Example 2: .007 kiloliters - 3 liters = ___ liters

1. .007 kiloliters = 7 liters
2. 7 - 3 = 4
   

Example 3: 16 grams - .005 kilograms = ___ gram(s)

1. .005 kilograms = 5 grams
2. 16 - 5 = 5

Example 4: 4 grams - 16 decigrams = ___ centigram(s)

1. 4 grams = 400 centigrams
2. 16 decigrams = 160 centigrams
3. 400 - 160 = 240

Example 5: 23 millimeters + 19 cm = ___ meters

1. 23 millimeters = .023 meters
2. 19 cm = .19 meters
3. .023 + .19 = .213

Example 6: If 1 cm = 0.39" then 10 meters = ___"

1. 10 meters = 10 x 100 cm = 1000 cm
2. 1000 x .39 = 390​

Middle School Number Sense Lesson 92: Converting Distance Units with Area--2 Dimensions

This concept is an extension of ConvDistance, but with more complexity.  These problems can involve either English units of measure (inches, feet, yards, etc.) or metric units.  This concept has appeared 3 times so far this year, with a median placement at question # 33.

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Number Dojo Level: 98
 
Like its easier counterpart, this concept has real-life application.  We recently decided we wanted to replace the worn-out carpet in our home, so we measured each of the carpeted rooms to see how many square feet of carpet we needed to buy.  When we got to the carpet store, we realized their carpet is priced by the square yard.  Let’s look at our calculation.
 
One of our living areas measures 15 feet by 18 feet, which is 270 square ft.  How many square yards of carpet do we need?

1. Convert each distance from feet to yards by dividing by 3 (twice).  15 ÷ 3 = 5, and 18 ÷3 = 6.
2. Multiply these distances to find the area in square yards.  5 x 6 = 30 square yards.

 
Notice that we need to use our conversion factor (1/3) once for each dimension of distance.  This understanding is key to having success with these number sense problems.
 
Example 1: 24 inches x 72 inches = ___ square ft

1. Going from inches to feet requires dividing by 12 in both directions.  24 ÷ 12 = 2, and 72 ÷ 12 = 6.
2. Now multiply to find the area.  2 x 6 = 12.

 
Example 2: 3 square yards = ___ square feet

1. Going from yards to feet requires multiplying by 3 in both directions.
2. 3 x 3 = 9; 9 x 3 = 27.

 
Example 3: 48 square inches = ___ square feet

1. Going from inches to feet requires dividing by 12 in both directions. 
2. 48 ÷ 12 = 4; 4 ÷ 12 = 1/3.

 
Example 4: 2.43 sq cm = ___ sq mm

1. There are 10 mm in each cm, so we’ll multiply by our conversion factor of 10, twice.
2. 2.43 x 10 x 10 = 243.

 
Example 5: 5 sq ft = ___ sq in

1. Going from feet to inches requires multiplying by 12 in both directions.
2. 5 x 12 = 60.  60 x 12 = 720.

 
Example 6: 1 square meter = ___ square cm

1. There are 100 centimeters in a meter, so multiply by 100 twice.
2. 1 x 100 x 100 = 10000.

High School Number Sense Lesson 91: Converting between Metric Units

This concept has not shown up as often recently, but it is an important life skill.  Also, other concepts build upon the principles learned from this one.  It showed up just 2 times on high school tests this year, with a median placement at question # 21.

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Number Dojo Level: 82

The beauty of the metric system is that the measurements are converted using powers of 10--as opposed to multiples of 2 (volume), 3 or 12 or 16.5 or 1760 (distance), or multiples of 16 or 2000 (weight) in the English system of measurement.

The base units of the metric system are the liter (volume), meter (distance), and gram (mass--or weight for us non-scientists).  There are many unit conversions to learn, but only 6 of them show up on number sense tests:  They are distinguished by their prefixes:

There's a great acronym to remember the descending order of these units: King Henry Died by Drinking Chocolate Milk​.

Elementary School Number Sense Lesson 5: Converting Time

Today's lesson will be pretty basic for most students; it involves converting between units of time.  The concept has appeared 5 times so far on elementary tests this year, with a median placement at question # 45.

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Number Dojo Level: 48

On elementary school tests, these problems are straightforward.  They require one or two basic calculations using the time conversion factors:

Like any other conversion, the key to these problems is knowing whether to multiply or divide, and by how much.

​Example 1: 2 hours = ___ minutes

1. 1 hour = 60 minutes
2. 2 x 60 = 120

Example 2: 14 weeks = ___ days

1. 1 week = 7 days
2. 14 x 7 = 98

Example 3: 15 days = ___ hours

1. 1 day = 24 hours
2. 15 x 24 = 360

Example 4: 9 minutes = ___ seconds

1. 1 minute = 60 seconds
2. 9 x 60 = 540

Example 5: 105 days = ___ weeks

1. 7 days = 1 week
2. 105 ÷ 7 = 15

Example 6: 1 hour = ___ seconds

1. 1 hour = 60 minutes
2. 1 minute = 60 seconds
3. 60 x 60 = 3600