Alpha School · Math · built 2 October 2026
This guide covers the nine kinds of questions a grade 4 math test asks, one section each. Every section starts with something you can already do, adds one new move, gives you the rule in a sentence, and names the trap to watch for. Then you practice.
Every question here is new. None of them is from a test. The numbers and stories are different on purpose, because the skill is what carries over, not the numbers.
How to use it: answer, press Check, then read the answer even when you were right. The answer tells you where each wrong choice comes from, and that is how you learn to spot the trap next time. Between sections there are mixed checks that do not tell you which skill to use. The mixed run at the end is like the test: no labels at all.
In this guide
What is the value of the 7 in 4,702?
4,702 is 4 thousands, 7 hundreds, 0 tens and 2 ones.
The 7 sits in the hundreds place, so its value is 700. Not 7. The digit is 7; the value is 700.
Now the same idea used two ways.
Clue puzzles. "The value of the 3 is 30,000. The value of the 8 is 800. The value of the 5 is 5." Put each digit where its value says: 3 in the ten-thousands, 8 in the hundreds, 5 in the ones. That gives 3_,8_5. Any number with that shape fits, such as 32,815.
Rounding. Round 8,462 to the nearest hundred. Underline the hundreds digit: 8,462. Look at the one digit to its right, the 6. It is 5 or more, so the 4 goes up to 5 and everything after it becomes zero.
8,462 rounded to the nearest hundred is 8,500.
Check: 8,462 sits between 8,400 and 8,500, and it is closer to 8,500.
A digit's value is the digit times its place.
To round: underline the place, look at the one digit to its right. 5 or more, round up. Less than 5, keep the digit and zero the rest.
Rounding to the wrong place. Asked for the nearest hundred, it is tempting to look at the ones digit and round to the nearest ten instead, giving 8,460. Underline the place the question names before you look anywhere else.
Digit instead of value. "The value of the 7" is 700, never 7. If the clue says the value of a digit is 40,000, that digit is in the ten-thousands place.
Practice
What is the value of the digit 6 in 56,318?
6,000
56,318 is 5 ten-thousands, 6 thousands, 3 hundreds, 1 ten, 8 ones. The 6 is in the thousands place, so its value is 6 × 1,000.
6 × 1,000 = 6,000.
If you wrote 6, that is the digit, not its value. If you wrote 60,000, you counted one place too far left.
A number is described by these clues.
Which number fits all three clues?
83,245
80,000 puts the 8 in the ten-thousands place. 200 puts the 2 in the hundreds place. 5 puts the 5 in the ones place. The number looks like 8_,2_5, and only 83,245 has that shape.
38,245 has the 8 in the thousands place, worth 8,000. 83,425 has the 2 in the tens place, worth 20. 83,254 has the 5 in the tens place, worth 50.
Round 6,249 to the nearest hundred.
6,200
Underline the hundreds digit: 6,249. The digit to its right is 4. That is less than 5, so the 2 stays and the rest become zeros.
6,249 is between 6,200 and 6,300, and 49 is less than 50, so it is closer to 6,200.
6,250 would be rounding to the nearest ten. 6,300 is what you get if you look at the 9 instead of the 4.
A stadium sold 35,972 tickets. What is 35,972 rounded to the nearest thousand?
36,000
Underline the thousands digit: 35,972. The digit to its right is 9, which is 5 or more, so the 5 goes up to 6.
35,972 rounded to the nearest thousand is 36,000.
35,000 ignores the 9 and keeps the 5. 35,970 is rounding to the nearest ten. 40,000 is rounding to the nearest ten-thousand.
A bar is cut into 10 equal parts and 3 of them are shaded. What fraction of the bar is shaded?
10 equal parts, 3 shaded: 3/10, three tenths.
Three tenths has a second name. The first place after the decimal point is the tenths place, so three tenths is written 0.3.
Cut each tenth into 10 smaller pieces and the bar has 100 pieces. Each piece is one hundredth. The hundredths place is the second place after the point.
3/10 = 0.3.
30/100 = 0.30.
0.3 and 0.30 name the same amount, because 3 tenths is 30 hundredths.
7/100 = 0.07. Only 7 hundredths, so the tenths place holds a 0.
On a number line from 0 to 1 cut into ten equal jumps, each jump is one tenth. The sixth mark after 0 is 0.6.
Tenths are the first place after the decimal point; hundredths are the second.
A fraction with 10 on the bottom fills one place. A fraction with 100 on the bottom fills two places.
Putting the digits straight after the point. 45/100 looks like it should be 0.045 or 4.5, but 100 on the bottom means exactly two places: 0.45.
Reading 0.07 as seven tenths. The 7 is in the second place, so it is seven hundredths. The 0 in the tenths place is doing a job: it says there are no tenths.
Practice
The large square is divided into 100 equal parts. What fraction of the square is shaded?
23/100
Count the shaded small squares: two full rows of 10 and 3 more, which is 23. There are 100 parts in all, so the fraction is 23/100.
As a decimal this is 0.23.
Which decimal names the same amount as 6/100?
0.06
100 on the bottom means two places after the point. Six hundredths has no tenths and 6 in the hundredths place.
0.6 is six tenths, which is 60/100. 6.0 is six wholes. 0.600 is six tenths written with extra zeros.
Point P is on the number line. Which decimal does point P represent?
0.7
From 0 to 1 there are ten equal jumps, so each jump is one tenth. Count the jumps from 0 to P: seven. Seven tenths is 0.7.
0.3 is what you get counting from 1 backwards instead of from 0 forwards. 0.07 would need 100 jumps between 0 and 1. 7 is seven wholes.
Dev ran 0.85 of a mile before school. Which fraction is equivalent to 0.85?
85/100
0.85 has two places after the point, so it is counted in hundredths: 85 hundredths.
85/10 is 8.5. 8/5 is more than 1. 85/1000 is 0.085, three places.
Mixed check 1
These two questions do not say which skill they use. Decide the method first, then answer.
Round 7,350 to the nearest thousand.
Method: Place value and rounding
7,000
Underline the thousands digit: 7,350. The digit to its right is 3, less than 5, so the 7 stays.
7,400 would be the nearest hundred. 8,000 comes from looking at the 5 instead of the 3.
A bar is divided into 10 equal parts and 9 of them are shaded. Which decimal shows the shaded part?
Method: Decimals and fractions
0.9
Nine of ten equal parts is nine tenths. Tenths are the first place after the point.
0.09 is nine hundredths. 9.0 is nine wholes. 0.19 is one tenth and nine hundredths.
What is 6 × 40?
6 × 4 tens is 24 tens.
24 tens is 240.
So 6 × 40 = 240. A fact you know, with a zero carried along.
Bigger numbers are the same move, done for every piece. Break each number into its places, multiply every piece by every piece, then add.
23 × 14. Break 23 into 20 + 3 and 14 into 10 + 4.
20 × 10 = 200.
20 × 4 = 80.
3 × 10 = 30.
3 × 4 = 12.
200 + 80 + 30 + 12 = 322.
Three digits times one digit works the same way. 348 × 6:
300 × 6 = 1,800.
40 × 6 = 240.
8 × 6 = 48.
1,800 + 240 + 48 = 2,088.
Break each number into its places. Multiply every part by every part. Add all the pieces.
Fronts and backs only. 23 × 14 is not 20 × 10 plus 3 × 4. That misses 20 × 4 and 3 × 10 and gives 212, which will be one of the choices. Four pieces for two-digit times two-digit, every time.
"Not here". When a choice list ends with "Not here", work the problem first, then look for your answer. If it is missing, and your check by estimating agrees with you, "Not here" is the answer.
Practice
What is 7 × 300?
2,100
7 × 3 hundreds is 21 hundreds.
21 hundreds is 2,100.
Which expression shows 36 × 24 broken into parts that can be multiplied one at a time?
(30 × 20) + (30 × 4) + (6 × 20) + (6 × 4)
36 is 30 + 6 and 24 is 20 + 4. Every part of 36 has to meet every part of 24, which makes four products.
(30 × 20) + (6 × 4) is fronts and backs only, two pieces missing. (36 × 20) + (36 × 40) breaks 24 into 20 and 40, which is 60, not 24. (30 + 20) × (6 + 4) adds the wrong pieces together.
What is 418 × 7?
2,926
400 × 7 = 2,800.
10 × 7 = 70.
8 × 7 = 56.
2,800 + 70 + 56 = 2,926.
Estimate to check: 418 is about 400, and 400 × 7 is 2,800, so 2,926 is reasonable.
A theater has 28 rows of seats. Each row has 32 seats. How many seats are in the theater?
896
20 × 30 = 600.
20 × 2 = 40.
8 × 30 = 240.
8 × 2 = 16.
600 + 40 + 240 + 16 = 896.
616 is fronts and backs only: 600 + 16. 840 is 28 × 30, forgetting the 2 in 32. 896 is in the list, so Not here is wrong this time.
What is 72 ÷ 8?
Think: 8 times what makes 72? 8 × 9 = 72.
So 72 ÷ 8 = 9.
Divide a bigger number place by place. 653 ÷ 4:
600 ÷ 4 = 150.
53 ÷ 4 is 13 with 1 left over, because 4 × 13 = 52.
150 + 13 = 163, remainder 1.
Check by multiplying back: 4 × 163 = 652, and 652 + 1 = 653.
Now the part that is really the skill: deciding what the remainder means. 653 pencils, 4 in each box.
Divide place by place, keep the remainder smaller than the divisor, and check by multiplying back.
Then read the question again and decide what the remainder means: drop it, round up, or report it.
Dropping the remainder when something still needs a place. Vans, boxes, tables and shelves all round up: one more is needed for the leftovers.
Reporting a remainder bigger than the divisor. 53 ÷ 4 is not 12 remainder 5. If the remainder is 4 or more you can divide once more.
Practice
What is 56 ÷ 7?
8
7 × 8 = 56, so 56 ÷ 7 = 8.
A baker has 275 muffins and packs 6 muffins in each box. Which expression can be used to find how many full boxes the baker packs?
275 ÷ 6
The muffins are being shared into equal groups of 6, so divide the total by the group size.
275 × 6 would be 275 boxes of 6. 275 − 6 takes away one box only. 6 ÷ 275 divides the wrong way round.
A library puts 936 books onto 4 shelves with the same number of books on each shelf. How many books go on each shelf?
234
800 ÷ 4 = 200.
136 ÷ 4 = 34.
200 + 34 = 234.
Check: 4 × 234 = 936. No remainder, because 936 shares evenly.
A school van holds 9 students. 203 students are going on a trip. How many vans are needed so that every student has a seat?
23
180 ÷ 9 = 20.
23 ÷ 9 is 2 with 5 left over.
20 + 2 = 22, remainder 5.
22 vans carry 198 students and 5 students are still standing. They need one more van, so 23 vans.
22 drops the remainder. 22 R5 is the division answer, not the number of vans. 5 is the remainder on its own.
Mixed check 2
These two questions do not say which skill they use. Decide the method first, then answer.
A rope 437 feet long is cut into pieces that are each 5 feet long. How many whole pieces can be cut?
Method: Divide with remainders
87
400 ÷ 5 = 80.
37 ÷ 5 is 7 with 2 left over.
80 + 7 = 87, remainder 2.
The question asks for whole pieces, so the 2 feet left over are not a piece. 87, not 88.
What is 32 × 15?
Method: Multiply big numbers
480
30 × 10 = 300.
30 × 5 = 150.
2 × 10 = 20.
2 × 5 = 10.
300 + 150 + 20 + 10 = 480.
310 is fronts and backs only, 300 + 10.
Naomi had $40 and spent $12. How much does she have left?
40 minus 12 equals 28. She has $28 left.
The whole was $40; one part is $12; the other part is $28.
Longer problems are the same whole-and-parts idea, used more than once.
Profit. Lin's supplies for a lemonade stand cost $18. She took in $65. Profit is what she took in minus what she spent: 65 minus 18 equals 47.
Change. Pay with a $50 bill for three things: add the three things first, then subtract that total from 50.
Writing the equation. Over three years a club collected 1,245 cans. Year one: 418. Year two: 506. Which equation finds n, the cans in year three? The parts add to the whole:
418 + 506 + n = 1,245.
A strip diagram shows the same thing: 1,245 across the top, and 418, 506 and n side by side underneath.
Find the whole and the parts first. The parts add up to the whole, so a missing part is the whole minus the known parts.
Right numbers, wrong operation. 1,245 + 418 + 506 = n uses all the right numbers and adds the whole to its own parts. Ask: is n a part or the whole?
Subtracting only one thing from the bill. If three items were bought, all three come off the $50, not just the first one.
Practice
A book costs $9 and a pen costs $4. What is the total cost?
$13
9 + 4 = 13.
Over three weeks Raul read 1,120 pages. He read 365 pages in week one and 410 pages in week two. Which equation can be used to find p, the number of pages Raul read in week three?
365 + 410 + p = 1,120
The whole is 1,120 pages. The three weeks are the parts, and the parts add to the whole.
1,120 + 365 + 410 = p adds the whole to its parts. 1,120 − 365 = p forgets week two. 365 + 410 = p is the two known weeks, not week three.
Priya spent $37 on supplies for a craft stall. She sold her crafts for a total of $112. What was Priya's profit?
$75
Profit is money taken in minus money spent.
112 minus 37 equals 75.
$149 adds the two amounts. $85 is a subtraction slip: 12 minus 7 is 5, then 11 minus 3 should be 7, not 8. $112 is the money taken in, before the cost comes off.
Ken pays with a $50 bill. He buys one game for $18, two posters for $6 each, and a snack for $3. How much change should Ken receive?
$17
Two posters: 2 × 6 = 12.
Total spent: 18 + 12 + 3 = 33.
Change: 50 minus 33 equals 17.
$23 counts one poster instead of two. $32 subtracts only the game. $27 is the total spent if you count one poster, not the change.
A pizza is cut into 8 equal slices and you eat 3. What fraction of the pizza did you eat?
3/8. Three of the eight equal slices.
The bottom number names the size of the slice. The top number counts slices.
Adding and subtracting with the same denominator is counting slices. 3 slices and 2 more slices is 5 slices, all eighths:
3/8 + 2/8 = 5/8.
The denominator does not change. You are still counting eighths.
Two friends finished 7/10 of a mural together. One of them did 4/10. The other did 7/10 − 4/10 = 3/10.
Comparing. Same denominator: more slices is more. Same numerator: the bigger denominator means smaller slices, so 5/8 is less than 5/6. Different everything: compare each to one half. 3/10 is less than a half; 2/3 is more than a half.
More than one whole. If five friends each drink one fourth of a jug, that is 5/4 of a jug, which is one whole jug and one more fourth.
Same-size pieces: add or subtract the counts and keep the piece name.
To compare, make the pieces alike, or compare each fraction to one half.
Adding the denominators. 3/8 + 2/8 is not 5/16. Sixteenths never appeared; you only had eighths. The answer 5/16 is smaller than 3/8 on its own, which cannot be right for a sum.
Bigger denominator, bigger fraction. 1/6 looks bigger than 1/4 because 6 is bigger than 4. But cutting into 6 pieces makes each piece smaller.
Practice
What is 3/8 + 2/8?
5/8
3 eighths and 2 eighths is 5 eighths. The denominator stays 8.
5/16 comes from adding the bottoms, which changes the size of the pieces.
Together, Ana and Beto painted 9/12 of a fence on Saturday. Ana painted 5/12 of the fence. Which expression can be used to find the fraction of the fence Beto painted?
9/12 − 5/12
The whole they painted together is 9/12. Ana's part is 5/12. Beto's part is the whole minus Ana's part.
9/12 + 5/12 adds the whole to a part. 12/12 − 5/12 uses the whole fence as the whole, but they did not paint the whole fence. 12/12 − 9/12 is the part of the fence nobody painted.
Which comparison is true?
5/6 is greater than 5/8
Both fractions have 5 pieces. Sixths are bigger pieces than eighths, so 5 sixths is more.
5/8 is greater than 5/6 is the bigger-denominator trap. 3/10 is less than half, because half of 10 is 5. 2/3 is more than half, because half of 3 is 1 and a half.
Five friends each drank one fourth of a jug of lemonade. The picture shows the two jugs with the parts they drank shaded. Which fraction shows how many jugs of lemonade the friends drank?
5/4
Each shaded part is one fourth of a jug. Five fourths were drunk: 5/4, which is one whole jug and one fourth more.
5/8 counts all eight parts of both jugs as the whole, but the question asks how many jugs, and one jug is 4 parts. 1/5 and 4/5 come from the number of friends, not the size of the parts.
Mixed check 3
These two questions do not say which skill they use. Decide the method first, then answer.
Which fraction is greater, 2/5 or 2/3?
Method: Fraction add and compare
2/3
Both have 2 pieces. Thirds are bigger pieces than fifths, so 2 thirds is more. Another check: 2/3 is more than one half and 2/5 is less than one half.
Jo had $60. She bought a shirt for $24 and socks for $7. How much money does Jo have left?
Method: Multi-step equations
$29
Total spent: 24 + 7 = 31.
60 minus 31 equals 29.
$36 is 60 minus 24 only, forgetting the socks. $91 adds everything.
How many degrees are in a right angle? In a straight angle?
A right angle, the corner of a square, is 90°. A straight angle, a flat line, is 180°.
Smaller than 90° is acute. Between 90° and 180° is obtuse.
Reading a protractor. A protractor has two rows of numbers. Put the center on the vertex and one ray along the zero line. Read the row of numbers that starts at 0 on that ray. If you only have the other row, the two rows always add to 180, so subtract from 180.
The ray crosses 40 on the row that does not start at 0 on your zero ray.
180 minus 40 equals 140, so the angle is 140°.
Check with your eyes: the angle opens past a right angle, so it must be more than 90°. 140° fits; 40° does not.
Naming triangles. Name a triangle by its biggest angle. One right angle: right triangle. One obtuse angle: obtuse triangle. All three acute: acute triangle.
Sides. Perpendicular sides meet at a right angle. Parallel sides never meet, like the long sides of a rectangle.
Symmetry. A line of symmetry is a fold that matches the two halves exactly. Check the vertical fold and the horizontal fold separately.
Read the protractor from the row that starts at 0 on your ray, then check: wider than a right angle means more than 90°.
A triangle takes its name from its biggest angle.
The wrong row. Reading 40 when the angle is 140. The eye check catches it every time: obtuse means more than 90.
"Acute, because it has two acute angles." Every triangle has at least two acute angles, so that reason proves nothing. Look at the biggest angle.
Practice
An angle opens wider than a right angle but less than a straight angle. Which word describes it?
Obtuse
More than 90° and less than 180° is obtuse. Acute is less than 90°. Right is exactly 90°. Straight is exactly 180°.
An angle is shown on the protractor. Which statement gives the measure of the angle?
140°, because 180 minus 40 equals 140
One ray lies on the zero line to the right. The other ray crosses 140 on the row that starts at 0 on that ray, and 40 on the other row. The two rows add to 180.
The angle opens past a right angle, so it must be more than 90°. 40° reads the wrong row. 50° and 220° use the wrong number to subtract from or add to.
A triangle has angles that measure 90°, 35° and 55°. Which term describes this triangle?
Right triangle, because one angle is a right angle
The biggest angle is 90°, a right angle, so the triangle is a right triangle.
Two acute angles is true of every triangle, so it is not a reason. 90° is not obtuse. Every triangle's angles add to 180°, so that reason does not pick out a right triangle.
Four figures are shown. Which figures appear to have both a horizontal line of symmetry and a vertical line of symmetry?
Figure 1 and Figure 4
The square folds to match top-to-bottom and left-to-right. The hexagon does too.
The triangle (Figure 2) folds left-to-right, but a top-to-bottom fold puts the point on the flat base, so it has only a vertical line. The slanted parallelogram (Figure 3) has no line of symmetry at all, even though it looks balanced.
A rectangle is 5 cm long and 3 cm wide. Add up all four sides.
5 + 3 + 5 + 3 = 16.
The distance around the rectangle is 16 cm. That is its perimeter.
Two sides are 5 and two are 3, so the perimeter is also 2 × (5 + 3). For a square all four sides match, so perimeter is 4 × side.
Area is a different question: how many unit squares fit inside. For a rectangle, length × width.
Perimeter of the 5 by 3 rectangle: 2 × (5 + 3) = 16 cm.
Area of the 5 by 3 rectangle: 5 × 3 = 15 square cm.
The unit tells you which one you found. Centimeters for around. Square centimeters for inside.
Measurement sense. A kilogram is about the mass of a thick book or a pair of shoes; a paper clip is about a gram. Fluid ounces measure liquid: a 60 fl oz jug fills seven 8 fl oz glasses with 4 fl oz left, because 60 ÷ 8 is 7 remainder 4.
Perimeter is the distance around: add the sides. Area is the squares inside: multiply length by width.
Check the unit. Plain units for around, square units for inside.
Answering the other one. The question says perimeter and the hand multiplies. Say the word "around" or "inside" to yourself before you write anything.
Squares. Side × side is area. Perimeter of a square is side × 4.
Practice
A square playground has sides that are each 9 yards long. What is the perimeter of the playground in yards?
36 yards
Four equal sides: 4 × 9 = 36.
81 would be 9 × 9, the area in square yards, not the distance around.
A rectangular garden is 14 meters long and 6 meters wide. Which expression gives the area of the garden in square meters?
14 × 6
Area is the squares inside, so multiply length by width.
2 × (14 + 6) and 14 + 6 + 14 + 6 are both the perimeter, 40 meters. 14 + 6 is half the perimeter.
A small watermelon has a mass of about 2 kilograms. Which object has a mass closest to 2 kilograms?
A bag of flour
A standard bag of flour is about 2 kilograms, close to a small watermelon.
A paper clip and a feather are about a gram each, a thousand times lighter. A car is about a thousand kilograms.
A spool holds 75 feet of ribbon. Mira cuts 8-foot pieces until there is not enough ribbon left to cut another whole piece. How many feet of ribbon are left on the spool?
3 feet
75 ÷ 8 is 9 with 3 left over, because 8 × 9 = 72 and 75 minus 72 equals 3.
9 pieces are cut and 3 feet remain.
9 feet is the number of pieces, not the leftover. 8 feet is one piece. 67 feet takes only one piece off the spool.
Mixed check 4
These two questions do not say which skill they use. Decide the method first, then answer.
A triangle has angles that measure 120°, 30° and 30°. Which term describes the triangle?
Method: Angles and shapes
Obtuse triangle
The biggest angle is 120°, which is more than 90°, so the triangle is obtuse.
Two acute angles does not make it acute; every triangle has two. There is no right angle. There is no such thing as a straight triangle.
A rectangular pen is 12 feet long and 7 feet wide. How many feet of fencing are needed to go all the way around the pen?
Method: Perimeter and area
38 feet
Fencing goes around, so this is perimeter.
2 × (12 + 7) = 38.
84 is 12 × 7, the area in square feet, which is the inside, not the fence.
In a stem-and-leaf plot, what number does the row "3 | 7" stand for?
The stem is the tens digit and the leaf is the ones digit. 3 | 7 is 37.
A row "2 | 0 3 3 7" holds four numbers: 20, 23, 23 and 27.
Greatest minus least. The least value is the first leaf of the first stem. The greatest is the last leaf of the last stem. Subtract.
Dot plots. One dot for each item, placed above its value. A table saying "4 meters: 3 boats" becomes 3 dots stacked above the 4.
Rule tables. A rule like "multiply the position by 3, then add 1" gives position 1 → 4, position 2 → 7, position 3 → 10. To check a rule against a table, test every row, not only the first one.
Position 1: 1 × 3 = 3, then 3 + 1 = 4.
Position 2: 2 × 3 = 6, then 6 + 1 = 7.
Position 3: 3 × 3 = 9, then 9 + 1 = 10.
Stem is the tens, each leaf is a one. In a dot plot, count the dots above each value.
A rule is only right if it works for every row of the table.
Reading the leaf alone. In "3 | 7" the number is 37, not 7. In "2 | 0" the number is 20.
A rule that fits the first row only. Many rules give the right first value. Wrong choices are built to pass row one and fail row two.
Practice
The stem-and-leaf plot shows the number of minutes nine students spent reading. What is the least number of minutes? What is the greatest?
| Minutes spent reading | |
|---|---|
| Stem | Leaf |
| 1 | 2 5 8 |
| 2 | 0 3 3 7 |
| 3 | 1 4 |
Key: 2 | 5 means 25
Least 12, greatest 34.
The first stem is 1 and its first leaf is 2, so the least value is 12. The last stem is 3 and its last leaf is 4, so the greatest is 34.
2 and 4 on their own are leaves, not values.
Using the same stem-and-leaf plot, what is the difference between the greatest and least number of minutes?
22
34 minus 12 equals 22.
If you got 2, you subtracted the leaves only, 4 minus 2. If you got 20, you subtracted the stems only.
The table shows how far some toy boats floated. If these data are shown in a dot plot, how many dots will be above 4? How many above 3?
| Distance (meters) | Number of boats |
|---|---|
| 2 | 3 |
| 3 | 1 |
| 4 | 4 |
| 5 | 2 |
Above 4: 4 dots. Above 3: 1 dot.
Each boat is one dot, placed above its distance. Four boats floated 4 meters, so four dots above 4. One boat floated 3 meters, so one dot above 3.
A wrong dot plot swaps the columns: 4 dots above 3 and 3 dots above 2. Check the number of dots in every column against the table.
The table shows the position of a number in a pattern and its value. Which rule describes the relationship?
| Position | Value |
|---|---|
| 1 | 2 |
| 2 | 6 |
| 3 | 10 |
| 4 | 14 |
Multiply the position by 4, then subtract 2
Position 1: 1 × 4 = 4, then 4 minus 2 equals 2.
Position 2: 2 × 4 = 8, then 8 minus 2 equals 6.
Position 4: 4 × 4 = 16, then 16 minus 2 equals 14.
Multiply the position by 2 works for row one only: 2 × 2 is 4, not 6. Multiply by 3 then subtract 1 also works for row one only: 2 × 3 minus 1 is 5, not 6. Add 4 fails even in row one.
Mixed run
Every skill in this guide, in no order, and nothing tells you which one to use. Name the method on your paper before you work it. That is exactly what the test asks.
A rectangular rug is 8 feet long and 5 feet wide. What is the area of the rug in square feet? What is the perimeter of the rug in feet?
Method: Perimeter and area
Area 40 square feet, perimeter 26 feet.
Area is inside: 8 × 5 = 40.
Perimeter is around: 2 × (8 + 5) = 26.
The two questions differ by one word. Say "inside" or "around" before you start.
Which list shows these numbers in order from least to greatest?
40,312 43,021 40,231 34,999
Method: Place value and rounding
34,999; 40,231; 40,312; 43,021
Compare the ten-thousands first: 34,999 has a 3, so it is least. The other three start with 4. Compare thousands: 43,021 has a 3, the others have 0, so 43,021 is greatest. Between 40,231 and 40,312, compare hundreds: 2 is less than 3.
The second choice swaps 40,231 and 40,312. The third is greatest to least. The fourth puts 34,999 in the middle because it looks long.
Which decimal is equivalent to 70/100?
Method: Decimals and fractions
0.7
70 hundredths is 0.70, and 0.70 is the same amount as 0.7, seven tenths.
0.07 is seven hundredths. 7.0 is seven wholes. 0.710 is 71 hundredths.
A florist makes 46 bouquets. Each bouquet uses 19 flowers. How many flowers does the florist use?
Method: Multiply big numbers
Not here
40 × 10 = 400.
40 × 9 = 360.
6 × 10 = 60.
6 × 9 = 54.
400 + 360 + 60 + 54 = 874.
874 is not in the list. Estimate to be sure: 46 × 19 is a little less than 46 × 20, which is 920, so 874 is right and the answer is Not here. 924 is a touch over 46 × 20; 784 and 814 come from dropped pieces.
A school receives 1,256 pencils to share equally among 7 classrooms. How many pencils does each classroom get? How many pencils are left over?
Method: Divide with remainders
Each classroom 179, left over 3.
700 ÷ 7 = 100.
490 ÷ 7 = 70.
66 ÷ 7 is 9 with 3 left over.
100 + 70 + 9 = 179, remainder 3.
Check: 7 × 179 = 1,253, and 1,253 + 3 = 1,256.
Marco had $75. He bought two concert tickets for $19 each and paid $8 for parking. Which equation can be used to find m, the amount of money Marco has left?
Method: Multi-step equations
75 − 19 − 19 − 8 = m
Start with the whole, $75, and take off every part: two tickets and the parking.
75 − 19 − 8 = m counts one ticket. 75 + 19 + 19 + 8 = m adds the spending to the money he had. 19 + 19 + 8 = m is the total spent, not the money left.
Sam says that 3/8 + 3/8 = 6/16. Is Sam right?
Method: Fraction add and compare
No, the sum is 6/8
3 eighths and 3 more eighths is 6 eighths. The pieces are still eighths.
6/16 adds the denominators, which makes the pieces smaller than the ones you started with. 9/8 multiplies the tops. 3/16 subtracts nothing and still changes the denominator.
Four angles are shown. Which angle does NOT appear to measure 90°?
Method: Angles and shapes
Angle 2
Angles 1, 3 and 4 each open the same amount as the corner of a square, even though they are tilted differently. Tilting an angle does not change its measure.
Angle 2 opens less than a square corner. It is acute, about 60°.
The stem-and-leaf plot shows the points scored by a team in six games. What is the difference between the greatest and least number of points?
| Points scored | |
|---|---|
| Stem | Leaf |
| 5 | 1 4 |
| 6 | 0 2 7 |
| 7 | 3 |
Key: 2 | 5 means 25
Method: Data and patterns
22
Least: stem 5, leaf 1, so 51. Greatest: stem 7, leaf 3, so 73.
73 minus 51 equals 22.
2 comes from subtracting the leaves, 3 minus 1. 20 comes from subtracting the stems only.
Read this once, cold. The first move of each skill, then a few habits.
| Skill | First move |
|---|---|
| Place value and rounding | Say the place the question names, underline that digit, then look one digit to its right. |
| Decimals and fractions | Count the places after the point: tenths is one place, hundredths is two. |
| Multiply big numbers | Break both numbers into their places. Every part meets every part. |
| Divide with remainders | Divide place by place, check by multiplying back, then decide what the remainder means. |
| Multi-step equations | Name the whole and the parts. A missing part is the whole minus the known parts. |
| Fraction add and compare | Same-size pieces: count the pieces, keep the name. Comparing: make pieces alike or compare to one half. |
| Angles and shapes | Read the protractor row that starts at 0 on your ray, then check: obtuse means more than 90. |
| Perimeter and area | Say "around" or "inside" first. Around adds the sides; inside multiplies them. |
| Data and patterns | Stem is tens, leaf is ones. Test a rule on every row of the table. |