← ISEE® Lower Level

Quantitative Reasoning: Word Problems and Rule Generation

Thirty-eight word problems in thirty-five minutes, with no quantitative comparisons at this level, where a wrong answer means you thought about it wrongly.

18 topics · 72 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

The Two Math Sections and How They Differ

  1. How it worksHow does a reasoning question hide its math so that you have to think before you compute?
  2. How it worksWalk me through one shopping story asked as a reasoning problem and then as a calculation problem.
  3. Common mistakeWhy isn't a wrong answer on Quantitative Reasoning usually caused by bad arithmetic?
  4. Going furtherWhat happens when a student treats every math question as something to be computed?
Your turneasy1 of 3
The Lower Level gives Quantitative Reasoning with 38 questions and Mathematics Achievement with 30. A student reaching question 7 of the first one notices she has barely had to compute anything yet. Why two math sections instead of one longer one?

Rule Generation: Find the Rule, Then Use It

  1. How it worksHow do you pull a rule out of a word problem before you touch any of the numbers?
  2. Worked exampleWalk me through this: a sphere weighs 2 pyramids, a cube weighs 4 pyramids, and the three together weigh 14 grams. Find one pyramid.
  3. Worked exampleFind the rule turning 20, 16 and 10 into 11, 9 and 6, then use it to get an output of 15.
  4. Common mistakeWhy isn't one matching pair of numbers enough to trust a rule you just guessed?
Your turneasy1 of 3
On a balance, one sphere weighs the same as 4 pyramids and one cube weighs the same as 6 pyramids. How many pyramids would balance a sphere, a cube and one extra pyramid together?

Information You Were Given but Do Not Need

  1. How it worksHow do you tell a number you actually need from a number that is only scenery?
  2. Worked exampleFind how many stickers go in each album when Maya has 3 sisters, 24 stickers and 4 albums.
  3. Common mistakeWhy doesn't every number in a word problem have to end up somewhere in your answer?
  4. Going furtherWhat happens to a hard problem when you cross out the extra facts before you start?
Your turneasy1 of 3
Maya has 4 sisters, 7 packs of 6 stickers, and 3 albums. She shares all the stickers equally among the albums. How many stickers go in each album?

Choosing the Operation a Story Needs

  1. How it worksHow does splitting a total into equal groups turn into a division sentence?
  2. How it worksWalk me through picking add, subtract, multiply or divide from the shape of a story.
  3. Worked exampleFind how many boxes Alyssa orders for 240 students when one box holds 12 shirts.
  4. Common mistakeWhy isn't the word altogether always a signal to add?
  5. Going furtherWhat changes about the operation when the total is the unknown instead of the group size?
Your turneasy1 of 3
Shirts come 10 to a box. Alyssa orders 21 boxes for the school. How many shirts is that?

Estimating, and Knowing When an Estimate Is Enough

  1. How it worksHow does rounding both numbers first tell you which answer choices are even possible?
  2. Worked exampleFind a quick estimate of 4.17 times 3.9 good enough to choose between 12, 16 and 20.
  3. Going furtherWhat happens to your estimate when you round both numbers up instead of one up and one down?
Your turneasy1 of 3
Round 69 to the nearest ten and round 71 to the nearest ten, then multiply the two rounded numbers. What do you get?

Comparing and Contrasting Quantities

  1. How it worksHow do you compare 3/8 of a pizza with 2/5 of the same pizza using only a picture?
  2. Worked exampleCompare 7 boxes of 6 pencils with 5 boxes of 9 pencils and find which pile is larger.
  3. Common mistakeWhy isn't the fraction with the bigger bottom number the bigger fraction?
  4. Going furtherWhat changes when the two amounts you are comparing came from different sized wholes?
Your turneasy1 of 3
One pile has 7 boxes of 6 pencils. Another pile has 5 boxes of 9 pencils. How many pencils is the difference between the two piles?

Numbers and Operations: Making Sense of the Numbers

  1. How it worksHow do you show one amount as a fraction, a decimal and a shaded picture all at once?
  2. Worked exampleFind how much Julie spent on cookies when bread cost 2.50, milk 4.75, eggs 1.50 and her total was 13.75 dollars.
  3. How it worksHow does a number line help you throw out an answer choice that could never be right?
  4. Ties togetherHow does number sense connect to estimating your way past a word problem you cannot finish?
Your turneasy1 of 3
Julie spent 250 cents on bread, 475 cents on milk and 150 cents on eggs, and the rest on cookies. Her total was 1700 cents. How many cents did the cookies cost?

Properties of Operations and What They Look Like

  1. How it worksHow does the distributive property break one hard multiplication into two easy ones?
  2. Worked exampleShow me why 15 x (8 + 17) equals (15 x 8) + (15 x 17) using rows of little squares.
  3. Common mistakeWhy doesn't the order you subtract in work the way the order you add in does?
  4. Going furtherWhat happens to the grouping rule when multiplication and addition share one line?
Your turneasy1 of 3
Work out 15 times (7 + 17) by breaking it into two easier multiplications. What is the answer?

Algebraic Concepts: Patterns and the Rule Behind Them

  1. How it worksHow do you build each term of a pattern out of the term just before it?
  2. Worked exampleFind the tenth shape in a pattern going 2 squares, 5 squares, 8 squares, 11 squares.
  3. Common mistakeWhy isn't every pattern built by adding the same amount each time?
  4. Going furtherWhat changes about a pattern rule when the numbers shrink instead of grow?
  5. Ties togetherHow does a growing pattern turn into an input and output table run by a single rule?
Your turneasy1 of 3
A pattern of shapes goes 2 squares, 5 squares, 8 squares, 11 squares, and keeps going the same way. How many squares are in shape number 15?

Turning a Story into an Equation

  1. How it worksHow does a price nobody told you become a letter you can carry through a sentence?
  2. Worked exampleBuild the equation for Jim's 22 dollar total after buying 2 boxes of nails at 6 dollars each and one hammer.
  3. Common mistakeWhy isn't 6 + 2h the right sentence when two boxes of nails cost 6 dollars each?
  4. Going furtherWhat happens to the equation when the unknown is the number of boxes instead of the price?
Your turneasy1 of 3
Jim buys 2 boxes of nails at 7 dollars a box and one hammer. His total is 26 dollars. How many dollars was the hammer?

Swapping Equal Things for Each Other

  1. How it worksHow does replacing every shape with the smallest one turn a puzzle into one division?
  2. Worked exampleFind the weight of one cube when 2 cubes balance 6 marbles and 12 marbles weigh 36 grams.
  3. Going furtherWhat happens to a balance when the thing you swap in weighs more than the thing you removed?
Your turneasy1 of 3
On a balance, one cube weighs the same as 2 marbles. One marble weighs 9 grams. How many grams does one cube weigh?

Geometry: Describing Shapes and Their Symmetry

  1. How it worksHow do you count the fold lines that lay a shape exactly onto itself?
  2. Worked exampleFind which letter has more lines of symmetry, H or C, by folding each one over.
  3. Worked exampleFind the shape made by joining the points (1,1), (1,5), (5,5) and (5,1) on a grid.
  4. Common mistakeWhy isn't a shape symmetric just because both halves look about the same size?
Your turneasy1 of 3
A regular shape has 8 equal sides and 8 equal angles. How many lines of symmetry does it have?

Measurement: What About an Object Can Be Measured

  1. How it worksHow do length, area and volume each measure a different thing about one box?
  2. Common mistakeWhy doesn't a longer fence around a garden always mean more ground inside it?
  3. Worked exampleFind the perimeter and the area of a 6 by 2 rectangle and show why they answer different questions.
  4. Going furtherWhat happens to the area of a rectangle when you double only its width?
Your turneasy1 of 3
A rectangle is 10 units long and 2 units wide. What is the distance all the way around it?

Working Out Area by Reasoning, Not Formula

  1. How it worksHow does taking pieces away from a big rectangle give you an odd shape's area?
  2. Worked exampleFind the area left when four 2-inch squares are cut from the corners of a 5-inch square.
  3. Going furtherWhat happens to the leftover area when the corner squares grow from 2 inches to 3 inches?
  4. Common mistakeWhy doesn't slicing a shape into pieces change how much area it has in total?
Your turneasy1 of 3
Four squares of side 3 inches are cut from the four corners of a square of side 11 inches. How many square inches of the big square are left?

Data Analysis: Reading Graphs and Drawing Conclusions

  1. How it worksHow do you read one value off a bar graph whose scale counts by fives?
  2. Worked exampleFind the two most popular foods from votes of pizza 74, meatloaf 31, burger 59 and turkey 36.
  3. Going furtherWhat happens to a bar graph's story when its scale starts at 50 instead of 0?
  4. Common mistakeWhy isn't a rising line on a graph a promise about what happens next?
Your turneasy1 of 3
A bar graph has a scale counting up in fives. One bar reaches 9 marks above the zero line. What value does that bar show?

Reasoning Out the Chance of an Event

  1. How it worksHow does a count of the markers you want turn into the chance of pulling one?
  2. Worked exampleFind which color has a 1 in 8 chance from a box of 2 red, 3 blue, 4 green, 6 yellow, 4 pink and 5 purple markers.
  3. Common mistakeWhy isn't a chance of 5 out of 24 the same as a chance of 1 out of 5?
  4. Going furtherWhat happens to the chance of drawing blue when six more yellow markers join the box?
Your turneasy1 of 3
A box holds 2 red, 3 blue, 4 green, 8 yellow, 4 pink and 5 purple markers. Any chance worked out from this box is a count of one colour over the whole box. What is that whole box count?

Arriving at a Statistical Answer

  1. How it worksHow does levelling uneven towers of blocks give you the mean of a set?
  2. Worked exampleFind the average number of books read when four friends read 3, 5, 6 and 10 books.
  3. Going furtherWhat happens to that average when a fifth friend turns up having read 26 books?
  4. Ties togetherHow does choosing between the mean and the middle value depend on what you want to show?
Your turneasy1 of 3
Four friends read an average of 9 books each over the summer. How many books did the four of them read in all?

Pacing, Ordered Answer Choices and Never Leaving a Blank

  1. How it worksHow does a math section that gets harder as it goes change where you spend your time?
  2. How it worksHow does knowing the four choices sit in order from least to greatest help you eliminate?
  3. Worked exampleFind the time you get per question when 38 problems must fit inside 35 minutes.
  4. Common mistakeWhy isn't leaving a hard question blank safer than guessing when nothing is deducted?
Your turneasy1 of 3
Quantitative Reasoning has 38 questions. A student has answered 20 of them and has 20 minutes left. How many seconds does each remaining question get?

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