NumeriveMATH
CurriculumFamily workspaceStudentMy characterAchievementsContact
← Prealgebra levels

Prealgebra / LEVEL 15 · DIFFICULTY 5/5

Problem-Solving Workshop

Choose, explain and check strategies across unfamiliar contexts.

3 stages · 24 practice problems · two 6-question assessment forms

Choose an island to read its lesson.

  1. MINI QUEST 15.1Patterns & Justified RulesRead the lesson
  2. MINI QUEST 15.2Tables, Diagrams & Exhaustive CasesRead the lesson
  3. MINI QUEST 15.3Working Backward & Checking ConstraintsRead the lesson
  4. LEVEL CHECKCastle challenge

    6 questions across this level.

    Sign in for the level check →

STAGE 15.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Patterns & Justified Rules

Useful preparation: Expressions & Substitution · Addition, Multiplication & Complements

Goal: Use a stated rule or structural argument rather than guessing from a short list.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Separate observation from a rule

A few starting terms can fit many different rules. When a rule is specified, use it consistently; when it is not, a continuation is a conjecture rather than a unique consequence.

finite examples do not determine a unique sequence

Track differences and ratios

A constant additive change gives an arithmetic sequence. A constant multiplicative change gives a geometric sequence. Connect the term number with how many changes have occurred.

aₙ = a₁ + (n − 1)d

Explain a visual growth step

Count what is added at each stage and what was already present. A formula should follow from the construction, then be checked on small cases.

first stage + repeated additions
WORKED EXAMPLE 1

A sequence starts at 7 and adds 4 each time. Find its 12th term.

  1. There are 11 additions after the first term.
  2. 7 + 11(4) = 51.

51

WORKED EXAMPLE 2

A row of n joined hexagons shares one full side between neighbors. How many exposed sides for n = 5?

  1. Five hexagons contribute 30 sides before joining.
  2. Four shared sides remove eight exposed sides.

22

Common pitfalls

Possible mix-up: The tenth term requires ten additions to the first term.

It requires nine additions.

Possible mix-up: A short numerical list has only one possible continuation.

Uniqueness requires an additional rule or constraint.

Explain it to yourself

Give two different rules that begin with the same three terms.

Preview the eight practice prompts
  1. A sequence starts at 6 and adds 5 each time. Find its 10th term.
  2. A sequence starts at 3 and doubles each time. Find its 6th term.
  3. A sequence is defined by aₙ = n² + 2. Find a₉.
  4. A sequence starts at 40 and subtracts 3 each time. Find its 8th term.
  5. Do the three terms 2, 4, 8 alone determine a unique next number? Enter yes or no.
  6. A pattern begins with 4 tiles and adds 3 tiles each stage. At what stage does it first have 31 tiles, counting the initial stage as 1?
  7. A row of n joined squares shares one complete side between neighbors. How many exposed unit sides does a row of 14 squares have? New context
  8. Seats are arranged in 12 rows. The first row has 9 seats, and each successive row has 2 more. How many seats are in the last row? New context
Open stage 15.1 in the student workspace →

STAGE 15.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Tables, Diagrams & Exhaustive Cases

Useful preparation: Casework, Pairs & Systematic Lists · Solving & Checking Linear Equations

Goal: Turn a story into a representation that makes constraints visible.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Define variables and a finite range

A list becomes reliable when it has a rule for starting, advancing and stopping. Use positivity, totals and parity to narrow the range first.

list each possible first value; determine the rest

A diagram records relationships

Use bars for equal shares, a number line for distance or a grid for possibilities. Mark only facts given or proved; a drawing need not be to scale.

representation supports reasoning; appearance is not proof

Check completeness and duplication

Explain why every valid solution occurs in the table and whether two rows describe the same outcome. A final count is justified only after these checks.

cover every case exactly once
WORKED EXAMPLE 1

How many nonnegative solutions of 4a + 7b = 28 are there?

  1. b is between 0 and 4.
  2. Testing these values leaves b = 0 or 4.
  3. These give a = 7 or 0.

2

WORKED EXAMPLE 2

A rectangle has integer side lengths and area 24. How many shapes exist if rotations count as the same?

  1. List factor pairs with the smaller side first.
  2. (1,24), (2,12), (3,8), (4,6).

4

Common pitfalls

Possible mix-up: A sketch proves two lengths equal because they look equal.

Use stated data or a theorem.

Possible mix-up: Stopping after several examples proves the list complete.

Use bounds to justify that no later case can work.

Explain it to yourself

What feature of your table proves you have reached its final possible row?

Preview the eight practice prompts
  1. How many nonnegative integer pairs (a,b) satisfy 3a + 5b = 15?
  2. How many rectangles with positive integer sides have area 18, counting rotations as the same?
  3. A farm has 9 animals, all chickens or goats, with 26 legs total. How many goats are there?
  4. Two quantities sum to 46 and differ by 12. Find the smaller quantity.
  5. How many positive integer triples (a,b,c) satisfy a + b + c = 5?
  6. A two-digit number has digit sum 9. How many such numbers are even?
  7. A booth sells $3 tickets for children and $7 tickets for adults. It sells 10 tickets for $46. How many adult tickets were sold? New context
  8. A rectangle has positive integer side lengths and perimeter 26. How many different shapes are possible if rotations count as the same? New context
Open stage 15.2 in the student workspace →

STAGE 15.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Working Backward & Checking Constraints

Useful preparation: Solving & Checking Linear Equations · Percent Increase, Decrease & Reverse Change

Goal: Reverse a sequence of changes and validate the recovered starting state.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Undo the final action first

A forward process builds layers of operations. To recover its input, apply inverse operations in the opposite order.

multiply, then add ⇔ subtract, then divide

Track the quantity at each stage

Words such as half of what remains refer to a new baseline. Draw a chain and write the amount between successive operations.

initial → first change → second change → final

Reverse operations may need domain checks

Squaring loses sign, and rounding loses information. Even after a numerical reversal, check positivity, whole-number requirements and the original story.

a candidate must pass the forward check
WORKED EXAMPLE 1

A number is tripled, then increased by 8, giving 50. Find the number.

  1. Undo the final addition: 50 − 8 = 42.
  2. Undo tripling: 42/3 = 14.
  3. Check 3(14) + 8 = 50.

14

WORKED EXAMPLE 2

After spending one third of a balance, then $12, a traveler has $28. Find the initial balance.

  1. Before the $12 spending, the balance was $40.
  2. That was two thirds of the initial amount.
  3. 40 ÷ (2/3) = 60.

60 dollars

Common pitfalls

Possible mix-up: Undo operations in the original order.

Reverse the order as well as the operations.

Possible mix-up: A reversed arithmetic calculation automatically fits the story.

Check the recovered values and all original restrictions.

Explain it to yourself

Give an operation that cannot be uniquely reversed without extra information.

Preview the eight practice prompts
  1. A number is doubled, then increased by 9, producing 35. Find the number.
  2. A number is reduced by 7, then multiplied by 4, producing 52. Find the number.
  3. Half a number is increased by 6, producing 19. Find the number.
  4. A positive number is squared, then 11 is added, producing 60. Find the number.
  5. After spending 1/4 of an amount, $45 remains. Find the initial amount.
  6. A quantity increases by 20%, then 8 is subtracted, leaving 40. Find the initial quantity.
  7. A jar loses half its marbles, then 5 more. It has 17 left. How many marbles were in the jar at first? New context
  8. A traveler spends 1/3 of the starting money, then $18, and finally half of what remains. The traveler ends with $21. What was the starting amount? New context
Open stage 15.3 in the student workspace →
Browse another level →