Explore · changing whole
A workshop starts with 56 tiles. It uses 1/4 of its original tiles on a wall. It then uses 1/2 of the remaining tiles on a table. How many tiles go on the table?
Great Inheritance Mathematics Term A · Independent learning
Word Problems — Part 2
Learning target: Solve multistage fraction problems, including fraction-of-remainder with changing whole.
Begin with the exact prerequisite checkpoint below. Afterward, explain why your strategy works, not merely which procedure to use.
Opening task
Lesson-specific prerequisite checkpoint
Do this without a worked example. If you cannot explain it, review the earlier skill before continuing.
A jar has 18 beads; 1/3 are removed. How many remain?
Understand the idea
Solve multistage fraction problems, including fraction-of-remainder with changing whole.
Some word problems intentionally change the reference whole. If one fraction is used first and another fraction is taken from the remainder, calculate and label the remainder before using the second fraction. A sequential two-bar model prevents applying both fractions to the original.
Vocabulary focus: fraction of a quantity, overlapping partitions, reciprocal. Make a small labeled model, work through the arithmetic, and explain the result in words.
Watch for: Confusing the fraction reference quantity after a change.
Model to study: Sequential before/after bar models.
Worked examples
Worked examples 1 · changing whole
A workshop starts with 80 tiles. It uses 1/4 of its original tiles on a wall. It then uses 3/4 of the remaining tiles on a table. How many tiles go on the table?
Solution: 45 tiles
Why: Original whole 80; wall uses 80×1/4=20; remaining 60. Table uses 60×3/4=45 tiles.
Worked examples 2 · changing whole
A workshop starts with 27 tiles. It uses 1/3 of its original tiles on a wall. It then uses 2/3 of the remaining tiles on a table. How many tiles go on the table?
Solution: 12 tiles
Why: Original whole 27; wall uses 27×1/3=9; remaining 18. Table uses 18×2/3=12 tiles.
Worked examples 3 · changing whole
A workshop starts with 60 tiles. It uses 2/5 of its original tiles on a wall. It then uses 1/2 of the remaining tiles on a table. How many tiles go on the table?
Solution: 18 tiles
Why: Original whole 60; wall uses 60×2/5=24; remaining 36. Table uses 36×1/2=18 tiles.
Check understanding
Check understanding 1 · changing whole
A workshop starts with 100 tiles. It uses 2/5 of its original tiles on a wall. It then uses 1/4 of the remaining tiles on a table. How many tiles go on the table?
Check understanding 2 · changing whole
A workshop starts with 40 tiles. It uses 4/5 of its original tiles on a wall. It then uses 1/2 of the remaining tiles on a table. How many tiles go on the table?
Guided practice
Guided practice 1 · changing whole
A workshop starts with 84 tiles. It uses 2/3 of its original tiles on a wall. It then uses 1/2 of the remaining tiles on a table. How many tiles go on the table?
Guided practice 2 · changing whole
A workshop starts with 72 tiles. It uses 1/3 of its original tiles on a wall. It then uses 1/2 of the remaining tiles on a table. How many tiles go on the table?
Guided practice 3 · changing whole
A workshop starts with 48 tiles. It uses 1/3 of its original tiles on a wall. It then uses 1/2 of the remaining tiles on a table. How many tiles go on the table?
Independent practice
Independent practice 1 · changing whole
A workshop starts with 60 tiles. It uses 3/5 of its original tiles on a wall. It then uses 1/2 of the remaining tiles on a table. How many tiles go on the table?
Independent practice 2 · changing whole
A workshop starts with 54 tiles. It uses 1/3 of its original tiles on a wall. It then uses 2/3 of the remaining tiles on a table. How many tiles go on the table?
Independent practice 3 · changing whole
A workshop starts with 18 tiles. It uses 2/3 of its original tiles on a wall. It then uses 1/2 of the remaining tiles on a table. How many tiles go on the table?
Independent practice 4 · changing whole
A workshop starts with 27 tiles. It uses 2/3 of its original tiles on a wall. It then uses 2/3 of the remaining tiles on a table. How many tiles go on the table?
Independent practice 5 · changing whole
A workshop starts with 160 tiles. It uses 1/5 of its original tiles on a wall. It then uses 1/4 of the remaining tiles on a table. How many tiles go on the table?
Challenge and error analysis
Challenge and error analysis 1 · worked-error diagnosis
Error-analysis notebook: Learner notebook (LESSON): “After using 1/4 of 24, I take 1/2 of 24 again, not of the remaining 18.” Identify exactly which written step or assumption fails, correct it, and independently solve this new task: A workshop starts with 24 tiles. It uses 1/4 of its original tiles on a wall. It then uses 1/2 of the remaining tiles on a table. How many tiles go on the table?
Challenge and error analysis 2 · reasoning transfer
Reasoning and transfer: Solve, then explain a different way to check your answer (inverse operation, estimate, model, or units as appropriate). A workshop starts with 36 tiles. It uses 1/3 of its original tiles on a wall. It then uses 2/3 of the remaining tiles on a table. How many tiles go on the table?
Exit check
Exit check 1 · changing whole
A workshop starts with 32 tiles. It uses 3/4 of its original tiles on a wall. It then uses 1/2 of the remaining tiles on a table. How many tiles go on the table?
Exit check 2 · changing whole
A workshop starts with 84 tiles. It uses 1/2 of its original tiles on a wall. It then uses 2/3 of the remaining tiles on a table. How many tiles go on the table?
Target for independent mastery: all exit-check problems correct plus a justified explanation. A instructor may reteach the concept if calculations are correct without meaningful reasoning.