Support
Answer and teaching response
9 · Each kit contains 2 + 1 objects, so 3 × (2 + 1) = 9. The equivalent expression (2 + 1) × 3 is also valid if the learner identifies the three complete groups.
Translate a story into a grouped numerical expression
Write numerical expressions from short supply stories and explain how parentheses distinguish repeating a whole group from adding an amount once.
Get student and teacher materialsPrepare the lesson
Learning goal: Translate the actions and grouping in a story into a numerical expression, interpret each part and explain why changing grouping or operation order can change its meaning.
You will need: Paper and pencil; optional counters in two colors. Use invented supplies rather than learner records. Write numerical expressions with +, -, ×, ÷ and parentheses; no letter variables or equation solving are needed. Learners write and interpret each expression before calculating its value.
Start with a diagnostic
Two bags each contain 3 green beads and 1 yellow bead. Write one numerical expression for all the beads, then enter its value. Underline the part that describes one bag.
Check the reasoning and choose a starting point
8 · One bag contains 3 + 1 beads, and there are two bags, so 2 × (3 + 1) = 8. The parentheses hold the complete one-bag amount. The expression 2 × 3 + 1 would include only one yellow bead in total.
Start with support: If the learner writes 2 × 3 + 1 or gives 7, draw two bag outlines and place 3 green and 1 yellow counter in each. Circle the complete contents of one bag, write (3 + 1), and then repeat that group twice. Recheck the learner's expression, not just a recounted total.
Ready for core: If the learner writes 2 × (3 + 1) or an equivalent grouped expression and explains the group, ask what different story 2 × 3 + 1 would describe. Continue to the two contrasting kit stories.
Work through an example
Six table kits each contain 4 pencils and 3 markers. Write a numerical expression for the total number of objects. How does it differ from a second story with 6 kits of 4 pencils and only 3 extra markers shared by the whole class?
The first story is 6 × (4 + 3), which has value 42 objects. The second is (6 × 4) + 3, which has value 27 objects. In the first, every kit includes all seven objects; in the second, the three markers are added once after counting the 24 pencils. Explain that grouping difference before evaluating.
- Read the first story and identify the repeated unit: one complete kit contains 4 pencils plus 3 markers. Write (4 + 3) and label it 'objects in one kit.' Ask learners to explain the parentheses before performing any arithmetic.
- Repeat the whole kit six times, writing 6 × (4 + 3). Connect the expression to six identical groups of seven objects. Its value is 42; multiplication applies to both kinds of objects in every kit.
- Read the second story and write (6 × 4) + 3. Circle six groups of four pencils, then a separate group of three markers added once. Its value is 27. Ask partners to point to the words that determine whether the three markers are inside or outside the repeated group.
- Check each operation against its verbal action, including its order. 'Subtract 3 from 12' means 12 - 3, not 3 - 12; 'share 12 among 3 equal groups' means 12 ÷ 3, not 3 ÷ 12. Reversing subtraction or division asks a different question. In a multi-action story, use parentheses to identify the quantity formed first and explain what the resulting group means.
Run the 40-minute lesson
- 5 minutes. Give the diagnostic and inspect the written expression before discussing totals. Follow the support or ready route, asking learners to identify the complete repeated group.
- 10 minutes. Model both kit stories side by side. Have learners write each expression before evaluating it, then label what every parenthesized part means. Use the subtraction and division examples to check how learners read 'from' and 'among.'
- 15 minutes. Support pairs complete p1 with group outlines and then p2 by acting out the sharing. Ready pairs complete p2 and p3, explaining each operation in words. Ask a partner to reconstruct a story from the expression and check it against the original; correct grouping before recalculating.
- 10 minutes. Give e1 independently for six minutes. Inspect its numerical expression, labels and value using the three-point rubric, then discuss the transfer contrast. Learners earning 0–2 points rebuild the repeated group with counters; learners earning 3 explain what story the changed grouping would represent.
A useful error to discuss
A learner might say: The expressions 6 × (4 + 3) and (6 × 4) + 3 describe the same supply story because both contain 6, 4 and 3.
Ask where the three markers are placed: in every kit or once for the whole class. Draw six kit outlines and put a marker group beside each for the first story; draw just one separate marker group for the second. Then match each drawing to its grouped expression before comparing values.
Practise at your level, then check independently
Use the diagnostic to choose support or core practice, then try the challenge. Complete the independent check without help. Enter a number, fraction or mixed number. The number check does not assess your drawing or explanation.
Core
Answer and teaching response
8 · One table first receives 18 ÷ 3 = 6 ribbons, then receives 2 more: (18 ÷ 3) + 2 = 8. The expression 18 ÷ (3 + 2) would share the original ribbons among five groups; it would not add ribbons to each of the three tables.
Challenge
Answer and teaching response
4 · The remaining collection is 24 - 8 = 16, so (24 - 8) ÷ 4 = 4 counters per group. By contrast, 24 - (8 ÷ 4) removes 2 from 24 and gives 22; it divides the removed amount rather than sharing the remaining collection.
Independent check
Answer and teaching response
52 · One pack contains 7 + 4 = 11 counters. Five complete packs contain 55, then removing 3 once leaves 52: (5 × (7 + 4)) - 3. The expression 5 × (7 + 4 - 3) removes 3 from every pack and would leave 40, a different story.
Transfer the idea
Without first evaluating, explain the difference between (24 - 8) ÷ 4 and (24 ÷ 4) - 8. Which describes removing 8 counters from the full collection before sharing the rest equally among four groups?
Discuss a possible response
(24 - 8) ÷ 4 removes 8 from the full collection and then shares what remains, so it matches. (24 ÷ 4) - 8 shares the full collection first and then tries to remove 8 from one group's share. That share is only 6, so the second expression does not give a possible remaining counter count for this story. The subtraction and division cannot simply exchange positions.
Choose the next teaching step
- Independent rubric, 1 point: writes (5 × (7 + 4)) - 3, or an equivalent numerical expression whose operations represent all five complete packs followed by one removal of 3.
- Independent rubric, 1 point: identifies 7 + 4 as one pack and explains that the subtraction happens once after combining, rather than once in each pack.
- Independent rubric, 1 point: obtains 52 counters with a calculation consistent with the written expression. Inspect grouping and interpretation separately; a correct numerical entry alone does not show the expression matches the story.
Support: Draw one container per repeated group and write the full contents inside it before introducing multiplication. Use arrows for 'then' actions after the groups combine. For sharing, label the total collection, number of equal groups and contents of one group so the dividend and divisor retain their meanings.
Extend: Write one story for 4 × (9 - 2) and a different story for (4 × 9) - 2. Explain the placement of the removed amount without calculating first, then evaluate to check your interpretations. Include the unit represented by each result.
Materials for your group
Student pages and teacher answers are separate. Keep the HTML files or print them in your browser.
Revisit a prerequisite
Repeated Fraction Groups connects multiplication to repeating a complete group, with fractional quantities. Use its group model if learners need help identifying the repeated unit. This investigation focuses on whole-number numerical expressions and how parentheses preserve a story's grouping.