Three Exit Tickets on 6 × [12 − (2 + 3)] + 4
Mr. Delgado's Grade 5 class is two weeks into writing and evaluating numerical expressions. Friday's exit ticket has one expression on the board: 6 × [12 − (2 + 3)] + 4. Of his 26 students, 19 wrote 46. Three papers represent the errors he keeps seeing.
- Amara: 2 + 3 = 5; 12 − 5 = 7; 6 × 7 = 42; 42 + 4 = 46.
- Devin: 12 − 2 = 10; 10 + 3 = 13; 6 × 13 = 78; 78 + 4 = 82.
- Cole: 6 × 12 = 72; 72 − 5 = 67; 67 + 4 = 71.
Mr. Delgado has 25 minutes of small-group time before the next lesson, in which students will interpret expressions without evaluating them. His goal is to choose one misconception to address with the whole class, and to plan a single follow-up item that would catch students who only copied the correct order of steps without understanding why the grouping symbols matter.
1. For each of Devin's and Cole's responses, name the misconception, state whether the error is about grouping symbols or about the meaning of the expression, and recommend which one Mr. Delgado should address with the whole class. Justify the choice.
Read the worked answer
Devin - the inner parentheses were ignored. He computed 12 − 2 first and then added 3, treating 12 − (2 + 3) as (12 − 2) + 3. The parentheses mark 2 + 3 as a single quantity, so the subtraction must be 12 − 5. A left-to-right habit overrode the grouping.
Cole - the multiplier was applied to only part of the bracket. He multiplied 6 × 12 and then subtracted 5, treating 6 × [12 − (2 + 3)] as 6 × 12 − 5. The bracket is one number (7), and the 6 multiplies that whole number.
Both errors share a root: a grouping symbol names one quantity that is computed first. Mr. Delgado should reteach that idea to the whole class, because it explains both papers and because Cole's version - distributing a factor across a bracket without multiplying every part - is the error most likely to reappear when students interpret expressions in the next lesson. Devin's paper can be repaired in a two-minute conference with a bar model showing 12 as a bar with 5 removed; Cole needs the class anchor, for example 6 groups of 7 drawn on a number line versus 6 groups of 12 with 5 taken off the end.
2. Write an expression for: add 4 and 5, then multiply the sum by 6. Then evaluate both (4 + 5) × 6 and 4 + 5 × 6, and explain what the two results show about why grouping symbols are necessary.
Read the worked answer
(4 + 5) × 6 names the sum first, then multiplies: 9 × 6 = 54.
4 + 5 × 6 without grouping uses the convention that multiplication is done before addition: 5 × 6 = 30, then 4 + 30 = 34.
Same numbers, same two operations, different results - 54 versus 34. That difference is the whole argument for grouping symbols: the order of operations cannot be inferred from the order the numbers are written, so the symbols must fix it. The parentheses bracket the sum as a single factor.
A useful check for students is to attach each expression to a story. (4 + 5) × 6 matches 'four dollars plus five dollars, then six times that total.' 4 + 5 × 6 matches 'four dollars plus six groups of five dollars.' Only one story fits each expression, which is why the notation has to be unambiguous.
3. A table of a pattern reads: term 1 → 4, term 2 → 10, term 3 → 16, term 4 → 22. Amara says the rule is 'add 6 each time.' Devin says the rule is 'multiply the term number by 6, then subtract 2.' Are both descriptions correct? Which is more useful for finding term 30, and what is that term?
Read the worked answer
Both describe the same table, and both can be checked against it.
- Amara's rule: 4, 10, 16, 22 - each term is 6 more than the one before. Correct for consecutive terms.
- Devin's rule: 6 × 1 − 2 = 4; 6 × 2 − 2 = 10; 6 × 3 − 2 = 16; 6 × 4 − 2 = 22. Correct for every term number.
The difference is efficiency and what each rule reveals. To reach term 30 with Amara's rule, a student must add 6 twenty-nine times: 4 + 29 × 6 = 178. With Devin's rule, term 30 is computed directly: 6 × 30 − 2 = 178. Devin's rule also explains why the step is 6 - every term is 2 less than a multiple of 6 - while 'add 6' only reports the step.
A good follow-up is to ask which rule finds term 100 faster and to have students test a rule on a term they have not listed yet (term 5 = 28 by either method), which is how the class confirms the two descriptions agree everywhere rather than only on the four printed rows.