Teaching Fractions in Grades 6 and 7
Teaching Fractions in Grades 6 and 7
Fractions are where arithmetic stops behaving. Multiplying can make a number smaller, dividing can make it bigger, and the whole changes from one question to the next. This page covers a grades 6 and 7 sequence built on the number line, with the misconceptions named as they arrive.
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The teaching problem
Fractions break the rules whole numbers taught
For five years students learn that a number grows as its digits grow, that multiplying makes a quantity larger and dividing makes it smaller. Fractions contradict all three, and nobody warns them. So 4/7 gets called larger than 3/5 because seven is larger than five, and 6 divided by 1/2 comes back as 3. The second difficulty is invisible: the whole. Half a pizza and half a class are the same fraction of very different things, and word problems switch the whole without announcing it. A third of the students in a class of thirty is not a third of the students in the school. Lessons have to make the whole explicit and put every fraction somewhere on a number line, or the procedures land on nothing.
A sequence that works
Five lessons from number line to division
Order matters more than usual here. Comparison comes before arithmetic, and multiplication and division are each introduced with a meaning attached before any rule gets written down.
- Fractions as numbers on a lineStudents place thirds, quarters and fifths on one number line, name the whole each time, and meet fractions greater than one before mixed number notation appears.
- Comparing without a common denominatorReasoning from benchmarks: nearer to zero, a half or one. Students justify 3/5 against 4/7 in words, then check with equivalence rather than starting there.
- Adding with a common unitEighths plus eighths works because the unit matches. Students build the common unit on the line, then meet the standard method as a shortcut for what they already did.
- Multiplying that makes things smallerTwo thirds of a half, treated as an operator on a quantity. Students predict whether the answer will be larger or smaller than each factor before calculating anything.
- Division as how many fitHow many quarters fit inside three? Students answer by measuring on the line first, then see why inverting the divisor gives the same result every time.
Where it goes wrong
The fraction errors worth planning for
Adding denominators is the famous one: 1/2 plus 1/3 becomes 2/5, an answer smaller than one of the parts. Ask for an estimate first and the error announces itself. Comparison by denominator size gives 4/7 as larger than 3/5. In multiplication, mixed numbers get split, so 2 1/3 times 3 turns into 6 1/3. Division prompts inverting the wrong number, or inverting both. And 0.3 gets treated as equal to 1/3 in conversion work. Two habits catch most of it: estimate before you calculate, and say aloud what the whole is. Mark the estimate as well as the answer for a few weeks.
What's in the download
Inside the files
Editable Word and PowerPoint plus print-ready PDFs, with answer keys throughout.
- Number line task sheets
- Fraction operation worksheets, four levels
- Diagnostic quiz flagging common errors
- Worked solutions for every task
- Slides for the comparison lesson
- Word problems with shifting wholes
Good to know
Frequently asked questions
The product says Years 6 to 7. Does it fit US grades 6 and 7?
Yes. The content covers comparing and ordering fractions, equivalence, the four operations with fractions and mixed numbers, and word problems where the whole shifts. That maps onto what US grade 6 and 7 classes do with number systems, and onto Years 6 and 7 in the UK and Australia. It is written to sit alongside those curricula rather than tied to one, so check your standards list against the contents page.
What do students need to know before starting?
Confident multiplication and division facts, and some experience of fractions as parts of shapes. The sequence does not assume equivalence or common denominators, since those are built in lessons two and three. If your class is shaky on times tables, the division lesson will be the sticking point, and a multiplication grid on the desk beats delaying the unit for a term.
Is there enough here for a class that has already met fractions?
Probably, though you would use it differently. Groups that have seen the procedures usually still compare by denominator and cannot say what dividing by a fraction means. Start with the diagnostic, teach lessons two, four and five, then use the rest as homework. Word problems where the whole changes tend to catch out students who have been getting the arithmetic right for a year.
Meaning first, then the rules
A student who can estimate the answer will not add denominators. Build that habit before the procedures arrive.
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