In this lesson native Marilyn Burns’s brand-new book teaching Arithmetic: class for introducing Fractions, qualities 4–5 (Math options Publications, 2001), students room asked to come up with as numerous different means as they have the right to to define why a fraction is indistinguishable to one-half. The class engages the students through algebraic reasoning as the techniques they indicate are translated to generalizations using variables to stand for numerators and denominators.

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After the course had experience comparing 2 fractions by utilizing one-half as a benchmark, I created on the board:


“Raise your hand if friend can describe why this portion is indistinguishable to one-half,” i said. Ns waited till every student had actually raised a hand. While the question was trivial for most students, ns planned to construct on their understanding and also have a course discussion about different ways to identify that a fraction is identical to one-half. Also, i planned come compile the explanations they will do offer right into a list and also represent castle algebraically, thus providing the students suffer with utilizing variables to define general numerical relationships.

Before calling on any kind of students come respond, I gave a direction. “Be certain to hear to what others say and also see if your idea is the very same or different. If you have a various way, then raise her hand again. Ns interested in seeing how many different methods we can come up through to describe what provides a portion equivalent come one-half.”

Jake report first. “You include the optimal twice and also see if it renders the bottom,” the said.

I knew, or thought that ns knew, what Jake was stating. Yet his solution gave me the opportunity to press for more clarity indigenous him.

“Tell me what you typical with the portion I composed on the board,” ns said.

“You go 6 plus six and you obtain twelve,” the said.

“Oh,” ns said. “You added six to itself.” Jake nodded his agreement.

“That works,” ns confirmed. “Can you say her idea again, yet this time use the indigenous numerator and also denominator instead of top and also bottom?”

Jake said, “You go molecule plus numerator and also see if the price is the denominator.” i then wrote on the board:

If the numerator plus the numerator equates to the denominator, climate the fraction is worth 1⁄2.

“Who has a different way to decision if a fraction is precious one-half?” i asked. I dubbed on Rosie.

She said, “Do the molecule times two and see if the answer is the denominator.” I wrote on the board:

If the molecule times 2 amounts to the denominator, then the fraction is precious 1⁄2.

“I have an additional way,” Donald said. “If the molecule goes right into the denominator 2 times, climate the portion is one-half.” i wrote:

If the molecule goes right into the denominator two times, then the fraction is precious 1⁄2.

I said to the class, “My hand gets exhausted writing under your ideas. Among the benefits of math is the we have symbols to explain ideas and also don’t have to use words every one of the time. The symbols are prefer shortcuts. Check out if girlfriend understand how I can record Jake’s, Rosie’s, and also Donald’s ideas in mathematics, not English.”

I turned to the board, explaining together I wrote. “Instead of composing numerator and denominator over and over, I’ll simply use a shortcut for each: n and d. That can describe why this renders sense?” I had actually written:


n = numerator

d = denominator

Carl answered, “You simply used the first letter.”

“Look at this fraction,” ns said, writing 3⁄4 on the board. “For this fraction, n is 3 and d is four.”

“But it’s not one-half,” Steven said.

“No, the not,” ns agreed.

“I don’t get why n is three and also d is four,” Gena said.

“Who have the right to explain?” ns asked. I dubbed on Peter.

“Because 3 is the top number in the fraction, therefore it’s the numerator,” the said. “And four is the bottom number, for this reason it’s the denominator.”

“But the fraction you created — n end d— no a real fraction,” Gena said.

The idea the algebraic variables was brand-new for these students and I tried come explain. “It’s no a details fraction,” ns said. “Suppose girlfriend went home and also told your mom that your teacher gave you homework. If it were math homework, then you’d be introduce to me. But if the were scientific research homework, or a book report, or something because that art, climate you’d it is in talking around a various teacher. ‘Teacher’ is a general description; ‘Ms. Burns’ would certainly be a certain description. In the same way, ‘n end d’ is a basic name that can mean any fraction, however ‘six-twelfths’ describes a specific fraction because girlfriend now recognize what number you’re reasoning of because that the numerator and denominator.”

I wasn’t certain if Gena understood, yet I pressed on. “Watch as I translate Jake’s idea into a mathematical shortcut. If it renders sense come you, it certain will conserve us some writing energy.” I created on the board alongside the sentence I had written to define Jake’s idea:

Jake: If n + n = d, then n⁄d = 1⁄2.

I continued, “And watch what I might write for Rosie’s idea.” i wrote:

Rosie: If n x 2 = d, then 2⁄d = 1⁄2.

“What can I compose for your idea, Donald, utilizing n’s and also d’s?” i asked.

Donald assumed for a moment and then asked, “Can ns come up and also write it?” i agreed. He come up and also used the notation for entirety number department to record. That wrote:


This notation for department isn’t standard to algebraic representation, which I wanted Donald and the others to know. However I likewise wanted to respect Donald’s contribution, which to be correct in concept, yet not in convention. “What friend wrote provides sense come me,” ns said. “See if this means also describes your idea. That the means you’ll commonly see your idea in math books.” i wrote:

Donald: 2


I then went back to the conversation of other ways to see if a fraction were equivalent to one-half. I called on Gena, who currently seemed much more confident.

Gena said, “If the numerator is half of the denominator, climate it works.” I wrote on the board:

Gena: If n 1⁄2 = the d, then n⁄d = 1⁄2.

George had one more idea. “If you deserve to divide the denominator by two and also get the numerator, climate it’s one-half.” i recorded:

George: If d ÷ 2 = n, climate n⁄d = 1⁄2.

I then said, “Raise her hand as soon as you have in mental another portion that’s equivalent to one-half.” ~ a moment, all however Jonathan and Connie had actually raised their hands.

“Do you have actually a fraction, Jonathan?” ns asked. He nodded and raised a hand. So did Connie.

I started approximately the room having actually students phone call me fractions, and I taped their suggestions on the board. When I dubbed on Addison, he said, “One hundred–two hundredths.” over there was an episode of giggles adhered to by a decision of other portion suggestions that led to even much more giggles: “Five hundred–one thousandths.” “One thousand–two thousandths.” “One million –two millionths.” I composed each of this on the board:

100⁄200 500⁄1,000 1,000⁄2,000 1,000,000⁄2,000,000

Ali took an additional direction as soon as it was she turn. “Seven and a half–fifteenths,” she said. I created on the board:


Others complied with with similar fractions.

I then asked, “So how plenty of fractions carry out you think you could write the are tantamount to one-half?”

“Infinity!” number of answered in ~ the exact same time.

I said, “I agree the there room an infinite variety of fractions that are tantamount to one-half.

Representing your principles algebraically together I did on the graph is a handy means to refer to many, numerous fractions.”

I then provided the class an assignment. “I’m going to write ten fractions on the board,” i explained. “Decide if every is equal to one-half, much less than one-half, or an ext than one-half. On your paper, describe your thinking for each.”

I noted on the plank the ten fountain the students to be to consider:

1. 4⁄8

2. 6⁄13

3. 3⁄5

4. 3⁄6

5. 7⁄10


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7. 25⁄50

8. 25⁄51

9. 25⁄49

10. 1,000⁄2,000

See figures 1 and 2 on the following pages for instances of exactly how students functioned on this assignment.