Wager Mage
Wager Mage
Photo: Sarah Chai
Each whole yields a two-thirds and one half of another two-thirds, therefore 3 sets of two-thirds can be made.
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Old Bill became the nickname for the Met police following the Great War after the fashion for wearing moustaches that looked very like the soldier...
Read More »Opening Question: How many 5âs are in 15? There are three sets of 5 in 15. Write this as a division problem: $$ 15div 5 = 3$$ You may want to do a couple of examples like this with whole numbers to familiarize students with thinking about division as how many of a number are in another number. How many halves are in 3? There are 6 half-sized pieces in 3 wholes. Write this as a division problem: $$3 div frac12 = 6$$ We can connect this to the algorithm for dividing fractions: $$3 div frac12 = 3 imes frac21 = 3 imes2$$ and we can see that there are 3 wholes with 2 halves in each whole, so there are $3 imes 2 = 6$ halves in $3$. How many sixths are in 4? There are 24 sixth-sized pieces in 4. Write this as a division problem: $$4 div frac16 = 24$$ We can connect this to the algorithm for dividing fractions: $$4 div frac16 = 4 imes frac61 = 4 imes 6 $$ and we can see that there are 4 wholes with 6 sixths in each whole, so there are $4 imes 6 = 24$ sixths in $4$. How many two-thirds are in 2? Each whole yields a two-thirds and one half of another two-thirds, therefore 3 sets of two-thirds can be made. Write this as a division problem: $$2 div frac23 = 3$$ We can connect this to the algorithm for dividing fractions: $$2 div frac23 = 2 imes frac32 = frac{2 imes 3}{2}$$ and we can see that there are 2 wholes with 3 thirds in each whole, so there are $2 imes 3$ thirds in $2$. Because we want to know how many two-thirds there are, we have to make groups of $2$ thirds, or divide the number of thirds we have by $2$. So there are $(2 imes 3)div 2 = 6div 2 = 3$ two-thirds in 2. How many three-fourths are in 2? 2 complete sets of three-fourths can be made and 2 of the 3 pieces need to make $frac34$ are left over, so we have another $frac23$ of a three-fourths. Write this as a division problem: $$2 div frac34 = 2 frac23$$ We can connect this to the algorithm for dividing fractions: $$2 div frac34 = 2 imes frac43 = frac{2 imes 4}{3}$$ and we can see that there are 2 wholes with 4 fourths in each whole, so there are $2 imes 4$ fourths in 2. Because we want to know how many three-fourths there are, we have to make groups of $3$ fourths, or divide the number of fourths we have by $3$. So there are $(2 imes 4)div 3 = 8div 3 = 2 frac23$ three-fourths in 2. How many $frac16$âs are in $frac13$? It takes two one-sixths to make a third. Write this as a division problem. $frac13 div frac16 = 2$ We can connect this to the algorithm for dividing fractions: $$frac13 div frac16 = frac13 imes frac61$$ and we can see that there are $1 imes 6 = 6$ sixths in a whole. The picture shows $frac13$ of those 6 pieces. There are $frac13 imes 6 = 2$ sixths in one-third.
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Read More »How many $frac16$âs are in $frac23$? It takes 4 one-sixth sized pieces to make the two-thirds. Write this as a division problem:$$frac23 div frac16 = 4$$ We can connect this to the algorithm for dividing fractions: $$frac23 div frac16 = frac23 imes frac61 $$ and we can see that there $1 imes 6 = 6$ sixths in a whole. The picture shows $frac23$ of those 6 pieces. There are $frac23 imes 6 = 4 $ sixths in two-thirds. How many $frac14$âs are in $frac23$? If the whole is divided into twelfths, four of them represent $frac13$ and three of them represent $frac14$ as shown in the picture above. In left picture, we have $frac23$, and in the right we have highlighted 2 twelfths that we are going to move to make it easier two see how many $frac14$ there are in $frac23$. On the left we can see where those 2 twelfths were moved to, and on the right we have colored them normally so we can focus on how many fourths there are. There are 2 fourths, and $frac23$ of another fourth. So all together, there are $2frac23$ fourths in $frac23$. Write this as a division problem: $$frac23 div frac14 = 2 frac23$$ We can connect this to the algorithm for dividing fractions: $$frac23 div frac14 = frac23 imes frac41 = frac{2 imes4}{3 imes1}$$ and we can see that there $2 imes 4 = 8$ twelfths in two-thirds. To find how many fourths there are, we need to see how many groups of 3 twelfths there are; in other words, there are $8div 3 = 2frac23 $ fourths in two-thirds.
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