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Alignments to Content Standards: 5.NF.B.6

A recipe for chocolate chip cookies makes 4 dozen cookies and calls for the following ingredients:

• $1\frac{1}{2}$ C margarine
• $1\frac{3}{4}$ C sugar
• $2$ t vanilla
• $3\frac{1}{4}$ C flour
• $1$ t baking powder
• $\frac{1}{4}$ t salt
• $8$ oz chocolate chips
1. How much of each ingredient is needed to make 3 recipes?
2. How much of each ingredient is needed to make $\frac34$ of a recipe?

## IM Commentary

This tasks lends itself very well to multiple solution methods. Students may learn a lot by comparing different methods. Students who are already comfortable with fraction multiplication can go straight to the numeric solutions given below. Students who are still unsure of the meanings of these operations can draw pictures or diagrams. Some students may find it easier to solve the second part by dividing a recipe for 12 dozen by 4. If they then compare this with multiplying by $\frac{3}{4}$ directly, it will give students another opportunity to make sense of what it means to multiply by $\frac{3}{4}$. Students who are having trouble even getting started with the problem can use concrete objects (actual measuring cups, or paper cut-outs, for example) to represent the quantities in the recipe.

This problem provides an opportunity to discuss unit conversion and rounding in a very realistic context. For example, students could talk about the fact that $\frac18$ cup is one tablespoon. Also, the recipe for 3 dozen cookies involves some sixteenths. We don't often measure in sixteenths in recipes, so an opportunity arises to have a useful discussion about what quantities we would actually use for 3 dozen cookies, and whether we would be able to tell the difference (by tasting the cookies) between two recipes that differ by, for example, $\frac{1}{16}$ t of salt.

## Attached Resources

• Lesson Plan: Cooking Time 3
• ## Solutions

Solution: Converting Mixed Numbers to Improper Fractions

1. To triple the recipe, you need to multiply the amount of every ingredient by 3.
2. To make $\frac{3}{4}$ of the recipe, you need to multiply the amount of every ingredient by $\frac{3}{4}$.
1 Recipea. 3 Recipesb. 3/4 Recipe
Margarine (cups)$1\frac{1}{2}$ \begin{align} 3 \times 1\frac{1}{2} =& \\ 3 \times \frac{3}{2} =& \\ \frac{9}{2} =& \\ & 4\frac{1}{2} \end{align} \begin{align} \frac{3}{4} \times 1\frac{1}{2} =& \\ \frac{3}{4} \times \frac{3}{2} =& \\ \frac{9}{8} =& \\ & 1\frac{1}{8} \end{align}
Sugar (cups)$1\frac{3}{4}$ \begin{align} 3 \times 1\frac{3}{4} =& \\ 3 \times \frac{7}{4} =& \\ \frac{21}{4} =& \\ & 5\frac{1}{4} \end{align} \begin{align} \frac{3}{4} \times 1\frac{3}{4} =& \\ \frac{3}{4} \times \frac{7}{4} =& \\ \frac{21}{16} =& \\ & 1\frac{5}{16} \end{align}
Vanilla (t)$2$ \begin{align} 3 \times 2 =& \\ & 6 \end{align} \begin{align} \frac{3}{4} \times 2 =& \\ \frac{6}{4} =& \\ & 1 \frac{1}{2} \end{align}
Flour (cups)$3\frac{1}{4}$ \begin{align} 3 \times 3\frac{1}{4} =& \\ 3 \times \frac{13}{4} =& \\ \frac{39}{4} =& \\ & 9\frac{3}{4} \end{align} \begin{align} \frac{3}{4} \times 3\frac{1}{4} =& \\ \frac{3}{4} \times \frac{13}{16} =& \\ \frac{39}{16} =& \\ & 2\frac{7}{16} \end{align}
Baking Powder (t)$1$ \begin{align} 3 \times 1 =& \\ & 3 \end{align} \begin{align} \frac{3}{4} \times 1 = \frac{3}{4} \end{align}
Salt (t)$\frac{1}{4}$ \begin{align} 3 \times \frac{1}{4} =& \\ & \frac{3}{4} \end{align} \begin{align} \frac{3}{4} \times \frac{1}{4} =& \\ & \frac{3}{16} \end{align}
Chocolate chips (oz)8 \begin{align} 3 \times 8 =& \\ & 24 \end{align} \begin{align} \frac{3}{4} \times 8 =& \\ \frac{24}{8} =& \\ & 6 \end{align}

Solution: Using the Distributive Property

1. To triple the recipe, you need to multiply the amount of every ingredient by 3.
2. To make $\frac{3}{4}$ of the recipe, you need to multiply the amount of every ingredient by $\frac{3}{4}$.
1 Recipea. 3 Recipesb. 3/4 Recipe
Margarine (cups)$1\frac{1}{2}$ \begin{align} 3 \times 1\frac{1}{2} =& \\ 3 \times (1 + \frac{1}{2}) =& \\ 3+\frac{3}{2} =& \\ & 4\frac{1}{2} \end{align} \begin{align} \frac{3}{4} \times 1\frac{1}{2} =& \\ \frac{3}{4} \times (1 + \frac{1}{2}) =& \\ \frac{3}{4} + \frac{3}{8} =& \\ & 1\frac{1}{8} \end{align}
Sugar (cups)$1\frac{3}{4}$ \begin{align} 3 \times 1\frac{3}{4} =& \\ 3 \times (1+\frac{3}{4}) =& \\ 3+\frac{9}{4} =& \\ & 5\frac{1}{4} \end{align} \begin{align} \frac{3}{4} \times 1\frac{3}{4} =& \\ \frac{3}{4} \times (1+\frac{3}{4}) =& \\ \frac{3}{4} + \frac{3}{16} =& \\ & 1\frac{5}{16} \end{align}
Vanilla (t)$2$ \begin{align} 3 \times 2 =& \\ & 6 \end{align} \begin{align} \frac{3}{4} \times 2 =& \\ \frac{6}{4} =& \\ & 1\frac{1}{2} \end{align}
Flour (cups)$3\frac{1}{4}$ \begin{align} 3 \times 3\frac{1}{4} =& \\ 3 \times (3+\frac{1}{4}) =& \\ 9+\frac{3}{4} =& \\ & 9\frac{3}{4} \end{align} \begin{align} \frac{3}{4} \times 3\frac{1}{4} =& \\ \frac{3}{4} \times (3+\frac{1}{4}) =& \\ \frac{9}{4}+\frac{3}{16} =& \\ & 2\frac{7}{16} \end{align}
Baking Powder (t)$1$ \begin{align} 3 \times 1 =& \\ & 3 \end{align} \begin{align} \frac{3}{4} \times 1 = \frac{3}{4} \end{align}
Salt (t)$\frac{1}{4}$ \begin{align} 3 \times \frac{1}{4} =& \\ & \frac{3}{4} \end{align} \begin{align} \frac{3}{4} \times \frac{1}{4} =& \\ &\frac{3}{16} \end{align}
Chocolate chips (oz)8 \begin{align} 3 \times 8 =& \\ & 24 \end{align} \begin{align} \frac{3}{4} \times 8 =& \\ \frac{24}{8} =& \\ & 6 \end{align}

#### lhwalker says:

over 4 years

If students struggle with this, one strategy could be to say, “When increasing the recipe from 4 dozen cookies to 12 dozen cookies, we multiplied by 3. What would we multiply to decrease the recipe from 4 dozen cookies to 3 dozen?”

I would write 4 x 3 = 12

          4x(3/?)= 3


This problem works out nice because 3 just happens to divide 12. I am wondering how many students would miss the logic details and come to the conclusion, "Since multiplying made the quantity bigger and dividing made it smaller, I guess I just divide."

The prior lesson had the students draw rectangles, and, in my opinion, this might be easier to understand. If each rectangle is a dozen and we want to have 3 dozen instead of 4, we can circle 3/4 of the dozens.

#### Kristin says:

over 4 years

Thanks, jzimba and math coach, I have (finally) made this change.

#### bridgette says:

over 4 years

I totally agree with you on this point...taking out the proportional relationships makes it more focused on the standards we want to work on.

#### Jason says:

over 5 years

I think this task shades beyond grade 5 into proportional relationships. The computations in the problem certainly take the form indicated in 5.NF.6, but the key insight that gets us started in the first place is that because you want three times as many dozens, you also want three times as much of any given ingredient. That seems like reasoning about a proportional relationship between two quantities.

For example, suppose the prompt said this instead:

A recipe for chocolate chip cookies makes 4 dozen cookies and calls for the following ingredients. ... (a) How much of each ingredient is needed to make three recipes? (b) How much of each ingredient is needed to make $\frac{3}{4}$ of a recipe?

Then the student is no longer being asked to reason about a proportional relationship in order to get started with the task.