Topic 7 Reteaching: Comprehensive Study Guide on Adding and Subtracting Fractions and Mixed Numbers

Set A: Estimating Sums and Differences of Fractions

To estimate a sum or difference, each fraction should be replaced with a benchmark value such as $0$, $\frac{1}{2}$, or $1$. A number line is a useful tool to determine which benchmark a fraction is closest to. For example, to estimate the sum of $\frac{5}{8} + \frac{4}{5}$, follow these steps:

Step 1: Identify that $\frac{5}{8}$ is close to $\frac{1}{2}$.

Step 2: Identify that $\frac{4}{5}$ is close to $1$.

Step 3: Add the benchmark values: $\frac{1}{2} + 1 = 1 \frac{1}{2}$. Therefore, the sum of $\frac{5}{8} + \frac{4}{5}$ is approximately $1 \frac{1}{2}$.

Similarly, to estimate $\frac{17}{12} - ᄉ\frac{1}{8}$, follow these steps:

Step 1: Identify that $\frac{17}{12}$ is close to $1 \frac{1}{2}$.

Step 2: Identify that $\frac{1}{8}$ is close to $0$.

Step 3: Subtract the values: $1 \frac{1}{2} - 0 = 1 \frac{1}{2}$. Thus, $\frac{17}{12} - \frac{1}{8}$ is about $1 \frac{1}{2}$.

Set B: Finding Common Denominators

To rename fractions with a common denominator, you must find a common multiple for the denominators. One method is to multiply the denominators. For the fractions $\frac{1}{9}$ and $\frac{1}{3}$, multiplying $9 \times 3$ results in $27$. Therefore, $27$ is a common denominator.

To rename the fractions, use the Identity Property:

$\frac{1}{3} = \frac{1 \times 9}{3 \times 9} = \frac{9}{27}$

$\frac{1}{9} = \frac{1 \times 3}{9 \times 3} = \frac{3}{27}$

As a result, the fractions are renamed as $\frac{9}{27}$ and $\frac{3}{27}$. Alternatively, check if one denominator is a multiple of the other. Since $9$ is a multiple of $3$, $9$ can also serve as the common denominator. In this case, $\frac{1}{3}$ becomes $\frac{3}{9}$.

Set C: Adding and Subtracting Fractions with Unlike Denominators

When subtracting fractions like $\frac{5}{6} - \frac{3}{4}$, a common denominator must be found by listing multiples. For $6$, the multiples are $6, 12, 18, 24, 30, 36, 42$. For $4$, the multiples are $4, 8, 12, 16, 20, 24, 28, 32$. Since $12$ is a common multiple, it is used as the common denominator.

Next, use the Identity Property to write equivalent fractions. Multiply the numerator and denominator by the same number:

$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$

$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

Finally, subtract the fractions: $\frac{10}{12} - \frac{9}{12} = \frac{1}{12}$.

In real-world applications, such as Teresa spending $\frac{1}{3}$ of her day at school and $\frac{1}{10}$ of her day eating, the fractions must be added to find the total portion of the day. The sum involves finding a common denominator for $3$ and $10$, which is $30$.

Set D: Estimating Mixed Number Sums and Differences

To round a mixed number to the nearest whole number, compare the fractional part to $\frac{1}{2}$. If the fractional part is less than $\frac{1}{2}$, round down to the nearest lesser whole number. If the fractional part is greater than or equal to $\frac{1}{2}$, round up to the nearest greater whole number. Benchmark fractions like $\frac{1}{4}$, $\frac{1}{3}$, $\frac{1}{2}$, and $\frac{2}{3}$ can also assist in estimation. The symbol $\approx$ is used to denote that a value is approximately equal.

Example estimation of $5 \frac{1}{3} + 9 \frac{5}{6}$:

• Since $\frac{1}{3} < \frac{1}{2}$, $5 \frac{1}{3}$ rounds to $5$.
• Since $\frac{5}{6} \ge \frac{1}{2}$, $9 \frac{5}{6}$ rounds to $10$.
• The estimated sum is $5 + 10 = 15$.

Application: A mark shows water depth at $4 \frac{5}{12}$ feet. High tide rises the level by $2 \frac{5}{6}$ feet. To find the approximate depth at high tide, round and add: $4 + 3 = 7$ feet.

Set E: Adding Mixed Numbers with Models

Step 1: Rename the fractions with a common denominator. Model the addends and add the fractional parts. Note that you may need to rename the resulting fraction as a mixed number.

Step 2: Add the whole numbers to the regrouped fractions. For example, if adding $1 \frac{3}{4} + 1 \frac{3}{4}$, the fractional parts sum to $\frac{6}{4}$. This $\frac{6}{4}$ is renamed as $1 \frac{2}{4}$. Adding the whole numbers ($1 + 1$) and the regrouped mixed number ($1 \frac{2}{4}$) results in $3 \frac{2}{4}$.

Set F: Subtracting Mixed Numbers with Modeling

To find the difference for $2 \frac{1}{4} - 1 \frac{3}{4}$, follow these modeling steps:

Step 1: Model the number you are subtracting from, which is $2 \frac{1}{4}$ (or equivalently $2 \frac{2}{8}$). Because $\frac{3}{4} > \frac{1}{4}$, one whole must be renamed. Rename the whole to allow subtraction: $1 \text{ whole} = \frac{4}{4}$, so $2 \frac{1}{4}$ becomes $1 \frac{5}{4}$.

Step 2: Cross out the amount being subtracted ($1 \frac{3}{4}$). The remaining amount in the model is the answer. In this case, $2 \frac{1}{4} - 1 \frac{3}{4} = \frac{2}{4}$ or $\frac{1}{2}$. The difference is the part of the model not crossed out.

Set G: Multi-Step Problems with Mixed Numbers

When solving problems involving multiple mixed numbers, rename fractional parts to have common denominators. Always perform operations within parentheses first.

Case Study: Gil had two lengths of wallpaper, $2 \frac{3}{8}$ yards and $1 \frac{5}{6}$ yards long. He used some and has $1 \frac{3}{4}$ yards left. To find out how much he used, use two steps:

Step 1: Add to find the total amount Gil had. $2 \frac{3}{8} = 2 \frac{9}{24}$$1 \frac{5}{6} = 1 \frac{20}{24}$ Total: $3 \frac{29}{24} = 4 \frac{5}{24}$

Step 2: Subtract the amount left from the total amount. $4 \frac{5}{24} = 3 \frac{29}{24}$$1 \frac{3}{4} = 1 \frac{18}{24}$ Subtraction: $3 \frac{29}{24} - 1 \frac{18}{24} = 2 \frac{11}{24}$ Gil used $2 \frac{11}{24}$ yards of wallpaper.

Set H: Thinking Habits and Mathematical Modeling

Modeling with math requires specific thinking habits. Ask yourself:

1. How can I use math I know to help solve this problem?
2. How can I use pictures, objects, or an equation to represent the problem?
3. How can I use numbers, words, and symbols to solve the problem?

A bar diagram is a valuable tool for writing addition or subtraction equations.

Example 1: Justin jogs $3 \frac{5}{6}$ miles in the morning and $4 \frac{1}{3}$ miles in the evening. A bar diagram can represent these two addends to find the total daily distance.

Example 2: Last year, Mia's tree was $5 \frac{11}{12}$ feet tall. This year it is $7 \frac{1}{2}$ feet tall. A bar diagram and a subtraction equation ($7 \frac{1}{2} - 5 \frac{11}{12}$) can be used to find out how many feet the tree grew.