- Replace \(1\frac{3}{4}\) with the nearby whole number given in the problem. - Multiply that benchmark by \(2\). - Choose the option closest to the estimate.

1. Use the given benchmark \(1\frac{3}{4}\approx2\). 2. Then \(1\frac{3}{4}\times2\approx2\times2=4\).

B) \(4\).

- Read the labeled endpoint in one panel. - Split that mixed number into its whole-number part and fractional part. - Keep the factor \(2\) outside the decomposed mixed number.

1. The endpoint in one panel is \(1\frac{1}{2}\), so the product represented is \(1\frac{1}{2}\times2\). 2. Rewrite the mixed number as \(1\frac{1}{2}=1+\frac{1}{2}\). 3. The equivalent expression is \(\left(1+\frac{1}{2}\right)\times2\), so choice A is correct.

A) \(\left(1+\frac{1}{2}\right)\times2\)

- Read the side lengths of panel a). - Read the side lengths of panel b). - Add the two area products because the rectangles are the two parts of the mixed-number side.

1. Panel a) has side lengths \(2\) and \(3\), so its area is represented by \(2\times3\). 2. Panel b) has side lengths \(\frac{1}{3}\) and \(3\), so its area is represented by \(\frac{1}{3}\times3\). 3. Adding the two partial areas gives choice A.

A) \(2\times3+\frac{1}{3}\times3\).

- Use the stated halfway convention when rounding the second factor. - Rewrite each mixed number as an improper fraction before finding the exact product. - Look for common factors to cancel before multiplying.

1. For a), \(4\frac{2}{3}\approx5\) and, using the stated halfway convention, \(1\frac{1}{2}\approx2\). The estimate is \(5\times2=10\). 2. For b), rewrite the mixed numbers: \(4\frac{2}{3}=\frac{14}{3}\) and \(1\frac{1}{2}=\frac{3}{2}\). 3. Multiply and simplify: \(\frac{14}{3}\times\frac{3}{2}=\frac{14}{2}=7\).

a) Estimate: \(10\) b) Exact product: \(7\)

- Rewrite each mixed number as an improper fraction. - Look for common factors between numerators and denominators before multiplying. - Simplify the result completely.

1. For a), \(1 \frac{1}{4}=\frac{5}{4}\). Then \(\frac{5}{4}\times\frac{8}{15}=\frac{2}{3}\) after canceling common factors. 2. For b), \(2 \frac{2}{3}=\frac{8}{3}\) and \(1 \frac{1}{8}=\frac{9}{8}\). Then \(\frac{8}{3}\times\frac{9}{8}=3\).

a) \(\frac{2}{3}\) b) \(3\)

- Rewrite each mixed number in a form that is easier to multiply. - Look for factors that can be simplified before multiplying. - Convert the final fraction to a mixed number if it is greater than \(1\).

1. Rewrite the mixed numbers: \(2\frac{2}{5}=\frac{12}{5}\) and \(1\frac{3}{4}=\frac{7}{4}\). 2. Multiply: \(\frac{12}{5}\times\frac{7}{4}=\frac{21}{5}\). 3. \(\frac{21}{5}=4\frac{1}{5}\).

\(\frac{12}{5}\times\frac{7}{4}=\frac{21}{5}=4\frac{1}{5}\) feet

- Rewrite both the mixed-number length and the \(2\frac{1}{2}\)-dollar unit price as improper fractions. - Look for common factors that cancel before multiplying. - Express the exact result in dollars and cents.

1. Rewrite the mixed-number length: \(1\frac{3}{5}=\frac{8}{5}\). Rewrite the unit price as \(\$2.50=2\frac{1}{2}=\frac{5}{2}\) dollars per yard. 2. Multiply in fraction form: \(\frac{8}{5}\times\frac{5}{2}=\frac{8}{2}=4\). 3. Therefore the cost is \(\$4.00\).

\(\frac{8}{5}\times\frac{5}{2}=4\), so the cost is \(\$4.00\).

- Convert each mixed number to an improper fraction. - Simplify any common factors before or after multiplying. - Convert an improper final fraction back to a mixed number.

1. Rewrite the mixed numbers as improper fractions: \(3\frac{1}{6}=\frac{19}{6}\) and \(2\frac{2}{3}=\frac{8}{3}\). 2. Multiply: \(\frac{19}{6}\times\frac{8}{3}=\frac{152}{18}=\frac{76}{9}\). 3. Convert the improper fraction to a mixed number: \(\frac{76}{9}=8\frac{4}{9}\).

\(8\frac{4}{9}\)

- The total distance is the length of one loop multiplied by the number of loops. - Rewrite mixed numbers as improper fractions and simplify common factors.

1. Rewrite the mixed numbers: \(1\frac{1}{2}=\frac{3}{2}\) and \(2\frac{2}{3}=\frac{8}{3}\). 2. Multiply: \(\frac{3}{2}\times\frac{8}{3}=4\).

The rider travels \(4\) miles.

- Convert the mixed numbers to improper fractions. - Look for a numerator and denominator with a common factor before multiplying.

1. Rewrite: \(4\frac{1}{2}=\frac{9}{2}\) and \(1\frac{1}{9}=\frac{10}{9}\). 2. Simplify the common factor \(9\): \(\frac{9}{2}\times\frac{10}{9}=\frac{10}{2}\). 3. \(\frac{10}{2}=5\).

Simplify \(\frac{9}{2}\times\frac{10}{9}\) by canceling the common factor \(9\), giving \(\frac{10}{2}=5\). The product is \(5\).

- Convert the mixed numbers to improper fractions. - Look for a common factor between a numerator and denominator before multiplying. - Convert the final improper fraction to a mixed number.

1. Rewrite: \(2\frac{3}{5}=\frac{13}{5}\) and \(3\frac{3}{4}=\frac{15}{4}\). 2. Simplify \(15\div5=3\), giving \(\frac{13\times3}{4}=\frac{39}{4}\). 3. \(\frac{39}{4}=9\frac{3}{4}\).

\(9\frac{3}{4}\)

- Convert the mixed numbers to improper fractions. - Look for a numerator and denominator that cancel completely. - Convert the final fraction to a mixed number.

1. Rewrite: \(1\frac{5}{6}=\frac{11}{6}\) and \(2\frac{4}{11}=\frac{26}{11}\). 2. Cancel \(11\) and simplify \(\frac{26}{6}=\frac{13}{3}\). 3. \(\frac{13}{3}=4\frac{1}{3}\).

\(4\frac{1}{3}\)

- Convert both mixed numbers to improper fractions. - Simplify common factors across the fractions before multiplying.

1. Rewrite: \(3\frac{3}{8}=\frac{27}{8}\) and \(1\frac{7}{9}=\frac{16}{9}\). 2. Simplify \(27\div9=3\) and \(16\div8=2\). 3. Multiply \(3\times2=6\).

\(\frac{27}{8}\times\frac{16}{9}\) simplifies to \(3\times2\), so the product is \(6\).

- Rewrite each mixed number in a form that is easier to multiply. - Look for factors that can be simplified before carrying out the multiplication. - Remember that multiplying two lengths gives an area unit.

1. Convert the side lengths: \(2\frac{1}{2}=\frac{5}{2}\) and \(1\frac{3}{5}=\frac{8}{5}\). 2. Multiply: \(\frac{5}{2}\times\frac{8}{5}=\frac{40}{10}=4\). 3. Area is measured in square yards.

\(\frac{5}{2}\times\frac{8}{5}=4\), so the area is \(4\) square yards.

- Rewrite both mixed numbers as fractions before multiplying. - Look for numerator and denominator factors that simplify cleanly. - The phrase “times as long” signals a multiplicative comparison.

1. Convert the mixed numbers: \(3\frac{1}{3}=\frac{10}{3}\) and \(1\frac{4}{5}=\frac{9}{5}\). 2. Multiply: \(\frac{10}{3}\times\frac{9}{5}\). 3. Simplify before multiplying to get \(2\times3=6\). 4. The light strip must be \(6\) feet long.

The light strip must be \(6\) feet long.

- Convert each mixed number to an improper fraction first. - Simplify common factors before multiplying numerators and denominators. - Convert an improper fractional result to a mixed number and attach an area unit.

1. Convert the side lengths: \(5\frac{5}{6}=\frac{35}{6}\) and \(1\frac{1}{7}=\frac{8}{7}\). 2. Multiply and simplify: \(\frac{35}{6}\times\frac{8}{7}=\frac{5\times8}{6}=\frac{20}{3}\). 3. Convert \(\frac{20}{3}=6\frac{2}{3}\). 4. The area is \(6\frac{2}{3}\) square feet.

The area is \(6\frac{2}{3}\) square feet.

- Change each mixed number to an improper fraction before multiplying. - Simplify the resulting fraction if possible. - Convert an improper fraction to a mixed number and use square units for area.

1. Rewrite the mixed numbers as improper fractions: \(4\frac{2}{5}=\frac{22}{5}\) and \(2\frac{3}{4}=\frac{11}{4}\). 2. Multiply and simplify: \(\frac{22}{5}\times\frac{11}{4}=\frac{121}{10}\). 3. Convert the improper fraction: \(\frac{121}{10}=12\frac{1}{10}\). 4. The platform area is \(12\frac{1}{10}\) square feet.

\(\frac{22}{5}\times\frac{11}{4}=\frac{121}{10}=12\frac{1}{10}\) square feet

- The phrase “times as long” calls for multiplication. - Rewrite both mixed numbers as improper fractions. - Look for factors that simplify before multiplying.

1. Convert the factors: \(6\frac{1}{4}=\frac{25}{4}\) and \(1\frac{3}{5}=\frac{8}{5}\). 2. Multiply and simplify: \(\frac{25}{4}\times\frac{8}{5}=5\times2=10\). 3. The replacement cable should be \(10\) feet long.

\(\frac{25}{4}\times\frac{8}{5}=10\), so the replacement cable should be \(10\) feet long.

- Rewrite each mixed number as an improper fraction. - Look for a numerator and denominator that are the same or share a large common factor. - Simplify before multiplying to keep the arithmetic small.

1. Convert the mixed numbers: \(2\frac{3}{4}=\frac{11}{4}\) and \(3\frac{7}{11}=\frac{40}{11}\). 2. Simplify \(\frac{11}{4}\times\frac{40}{11}\) by canceling the factor \(11\) and reducing \(40\div4=10\). 3. The product is \(10\).

\(\frac{11}{4}\times\frac{40}{11}\) simplifies by canceling \(11\) and reducing \(40\div4\), leaving \(10\).

- Decide which operation connects the two side lengths to the area. - Rewrite the mixed numbers so the multiplication can be simplified before you finish it. - Include the correct square unit in the final answer.

1. The area is the product of the side lengths: \(3\frac{4}{7}\times1\frac{3}{4}\). 2. Rewrite the factors: \(3\frac{4}{7}=\frac{25}{7}\) and \(1\frac{3}{4}=\frac{7}{4}\). 3. Multiply and simplify: \(\frac{25}{7}\times\frac{7}{4}=\frac{25}{4}=6\frac{1}{4}\).

\(6\frac{1}{4}\,\text{ft}^2\)

- Rewrite both mixed numbers as improper fractions. - Look for factors that can be reduced before multiplying. - The simplified factors may make the final product a whole number.

1. Convert the mixed numbers: \(4\frac{1}{6}=\frac{25}{6}\) and \(2\frac{2}{5}=\frac{12}{5}\). 2. Simplify across the product: \(\frac{25}{6}\times\frac{12}{5}=5\times2\). 3. The product is \(10\).

\(\frac{25}{6}\times\frac{12}{5}\) simplifies to \(5\times2\), so the product is \(10\).

- Use the rectangle area formula with the two side lengths. - Rewrite both mixed numbers as improper fractions. - Convert the final improper fraction to a mixed number and include square feet.

1. The area is \(2\frac{1}{2}\times3\frac{1}{4}\). 2. Rewrite the mixed numbers: \(2\frac{1}{2}=\frac{5}{2}\) and \(3\frac{1}{4}=\frac{13}{4}\). 3. Multiply: \(\frac{5}{2}\times\frac{13}{4}=\frac{65}{8}\). 4. Convert to a mixed number: \(\frac{65}{8}=8\frac{1}{8}\).

\(\frac{5}{2}\times\frac{13}{4}=\frac{65}{8}=8\frac{1}{8}\,\text{ft}^2\)

- Use the rectangle area formula with the two side lengths. - Rewrite each mixed number as an improper fraction. - Convert the final improper fraction to a mixed number and include square feet.

1. The area is \(5\frac{1}{2}\times1\frac{3}{4}\). 2. Rewrite the mixed numbers: \(5\frac{1}{2}=\frac{11}{2}\) and \(1\frac{3}{4}=\frac{7}{4}\). 3. Multiply: \(\frac{11}{2}\times\frac{7}{4}=\frac{77}{8}\). 4. Convert to a mixed number: \(\frac{77}{8}=9\frac{5}{8}\).

\(\frac{11}{2}\times\frac{7}{4}=\frac{77}{8}=9\frac{5}{8}\,\text{ft}^2\)

- Identify the original length and the factor that tells how many times as long the new route is. - Rewrite the mixed numbers before multiplying. - Look for a common factor that can simplify the multiplication.

1. Multiply the loop length by the scale factor: \(3\frac{1}{2}\times2\frac{5}{7}\). 2. Rewrite the factors: \(3\frac{1}{2}=\frac{7}{2}\) and \(2\frac{5}{7}=\frac{19}{7}\). 3. Multiply and simplify: \(\frac{7}{2}\times\frac{19}{7}=\frac{19}{2}=9\frac{1}{2}\).

\(9\frac{1}{2}\) miles

- Identify the original width and the scale factor. - Rewrite both mixed numbers as improper fractions. - Convert the product to a mixed number and keep the length unit.

1. Multiply the original width by the scale factor: \(1\frac{3}{4}\times1\frac{1}{2}\). 2. Rewrite the factors: \(1\frac{3}{4}=\frac{7}{4}\) and \(1\frac{1}{2}=\frac{3}{2}\). 3. Multiply: \(\frac{7}{4}\times\frac{3}{2}=\frac{21}{8}\). 4. Convert to a mixed number: \(\frac{21}{8}=2\frac{5}{8}\).

\(\frac{7}{4}\times\frac{3}{2}=\frac{21}{8}=2\frac{5}{8}\,\text{ft}\)

- Use the rectangle area formula with the two side lengths. - Rewrite each mixed number as an improper fraction. - Convert the final improper fraction to a mixed number and include square inches.

1. The area is \(2\frac{5}{8}\times1\frac{1}{2}\). 2. Rewrite the mixed numbers: \(2\frac{5}{8}=\frac{21}{8}\) and \(1\frac{1}{2}=\frac{3}{2}\). 3. Multiply: \(\frac{21}{8}\times\frac{3}{2}=\frac{63}{16}\). 4. Convert to a mixed number: \(\frac{63}{16}=3\frac{15}{16}\).

\(\frac{21}{8}\times\frac{3}{2}=\frac{63}{16}=3\frac{15}{16}\,\text{in.}^2\)

- Read each rectangle's side lengths as one pairing of the decomposed mixed-number parts. - Find each partial area separately before combining them. - Use a common denominator when adding the fractional partial areas.

1. Decompose the factors as \(1+\frac{3}{4}\) and \(2+\frac{1}{2}\). 2. The four partial areas are \(1\times2=2\), \(1\times\frac{1}{2}=\frac{1}{2}\), \(\frac{3}{4}\times2=\frac{3}{2}\), and \(\frac{3}{4}\times\frac{1}{2}=\frac{3}{8}\). 3. Add: \(2+\frac{1}{2}+\frac{3}{2}+\frac{3}{8}=4+\frac{3}{8}=4\frac{3}{8}\).

The partial areas are \(2\), \(\frac{1}{2}\), \(\frac{3}{2}\), and \(\frac{3}{8}\). Their sum is \(4\frac{3}{8}\).

- Pick easy nearby numbers for both factors and use them to get a rough product. - Think of \(1\frac{1}{3}\) as one full copy plus one-third of a copy. - Compare the exact result with your estimate at the end.

1. Use easy nearby numbers: \(2\frac{1}{4}\approx2\) and \(1\frac{1}{3}\approx1\frac{1}{2}\). Then \(2\times1\frac{1}{2}=3\), so an estimate near \(3\) is reasonable. 2. Use the requested decomposition: \(2\frac{1}{4}\times\left(1+\frac{1}{3}\right)=2\frac{1}{4}+\left(2\frac{1}{4}\times\frac{1}{3}\right)\). 3. One third of \(2\frac{1}{4}=\frac{9}{4}\) is \(\frac{3}{4}\). 4. Add: \(2\frac{1}{4}+\frac{3}{4}=3\). The exact result agrees with the estimate.

An estimate is about \(3\): \(2\frac{1}{4}\approx2\) and \(1\frac{1}{3}\approx1\frac{1}{2}\). Using \(1\frac{1}{3}=1+\frac{1}{3}\), \(2\frac{1}{4}\times1\frac{1}{3}=2\frac{1}{4}+\frac{3}{4}=3\). The exact answer agrees with the estimate.

- Rewrite both mixed numbers as improper fractions. - Look for factors in a numerator and denominator that can cancel. - Notice whether any fractional part remains after simplifying.

1. Rewrite the factors: \(2\frac{1}{4}=\frac{9}{4}\) and \(1\frac{1}{3}=\frac{4}{3}\). 2. Multiply and simplify: \(\frac{9}{4}\times\frac{4}{3}=\frac{9}{3}=3\). 3. The factors \(4\) cancel completely, and \(9\) is divisible by \(3\), so no fractional part remains.

The product is \(3\). It is a whole number because the factor \(4\) cancels completely and \(9\div3=3\), leaving no fractional part.

- Rewrite the mixed numbers as improper fractions. - Test each proposed factor and compare the product with the target value.

1. Rewrite \(1\frac{1}{2}=\frac{3}{2}\) and test the choices. 2. \(\frac{3}{2}\times2=3\), \(\frac{3}{2}\times\frac{5}{2}=\frac{15}{4}=3\frac{3}{4}\), and \(\frac{3}{2}\times3=4\frac{1}{2}\). 3. Choice B makes the equation true.

B) \(2\frac{1}{2}\), because \(\frac{3}{2}\times\frac{5}{2}=\frac{15}{4}=3\frac{3}{4}\).

- Estimate the product using nearby whole numbers first. - Rewrite the mixed numbers in a form that can be multiplied exactly. - Compare the simplified result with both endpoints of the interval.

1. Rewrite the factors: \(5\frac{1}{4}=\frac{21}{4}\) and \(1\frac{2}{7}=\frac{9}{7}\). 2. Multiply and simplify: \(\frac{21}{4}\times\frac{9}{7}=\frac{27}{4}=6\frac{3}{4}\). 3. Since \(6<6\frac{3}{4}<7\), the product is between \(6\) and \(7\).

Yes. The product is \(6\frac{3}{4}\).

- Match each rectangle to one part of the first factor times one part of the second factor. - List every product made by pairing one part of the first factor with one part of the second factor. - Use improper fractions to verify the complete product.

1. The four partial products are \(2\times3\), \(2\times\frac{1}{3}\), \(\frac{1}{7}\times3\), and \(\frac{1}{7}\times\frac{1}{3}\). Lena used only the first and last products. 2. She omitted these two partial products: \(2\times\frac{1}{3}=\frac{2}{3}\) and \(\frac{1}{7}\times3=\frac{3}{7}\). 3. Rewrite the mixed numbers: \(2\frac{1}{7}=\frac{15}{7}\) and \(3\frac{1}{3}=\frac{10}{3}\). Then \(\frac{15}{7}\times\frac{10}{3}=\frac{50}{7}=7\frac{1}{7}\).

Lena omitted \(2\times\frac{1}{3}\) and \(\frac{1}{7}\times3\). The correct product is \(7\frac{1}{7}\).

- Rewrite each mixed number as an improper fraction. - Look for factors that cancel before multiplying. - Compare the simplified products rather than estimating from the mixed numbers alone.

1. Convert the first pair: \(1\frac{1}{2}=\frac{3}{2}\) and \(2\frac{2}{3}=\frac{8}{3}\). Their product is \(\frac{3}{2}\times\frac{8}{3}=4\). 2. Convert the second pair: \(1\frac{3}{4}=\frac{7}{4}\) and \(2\frac{2}{7}=\frac{16}{7}\). Their product is \(\frac{7}{4}\times\frac{16}{7}=4\). 3. The products are equal.

The products are equal; each is \(4\).

- Use nearby whole numbers to make a rough size prediction for the product. - Rewrite the mixed numbers as improper fractions for the exact calculation. - Look for factors that cancel completely before multiplying.

1. Both factors are greater than \(2\) and \(4\), so the product is greater than \(8\). Also, \(4\frac{1}{8}<4\frac{1}{2}\), and \(2\frac{2}{3}\times4\frac{1}{2}=12\), so the given product is less than \(12\). 2. Convert the factors: \(2\frac{2}{3}=\frac{8}{3}\) and \(4\frac{1}{8}=\frac{33}{8}\). 3. Multiply and simplify: \(\frac{8}{3}\times\frac{33}{8}=\frac{33}{3}=11\). 4. The exact product \(11\) is between \(8\) and \(12\).

The product is \(11\), which is between \(8\) and \(12\).

- Use the whole-number parts to think about a reasonable size for the product. - Rewrite each mixed number in an equivalent form that is easier to multiply. - Compare the exact product with the prediction after simplifying it.

1. Rewrite the factors: \(3\frac{5}{6}=\frac{23}{6}\) and \(2\frac{2}{5}=\frac{12}{5}\). 2. Multiply and simplify: \(\frac{23}{6}\times\frac{12}{5}=\frac{46}{5}=9\frac{1}{5}\). 3. Since \(9\frac{1}{5}>9\), the prediction is correct.

Yes. The exact product is \(9\frac{1}{5}\), which is a little more than \(9\).

- First find the whole-number product used for comparison. - Rewrite the mixed numbers so you can multiply them exactly. - Simplify common factors before deciding whether the claim is true.

1. The comparison product is \(2\times3=6\). 2. Rewrite the mixed numbers: \(1\frac{7}{8}=\frac{15}{8}\) and \(3\frac{1}{5}=\frac{16}{5}\). 3. Multiply and simplify: \(\frac{15}{8}\times\frac{16}{5}=6\). 4. The claim is true because both products equal \(6\).

The claim is true. \(1\frac{7}{8}\times3\frac{1}{5}=6\).

- First think about a nearby whole-number benchmark for the product. - Rewrite both factors in a form that can be multiplied directly. - Use the exact result to check your size comparison.

1. For a size check, \(3\frac{3}{8}<3\frac{1}{2}\) and \(2\frac{2}{9}<2\frac{1}{4}\). The larger benchmark product is \(3\frac{1}{2}\times2\frac{1}{4}=\frac{63}{8}=7\frac{7}{8}<8\), so the original product must also be less than \(8\). 2. For the exact value, rewrite the factors as \(\frac{27}{8}\) and \(\frac{20}{9}\). 3. Multiply and simplify: \(\frac{27}{8}\times\frac{20}{9}=\frac{15}{2}=7\frac{1}{2}\).

The product is less than \(8\). It is exactly \(7\frac{1}{2}\).

- Estimate the size of the product before finding it exactly. - Rewrite the mixed numbers in a form that is convenient for multiplication. - Use the whole-number part of the exact result to identify the interval.

1. Rewrite \(2\frac{1}{2}=2+\frac{1}{2}\). 2. Multiply \(4\frac{4}{9}\) by each part: \(4\frac{4}{9}\times2=8\frac{8}{9}\) and \(4\frac{4}{9}\times\frac{1}{2}=2\frac{2}{9}\). 3. Add the partial products: \(8\frac{8}{9}+2\frac{2}{9}=11\frac{1}{9}\). 4. Therefore the product lies between \(11\) and \(12\).

The product is \(11\frac{1}{9}\), so it lies between \(11\) and \(12\).

- Find the comparison product first. - Rewrite the mixed-number product in an equivalent form that can be simplified. - Look for factors that undo each other before multiplying everything.

1. The comparison value is \(3\times3=9\). 2. Rewrite the mixed numbers: \(2\frac{7}{10}=\frac{27}{10}\) and \(3\frac{1}{3}=\frac{10}{3}\). 3. Multiply and simplify: \(\frac{27}{10}\times\frac{10}{3}=9\). 4. The two products are equal.

They are equal; both products are \(9\).

- Replace the box with the number on the card. - Rewrite both mixed numbers in equivalent fractional form. - Check whether the exact product matches the right side of the equation.

1. Substitute the card value: \(2\frac{4}{5}\times2\frac{1}{7}\). 2. Rewrite the factors: \(2\frac{4}{5}=\frac{14}{5}\) and \(2\frac{1}{7}=\frac{15}{7}\). 3. Multiply and simplify: \(\frac{14}{5}\times\frac{15}{7}=6\). 4. The card makes the equation true.

Yes. \(2\frac{1}{7}\) correctly completes the equation because the product is \(6\).

- Decompose each mixed number into a whole-number part and a fractional part. - Check whether every part of the first factor has been paired with every part of the second factor. - Use the four rectangles to identify the two cross-products before adding all partial areas.

1. The factors decompose as \(3+\frac{1}{4}\) and \(1+\frac{2}{3}\). 2. Panel a) is \(3\times1\), and panel d) is \(\frac{1}{4}\times\frac{2}{3}\). 3. Luca omitted panel b), \(3\times\frac{2}{3}=2\), and panel c), \(\frac{1}{4}\times1=\frac{1}{4}\). 4. Add all four partial areas: \(3+2+\frac{1}{4}+\frac{1}{6}=5+\frac{5}{12}=5\frac{5}{12}\).

Luca omitted \(3\times\frac{2}{3}\) and \(\frac{1}{4}\times1\), shown by panels b) and c). The correct product is \(5\frac{5}{12}\).

- Imagine splitting the rectangle into four smaller rectangles by separating each side length into a whole-number part and a fractional part. - How do you rewrite a mixed number as an improper fraction? - Look for common factors to cancel before multiplying.

1. For a), Jordan counted only two of the four partial areas. Splitting the side lengths into whole-number and fractional parts creates two additional positive products, \(2\times\frac{1}{5}\) and \(\frac{1}{2}\times3\), so the claimed area is too small. 2. For b), rewrite the side lengths: \(2 \frac{1}{2}=\frac{5}{2}\) and \(3 \frac{1}{5}=\frac{16}{5}\). 3. Multiply and simplify: \(\frac{5}{2}\times\frac{16}{5}=\frac{16}{2}=8\). 4. The actual area is \(8\,\text{cm}^2\).

a) Jordan omitted the two partial areas formed by multiplying a whole-number part by a fractional part, so the result is too small. b) \(8\,\text{cm}^2\)

- Write a mixed number as a sum of a whole number and a fraction. - Apply the distributive property when one factor is a whole number. - For two mixed numbers, expand \(\left(2 + \frac{1}{3}\right)\left(1 + \frac{1}{2}\right)\) and count the partial products.

1. Multiplying by a whole number: The method works because of the distributive property. For example, \(4 \times \left(2 + \frac{1}{3}\right) = 4 \times 2 + 4 \times \frac{1}{3} = 8 + \frac{4}{3} = 9\frac{1}{3}\). 2. Check by converting the mixed number: \(\frac{7}{3} \times 4 = \frac{28}{3} = 9\frac{1}{3}\). 3. Multiplying two mixed numbers: Multiplying only the whole-number parts and only the fractional parts does not work because it omits two partial products. 4. For example, the incorrect shortcut gives \(2 \times 1 + \frac{1}{3} \times \frac{1}{2} = 2\frac{1}{6}\) for \(2\frac{1}{3} \times 1\frac{1}{2}\). 5. The correct product is \(\frac{7}{3} \times \frac{3}{2} = \frac{21}{6} = 3\frac{1}{2}\). Since \(2\frac{1}{6} \ne 3\frac{1}{2}\), the shortcut is invalid for two mixed numbers.

The claim is true when one factor is a whole number. It is false for two mixed numbers because multiplying only the matching parts omits the cross-products. For example, \(2\frac{1}{3} \times 1\frac{1}{2} = 3\frac{1}{2}\), not \(2\frac{1}{6}\).