Question 1: Using the grid, shade the area representing 2/3 of a whole, then shade 3 times this amount. What is the total shaded area?

Answer: 2

Question 2: Construct a visual model showing 4 × 1/4 on the grid. How many small squares are shaded?

Answer: 4

Question 3: Shade the grid to represent 3/5 multiplied by 2. How many small sections are shaded?

Answer: 6

Question 4: Draw a visual model for 1/2 × 3 and identify any errors in your representation.

Answer: Correct model shows 3 halves shaded; common mistake is not dividing the whole into halves properly.

Question 1: A student claims that shading 2/3 of a grid three times gives 2/3 × 3 = 2. Is this correct? Use a visual model to support your answer.

Answer: No, the total shaded area is 2/3 of the grid three times, which results in a total of 2, not 2/3. The misconception is multiplying the fraction by the number of copies without considering total area.

Question 2: Construct a visual model for 3/4 × 2. Explain any common mistakes in similar models.

Answer: Correct model shows 3/4 of each 1/2 shaded; common mistake is misrepresenting the fractional parts or multiplying the numerator and denominator directly.

Question 3: Identify the error in a visual model where a student shaded 2/5 and then shaded 2 more parts, claiming it's 2/5 × 2. Correct the mistake.

Answer: The mistake is adding fractions instead of multiplying. Correct approach is to shade 2/5 twice, resulting in 4/5 if they are separate parts, but the key is the multiplication process.

Question 1: A garden bed is 3/4 filled with soil. If a gardener adds 2 times this amount of soil using visual stacking, how much of the bed is filled?

Answer: 3/4 × 2 = 3/2 or 1 1/2 (exceeds 1, so the bed is completely filled plus half).

Question 2: A cake is cut into 8 equal slices. If you take 3 slices and then make 2 identical layers of this portion using visual stacking, how many slices does this represent?

Answer: 3/8 × 2 = 6/8 or 3/4 of the cake.

Question 1: Construct a visual model for 5/6 × 4. Show the model and explain why multiplying the numerator and denominator directly is incorrect or correct.

Answer: The model shows 5/6 shaded four times; multiplying numerator and denominator directly gives 20/6, which is over 3, but the correct method multiplies numerator by integer: 5×4=20, denominator stays 6, resulting in 20/6.

Question 2: A visual model indicates 2/3 of a rectangle is shaded, and then the entire rectangle is shaded twice. Explain the misconception and correct representation.

Answer: The misconception is thinking adding multiple whole rectangles is the same as multiplying a fraction. Correctly, 2/3 shaded twice equals total shading of 4/3, exceeding one whole.

Question 1: Shade two copies of 1/4 on the grid and then shade 3/4 of this combined area. What is the total shaded area?

Answer: 3/4 of 2/4 = (2×3)/(4×4) = 6/16 or 3/8.

Question 2: A student incorrectly claims that 1/2 × 2/3 equals 2/6. Using a visual model, explain the mistake and find the correct answer.

Answer: Mistake is multiplying numerators and denominators directly without understanding the process. Correct: 1/2 × 2/3 = (1×2)/(2×3) = 2/6, which simplifies to 1/3.