Question

key idea
how can i use a model to multiply a whole number by a unit fraction?
the ? tells me how many equal parts to shade. the ? is the total amount shaded.

Explanation:

Step1: Analyze the first blank

When multiplying a whole number by a unit fraction (e.g., \(n\times\frac{1}{d}\), where \(n\) is the whole number and \(d\) is the denominator of the unit fraction), the denominator of the unit fraction tells us how many equal parts to divide the whole into, and when we multiply by the whole number, the numerator (which is 1 for a unit fraction) combined with the whole number operation: actually, for the first blank, when using a model (like area models or number lines) to multiply a whole number by a unit fraction \(\frac{1}{d}\), the "whole number" tells us how many times we take the unit fraction, but wait, no—wait, let's think again. Wait, the unit fraction is \(\frac{1}{d}\), so the denominator \(d\) is about the number of parts, but when multiplying a whole number \(n\) by \(\frac{1}{d}\), the process is like taking \(n\) copies of \(\frac{1}{d}\). But in terms of shading: if we have a model (like a rectangle divided into \(d\) parts, each \(\frac{1}{d}\)), and we multiply by a whole number \(n\), then the "whole number" tells us how many of those \(\frac{1}{d}\) parts to shade? Wait, no, maybe the first blank is "unit fraction's numerator" but no, unit fraction has numerator 1. Wait, maybe the first blank is "whole number"—wait, let's recall the concept. When multiplying a whole number \(n\) by a unit fraction \(\frac{1}{d}\), the model: for example, if we have \(n\) groups, each group is \(\frac{1}{d}\) of a whole. So the "whole number" tells us how many equal parts (of size \(\frac{1}{d}\)) to shade. Then the "product" (or the result of \(n\times\frac{1}{d}=\frac{n}{d}\)) is the total amount shaded. Wait, but the second blank is "the total amount shaded", which is the product. But let's check the sentence: "The [first blank] tells me how many equal parts to shade. The [second blank] is the total amount shaded." So when multiplying \(n\times\frac{1}{d}\), the whole number \(n\) tells us how many of the \(\frac{1}{d}\) parts to shade (since each part is \(\frac{1}{d}\), and we take \(n\) of them). Then the total amount shaded is the product \(n\times\frac{1}{d}=\frac{n}{d}\), but maybe the second blank is "product" or "whole number times unit fraction". Wait, maybe the first blank is "unit fraction's denominator" no, that's about the number of parts per whole. Wait, no—let's take an example: multiply 3 by \(\frac{1}{4}\). So we have 3 groups, each group is \(\frac{1}{4}\) of a whole. So we shade 3 parts, each of size \(\frac{1}{4}\). So the "whole number (3)" tells us how many equal parts (each \(\frac{1}{4}\)) to shade. Then the total amount shaded is \(3\times\frac{1}{4}=\frac{3}{4}\), which is the product. But maybe the first blank is "unit fraction" no, unit fraction is \(\frac{1}{d}\). Wait, maybe the first blank is "whole number" and the second blank is "product" (or "whole number times unit fraction"). Let's confirm the concept: When using a model to multiply a whole number by a unit fraction, the whole number tells you how many of the unit - fraction parts to shade. The product (or the result of the multiplication) is the total amount shaded. So first blank: whole number; second blank: product (or the result, which is \(\frac{\text{whole number}}{\text{denominator of unit fraction}}\)).

Step2: Analyze the second blank

After determining how many parts to shade (based on the whole number), the total amount shaded is the result of the multiplication, which is the product of the whole number and the unit fraction. So the second blank should be the "product" (or the value \(n\times\frac…

Answer:

First blank: whole number; Second blank: product (or the result of the multiplication, e.g., \(\frac{n}{d}\) where \(n\) is the whole number and \(d\) is the denominator of the unit fraction)

Wait, but maybe the exact terms are: first blank "unit fraction" no, no. Wait, let's check standard teaching: When multiplying a whole number by a unit fraction, the model: The whole number tells me how many equal parts (each being the unit fraction) to shade. The product (or the result, which is \(\frac{\text{whole number}}{\text{denominator}}\)) is the total amount shaded. So the first blank is "whole number", the second blank is "product" (or "total" or "result").