Readiness Check

Basic units for measuring length are introduced in grade 2 and for volume in grade 3. In grades 4 and 5, students learn about more units and how to convert between them. This problem elicits some of the basic vocabulary that students will use in this unit.

If most students struggle with this item, do the optional Activity 2, “What Do They Measure?”, in Lesson 1. The activity prompts students to classify units of measurement by the attribute that they measure, reinforcing the vocabulary for length, volume, and weight.

In grade 4, students developed familiarity with the relative size of different units of measurement within a single system of measurement (for instance, 12 inches per 1 foot). In this problem, students estimate the mass or weight of objects and express them in units of their choice. Students may choose units from either the metric or customary system for all objects or use units from both systems.

For each object listed, a range of measurements is expected. Look for reasonableness in terms of order of magnitude and units of measurement. Some students may be familiar with pounds, for instance, but not know units small enough to describe the weight of an insect. This problem is a good opportunity to gather information about students’ knowledge of different units of measurement.

If most students struggle with this item, do the optional Activity 3, “Cutting String,” in Lesson 1 to allow students to gain a stronger sense of various units of length. To build students’ grasp of units of mass or weight, make available several physical objects that students can hold when doing Activity 4, “Card Sort: Measurement Benchmarks.”

To do this problem, students need to recognize a given fraction as being greater than 1, and realize that multiplying by a number greater than one results in a larger number. In this unit, students encounter fractional unit rates. Understanding that multiplying by a fraction is scaling or resizing helps students use unit rates appropriately to solve problems, such as when converting units of measurement, scaling a recipe, and comparing rates. It can also help students make sense of percentages as multiplicative comparisons and help them interpret percentages that are less than and greater than 100%.

If most students struggle with this item, use a measurement conversion problem in Lesson 3 to clarify how the size of a fractional factor in a multiplication relates to its product. Take the last problem in Activity 3, for example. After discussing how 8 kilograms can be converted to pounds by finding $8 \boldcdot \frac{22}{10}$ and before performing the calculation, ask students whether the result will be less than or greater than 8 and how they know. Consider comparing that product with the result of multiplying 8 by a fraction that is less than 1 (such as $\frac{7}{10}$) and a fraction that is equivalent to 1 (such as $\frac{10}{10}$).

Students will need to perform division when calculating unit rates and percentages. Look for students who reverse the dividend and divisor in their calculation or who misplace the decimal point.

If most students struggle with this item, use items from the Practice Problems in Lessons 1 and 2 to review division with decimal solutions before doing Lesson 4. In that lesson, Activity 3 will offer more practice using a shopping context.

The first two parts of this problem involve multiplying a fraction by a whole number. The second two parts require keeping track of place value in decimal multiplication. Students will need to perform these types of calculations when solving problems involving unit rates and percentages.

If most students struggle with the first part of this item, revisit the relationship between multiplication by a unit fraction and division in Lesson 3, Activity 1. In that activity, a Math Talk Warm-up, also highlight ways to use the structure of a simpler problem (such as finding $\frac{1}{4}$ of 32) to reason about related problems (such as finding $\frac{3}{4}$ of 32).

To solve problems about unit rates starting in Lesson 5 and about percentages in Lesson 15, students may multiply a whole number by a decimal, among other strategies. If students struggle with the second part of this item, help them make connections between multiplication and other strategies that rely on repeated addition. Also look for opportunities to reiterate reasoning strategies based on place value. For instance, to find $(0.3) \boldcdot 50$, we can first find $3 \boldcdot 50$, which is 150. Since 0.3 is one-tenth of 3, we can find one-tenth of 150.

This problem prompts students to use diagrams to represent fractions. In the first part, a blank tape diagram is given. In the last two parts, students create their own diagram to represent a fraction greater than 1. A variety of diagrams are possible, but a complete diagram should indicate the portion that corresponds to 1 whole.

If most students struggle with this item, revisit the meaning of fractions, and ways to represent them, in the synthesis of Lesson 12 Activity 1. Discuss with students what the numerator and denominator tell us and how they can be depicted on a diagram. Also draw attention to how 1 whole is shown, as this will help students identify 100% when representing percentages.

One of the fractions has a denominator of 100 and a numerator that is a multiple of 10, encouraging students to partition the given blank diagram into 10 equal parts instead of 100. If students partition it into 100 parts or indicate small parts that represent $1 \over 100$ each, ask students about equivalent fractions with a smaller denominator.