Students interpret, represent, and solve multiplicative comparison problems, using an understanding of the relationship between multiplication and division. They use this thinking to convert units of measure, within a given system, from larger to smaller units.
Unit Narrative
In this unit, students make sense of multiplication as a way to compare quantities. They use this understanding to solve problems about measurement.
In earlier grades, students related two quantities and made an additive comparison, where the key question was “How many more?” Here they make a multiplicative comparison, in which the underlying question is “How many times as many?” For example, if Mai has 3 cubes and Tyler has 18 cubes, we can say that Tyler has 6 times as many cubes as Mai does.
Initially, students reason, using concrete manipulatives and discrete images. Later, they reason more abstractly, using tape diagrams and equations. Comparative language, such as “_____ times as many (or much) as ____” is emphasized, offering students opportunities to attend to precision as they communicate mathematically (MP6).
Write a multiplication equation to compare
the pages read by Elena and Clare.
Use a symbol to represent the unknown.
Next, students use the idea and language of multiplicative relationships to learn about various units of length, mass, capacity, and time, and to convert from larger units to smaller units, within the same system of measurement. For example, they describe 1 kilometer as 1,000 times as long as a meter. Students then use their new knowledge to solve measurement problems.
Elena’s disc went 3 times as far as Clare’s did.
Andre’s disc went 4 times as far as Tyler’s did.
Analyze, describe, and represent multiplicative comparison situations.
Solve one-step and two-step problems involving multiplicative comparison.
Section Narrative
In this section, students learn to compare two quantities in terms of multiplication and to solve multiplicative comparison problems.
In earlier grades, students made comparisons in terms of addition or subtraction. To describe the number of cubes in the image, they may say, “Han has 3 more cubes than Andre,” or “Andre has 3 fewer cubes than Han.” Here they make this comparison by saying “Han has 2 times (or twice) as many cubes as Andre.”
Students begin with comparisons that involve lesser factors and familiar situations (such as comparing blocks), using familiar multiplicative comparison language (such as “twice,” or “twice as many”). They progress from using concrete representations (actual cubes) to discrete diagrams (showing cubes, or showing sections that each represent single objects). As they encounter greater factors and more-abstract situations, students interpret and use diagrams in which each section represents any quantity.
Diego has 5 times as many cubes as Kiran.
Lin read 7 books. Diego read 8 times as many books as Lin. How many books did Diego read?
Students write multiplication equations to express comparisons. As the problems become more complex, they reason, with given diagrams (or diagrams they draw), and use division to find an unknown factor.
Jada read some pages. Han read 60 pages altogether. The diagram shows how their pages compare.
How many times the number of Jada's pages did Han read?
diagram. two rectangles. bottom rectangle, Han's pages. partitioned into 3 equal parts each labeled question mark, total length 60. Top rectangle, Jada's pages. Same size as one of the 3 parts of bottom rectangle, labeled question mark.
Solve multi-step problems involving multiplicative comparison and measurement.
Section Narrative
In this section, students use multiplicative comparison and measurement conversion strategies to solve multi-step problems. As they convert customary and metric units of length, mass, and capacity, they continue to develop their understanding of relative sizes of units within the same system.
The problems here involve measurement units introduced in the previous section (pounds, ounces, kilometers, meters, centimeters), some from previous grades (yards, feet, and inches), as well as some new ones (gallons, quarts, and cups). As they make sense of situations, create representations, and write equations to solve problems, students practice reasoning quantitatively and abstractly (MP2).
Students also explore multiplicative relationships in geometric contexts. They analyze the relationship between the side lengths and the perimeters of quadrilaterals, performing unit conversions along the way.
Find the perimeter of Figure B and the missing side length of Figure D.
The perimeter of B is how many times the perimeter of D?
C
Perimeter = ______________
D
Perimeter = 12 inches
The section ends with an optional lesson in which students apply the understandings from this unit to make sense of measurements related to animals, and analyze statements about them.
Convert from larger units to smaller units, within a given system of measurement.
Solve multi-step problems involving multiplicative comparison and measurement.
Understand the relative sizes of kilometers, meters and centimeters, liters and milliliters, kilograms and grams, and pounds and ounces.
Section Narrative
Students have encountered units of measurement in earlier grades and in their daily lives. They have measured and estimated lengths in centimeters and in meters, recognized the number of minutes in an hour and measured intervals of time, and solved problems involving capacity and mass.
In this section, students expand on these concepts to convert measurements, within the same system (metric or customary), from larger units to smaller units. These conversions require an understanding of the multiplicative relationship between units.
Students begin by exploring lengths in metric units. To develop a sense of the multiplicative relationship between centimeters and meters, students build a length of 1 meter from centimeter grid paper. They recognize that 1 meter is 100 times as long as 1 centimeter, and use this reasoning to convert meters to centimeters. Later, they make sense of 1 kilometer by relating it to multiples of shorter measurements, such as the length of a basketball court or a soccer field.
Later, students learn the relationships between grams and kilograms, milliliters and liters, ounces and pounds, and hours, minutes, and seconds. As they solve problems and use multiplication to perform conversion, they develop a sense of the relative sizes of the units.
Put the animals and their travel distances in order, from the shortest to the longest.
