Students solve multi-step problems involving measurement conversions, line plots, and fraction operations, including addition and subtraction of fractions with unlike denominators. They also explain patterns when multiplying and dividing by powers of 10 and interpret multiplication as scaling, by comparing products with factors.
Unit Narrative
In this unit, students deepen their understanding of place-value relationships of numbers in base ten, unit conversion, operations on fractions with unlike denominators, and multiplicative comparison. The work here builds on several important ideas from grade 4.
In grade 4, students learned the value of each digit in a whole number is 10 times the value of the same digit in a place to its right. Here, they extend that insight to include decimals to the thousandths. Students recognize that the value of each digit in a place (including decimal places) is the value of the same digit in the place to its left.
This idea is highlighted as students perform measurement conversions in metric units.
Previously, students learned to convert from a larger unit to a smaller unit. Here, they learn to convert from a smaller unit to a larger unit. They observe how the digits shift when multiplied or divided by a power of 10 and learn to use exponential notation for powers of 10 to represent large numbers.
L
mL
5
6.3
0.95
800,000
65
Next, students turn their attention to fractions. In earlier grades, students made sense of equivalent fractions, added and subtracted fractions with the same denominator, and added tenths and hundredths. In this unit, they add and subtract fractions with different denominators. They see that the key is to find a common denominator and analyze different techniques for doing so.
Students then solve problems that involve measurement data (in halves, fourths, and eighths) that are displayed on line plots.
In the final section, students reason about the size of a product of fractions and the sizes of the factors. This work builds on the multiplicative comparison work in grade 4, in which students compared a whole number as “_____ times as many (or as much) as” another whole number. Here, students reason about products of a whole number and a fraction, without finding the value of each product. They use diagrams and expressions to support their reasoning.
Write , , or in each blank to make true statements.
Add and Subtract Fractions with Unlike Denominators
Section Goals
Add and subtract fractions with unlike denominators.
Create line plots to display fractional measurement data, and use the information to solve problems.
Solve problems involving addition and subtraction of fractions.
Section Narrative
In this section, students learn to add and subtract fractions (including mixed numbers) with unlike denominators and apply this learning to solve problems.
Students begin to add and subtract fractions, using strategies and diagrams that make sense to them, relying on what they know about adding and subtracting fractions with like denominators and with equivalent fractions. They then consider ways to write equivalent fractions so that the fractions in an expression have the same denominator. Later, they analyze and then use numerical strategies for finding common denominators, such as multiplying the denominators and finding a common multiple.
At the end of the section, students create line plots to display measurement data in fractional units (halves, fourths, and eighths), interpret the data on line plots, and use all four fraction operations to solve problems involving fractional measurements.
Do all of Mai’s apricots together weigh more or less than a pound?
Dot plot titled Mai's Apricots from 0 to 3 by 1’s. Hash marks by eighths. Horizontal axis, apricot weights, in ounces. Beginning at 7 eighths, the number of X’s above each eighth increment is 1, 0, 2, 0, 0, 2, 5, 3, 1, 0, 0, 0, 1.
In this section, students extend their understanding of place value and apply it to perform conversions between different, mostly metric, units.
Students begin by observing that the value of the digit in each place is 10 times the value of the same digit in the place to its right and the value of the same digit in the place to its left. They see that this applies not only to whole-number places but also to decimal places. Students then learn to use exponential notation for powers of 10 and to represent very large numbers, such as 1 million or 1 billion.
Next, students reason about measurement conversions in metric and customary units. Conversion in metric units further highlights place-value relationships between numbers in base ten. For example, this table shows some distances in centimeters, meters, and kilometers.
centimeters (cm)
meters (m)
kilometers (km)
1,500
15
0.015
15,000
150
0.15
150,000
1,500
1.5
Students notice that multiplying or dividing by a power of 10 shifts the position of the digits in a decimal number to the right or the left.
As they perform conversions from a larger unit to a smaller unit and the other way around, students apply what they learned about performing operations on whole numbers and decimals.
Make generalizations about multiplying a whole number by a fraction greater than 1, a fraction less than 1, and a fraction equal to 1.
Section Narrative
In this section, students build on their understanding of multiplication to include the concept of scaling. They interpret multiplication expressions as a quantity that is resized or scaled by a factor. This idea builds on the multiplicative comparison work students did with whole numbers in grade 4.
To develop an understanding of this concept, students compare the values of multiplication expressions, without performing the multiplication. Early in the section, the expressions are such that one factor among them is the same and the other factor is different.
Which expression represents the greatest product?
For example, they reason that is greater than and because in each expression, 4 is being multiplied by a fraction, and is the greatest of the three.
Students use visual representations to help them compare products. For instance, the shaded regions of the following diagrams can represent and .
Students also reason about products with one unknown factor, which prompts them to make the comparisons, based on the size of the other factor.
