Unit 3: Unit Rates and Percentages Lesson 3: Measuring with Different-Sized Units …
Unit 3: Unit Rates and Percentages Lesson 3: Measuring with Different-Sized Units
This lesson develops students’ familiarity with standard units of length, volume, weight, and mass through the tactile experiences of measuring objects. The main idea is that it takes more of a smaller unit and less of a larger unit to measure the same quantity. This idea is an important foundation for converting units of measurement using ratio reasoning in the next lesson (MP7).
Unit 8: Data Sets and Distributions Lesson 13: The Median of a …
Unit 8: Data Sets and Distributions Lesson 13: The Median of a Data Set
In this lesson, students consider another measure of center, the median, which divides the data into two groups with half of the data greater and half of the data less than the median. To find the median, they learn that the data are to be arranged in order, from least to greatest. They make use of the structure of the data set (MP7) to see that the median partitions the data into two halves: one half of the values in the data set has that value or smaller values, and the other half has that value or larger. Students learn how to find the median for data sets with both even and odd number of values.
Students engage in MP2 as they find the median of a numerical data set and interpret it in context. They begin to see that, just like the mean, the median can be used to describe what is typical in a distribution, but that it is interpreted differently than is the mean.
Unit 5: Arithmetic in Base Ten Lesson 6: Methods for Multiplying Decimals …
Unit 5: Arithmetic in Base Ten Lesson 6: Methods for Multiplying Decimals
In this lesson, students continue to develop methods for computing products of decimals, including using area diagrams. They multiply decimals by expressing them as fractions, or by interpreting each decimal as a product of a whole number and a power of 10 and (1/10). To multiply (0.25) x (1.6), for example, students may first multiply 0.25 by 100 and 1.6 by 10 to have whole numbers 25 and 16, multiply the whole numbers to get 400, and then multiply 400 by 1/1000 to invert the initial multiplication by 1,000. They may also think of 0.25 and 1.6 as 25/100 and 16/10, multiply the fractions, and then express the fractional product as a decimal.
In earlier grades, students used the area of rectangles to represent and find products of whole numbers and fractions. Here they do the same to represent and find products of decimals. They see that a rectangle that represents , for instance, can also be used to reason about (0.4) x (0.2), (0.004) x (0.002), or 40 x 20 because they all share a common structure. In this lesson, students extend their understanding of multiplication of fractions and multiplication using area diagrams by using previous methods to multiply any pair of decimals.
All work on probability is based on ideas of randomness, an idea …
All work on probability is based on ideas of randomness, an idea which has precise mathematical meaning, while being informally used in everyday life. This Nrich activity will lead to discussions of tricky ideas should challenge students' understanding.
This Nrich problem gives a clear context in which fractions, ratio and …
This Nrich problem gives a clear context in which fractions, ratio and proportion can be investigated. When using the interactivity, students can develop strategies for comparing fractions or ratios while thinking about which strategies are most useful for different cases.
This Nrich problem gives learners the opportunity to practice addition, subtraction, multiplication …
This Nrich problem gives learners the opportunity to practice addition, subtraction, multiplication and division of money, while it includes calculating with percentage. It is also a good context for developing a recording system and a systematic approach.
This Nrich activity gives pupils the opportunity to practice ordering and comparing …
This Nrich activity gives pupils the opportunity to practice ordering and comparing fractions. It extends the idea of finding a fraction of a whole to finding a fraction of 2. Working together on this task will encourage pupils to build their fraction vocabulary.
Unit: Area and Surface Area Lesson 14: More Nets, More Surface Area …
Unit: Area and Surface Area Lesson 14: More Nets, More Surface Area
This lesson further develops students’ ability to visualize the relationship between nets and polyhedra and their capacity to reason about surface area.
Previously, students started with nets and visualized the polyhedra that could be assembled from the nets. Here they go in the other direction—from polyhedra to nets. They practice mentally unfolding three-dimensional shapes, drawing two-dimensional nets, and using them to calculate surface area. Students also have a chance to compare and contrast surface area and volume as measures of two distinct attributes of a three-dimensional figure.
Unit 6: Expressions and Equations Lesson 18: More Relationships This lesson is …
Unit 6: Expressions and Equations Lesson 18: More Relationships
This lesson is optional. If time permits, it offers opportunities to look at multiple representations (equations, graphs, and tables) for some different contexts. Consider offering students a choice about which one they work on.
This final lesson on relationships between two quantities examines situations of constant area, constant volume, and a doubling relationship. Students have an opportunity to engage in MP7 as they notice the similar structures of the situations in the Making a Banner and Cereal Boxes activities, as well as connecting the Multiplying Mosquitoes activity to prior work with exponents and the Genie's coins situation from earlier in the unit. They may use those observations and knowledge to more easily solve the problems in the activities.
This Nrich problem follows on from Twisting and Turning, in which students …
This Nrich problem follows on from Twisting and Turning, in which students are introduced to an intriguing trick which provides a context for practicing manipulation of fractions. The trick is a hook to engage students' curiosity, leading to some intriguing mathematics to explore and explain, and ultimately generalize and prove.
Unit 3: Unit Rates and Percentages Lesson 8: More about Constant Speed …
Unit 3: Unit Rates and Percentages Lesson 8: More about Constant Speed
This lesson allows students to practice working with equivalent ratios, tables that represent them, and associated unit rates in the familiar context of speed, time, and distance. Students use unit rates (speed or pace) and ratios (of time and distance) to find unknown quantities (e.g., given distances and times, find a constant speed or pace; and given a speed or pace, solve problems about distance and time).
Unit 9: Putting It All Together Lesson 5: More than Two Choices …
Unit 9: Putting It All Together Lesson 5: More than Two Choices
This lesson is optional. It is the second of three lessons that explores the mathematics of voting. The activities in this lesson build on each other and on the previous lesson. As with all lessons in this unit, all related standards have been addressed in prior units; this lesson provides an optional opportunity to go more deeply and make connections between domains.
The five activities in this lesson deal with elections in which there are more than two choices. For example, if there are three choices, then the top vote getter might be approved by only 34% of the voters. Students explore several different rules for determining the winner: plurality, runoff, and instant runoff, and discover that the rules can give different results from the same set of voter preferences. They think about which voting rule more fairly represents the opinions of the voters. The mathematics in these activities emphasizes quantitative reasoning in a real-world situation (MP2 and MP4).
Most of the activities use students’ skills from earlier units to reason about ratios and proportional relationships (MP2) in the context of real-world problems (MP4). While some of the activities do not involve much computation, they all require serious thinking.
Most importantly, this lesson addresses topics that are important for citizens in a democracy to understand. Teachers may wish to collaborate with a civics/government teacher to learn how the fictional middle-school situations in this lesson relate to real-world elections.
A young girl has a wonderful idea to make the most MAGNIFICENT …
A young girl has a wonderful idea to make the most MAGNIFICENT thing! But making her magnificent thing is anything but easy, and the girl repeatedly tries and fails. Eventually, she quits, but a walk with her dog and time to think, she comes back to her project with renewed enthusiasm and manages to get it just right. The resource includes a lesson plan/book card, a design challenge, and copy of a design thinking journal that provide guidance on using the book to inspire students' curiosity for design thinking. Maker Challenge: Create small groups. Pass out one of the challenges listed in the lesson plan/book card to each group for them to come up with an invention that will solve the problem at hand.
A document is included in the resources folder that lists the complete standards-alignment for this book activity.
By the end of second grade, second graders should know their multiplication …
By the end of second grade, second graders should know their multiplication facts. Students need to practice and practice until they have them memorized. There are multiple ways to help them learn their facts. Learn about several digital and hands-on ways to teach them their times tables.
This Nrich problem provides the children with an opportunity to practise multiplying …
This Nrich problem provides the children with an opportunity to practise multiplying a single digit number by a multiple of 10. It also reinforces learning about equations being balanced and may lead to conversations about common factors. It encourages children to record their results, notice patterns and make predictions.
This problem provides the children with an opportunity to practice multiplying a …
This problem provides the children with an opportunity to practice multiplying a single digit number by a multiple of 100. It also reinforces learning about equations being balanced and may lead to conversations about common factors. It encourages children to record their results, notice patterns and make predictions.
This Nrich problem provides the children with an opportunity to practice multiplying …
This Nrich problem provides the children with an opportunity to practice multiplying a multiple of 10 by another multiple of 10. It also reinforces learning about equations being balanced and may lead to conversations about common factors. It encourages children to record their results, notice patterns and make predictions.
Unit 2: Introducing Ratios Lesson 12: Navigating a Table of Equivalent Ratios …
Unit 2: Introducing Ratios Lesson 12: Navigating a Table of Equivalent Ratios
The purpose of this lesson is to develop students’ ability to work with a table of equivalent ratios. It also provides opportunities to compare and contrast different ways of solving equivalent ratio problems.
Students see that a table accommodates different ways of reasoning about equivalent ratios, with some being more direct than others. They notice (MP8) that to find an unknown quantity, they can:
Find the multiplier that relates two corresponding values in different rows (e.g., “What times 5 equals 8?”) and use that multiplier to find unknown values. (This follows the multiplicative thinking developed in previous lessons.) Find an equivalent ratio with one quantity having a value of 1 and use that ratio to find missing values.
All tasks in the lesson aim to strengthen students’ understanding of the multiplicative relationships between equivalent ratios—that given a ratio , an equivalent ratio may be found by multiplying both and by the same factor. They also aim to build students’ awareness of how a table can facilitate this reasoning to varying degrees of efficiency, depending on one’s approach.
Ultimately, the goal of this unit is to prepare students to make sense of situations involving equivalent ratios and solve problems flexibly and strategically, rather than to rely on a procedure (such as “set up a proportion and cross multiply”) without an understanding of the underlying mathematics.
To reason using ratios in which one of the quantities is 1, students are likely to use division. In the example above, they are likely to divide the 90 by 5 to obtain the amount earned per hour. Remind students that dividing by a whole number is the same as multiplying by its reciprocal (a unit fraction) and encourage the use of multiplication (as shown in the activity about hourly wages) whenever possible. Doing so will better prepare students to: 1) scale down, e.g., to find equivalent ratios involving values that are smaller than the given ones, 2) relate fractions to percentages later in the course, and 3) understand division of fractions (including the “invert and multiply” rule) in a later unit.
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