In this culminating lesson on multiplication, students continue to use the structure of base-ten numbers to make sense of calculations (MP7) and consolidate their understanding of the themes from the previous lessons. They see that multiplication of decimals can be accomplished by:thinking of the decimals as products of whole numbers and fractions;writing the non-zero digits of the factors as whole numbers, multiplying them, and moving the decimal point in the product; representing the multiplication with an area diagram and finding partial products; andusing a multiplication algorithm, the steps of which can be explained with the reasonings above.

This lesson serves two purposes. The first is to show that we can divide a decimal by a whole number the same way we divide two whole numbers. Students first represent a decimal dividend with base-ten diagrams. They see that, just like the units representing powers of 10, those for powers of 0.1 can also be divided into groups. They then divide using another method—partial quotients or long division—and notice that the principle of placing base-ten units into equal-size groups is likewise applicable.The second is to uncover the idea that the value of a quotient does not change if both the divisor and dividend are multiplied by the same factor. Students begin exploring this idea in problems where the factor is a multiple of 10 (e.g. 8÷1=80÷10). This work prepares students to divide two decimals in the next lesson.

In the previous lesson, students learned how to divide a decimal by a whole number. They also saw that multiplying both the dividend and the divisor by the same power of 10 does not change the quotient. In this lesson, students integrate these two understandings to find the quotient of two decimals. They see that to divide a number by a decimal, they can simply multiply both the dividend and divisor by a power of 10 so that both numbers are whole numbers. Doing so makes it simpler to use long division, or another method, to find the quotient. Students then practice using this principle to divide decimals in both abstract and contextual situations.

Students use a geometric model to investigate common factors and the greatest common factor of two numbers.Key ConceptsA geometric model can be used to investigate common factors. When congruent squares fit exactly along the edge of a rectangular grid, the side length of the square is a factor of the side length of the rectangular grid. The greatest common factor (GCF) is the largest square that fits exactly along both the length and the width of the rectangular grid. For example, given a 6-centimeter × 8-centimeter rectangular grid, four 2-centimeter squares will fit exactly along the length without any gaps or overlaps. So, 2 is a factor of 8. Three 2-centimeter squares will fit exactly along the width, so 2 is a factor of 6. Since the 2-centimeter square is the largest square that will fit along both the length and the width exactly, 2 is the greatest common factor of 6 and 8. Common factors are all of the factors that are shared by two or more numbers.The greatest common factor is the greatest number that is a factor shared by two or more numbers.Goals and Learning ObjectivesUse a geometric model to understand greatest common factor.Find the greatest common factor of two whole numbers equal to or less than 100.

Lesson OverviewStudents use a geometric model to investigate common multiples and the least common multiple of two numbers.Key ConceptsA geometric model can be used to investigate common multiples. When congruent rectangular cards with whole-number lengths are arranged to form a square, the length of the square is a common multiple of the side lengths of the cards. The least common multiple is the smallest square that can be formed with those cards.For example, using six 4 × 6 rectangles, a 12 × 12 square can be formed. So, 12 is a common multiple of both 4 and 6. Since the 12 × 12 square is the smallest square that can be formed, 12 is the least common multiple of 4 and 6.Common multiples are multiples that are shared by two or more numbers. The least common multiple (LCM) is the smallest multiple shared by two or more numbers.Goals and Learning ObjectivesUse a geometric model to understand least common multiples.Find the least common multiple of two whole numbers equal to or less than 12.

Goals and Learning ObjectivesFind the greatest common factor of two whole numbers equal to or less than 100.

Goals and Learning ObjectivesFind the least common multiple of two whole numbers equal to or less than 12.

In this lesson, students apply what they have learned about factors and multiples to solve a variety of problems. In the first activity, students to use what they have learned about common factors and common multiples to solve less structured problems in context (MP1).

Students will learn to use the distributive property to rewrite each sum as a product.  Visual representations of the areas of rectangles and their respective measurements (length and width) will be used. 

This is the first of two lessons that develop the idea of equivalent ratios.

In this lesson, they work with equivalent ratios more abstractly, both in the context of recipes and in the context of abstract ratios of numbers. They understand and articulate that all ratios that are equivalent to a:b can be generated by multiplying both aand b by the same number (MP6).By connecting concrete quantitative experiences to abstract representations that are independent of a context, students develop their skills in reasoning abstractly and quantitatively (MP2). They continue to use diagrams, words, or a combination of both for their explanations. The goal in subsequent lessons is to develop a general definition of equivalent ratios.

In this lesson, students use collections of objects to make sense of and use ratio language. Students see that there are several different ways to describe a situation using ratio language. For example, if we have 12 squares and 4 circles, we can say the ratio of squares to circles is 12:4 and the ratio of circles to squares is 4 to 12. We may also see a structure that prompts us to regroup them and say that there are 6 squares for every 2 circles, or 3 squares for every one circle (MP7).Expressing associations of quantities in a context—as students will be doing in this lesson—requires students to use ratio language with care (MP6). Making groups of physical objects that correspond with “for every” language is a concrete way for students to make sense of the problem (MP1).

Students work with a set of cards showing different ways of expressing ratios, including both part-part statements and part-whole statements. They group the cards that show the same ratio of boys to girls, but without the explicit use of the term equivalent.Key ConceptsRatios can be represented in a:b form, as fractions, as decimals, as factors, and in words; they can be expressed in part-part statements or in part-whole statements.Goals and Learning ObjectivesGroup cards showing ratios that are equivalent but expressed in different forms.

In this activity students will practice ordering food, discussing their meal, and paying for their meal.

Students will play a written version of the game telephone, and will determine what sorts of communication is effective with limited information, if any.  This lesson is part of a media unit curated at our Digital Citizenship website, "Who Am I Online?". 

To pique students’ curiosity and anchor the learning for the unit in the visible and concrete, students start with an experience of observing and analyzing a bath bomb as it fizzes and eventually disappears in the water. Their observations and questions about what is going on drive learning that digs into a series of related phenomena as students iterate and improve their models depicting what happens during chemical reactions. By the end of the unit, students have a firm grasp on how to model simple molecules, know what to look for to determine if chemical reactions have occurred, and apply their knowledge to chemical reactions to show how mass is conserved when atoms are rearranged.

In this 21-day unit, students are introduced to the anchoring phenomenon—a flameless heater in a Meal, Ready-to-Eat (MRE) that provides hot food to people by just adding water. Students explore the inside of an MRE flameless heater, then do investigations to collect evidence to support the idea that this heater and another type of flameless heater are undergoing chemical reactions as they get warm. Students have an opportunity to reflect on the engineering design process, defining stakeholders, and refining the criteria and constraints for the design solution.

This unit is part of the OpenSciEd core instructional materials for middle school.

This unit on metabolic reactions in the human body starts out with students exploring a real case study of a middle-school girl named M’Kenna, who reported some alarming symptoms to her doctor.

Students investigate data specific to M’Kenna’s case in the form of doctor’s notes, endoscopy images and reports, growth charts, and micrographs. They also draw from their results from laboratory experiments on the chemical changes involving the processing of food and from digital interactives to explore how food is transported, transformed, stored, and used across different body systems in all people.