This task requires students to study the make-a-ten strategy that they should already know and use intuitively. In this strategy, knowledge of which sums make a ten, together with some of the properties of addition and subtraction, are used to evaluate sums which are larger than 10.

Making a 10 provides a technique to help students master single digit addition. The task is designed to help students visualize where the 10's are on a single digit addition table and explain why this is so. This knowledge can then be used to help them learn the addition table.

This task provides three types of comparison problems: Those with an unknown difference and two known numbers; those with a known difference and a bigger unknown number; and those with a known difference and smaller unknown number. Students may solve each type using addition or subtraction, although the language in specific problems tends to favor one approach over another.

This short video and interactive assessment activity is designed to teach fourth graders about matching the time to the clock.

Lesson OverviewStudents solve a classic puzzle about finding a counterfeit coin. The puzzle introduces students to the idea of a scale being balanced when the weight of the objects on both sides is the same and the scale being unbalanced when the objects on one side do not weigh the same as the objects on the other side.Key ConceptsThe concept of an inequality statement can be modeled using an unbalanced scale. The context—weighing a set of coins in order to identify the one coin that weighs less than the others—allows students to manipulate the weight on either side of the scale. In doing so, they are focused on the relationship between two weights—two quantities—and whether or not they are equal.Goals and Learning ObjectivesExplore a balance scale as a model for an equation or an inequality.Introduce formal meanings of equality and inequality.

Lesson OverviewStudents represent real-world situations using inequality statements that include a variable.Key ConceptsInequality statements tell you whether values in a situation are greater than or less than a given number and also tell you whether values in the situation can be equal to that number or not.The symbols < and > tell you that the unknown value(s) in a situation cannot be equal to a given number: the unknown value(s) are strictly greater than or less than the number. The inequality x < y means x must be less than y. The inequality x > y means x must be greater than y.The symbols ≤  and ≥ tell you that the unknown value(s) in a situation can also be equal to a given number: the unknown value(s) are less than or equal to, or greater than or equal to, the number. The inequality x ≤ y means x is less than or equal to y. The inequality x ≥ y means x is greater than or equal to y.Goals and Learning ObjectivesUnderstand the inequality symbols <, >, ≤, and ≥.Write inequality statements for real-world situations.ELL: When writing the summary, provide ELLs access to a dictionary and give them time to discuss their summary with a partner before writing, to help them organize their thoughts. Allow ELLs who share the same primary language to discuss in their native language if they wish.

Lesson OverviewStudents represent inequalities on a number line, find at least one value that makes the inequality true, and write the inequality using words.SWD:When calling on students, be sure to call on ELLs and to encourage them to actively participate. Understand that their pace might be slower or they might be shy or more reluctant to volunteer due to their weaker command of the language.SWD:Thinking aloud is one strategy for making learning visible. When teachers think aloud, they are externalizing their internal thought processes. Doing so may provide students with insights into mathematical thinking and ways of tackling problems. It also helps to model accurate mathematical language.Key ConceptsInequalities, like equations, have solutions. An arrow on the number line—pointing to the right for greater values and to the left for lesser values—can be used to show that there are infinitely many solutions to an inequality.The solutions to x < a are represented on the number line by an arrow pointing to the left from an open circle at a.Example: x < 2The solutions to x > a are represented on the number line with an arrow pointing to the right from an open circle at a.Example: x > 2The solutions to x ≤ a are represented on the number line with an arrow pointing to the left from a closed circle at a.Example: x ≤ 2The solutions to x ≥ a are represented on the number line with an arrow pointing to the right from a closed circle at a.Example: x ≥ 2Goals and Learning ObjectivesRepresent an inequality on a number line and using words.Understand that inequalities have infinitely many solutions.

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.

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.

Students explore methods of dividing a fraction by a unit fraction.Key ConceptsIn this lesson and in Lesson 5, students explore dividing a fraction by a fraction.In this lesson, we focus on the case in which the divisor is a unit fraction. Understanding this case makes it easier to see why we can divide by a fraction by multiplying by its reciprocal. For example, finding 34÷15 means finding the number of fifths in 34. In this lesson, students will see that this is 34 × 5.Students learn and apply several methods for dividing a fraction by a unit fraction, such as 23÷14.Model 23. Change the model and the fractions in the problem to twelfths: 812÷312. Then find the number of groups of 3 twelfths in 8 twelfths. This is the same as finding 8 ÷ 3.Reason that since there are 4 fourths in 1, there must be 23 × 4 fourths in 23. This is the same as using the multiplicative inverse.Rewrite both fractions so they have a common denominator: 23÷14=812÷312. The answer is the quotient of the numerators. This is the numerical analog to modeling.Goals and Learning ObjectivesUse models and other methods to divide fractions by unit fractions

Students use models and the idea of dividing as making equal groups to divide a fraction by a whole number.SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. Students can highlight the critical features or concepts in order to help them pay close attention to salient information.Key ConceptsWhen we divide a whole number by a whole number n, we can think of making n equal groups and finding the size of each group. We can think about dividing a fraction by a whole number in the same way.8 ÷ 4 = 2 When we make 4 equal groups, there are 2 wholes in each group.89÷4=29  When we make 4 equal groups, there are 2 ninths in each group.When the given fraction cannot be divided into equal groups of unit fractions, we can break each unit fraction part into smaller parts to form an equivalent fraction.34 ÷ 6 = ?     68 ÷ 6 = ?     68 ÷ 6 = 18  Students see that, in general, we can divide a fraction by a whole number by dividing the numerator by the whole number. Note that this is consistent with the “multiply by the reciprocal” method.ab÷n=a÷nb=anb=an×1b=an×b=ab×1nGoals and Learning ObjectivesUse models to divide a fraction by a whole number.Learn general methods for dividing a fraction by a whole number without using a model. 

Students solve word problems that require dividing and multiplying with fractions and mixed numbers.Key ConceptsStudents apply their knowledge about multiplying and dividing fractions to solve word problems. This includes applying the general methods for dividing fractions learned in previous lessons:Rewrite the dividend and the divisor so they have a common denominator. The answer to the original division will be the quotient of the numerators.Multiply the dividend by the reciprocal of the divisor.Goals and Learning ObjectivesApply knowledge of fraction multiplication and division to solve word problems.

Gallery OverviewAllow students who have a clear understanding of the content in the unit to work on Gallery problems of their choosing. You can use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionTiling a FloorStudents determine which size tiles are cheaper to use to tile a floor with given dimensions.Adam's HomeworkStudents find and correct an error in a whole number division problem.Then and NowStudents solve comparison problems involving census data from 1940 and 2010.Graphical MultiplicationGiven points m and p on a number line, students must locate m × p.When Does Zero Matter?Students must determine how the placement of 0 affects the value of a number.

Students explore whether multiplying by a number always results in a greater number. Students explore whether dividing by a number always results in a smaller number.Key ConceptsIn early grades, students learn that multiplication represents the total when several equal groups are combined. For this reason, some students think that multiplying always “makes things bigger.” In this lesson, students will investigate the case where a number is multiplied by a factor less than 1.Students are introduced to division in early grades in the context of dividing a group into smaller, equal groups. In whole number situations like these, the quotient is smaller than the starting number. For this reason, some students think that dividing always “makes things smaller.” In this lesson, students will investigate the case where a number is divided by a divisor less than 1.Goals and Learning ObjectivesDetermine when multiplying a number by a factor gives a result greater than the number and when it gives a result less than the number.Determine when dividing a number by a divisor gives a result greater than the number and when it gives a result less than the number.

Students explore methods of dividing a fraction by a fraction.Key ConceptsStudents extend what they learned in Lesson 4 to divide a fraction by any fraction. Students are presented with two general methods for dividing fractions:Rewrite the dividend and the divisor so they have a common denominator. The answer to the original division will be the quotient of the numerators.Multiply the dividend by the reciprocal of the divisor.These two methods will work for all cases, including cases in which one or both of the numbers in the division is a fraction or whole number.Goals and Learning ObjectivesUse models and other methods to divide fractions by fractions.

Students use estimation or other methods to place the decimal points in products and quotients. They review the algorithms for the four basic decimal operations and solve multistep word problems involving decimal operations.Key ConceptsThe algorithms for whole-number operations can be extended to decimal operations. Students learned the algorithms for decimal operations in Grade 5. By the end of Grade 6, they should be fluent with these operations.For decimal addition and subtraction, once the decimal points of the addends are aligned (which aligns like place values), the algorithms are the same as for whole numbers. The decimal point in the sum or difference goes directly below the decimal point in the numbers that were added or subtracted.For decimal multiplication and division, one method is to ignore the decimal points and apply the whole-number algorithms. Then use estimation or some other method to place the decimal point in the answer.Goals and Learning ObjectivesReview and practice the algorithms for all four decimal operations.Solve real-world problems involving decimal operations.

Students explore methods for dividing a whole number by a fraction.Key ConceptsIn earlier grades, students learned to think of a whole number division problem, such as 8 ÷ 4, in terms of two types of equal groups.Divisor as the Number of Groups Divide 8 into 4 equal groups and find the size of each group.Divisor as the Group Size Divide 8 into groups of 4 and find the number of groups.To divide a fraction by a whole number in Lesson 2, students used the first interpretation. For example, to find 89 ÷ 4, they divided 8 ninths into 4 equal groups and found that there were 2 ninths in each group.To divide a whole number by a fraction, the second interpretation is most helpful. For example, to find 3 ÷ 34, we find the number of groups of 3 fourths in 3 wholes. The diagram in the Opening shows that there are 4 groups, so 3 ÷ 34 = 4.Just as with whole number division, the quotient when a whole number is divided by a fraction is not always a whole number. Below is a model for 2 ÷ 35. The model shows that there are 3 groups of 3 fifths in 2 wholes plus 13 of another group (13 of a group of 3 fifths is 1 fifth). Therefore, 2 ÷ 35 = 313. Notice that once we have divided the 2 wholes into fifths, we are finding the number of groups of 3 fifths in 10 fifths. This is simply 10 ÷ 3.These models can help explain that the “multiply by the reciprocal” method of dividing a whole number by a fraction works. To find 2 ÷ 35, we can multiply 2 by 5 to find the total number of fifths in 2 and then divide the result (10) by 3 to find the number of groups of 3 of these fifths in 2. So, 2÷35=2×53=2×53.ELL: Encourage students to verbalize their explanations. To help students gain confidence and increase their understanding, allow those that share the same language of origin to speak in small groups using their prefered language.Goals and Learning ObjectivesUse models and other methods to divide a whole number by a fraction.

Gallery OverviewAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionRepresent a Math ProblemStudents explore the number line tool and the double number line tool. They use the number line tool to solve a problem about the weights of a cheetah and a fisher cat.Research ExpressionsStudents learn the difference between numerical expressions and variable expressions. They watch video tutorials, review worked examples, use the Glossary, and explore other resources.Fish TankStudents create diagrams and use text and images as they solve a problem about the size of a fish tank.

In this lesson, students are introduced to rate in the context of music. They will explore beats per minute and compare rates using mathematical representations including graphs and double number lines.Key ConceptsBeats per minute is a rate. Musicians often count the number of beats per measure to determine the tempo of a song. A fast tempo produces music that seems to be racing, whereas a slow tempo results in music that is more relaxing. When graphed, sets with more beats per minute have smaller intervals on the double number line and steeper lines on the graph.Goals and Learning ObjectivesInvestigate rate in music.Find beats per minute by counting beats in music.Represent beats per minute on a double number line and a graph.

In this lesson, students explore rate in the context of grocery shopping. Students use the unit price, or price per egg, to find the price of any number of eggs.Key ConceptsA unit price is a rate. The unit price tells the price of one unit of something (for example, one pound of cheese, one quart of milk, one box of paper clips, one package of cereal, and so on).The unit price can be found by dividing the price in dollars by the number of units.The unit price can be used to find the price of any quantity of something by multiplying the unit price by the quantity.Goals and Learning ObjectivesInvestigate rate as a unit price.Find a unit price by dividing the price in dollars by the number of units.Find the price of any quantity of something by multiplying that quantity by the unit price.