Grade 4 - richland.k12.la.us



Grade 4

Mathematics

Unit 4: Solving Algebra and Pattern Problems

Time Frame: Approximately three weeks

Unit Description

In this unit, students build their understanding of the concept of equality through the exploration of equality relationships. Students apply algebraic thinking to real-life scenarios through the use of addition, subtraction, multiplication, and division including one- and two-step problems. Students evaluate and generate rules and features of patterns.

Student Understandings

Students will understand the type of equality relationship in a given equation. Students will understand which operation(s) and steps are needed to solve real-life scenarios. Students will analyze repeating and nonrepeating patterns, evaluating the pattern’s rules and features.

Guiding Questions

1. Can students create a number sentence for a given real-life scenario?

2. Can students apply addition, subtraction, multiplication, and division facts and algorithms to a given scenario?

3. Can students use mental math to solve for a variable in an equation?

4. Can students solve one- and multi-step word problems?

5. Can students find the missing number in a sequence?

6. Can students find the rule to a given pattern?

7. Can students apply the rule of a given pattern to find numbers in the pattern’s sequence not listed?

8. Can students explain features of a pattern that do not include the rule?

9. Can students create a pattern from a given rule?

Unit 4 Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS)

|Grade-Level Expectations |

|GLE # |GLE Text and Benchmarks |

|Number and Number Relations |

|10. |Solve multiplication and division number sentences including interpreting remainders (N-4-E) (A-3-E) |

|Algebra |

|15. |Write number sentences or formulas containing a variable to represent real-life problems (A-1-E) |

|19. |Solve one-step equations with whole number solutions (A-2-E) (N-4-E) |

|Patterns |

|43. |Identify missing elements in a number pattern (P-1-E) |

|CCSS for Mathematical Content |

|CCSS # |CCSS Text |

|Operations and Algebraic Thinking |

|4.OA.3 |Solve multistep word problems posed with whole numbers and having whole-number answers using the four |

| |operations, including problems in which remainders must be interpreted. Represent these problems using |

| |equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental|

| |computation and estimation strategies including rounding. |

|4.OA.4 |Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that|

| |were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, |

| |generate terms in the resulting sequence and observe that the terms appear to al-ternate between odd and |

| |even numbers. Explain informally why the numbers will continue to alternate in this way. |

|ELA CCSS |

|CCSS # |CCSS Text |

|Writing Standards |

|W.4.2d |Write informative/explanatory texts to examine a topic and convey ideas and information clearly. Use precise|

| |language and domain-specific vocabulary to inform about or explain the topic. |

|Speaking and Listening Standards |

|SL.4.1 |Engage effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with |

| |diverse partners on grade 5 topics and texts, building on others’ ideas and expressing their own clearly. |

| |Follow agreed-upon rules for discussions and carry out assigned roles. |

| |Pose and respond to specific questions by making comments that contribute to the discussion and elaborate on|

| |the remarks of others. |

| |Review the key ideas expressed and draw conclusions in light of information and knowledge gained from the |

| |discussions. |

Sample Activities

Activity 1: Algebra Vocabulary Cards (CCSS: W.4.2d)

Materials List: paper, pencil, index cards, zip lock bag or envelope for the vocabulary cards.

Have students create algebra vocabulary cards, (view literacy strategy descriptions) for the following terms: equality, expression, equation, variable, and inverse operations. Vocabulary knowledge is one of the most essential pieces of understanding mathematics. Vocabulary cards will help students learn the content-specific terminology necessary for higher order understanding. Each vocabulary card has four parts: the definition, characteristics, an example, and an illustration. These vocabulary cards will be used throughout the unit to review the key terms for algebra and to serve as future reference cards to deepen the understanding of algebra.

Vocabulary card example:

|Definition |

|Characteristics |

|A statement that states It has an expression |

|that two expressions are on both sides of the |

|equal. equal sign.|

| |

| |

|equation |

| |

|Illustration Example |

| |

|EQUAtion – |

|An equation has the 4 + 5 = 9 |

|beginning of the word, |

|equal, in it |

| |

Activity 2: Equality Relationships (GLEs: 10, 15, 19)

Materials List: counters or ones cubes, paper, pencils

In this activity, students explore how the equality relationship is represented in different types of equations. The activity will provide students with an opportunity to study how both sides of an equation must be equal regardless of the operation or the number of parts. Provide students with counters or ones cubes. Model the four equality relationships using counters or ones cubes and write the corresponding equation.

|Whole to Whole |o o o = o o o |3 = 3 |

|Part-Part to Whole |o + o = o o |1 + 1 = 2 |

|Whole to Part-Part |o o o o o = o o + o o o |5 = 2 + 3 |

|Part-Part to Part-Part |o + o o o o o = o o o + o o o |1 + 5 = 3 + 3 |

Give students other equalities involving addition for them to model with the counters. At times, show the counters and ask students to write an equation. Modeling the operations of subtraction, multiplication, and division can become very cumbersome with counters.

As they explore whole-to-whole relationships, part-part to whole relationships, whole to part-part relationships, and part-part to part-part relationships, discuss with students what an expression is. An expression is a mathematical phrase that involves numbers and/or symbols. For example, 2 is an expression and so is 1 + 1 even though 1 + 1 contains an operation symbol. When two expressions name the same amount, they can be connected with an equal sign and the sentence is now called an equation. Have students give examples of expressions and equations.

Have students show these equality relationships with all 4 operations using a modified version of split-page notetaking (view literacy strategy descriptions). This is a modified version of split-page notes because it employs a four-column format instead of the standard two-column format. Have students use the split page notetaking to give examples (in the form of equations) for the 4 relationships for all operations. Examples of what they might show are shown in the table below. Note that the whole to whole relationship will be the same for each operation, although the numbers may be different.

|Whole to Whole |3 = 3 |6 = 6 |5 = 5 |10 = 10 |

|Part-Part to |1 + 1 = 2 |4 – 1 = 3 |5 × 5 = 25 |6 ( 3 = 2 |

|Whole | | | | |

|Whole to |5 = 2 + 3 |7 = 9 - 2 |16 = 8 × 2 |7 = 49 ( 7 |

|Part-Part | | | | |

|Part-Part to |1 + 5 = 3 + 3 |4 – 2 = 3 - 1 |3 × 4 = 6 × 2 |9 ( 3 = 6 ( 2 |

|Part-Part | | | | |

Extend the discussion to include equations that have different operations in the expressions (3 × 5 = 13 + 2). Emphasize that the expressions are still equal to each other despite their different operations. Give many examples of these types of equations as they are often very difficult for students.

Give student additional equations to categorize using their split-page notes as a guide. Have students reference their notes to study independently or with a partner for other class activities, homework, or quizzes.

Activity 3: Properties of Equality (GLEs: 10, 15, 19; CCSS: SL.4.1c)

Materials List: balances, pencil, paper

Provide pairs of students with balances. Have them use the balances to show the different equality relationships from the previous activity. For example, have them use the balance to demonstrate 3 = 3, 1 + 1 = 2, 5 = 2 + 3, and 1 + 5 = 3 + 3. The addition property of equality states that performing addition to both expressions preserves the equality relationship. For example, if the original equation is 4 = 4, and 1 is added to the left expression, 1 also needs to be added to the right side of expression, 4 + 1 = 4 + 1. This keeps the equation balanced. Summarize this for students as the following: If you add the same number to both sides of an equation, the two sides will remain equal or balanced. If 4 = 4 and you add 10 to both sides and 4 + 10 = 4 + 10, the two sides are still equal because 14 = 14.

Tell students that the idea holds true for the other operations. If you subtract the same number from both sides, the equation is still balanced. Give students the equation 5 = 5. Have them subtract 2 from the left expression (5 – 2 ? 5). Discuss if the two sides are equal. (No.) Ask, what must be done to get the expressions to balance. (Subtract 2 from the right expression.)

Tell students that this idea holds true for multiplication and division. Give the students the equation 4 = 4. For multiplication, ask students to multiply the right expression by 8 (4 ? 4 × 8). Ask if the equation is still equal. (No.) Ask, what must be done to the left expression to make the equation equal. (Multiply 4 × 8.) For division, have the students divide both sides of the equation 4 = 4 by 2. Ask if the equation is balanced. (Yes.)

Activity 4: Solving One-Step Equations (GLE: 10, 15, 19; CCSS: SL.4.1.d)

Materials List: beans, small cups, Equation Mats BLM

Have students work in pairs to create equations using beans, small cups, and the Equation Mats BLM. Using cups to represent the variable, model addition and multiplication equations. Subtraction and division are very complicated using the cups and beans. Model an addition problem such as n + 3 = 5. Have students place one cup and 3 beans on one side of the Equation Mat and five beans on the other side. Ask students how they could find the number of beans in the cup in order for the equation to be true. They could subtract 3 beans from both sides of the equation to find out that the cup would hold 2 beans. Do more examples with the students. They need to realize that if given an addition equation, they can solve it by doing the opposite, or inverse, operation of subtraction. They are undoing the addition and finding out what the variable, n, represents.

For multiplication problems such as 4n = 8, place 4 cups on one side of the Equation Mat and 8 beans on the other side. Divide your eight beans equally for the 4 cups to show that division is used to solve for n. Each cup would equal 2 beans. Do more examples with the students. They need to realize that if given a multiplication equation, they can solve it by doing the opposite, or inverse, operation of division. They are undoing the multiplication and finding out what the variable, n, represents.

Tie this idea to subtraction and division. For an equation such as n – 4 = 6, ask students what the inverse operation of subtraction is. (Addition.) Have students add 4 to both expressions and solve to find n. (n = 10) For an equation such as n ( 5 = 3, ask students what the inverse operation of division is. (Multiplication.) Have students multiply 5 to both expressions to solve for n. (n =15).

Extend the activity to real-life scenarios. Give students a variety of word problems that contain variables. Have them record the equation for each problem and explain why that equation matched the word problem.

For example: There are 5 rows of seats at a basketball game. There are 20 seats total. How many seats are in each row? (5n = 20; n = 4 or 20 ( 5 = n; n = 4)

Activity 5: Solving One-Step Equations using Mental Math (GLE: 10, 19; CCSS: SL.4.1b)

Materials List: Black Card BLM, notecards, paper, pencils

Review fact families with students for addition and subtraction facts and multiplication and division facts. Provide students with an equation with a variable such as 8 + n = 12. Write the equation on the board and use one of the black cards from the Black Card BLM to block out the variable. Think aloud, “What do I have to do add to 8 to get 12?” Discuss various methods for finding n such as counting up from 8 to 12 or counting down from 12 to 8. The count is 4. The black card, or n, represents 4. For subtraction, provide students with an equation with a variable such as 9 – n = 4. Write the question on the board and use a black card to block out the variable. Think aloud, “What do I have to subtract from 9 to get 4?” Discuss various methods of finding n such as counting up from 4 until you get to 9. This count is 5. The black card, or n, represents 5. Have students solve addition and subtraction equations using this process.

When students become comfortable solving addition and subtraction equations, move on to multiplication and division equations. Provide students with an equation with a variable such as 9 × n = 36. Write the equation on the board and cover the n with a black card. Think aloud, “What do I need to multiply 9 by to get 36? I’m going to think through my 9 times tables and see what multiplied by 9 gives me 36. I know that 9 × 4 = 36 so n = 4.” Provide students with an equation with a variable such as 8 ( n = 2. Write the equation on the board and cover n with a black card. Think aloud, “What do I need to divide 8 by to get 2? I know that multiplication is the opposite of division so I could say, ‘2 times what equals 8? 2 × what = 8. I know that 2 × 4 = 8 so 8 ( 2 = 4. n = 4.’”

Write different equations with variables on notecards. Choose one student to come up to be “The Professor” to play professor know-it-all (view literacy strategy descriptions). “The Professor” will draw a card with an equation like the ones above. The student will read the problem aloud and write the equation on the board. Have the other students in the class complete the problem while the “professor” works the problem himself/herself. Have students question the “professor” about how he/she used mental math to solve the equation. Repeat the game again with other students and other problems. Use this strategy later in the unit to review for quizzes or other assignments.

2013-14

Activity 6: Solving Multi-Step Word Problems (CCSS: 4.OA.3, SL.5.1c)

Materials List: Multi-Step Questioning the Content BLM, base-10 blocks, pencils

Explain that students will be solving multi-step word problems that include multiple operations. Tell them that before they begin using algorithms, they must first identify what the problem is asking and which operations to use. Engage students in the questioning the content (view literacy strategy descriptions) or QtC process to help them solve multi-step word problems. In the QtC process, students are given the types of questions they are expected to ask about the content. The questions can be given to students in the form of a handout, poster, or projected in the classroom, but students should have access to the questions whenever they are needed. Model for students how to use the QtC process with multi-step word problems. Put students in pairs to practice questioning the word problems together while you monitor, providing additional clarification and modeling. An example of a QtC process for multiplication is below:

Damien bought clothes for school. He bought 3 shirts for $12 each and a pair of pants for $14. How much money did Damien spend on his new school clothes?

|Initiate discussion |What is the problem about? |Damien buying clothes for school |

| |What is the question asking? What do you need|How much Damien spent on his new clothes |

| |to find? | |

|Focus on the content’s message |How can you draw a picture of the problem? | |

| | | |

| | | |

| | | |

| | |$12 $12 $12 $14 |

| |What operation(s) would you use to solve this|Multiplication and Addition |

| |problem? |OR |

| | |Repeated Addition |

| |What parts of the content told you to use |“3 shirts for $12 each” is multiplication |

| |those operations? |because there are 3 groups of 12. “A pair of |

| | |pants for $14” is addition because pants are |

| | |different than shirts and need to be included|

| | |in the total. |

| |What would be the equation for this problem? |3 × $12 + $14 = n |

|Link information |What would this problem look like with | |

| |base-10 blocks? | |

| | | |

| | | |

| | | |

| | | |

|Identify problems with understanding |How can you solve the problem? |$12 × 3 = $36 |

| | |$36 + $14 = $50 |

| |If dividing, what do you do with the |Not applicable |

| |remainder? How do you know what to do? | |

| |How do you know you are correct? |I double checked my arithmetic and also |

| | |looked at my base-10 blocks. |

| |How would you answer the question using |Damien spent $50 on his new school clothes. |

| |words? | |

Give students a variety of other multi-step word problems including problems where students have to interpret remainders. For example, Kim is making candy bags. There will be 5 pieces of candy in each bag. She has 53 pieces of candy. She ate 14 pieces of candy. How many candy bags can Kim make now? (53 – 14 = 39. 39 ( 5 = 7 bags with 4 left over) When students use the questioning the content process for these types of word problems, discuss with them how to interpret the remainder. Use word problems where the remainder is not included in the final answer and problems where the remainder is included.

Use the questioning the content process to help students determine the operations they need to use to solve the problem and why they are using those operations. Use the process to review for quizzes, homework, and other assignments.

Activity 7: Investigating Repeating Patterns (GLE: 43; CCSS: 4.OA.5)

Materials List: paper, pencils

Give students the pattern A-B-C-A-B-C… and the following questions: Imagine extending the pattern. What will be the next letter? (A) What will be the 9th letter? (C) What will be the 15th letter? (C) What will be the 32nd letter? (B) If you examine the first 42 letters in the pattern, how many of the numbers will be Bs? (14) Engage students in a discussion of how they came to their answers for the missing letters. Record the students’ generalizations. Were students dividing by three and then using the remainder to figure out their answer? Were students writing out the pattern until they got enough letters to answer the question? Provide students with other repeating patterns (both numerical and symbolic) and similar questions to get them to make generalizations about repeating patterns.

Activity 8: Growing Patterns (GLE: 43; CCSS: 4.OA.5, SL.4.1c)

Materials List: Growing Patterns BLM, pencils

Prepare several growing patterns for students such as 2, 4, 6, 8, 10,… or 7, 10, 13, 16, 19, 22,… Growing patterns can decrease or grow smaller. Explain to students that the change from number to number is what is added, subtracted, multiplied, or divided from the previous number in the sequence. Tell them that the goal of working with growing patterns is to identify the change from number to number, determine missing numbers in the pattern, and make generalizations about the pattern. Engage students in the questioning the content (view literacy strategy descriptions) or QtC process. Model for students how to use the QtC process with patterns. Put students in pairs to practice questioning the patterns together while you monitor, providing additional clarification and modeling. An example of a QtC process for patterns is below:

4, 8, 16, 32,…

|Initiate discussion |What are the changes from number to number? |+ 4, + 8, + 16 or |

| | |× 2, × 2, × 2 |

|Focus on the content |Are the changes from number to number the same or |Different (for addition) |

| |different? |Same (for multiplication) |

| |What operation(s) is being used? |Addition or multiplication |

|Link information |Do the changes from number to number have any patterns? |The addition changes are doubling each time. The|

| | |multiplication changes are the same every time. |

| |What number would extend the pattern? |64 |

|Identify problems |Is the change from your extended number to the last |Yes. For addition, it continues the pattern of |

| |number the same as the previous changes? |doubling. For multiplication, it continues the |

| | |pattern of × 2. |

| |Does the pattern make sense? |Yes. Doubling is the same thing as multiplying |

| | |by 2. The pattern stays the same. |

| |How can you explain the pattern clearly? |The next number in the pattern is double the |

| | |previous number in the pattern. |

Use the questioning the content process for other nonrepeating patterns. Have students use it to review for quizzes, homework, or other pattern applications.

Activity 9: Explain the Rule (GLE: 43; CCSS: 4.OA.5)

Materials List: Explain the Rule BLM, calculator, pencils

Write these open-ended number sequences on the board.

• 4, 8, 12, 16, _____, _____, _____

Rule: __________________________________

• 3, 6, 12, 24, _____, _____, ______

Rule: __________________________________

• 201, 193, 185, 177, _____, _____, ______

Rule: __________________________________

Have students find the next three numbers in each pattern, and describe the rule for finding the next number. Have students determine if the numbers are increasing (adding or multiplying) or decreasing (subtracting or dividing).

Have the students work in pairs to complete the Explain the Rule BLM. When they have completed this activity, have students generate their own open-ended number sequence for a partner to solve by using the calculator. Have them choose a starting number and put in “the rule” ( + 5, or ( 3, or - 4, or ( 6, etc.). Have them write down the first four numbers of their number sequence. Their partner tries to discover the rule and writes down the 5th, 6th, and 7th number in the pattern.

Activity 10: Features of Growing Patterns (GLEs: 43; CCSS: 4.OA.5, SL.4.1b)

Materials List: paper, pencils

Have students extend their thinking with patterns to include identifying features within patterns. Give the students the pattern: 3, 8, 13, 18, 23,…. Have students find the rule for the next number in the pattern (start with 3, add 5). Have students discuss the features of the pattern. Students might respond that it is increasing or that it uses addition. Challenge them to see other features such as the numbers alternatively end with 8 or 3 and they alternate between odd and even numbers. Have students work in pairs to explore other patterns to find the rules and features of each pattern.

Extend by giving students the rule and having them create the pattern. For example, create a pattern that starts at 1 and multiplies each number by 3. Stop when you have 6 numbers. Have students discuss the features of this pattern.

Sample Assessments

General Assessments

• Maintain portfolios containing student work.

• Record anecdotal notes on students as they complete tasks.

• Give prompts such as the one that follows, for students to record their thoughts in their personal math learning logs.

o Ask students to demonstrate comprehension of addition and multiplication concepts in real-world problems.

Activity-Specific Assessments

• Activities 2, 3: Give the students an equation such as 8 × 4 = 32. Have students identify the type of equality relationship in the problem and determine if the equation is true. Have students demonstrate the different properties of equality. For example, 8 × 4 + 9 = 32 + 9. Have students simplify the equations to ensure equality.

• Activity 6: Have the students write their own word problems using the same format as in activity 6. Have them write the equations and the solutions for each of the problems.

Example: There are 23 students in Ms. Diaz’s class, 20 students in Mrs. Kauffman’s class, and 19 students in Mrs. Maurin’s class. All of the students are going on a fieldtrip in vans that can only hold 9 students. How many vans are needed to bring all the students on the fieldtrip?

• Activity 9: Have the students complete various missing number pattern problems such as 2, 4, 8, 16, __, __, __. Have them identify the rule (double the previous number in the pattern) and any features of the pattern (all even, multiples of the previous numbers in the pattern). Give them various rules and have them create the pattern and identify any features such as, start at 3 and add 2 to the previous number.

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