CC8.EE Analyze and solve linear equations and pairs of simultaneous linear equations. Prioritized Common Core Mathematical Practice Standard(s)

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1 Math Solutions Lesson from the Classroom Cats and Birds A Lesson for Grades 6 8 This lesson was adapted from the Math Solutions publication Supporting English Language Learners in Math Class, Grades 6 8 by Kathy Melanese, Luz Chung, and Cheryl Forbes. Featured in Math Solutions Online Newsletter, Issue 44 Math Goal Students will use a variety of problem-solving strategies that involve reasoning with algebraic expressions, constructing simple equations, and working with two linear equations using two variables. Time Two class periods of 45 minutes each Common Core Content Standard Correlation CC6.EE Apply and extend previous understanding of arithmetic to algebraic expressions. Write, read and evaluate expressions in which letters stand for numbers. CC7.EE Use properties of operations to generate equivalent expressions. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. CC8.EE Analyze and solve linear equations and pairs of simultaneous linear equations. Prioritized Common Core Mathematical Practice Standard(s) MP1 Make sense of problems and persevere in solving them. MP7 Look for and make use of structure. Supporting Common Core Mathematical Practice Standards MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. Language Goal Students will make inferences orally and in writing about the possible solutions to the math problems. They will describe their process for discovering the final solution in their small groups and with the whole class. Vocabulary altogether, clues, divisible, expression, common factors, multiple, possible, solutions, times Materials Frayer Model Vocabulary Worksheet, 1 per group of 4 students [HO-1] Cats and Birds: A Quick Review of Some Algebraic Expressions Sets of Clue Cards, 1 set per group of 4 students [HO-2] Poster paper, 1 per group of 4 students Markers for each group of students 1

2 Sentence Frames Beginning Our guess is cats and birds. We think we are right because we. We know there are cats and birds. Our solution shows. Intermediate/Advanced Our guess is that the possible number of cats is and the possible number of birds is. We think this because. The number of cats must be and the number of birds must be. We can prove that our solutions is correct because. Anticipating Confusion It is common for students to misrepresent a verbal expression such as There are three times as many apples as there are oranges by recording symbolically in the same order as the words, for example, 3 x a = o. If this occurs, the teacher can discuss with the students which fruit has more, apples or oranges, and then assign a value, such as 5 oranges, to figure out how many apples, and then record this symbolically to compare with the erroneous recording. On Day 2, some students may be confused by the word pet being used both as a noun and as a verb in the clues. Demonstrate use of the word both ways before distributing the clue cards, for example, My pet is a dog. I also have a pet fish. I pet my dog on the head when I get home; but my fish does not want me to pet it!. Additionally, some students may not be familiar with the use of the word paw as a synonym for the foot of a dog and may confuse it with the homonym pa, meaning father. Planning Questions Ask students these questions to engage them in talking about what they understand or to reveal misconceptions. Which is greater, the number of apples or the number of oranges? What are three numbers that are multiples of 5? How did you figure it out? What are the factors of 12? Of 18? What factors do they share, or have in common? What are the common factors of 16 and 20? What are some numbers that are divisible by three? What happens if a number is not divisible by three? Give and example and explain what happens if that number is divided by three. 2

3 Lesson Implementation Minilesson 1. This lesson begins with a review of the important vocabulary and algebraic expressions that the students will encounter when solving the problem clues. Taking the time to explore key vocabulary is essential in order to ensure that students have access to the lesson s math content by tapping prior knowledge and clarifying any confusion. Begin by displaying a large diagram of a Frayer model. Explain to students that this is called a Frayer model and is very useful for understanding and explaining important vocabulary. In the center under Vocabulary Word record the word Times. Tell students to talk in their groups about what the math word times means. 2. Ask students to share an example of times that they discussed in their groups and record their examples in the lower left section of the model (e.g., two times three is six or 2 x 3 = 6). Now ask them to think of a nonexample you can record (e.g., 10 5 = 2). 3. Ask students to suggest a definition for the word times and record that in the Definition section of the model. Students may say something like, Times means multiplication. Explain that the top right section can be used to draw an example, to use the word in a sentence, or to tell more information or facts about the word s meaning. Record any suggestions the students have for this section (e.g., times always has equal sized groups, or a drawing of two rows of four stars in a row ). 4. Now give each group one handout of the Frayer model and assign one vocabulary word from the lesson to each group. Give students about 10 to 15 minutes to work on this. Tell them that they will be presenting their models to the class. 5. When all groups have finished, invite each group to share their results. Include opportunities for other students to add ideas that have not been mentioned. While groups are presenting, listen for the big idea about each vocabulary word and capture this on a Math Vocabulary Chart for students to refer to later in the lesson. Introduction 6. Explain to the students that they will be solving a problem about the number of cats and birds that a woman named Ms. Lang keeps as pets. The clues they will use have sentences that describe algebraic expressions. To make sure they understand what these same expressions mean they will look at some examples about apples and oranges. Give each student a copy of the handout called A Quick Review of Some Algebraic Expressions. Assign partners to collaboratively discuss what each means and use their own words or drawings to represent the expressions on the sheet. 7. When groups have finished, facilitate a whole-class discussion that invites students to share the variety of ways they have explained each expression, encouraging and modeling the value of flexible thinking. Record students thinking where everyone can see it. Listen for erroneous thinking and use it as an opportunity to explore that misconception. Support students in thinking through the arithmetic operations behind the expressions, before testing the idea with numbers and concrete examples, such as apples and oranges, to see if it works. Encouraging multiple ways to express the same idea supports all students in their academic language development as well as validating their thinking and reason process. 3

4 8. Conclude the minilesson by telling students that they are ready to solve a problem about cats and birds using the vocabulary they just reviewed. Make sure the students retain their worksheets for reference during the next lesson. Exploration 9. Display Cats and Birds: Clue Card 1. Ask a student to read the first clue. Then ask what information the clue gives, and record this on the board (e.g., We know there are two kinds of pets; we know she has cats and birds. We know there are a total of 25 pets.) Read the question and ask students to think of one way there could be 25 pets of two kinds of animals. Record one possibility they suggest on the board using the following sentence frames: We think a possible solution is cats and birds. Our proof is + = 25 pets. 10. Have students work in groups of four to find multiple solutions that tell how many cats and how many birds for a total of 25 pets. The important idea at this step is that students conclude that there are many possible solutions and insufficient information has been provided so far in order to determine exactly how many pets of each kind Ms. Lang has. 11. Call the class back together and invite each group to share a different possible solution, recording each on the board. Emphasize the use of the word possible in order to validate (1) there are many combinations of two pets that can total 25, and (2) more information is needed to determine a final, correct, or single solution. Explain that all their possible solutions for this clue can be written as an algebraic expression: c + b = 25; c = number of cats b = number of birds. 12. Tell the students that each group will receive a set of five more clues about the same problem. Groups can choose to work in partners on different clues and then share their work with the rest of the group, or work as a whole group deciding which clue to start with, exploring the information of that clue, and then continuing in this way with the rest of the clues. Tell students to keep track of their work, using words, numbers, and algebraic expressions that represent each clue. Remind students to use the class vocabulary charts they made in the minilesson if they need help understanding a clue. 13. As groups are working, circulate among them, listening to conversations, supporting students with math academic vocabulary that helps them express their ideas, and offer suggestions for organizing or keeping track of their thinking when helpful. Encourage students to explain their reasoning to their group members and remind them of sentence frames that could help with this. Take note of any continued misconceptions that arise as well as explanations that would be useful for the rest of the class to hear when summarizing the problem at the end of the lesson. 4

5 14. When groups have had the opportunity to explore all six clues, provide each group with a piece of chart paper. The chart should have the group s solution represented with sentence frames, algebraic expressions, and equations that justify their solution. Tell them they will need to show their work for each clue using both numbers and words that prove their solution. Write the following sentence frame on the board for them to use: We think the correct solution is cats and birds. We can prove this by showing that. 15. As you explain your expectations for their posters, record the expectations, For example: Summary Your poster should have: Title of the problem Your solution explained in a sentence Your work for each clue that proves your solution using: o equations o expressions o use of important vocabulary words Names of group members 16. Begin the lesson summary by telling the class that each group will have an opportunity to share their work for one clue. Assign a clue to each group, giving them a couple minutes to decide what each group member will share. Suggest and record sequencing sentence frames that can support their explanations. For example: First we. Next we [found out, tried, made a list of, etc.]. These are the numbers we found that worked. Then we checked. Last [finally] we. Our solution is cats and birds. 17. After each group shares their work, invite other class members to add anything else their group found for that clue. During this part of the lesson summary, listen to how students use math vocabulary and academic language, algebraic thinking, and demonstrate understanding of key math concepts of multiples, divisibility, common factors, and solving linear equations. 18. Conclude the lesson by verifying that all are in agreement about a single solution to the problem. Validate the students work by telling them they solved a problem involving algebra by using what they know, working together, discussing their thinking, and listening to the reasoning and ideas of others. Find more classroom lessons online at mathsolutions.com. Visit the Educator Tools section and click Free Classroom Lessons. 5

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7 Definition Facts/ Drawing/Use in a sentence Vocabulary Word Example Nonexample