# Barnyard Buddies

A farmer has 8 cows and 10 chickens. The farmer counts all the cow and chicken legs. How many legs are there all together? Show all your mathematical thinking.

## Plan

#### Underlying Mathematical Concepts

#### Possible Problem-Solving Strategies

#### Formal Mathematical Language and Symbolic Notation

- Model
- Tally chart
- Table
- Diagram/Key
- Number/Hundred chart
- Number line
- Ten frames
- Pair
- Per
- Pattern
- Multiple
- Equation
- Unknown quantity
- Difference
- Amount
- Addition
- Total/Sum
- Double
- Place value
- Ones, tens
- More than (>)/Greater than (>)/Less than (<)
- Most/Least
- Equivalent/Equal to
- Dozen
- Halves, half of, two halves, one-half
- Odd/Even

A student may independently select a printed number line, number chart, ten frames, graph paper, etc. as they work on a task.

### Suggested materials

Engagement Image:

Teachers may project the image below to launch this task for their students, define nouns, promote discussion, access prior knowledge, and inspire engagement and problem solving.

### Possible Solutions

## Solutions tabs

### Assess

### Anchor Papers

#### Novice 1

### Scoring Rationale

#### Problem Solving

#### Apprentice

The student's strategy of using a diagram to show four legs for each cow leads to solving only part of the task. The student's flaw in reasoning is that they do not include four legs on each cow and fails to include ten chickens in total legs. The student's answer, "32 legs," is not correct.

#### Reasoning & Proof

#### Apprentice

The student's reasoning has gaps. The student does not define the cows, and the counting of the legs by four does not match the indicated legend—the numbers written in the cow's heads. You have to assume that the last cow is missing two legs and is not a chicken.

#### Communication

#### Novice

The student uses no mathematical language.

#### Connections

#### Novice

The student solves the task and does not make a connection.

#### Representation

#### Apprentice

The student's diagram is appropriate to the task but not accurate. Cows three and five are missing a leg. Cow eight is missing two legs. There is no key, label or scribing to indicate that the student is diagramming cows.

#### Overall Achievement Level:

#### Novice

#### Apprentice 1

### Scoring Rationale

#### Problem Solving

#### Apprentice

The student's strategy of using a diagram to show four legs for each cow and two legs for each chicken works to solve this task. The student does not clearly state a correct answer. Most teachers have students highlight, underline, circle, box, etc. the answer.

#### Reasoning & Proof

#### Practitioner

The student's diagrams support their understanding of the underlying mathematical concepts of the task.

#### Communication

#### Apprentice

The student correctly uses the mathematical term *key*.

#### Connections

#### Apprentice

The student attempts to make a connection. The student attempts four tables but they are incorrect. It is difficult to determine what information the student is trying to represent.

#### Representation

#### Practitioner

The student's diagrams are appropriate to the task and accurate. A key defines "ch" as chickens. Each cow is labeled. The student's tables are not appropriate or accurate to the task. Only one representation has to be appropriate and accurate to earn a Practitioner.

#### Overall Achievement Level:

#### Apprentice

#### Apprentice 2

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of making tables of the cow and chicken legs for a total of 52 legs works to solve the task. The student states a correct answer, 52 chicken's and cow's legs.

#### Reasoning & Proof

#### Practitioner

The student's solution clearly shows their reasoning and proof. The representations are linked to the text and the presentation is in logical order.

#### Communication

#### Apprentice

The student correctly uses one mathematical term *table*. The formal term for "all together" is total.

#### Connections

#### Novice

The student does not make any mathematically relevant observation about their solution.

#### Representation

#### Practitioner

The student's tables are appropriate to the task and accurate. Both tables have the necessary labels.

#### Overall Achievement Level:

#### Apprentice

The overall achievement level for this piece of student work falls under Exemplars exception to the rule category. If a student has all Apprentice scores or above, but a Novice in "Connections," the student may still receive an achievement level score of Apprentice. To learn more about Exemplars scoring, please refer to the section of your dashboard called "Tools for Success" and click on the link for “Using the Assessment Rubric.”

#### Apprentice 3

### Scoring Rationale

#### Problem Solving

#### Apprentice

The student's strategy of using ten frames to show ten chickens, 20 legs, and tally marks in a diagram to show eight cows and 32 legs, works to solve part of the task. The student does not continue to find the total legs the farmer counted.

#### Reasoning & Proof

#### Apprentice

Arguments are constructed with adequate mathematical basis for part of the task. The student applies the add two and add four patterns and computes the total number of legs for each animal. The student does not find the combined total of legs.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms— "Totle," *ten frames*, *tally*, *pair* and *key*.

#### Connections

#### Practitioner

The student makes the mathematically relevant connection, "2 is a pair."

#### Representation

#### Practitioner

The student uses appropriate and accurate ten frames to represent the legs of ten chickens. All labels are included. The student uses an appropriate and accurate diagram with tally marks to represent the legs of eight cows. A key defines the cows and cow legs.

#### Overall Achievement Level:

#### Apprentice

#### Practitioner 1

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of using diagrams to show the eight cows and 10 chickens works to solve the task. The student states a correct answer, "There are 52 legs in all."

#### Reasoning & Proof

#### Practitioner

Arguments are constructed with adequate mathematical basis. The student's correct reasoning is evident in the diagrams and key.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms *diagram* and *key*.

#### Connections

#### Practitioner

The student makes the mathematically relevant observations, "cows count by 4 and chickens count by 2s."

#### Representation

#### Practitioner

The student uses appropriate and accurate diagrams to represent the eight cows and 10 chickens. A key is used to define the cows and chickens.

#### Overall Achievement Level:

#### Practitioner

#### Practitioner 2

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of making diagrams to show the number of legs on eight cows and 10 chickens works to solve the task. The student also makes an equation to find a total of 52 legs. The student does not earn Expert for verification because they do not compare the two totals of 52 legs.

#### Reasoning & Proof

#### Practitioner

The solution clearly shows the student's reasoning. The representations are linked to the text, results of computation are present, the answer is indicated, and the presentation is in logical order.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms * key* and *diagram*.

#### Connections

#### Practitioner

The student solves the task and then uses a different process to solve the same task. The student uses a number sentence to find a total of 52 legs.

#### Representation

#### Practitioner

The student's diagrams are appropriate to the task and accurate. A key defines the cow and chicken legs.

#### Overall Achievement Level:

#### Practitioner

#### Practitioner 3

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of using two number lines to determine the number of legs on eight cows and ten chickens and then finding the total legs works to solve the task. The student's answer, "52 legs," is correct.

#### Reasoning & Proof

#### Practitioner

The student's solution is constructed with adequate mathematical basis. The student shows understanding of counting by a two and four pattern and then finding a total using addition.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms— *total*, *number lines*, "patern."

#### Connections

#### Practitioner

The student solves the task and makes a mathematically relevant observation. The student states, "the paterns for legs chickens +2, cows +4."

#### Representation

#### Practitioner

The student's number lines are appropriate to the task and accurate. All necessary labels are included and the "jumps" are correct.

#### Overall Achievement Level:

#### Practitioner

#### Expert 1

### Scoring Rationale

#### Problem Solving

#### Expert

The student's strategy of making two tables to show the number of legs on eight cows and ten chickens and finding a total of 52 legs works to solve the task. The student's answer, "52 is the sum of the legs," is correct. The student then uses an alternative strategy, number lines, to verify that 52 legs is the correct answer.

#### Reasoning & Proof

#### Expert

The student justifies and supports their decisions made and conclusions reached. The student uses two number lines to verify the add four and add two patterns and a correct answer.

#### Communication

#### Expert

The student correctly uses the mathematical terms— "totle," *tables*, *patterns*, *double*, *sum*, *number lines*. The student correctly uses the Expert mathematical notation, `sf(1/2)`.

__Note__: The standards do not expect second grade students to use fractional notation. Fractional notation is introduced in grade 3. Students are only required to use the terms halves, thirds, fourths, whole, in second grade.

#### Connections

#### Expert

The student makes the following Practitioner connections, "patterns-cow +1, cow legs +4, chickens +1, chicken legs +2, cow legs double chicken legs, so 10 cows is 20 + 20 = 40 legs, chicken legs is `sf(1/2)` cow legs so 4 cow legs = 2 chickens like on the tables." The student uses number lines to verify that the data and total legs found using the two tables is correct. The student states, "50 + 2 = 52, same answer agin. It is correct."

#### Representation

#### Expert

The student's two tables are appropriate to the task and accurate. All necessary labels are correct. The student's two number lines are appropriate to the task and accurate. Each is correctly labeled and the "jumps" are accurate. The student uses the two number lines to verify their answer.