Domainoa: Operations and Algebraic Thinking

Grade Tasks12/2/2012

1.OA At the Park

Alignment 1: 1.OA.A.1

DomainOA: Operations and Algebraic Thinking

ClusterRepresent and solve problems involving addition and subtraction.

StandardUse addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.See Glossary, Table 1.

There were 7 children at the park. Then 4 more showed up. How many children were at the park all together?
There were 7 children at the park. Some more showed up. Then there were 11 children in all. How many more children came?
There were some children at the park. Four more children showed up. Then there were 11 children at the park. How many children were at the park to start with?

Commentary:

This task includes three different problem types using the "Add To" context with a discrete quantity; see "1.OA The Pet Snake" for an "Add To" problem with a continuous quantity. Table 1 in the glossary of the CCSSM offers a succinct overview of all addition and subtraction problem types.

Although students should experience and practice with all three problem types, they would not necessarily be introduced at the same time. Please see the K, Counting and Cardinality; K–5, Operations and Algebraic Thinking Progressions Document for in-depth information about issues related to students’ learning of these kinds of problems.

While students are expected to add and subtract fluently within 10 in first grade (1.OA.6), they are not expected to add and subtract fluently within 20 until second grade (2.OA.2).

Solution:Classifications included

Students may use objects, pictures, or equations to represent their solutions. The solutions show equations with a question mark representing the unknown value, but other symbols are often used. For example, 4 + ? = 11 might also be written
4 + ____ = 11 or 4 + ☐ = 11.

Total Unknown: There were 11children in all.
Possible equation: 7+4=?
Addend Unknown:4more children came.
Possible equation: 7+?=11
Start Unknown: There were 7children in the park to start with.
Possible equation: ?+4=11

1.OA Boys and Girls, Variation 1

Alignment 1: 1.OA.A.1

DomainOA: Operations and Algebraic Thinking

ClusterRepresent and solve problems involving addition and subtraction.

9 boys and 8 girls were in the class. How many children were in the class in all?
17 children were in the class. 9 were boys and the rest were girls. How many girls were in the class?
17 children were in the class. There were some boys and 8 girls. How many boys were in the class?

Commentary:

These tasks types represent the Put Together/Take Apart contexts for addition and subtraction (see Table 1 in the glossary of the CCSSM for all all addition and subtraction problem types). Students may use either addition or subtraction to solve these types of word problems, with addition related to the action of putting together and subtraction related to the action of taking apart. Depending on how students think about these word problems, either is appropriate for the “addend unknown” problems. Seeing it both ways emphasizes the relationship between addition and subtraction.

The use of “in all” or “altogether” makes result unknown problems significantly easier for students, since these words are such strong cues to put sets together. Students need experiences with problems both with and without such cues. When working with categories like boys-girls-children, it is important to be sure that students understand the contexts. Please see page 10 of the K, Counting and Cardinality; K–5, Operations and Algebraic Thinking Progressions Document for in-depth information about issues related to students’ learning of these kinds of problems.

Note that while students are expected to add and subtract fluently within 10 in first grade (1.OA.6), they are not expected to add and subtract fluently within 20 until second grade (2.OA.2).

Solution:Classifications included

Please note that students may use objects, pictures, or equations to represent their solutions. Furthermore, the solutions show equations with a question mark representing the unknown value, but other symbols are often used. For example, 9+?=17might also be written
9+____ =17or 9+☐=17. Note that it is important to include examples with the total written on both sides of the equation.

Total Unknown: 17 children were in the class.
Possible equations:
9+8=?
8+9=?
? = 9+8
? = 8+9

Addend Unknown: There were 8 girls.
Possible equations:
17=9+?
17=? + 9
9+?=17
? +9=17
17–9=?

Addend Unknown: There were 9 boys.
Possible equations:
17=8+?
17=? + 8
8+?=17
? +8=17
17–8=?

1.OA Boys and Girls, Variation 2

Alignment 1: 1.OA.A.1

DomainOA: Operations and Algebraic Thinking

ClusterRepresent and solve problems involving addition and subtraction.

9 children were in the class. How many boys and how many girls could have been in the class?

Solve the problem. Write an equation. Draw a picture and use it to explain your answer.

Commentary:

This task represents the Put Together/Take Apart with both addends unknown context for addition and subtraction (see Table 1 in the glossary of the CCSSM for all addition and subtraction problem types). Once a student finds one correct answer, he/she can be encouraged to find another. Ask the student to use objects, pictures, or equations to represent each answer.

Please see the K, Counting and Cardinality; K–5, Operations and Algebraic Thinking Progressions Document for in-depth information about issues related to students’ learning of these kinds of problems.

Solution:Answers

Listing the possible pairings of boys and girls in a systematic way might help the student show that s/he has found all of the possible pairings.

There are 10 possible solutions. Students can select a number between 0 and 9 to represent the number of boys (or girls) and then find the number of girls (or boys, resp).
Possible equations:

9=0+9
9=1+8
9=2+7
9=3+6
9=4+5
9=5+4
9=6+3
9=7+2
9=8+1
9=9+0

Note that students may write the total on either side of the equation.

1.OA Finding a Chair

Alignment 1: 1.OA.A.1

DomainOA: Operations and Algebraic Thinking

ClusterRepresent and solve problems involving addition and subtraction.

There are 8 children and 6 chairs. A child sits in each chair. How many children won’t have a chair?
There are 8 children and some chairs. A child sits in each chair. 2 children don’t have a chair. How many chairs are there?
There are some children and 6 chairs. A child sits in each chair. 2 children don’t have a chair. How many children are there?
There are 8 children and 10 chairs. A child sits in each chair. How many empty chairs are there?
There are 8 children and some chairs. A child sits in each chair. Two chairs are empty. How many chairs are there?
There are some children and 10 chairs. A child sits in each chair. Two chairs are empty. How many children are there?

Commentary:

These tasks types represent Compare contexts for addition and subtraction (see Table 1 in the glossary of the CCSSM for all addition and subtraction problem types).

These problems explicitly describe one-to-one correspondences without using comparison language. Such problems are easier for students to solve than problems that use comparison language such as “How many more?” or “How many fewer.” Please see the K, Counting and Cardinality; K–5, Operations and Algebraic Thinking Progressions Document for in-depth information about issues related to students’ learning of these kinds of problems.

Solution:Answers

2 children will not have a chair.
There are 6 chairs.
There are 8 children.
There are 2 empty chairs.
There are 10 chairs.
There are 8 children.

1.OA Maria’s Money

Alignment 1: 1.OA.A.1

DomainOA: Operations and Algebraic Thinking

ClusterRepresent and solve problems involving addition and subtraction.

Ali had $9. Maria had $5. How many more dollars did Ali have than Maria?
Ali had $9. Maria had $5. How many fewer dollars did Maria have than Ali?

Ali had $4 more than Maria. Maria had $5. How many dollars did Ali have?
Maria had $4 less than Ali. Maria had $5. How many dollars did Ali have?

Ali had $4 more than Maria. Ali had $9. How many dollars did Maria have?
Maria had $4 less than Ali. Ali had $9. How many dollars did Maria have?

Commentary:

This task includes problem types that represent the Compare contexts for addition and subtraction (see Table 1 in the glossary of the CCSSM for all all addition and subtraction problem types). There are 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. Each of these problem types can be solved using addition or subtraction, although the language in specific problems tends to favor one approach over another.

Students benefit from encountering one problem type limited to small numbers and to develop strategies for that type of problem before encountering mixed sets of problems and larger numbers that distract the student from the problem itself. Over time they will be able to distinguish between types of problems in mixed sets and apply the appropriate strategy to solve each.

Solution:Classifications included (need solutions written for other three Qs)

This solution is written in teacher language. Students may use objects, pictures, or equations to represent their solutions. While students are expected to add and subtract fluently within 10 at grade 1 (1.OA.6), they are not expected to add and subtract fluently within 20 until second grade; see 2.OA.2.

The solutions show equations with a question mark representing the unknown value, but other symbols are often used. For example, 4 + ? = 9 might also be written 4 + ____ = 9 or 4 + ☐ = 9.

Difference Unknown:
Ali had $4 more than Maria. (or)
Maria had $4 less than Ali.
Possible equations: 5 + ? = 9; 9 – 5 = ?
Bigger Unknown: Ali had $9.
Possible equations: 5 + 4 = ?; ? - 4 = 5
Smaller Unknown: Maria had $5.
Possible equations: ? + 4 = 9; 9 – 4 = ?

1.OA Measuring Blocks

Alignment 1: 1.OA.A.1

Domain OA: Operations and Algebraic Thinking

Cluster Represent and solve problems involving addition and subtraction.

Standard Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.See Glossary, Table 1.

Have two or more blocks of different lengths on hand and paper clips to use to measure them. The blocks need to measure a whole number of paper clips whose combined length is less than or equal to 20 paper clips.

Have students work in pairs. Give each pair a block to measure using paper clips. After they have measured their block, say,

Find someone who measured the other block. Ask them how many paper clips long it is. How long will the two different blocks be together if they are laid end-to-end? First try to figure this out. Then put the blocks end-to-end and measure it to check your answer.

Ask students to explain how they solved the problem and whether their answer checked out correctly. Even if students added correctly, they may not have lined up the paperclips very carefully and could get different lengths. This is a good opportunity to talk about how important it is to be careful when measuring.

Next, ask the students,

Imagine you put another block end-to-end with the first one you measured. Together, they measure 18 paper clips. How long is the new block? Draw a picture to explain how you know.

Ask students to explain how they solved the problem and whether their answer checked out correctly. Finally, ask the students to write equations to represent their work.

Commentary:

This task is a kernel for an instructional task that could be elaborated with commentary about teaching strategies and examples of student work. When this website is fully functional, teachers will be able to submit tasks and related materials for review. The Illustrative Mathematics Project invites teachers to begin this work.

1.OA School Supplies

Alignment 1: 1.OA.A.1

Domain OA: Operations and Algebraic Thinking

Cluster Represent and solve problems involving addition and subtraction.

Standard Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.See Glossary, Table 1.

Pia takes some money to the store to buy school supplies. She buys some paper for $3 and a pen for $2. After she buys these supplies, she has $7 left. How much money did Pia bring to the store?

Commentary:

This task could be used for either instructional or assessment purposes, depending on where students are in their understanding of addition and how the teacher supports them. The solution shown is very terse; students' solution strategies are likely to be much more varied.

Solution:Tape diagram

Students who are familiar with tape diagrams might use them to solve the problem by first drawing a picture:

and then reasoning, "If she spent $2 + $3 dollars and had $7 left over, then she had $2+$3+$7=$12to start."

There are many more strategies that students might use.

Solution:Using Addition with Objects

Students could use objects to add

2+3=5 5+7=12

Students who have mastered fluent addition for numbers up to 12 could also add these two quantities in their head.

1.OA Sharing Markers

Alignment 1: 1.OA.A.1

DomainOA: Operations and Algebraic Thinking

ClusterRepresent and solve problems involving addition and subtraction.

Char had 10 markers. She gave 3 to a friend. How many did she have left?
Char had 10 markers. She gave some to a friend. Now she has 7 left. How many markers did she give to her friend?
Char had some markers. She gave 3 to a friend. Then she had 7 left. How many markers did she have to start with?

Commentary:

These tasks types represent the Take From contexts for addition and subtraction (see Table 1 in the glossary of the CCSSM for all all addition and subtraction problem types). This task includes the three different problem types using the Take From context: result unknown, change unknown, and start unknown. Students need experience and practice with all three types.

Result unknown problems (a) and change unknown problems (b) are both fairly easy for most students, since they can be acted out directly. Start unknown problems (c) are the most difficult of the three for most children because they involve thinking about a situation in reverse (“undoing” the action in a situation). Guessing an initial amount then trying it out to see if it works is another possible strategy.

The solutions below are written in teacher language. Students may use objects, pictures, or equations to represent their solutions. While students are expect to add and subtract fluently within 10 at grade 1 (1.OA.6), they are not expected to add and subtract fluently within 20 until second grade; see 2.OA.2.

The solutions show equations with a question mark representing the unknown value, but other symbols are often used. For example, 10 - ? = 7 might also be written 10 - ____ = 10 or 10 - ☐ = 7.

Solution:Classifications included

Result Unknown: Char had 7 markers left.
Possible equation: 10 – 3 = ?
Change Unknown: Char gave 3 markers to her friend.
Possible equation: 10 – ? = 7
Start Unknown: Char had 10 markers to start with.
Possible equation: ? – 3 = 7

1.OA The Pet Snake