Cuisenaire number sentences
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Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about using Cuisenaire rods to make number sentences.
You will be given a Cuisenaire rod representing one of the following numbers:
7 (black), 8 (brown), 9 (blue) or 10 (orange). You will also receive rods representing smaller numbers.
Your task is to:
- Use some of the smaller Cuisenaire rods to make up the target value of the rod you have been given.
- Record each finding as a number sentence below.
Task administration:
This task is completed with pencil and paper, and other equipment.
Equipment
Students will need sets of Cuisenaire rods which include the target value they are making number sentences for, i.e., 7, 8, 9, or 10, and a mixture of rods smaller than the target value.
A minimum of: 10 white, 5 red, 3 light green, 2 each of magenta, yellow and dark green for all target values:
- for target value 8 include1 black,
- for target value 9 include 1 black and 1 brown
- for target value 10 include 1 each of black, brown and dark blue.
These could be placed in small plastic bags.
Alternatively, Animal strips can be used for this task. Provide similar quantities of each strip as you would for Cuisenaire rods.
- Give students their bag of mixed rods, and then give them their target rod, stating its value as you do so.
- Modelling the way in which the students are to make and record their findings can be done using a rod of lesser value than those in the task. For example, use the light green (3) rod as the target value. Place a red (2) rod and a white (1) rod together underneath:
- Record this as 3 = 2 + 1.
- Students can work in groups, pairs, or individually, if there are sufficient Cuisenaire rods.
- If working in groups, students could take turns at making and recording a number sentence for the target value, using a single recording sheet and pencil. (Roundtable^{1})
- Students who complete the task quickly can be given a different target value.
^{1} Kagan, S. (1994). Cooperative Learning. San Clemente, CA: Kagan Cooperative Learning, p. 8:9.
Level:
2
Curriculum info:
Keywords:
Description of task:
Students use Cuisenaire rods or Animal strips to make up addition number sentences and record them.
Curriculum Links:
This resource can be used to help to identify students' ability to partition whole numbers and record their results as an equation.
Learning Progression Frameworks
This resource can provide evidence of learning associated with within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.Answers/responses:
Answers will vary. Look for students giving a range of answers (there are up to 40 possibilities in some cases) and for the position of the equals sign – it should not always be at the end of the number sentence.
Sample responses from a Year 4 class:
9 | 4 + 2 + 1 + 1 + 1 = 9 | 9 = 5 + 1 + 3 | 3 + 3 + 3 = 9 |
10 | 10 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 | 10 = 4 + 3 + 3 | 10 = 7 + 2 + 1 |
8 | 8 = 2 + 1 + 1 + 1 + 1 + 1 + 1 | 8 = 1 + 2 + 5 | 3 + 1 + 1 + 1 + 1 + 1 = 8 |
Based on classroom-based (Year 4) research on algebraic thinking, December 2005.
Teaching and learning:
This resource explores the meaning of the equals sign and the concept of equality. It seeks to reinforce the idea that number sentences are not always in the format a + b = c. It can be used to initiate or follow up discussions about the idea of equality meaning that the quantity that is on one side of the equals sign is equivalent to the quantity on the other side.
Students could be encouraged to verbalise their number sentences using the phrase “is the same as” to emphasize this idea of equality. Further information about the concept of equality, refer to the Algebraic thinking concept map.
Students could be encouraged to verbalise their number sentences using the phrase “is the same as” to emphasize this idea of equality. Further information about the concept of equality, refer to the Algebraic thinking concept map.
(This resource has been adapted from an idea in Stacey, K., & MacGregor, M. (1997) Building foundations for algebra. Mathematics Teaching in the Middle School, 2(4), 252-260.)
Diagnostic and formative information:
Most students involved in our classroom research had little or no difficulty generating number sentences. Once given the model 6 =, about a third of students solely used this format to record their number sentences. Just over a third were able to use both this format as well as the a + b = c format. A few students continued to use only the a + b = c format. Some students went outside the parameters of the exercise and used operations other than addition or generated number sentences for numbers larger than the ones based on the Cuisenaire rods, e.g., 10 + 9 + 1 = 20, 9 = 36 ÷ 4, 9 = 3 × 3.
Next steps:
Students who record their number sentences solely in the format a + b = c may not yet be comfortable with the idea of equality meaning that the quantity that is on one side of the equals sign is equivalent to the quantity on the other side and may still have an arithmetic understanding of the equals sign as meaning “the answer comes next”. Exploring number sentences in different formats such as 6 = 6 or 10 = 8 + 2 can open up discussion about the algebraic meaning of the equals sign. Resource What is equal? can be used to initiate this discussion.
If students are comfortable using practical equipment to demonstrate equality then they can be given other numbers to work with and encouraged to write as many equivalent number sentences as they can without using equipment. Students could be asked questions like "My number is 12, how many different ways can you write it?" Addition is the only operation used in this resource. Students can be extended further by being encouraged to use other operations in their number sentences.
If students are comfortable using practical equipment to demonstrate equality then they can be given other numbers to work with and encouraged to write as many equivalent number sentences as they can without using equipment. Students could be asked questions like "My number is 12, how many different ways can you write it?" Addition is the only operation used in this resource. Students can be extended further by being encouraged to use other operations in their number sentences.
For further information and examples of student generated number sentences, refer to the Algebraic thinking concept map
Another way to extend this task is to explore number sentences of the form a + b = c + d. Use two rods to represent a number sentence, e.g., 5 + 2. Have students use at least two other rods to generate more equivalent number sentences.
These can then be recorded as 5 + 2 = 4 + 3, 5 + 2 = 1 + 2 + 2 + 2, etc.
The idea of equality as balance can be introduced using balance pans and blocks which can lead on to solving simple problems presented as open number sentences. See Balance pans for balance pan ideas and Equal number sentences for open number sentences. After these discussions and activities about equality, have students record their understanding of the equals sign in a journal using a starter statement e.g., “I think the equals sign means…” For more information about journalling refer to Journalling in mathematics.
These can then be recorded as 5 + 2 = 4 + 3, 5 + 2 = 1 + 2 + 2 + 2, etc.
The idea of equality as balance can be introduced using balance pans and blocks which can lead on to solving simple problems presented as open number sentences. See Balance pans for balance pan ideas and Equal number sentences for open number sentences. After these discussions and activities about equality, have students record their understanding of the equals sign in a journal using a starter statement e.g., “I think the equals sign means…” For more information about journalling refer to Journalling in mathematics.
For further information and examples of student generated number sentences, refer to the Algebraic thinking concept map
Numeracy Links
This task involves students recalling addition basic facts and recording written equations (The Number Framework – Knowledge, Stage 4).
Equality as balance: refer to A Balancing Act (Book 5: Teaching Addition, Subtraction, and Place Value, page 40) and
The Equals Sign Again (Book 8: Teaching Number Sense and Algebraic Thinking, page 12