Māra kai

Māra kai

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about showing how to solve multiplication problems.
Some whānau are setting up a māra kai. 
 
a)		  
Hohepa planted 9 rows with 8 tomato plants in each row. Show how to work out how many tomato plants he planted altogether.
 
 
 
 
 

Answer: __________  
 
b)		  
Paul planted 7 rows with 20 carrots in each row. Show how to work out how many carrots he planted altogether.
 
 
 
 
 

Answer: __________  
 
c)		  
Iritana planted 5 rows with 14 lettuces in each row. Show how to work out how many lettuces she planted altogether.
 
 
 
 
 

Answer: __________  
 
d)		  
Dai planted 6 rows with 15 strawberry plants in each row. Show how to work out how many strawberry plants he planted altogether.
 
 
 
 
 

Answer: __________  
Task administration: 
This task is completed with pencil and paper only.
Level:
3
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Keywords: 
multiplication
Description of task: 
Students answer maths problems involving multiplication and show their strategies.
Curriculum Links: 
This resource can be used to help to identify students' understanding about multiplication. Students' strategies give a much better indication of curriculum level and progression than their answer. 
A possible progression for understanding multiplication could involve:
 
  • Using grouping diagrams.
  • Using only array diagrams (Number Framework Stage 3).
  • Using only repeated addition (Number Framework Stage 5). Skip counters were of similar mean ability as repeated adders for this resource.
  • Using doubling strategies involving some addition (Number Framework Stage 6).
  • Using fully multiplicative doubling and halving, or a mix of multiplicative and additive strategies (Number Framework Stage 6).
  • Fully multiplicative basic facts partitioning attracted students of equal ability to place value partitioning.
  • ​Fully multiplicative partitioning strategies (Number Framework Stage 7).
Key competencies

This resource involves recording the strategies students used to solve mulitiplcation problems. This relates to the Key Competency: Using language, symbols and text.

Learning Progression Frameworks
This resource can provide evidence of learning associated with
Additive thinking, sets 6-7
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y7 (03/2010)
a) 72
Any 1 of the following methods of solution for 9 × 8:

  • Basic fact of 10 and compensation, e.g., 10 × 8 = 80; 80 – 8 = 72
  • Basic fact not of 10 and compensation, e.g., 9 × 9 = 81; 81 – 9 = 72
  • Combination of basic facts, e.g., 9 × 5 = 45; 9 × 3 = 27; 45 + 27 = 72
  • Basic fact and doubling, e.g., 9 × 4 = 36; 36 × 2 = 72
  • Basic fact, doubling and compensation, e.g., 8 × 4 = 32; 32 × 2 = 64;
    64 + 8 = 72
  • Skip counting, e.g. 8, 16, 24, 36, …, 72 or 9, 18, 27, 36 , … , 72
  • Repeated addition e.g., 8 + 8 = 16; 16 + 8 = 24;  …. ; 64 + 8 = 72
  • Successive doubling and compensation, e.g. 8 × 2 = 16; 16 × 2 = 32;
    32 × 2 = 64; 64 + 8 = 72
  • Repeated doubling by addition e.g. 8 + 8 = 16; 16 + 16 = 32; 32 + 32 = 64;
    64 + 8 = 72
  • Draws an array of elements, (and may count them all)
  • Draws grouping diagram with 9 groups with 8 in each one (or vice versa).
 easy
moderate*
b) 140
Any 1 of the following methods of solution for 7 × 20:
Any of the above methods

  • Place value partitioning, e.g. 7 × 2 = 14 so 70 × 2 = 140. [Accept if they say 7 × 2 = 14 so add a zero to get 140. Question these students as why they add a 0.]
  • Doubling and halving, e.g. 7 × 2 = 14; 20 ÷ 2 = 10; 14 × 10 = 140
  • Vertical algorithm (no "carrying" will be present).
 easy
easy
c) 70
Any 1 of the following methods of solution for 5 ×14:
Any of the above methods

  • Place value partitioning, e.g. (5 × 10) + (5 × 4) = 50 + 20 = 70
  • Vertical algorithm (generally with "carrying" present).
 easy
easy
d) 90
Any of the above methods for 6 × 15

  • Successive doubling/trebling , e.g., 15 × 2 = 30 and 3 × 2 = 6: 30 × 3 =90
 easy
easy
Based on a representative sample of 209 students.
 
NOTE:

(*) Students are more likely to just state the result (i.e., 9 × 8 = 72) because they may know it as a basic fact. This means fewer students in part a) give acceptable strategies. If students just state the answer (or use the vertical algorithm), ask them how they got their answer (or how the algorithm works). If they laid out part a) as a vertical algorithm and gave the correct answer, we treated this as equivalent to just stating the answer.

Diagnostic and formative information: 
  Common error Likely misconception
a)
b)
c)
d)		 17
27
19
21		 Adds instead of multiplying
a)
b)
c)
d)		 Close to 72 (70 – 74)
Close to 140 (138 –142)
Close to 70 (68 – 72)
Close to 90 (88 - 92)		 Counts all the objects that they draw in a diagram
a)
b)		 64, 80 or 81
120 or 160		 Skip counting or repeated addition error
Counts one group of objects short or over
Next steps: 

Adds instead of multiplying
Students need to re-read the problem. They could then draw a diagram of what the garden looks like for part a). This can then lead to a discussion that the answer is 8 + 8 + … + 8, i.e. the repeated addition model. If they are not ready for this, then they can count all the objects.

Counts all the objects that they draw in a diagram
These students are still at the counting stage. They need to move onto skip counting or repeated addition. This could either be of the equal sets of a grouping diagram, or the rows (or columns) of an array diagram. If they display a rectangular array, then they do have a physical model of multiplication.

Skip counting or repeated addition errors - Counts one group of objects short or over.
These students need to keep track of the number of additions or skips they have done. This could be done by recording these. For simple problems such as these, students could even tally the number of skips on their fingers.

Use of multiple strategies

This was analysed for parts b) – d) only as many student knew 9 × 8 as a basic fact.

  • Almost three-quarters of students (153 of 209) gave at least one acceptable strategy.
  • Almost half the 153 students who gave correct strategies used more than one strategy for parts b) – d), with about a third using two strategies, and a tenth using three different strategies.
  • Students only rarely mixed certain types of strategies:

- They rarely mixed numerical and diagrammatic strategies;
- Only 1 student used both skip counting and repeated addition;
- Generally, students who used the vertical algorithm gave no other strategies.

  • Many students used a mix of fully multiplicative strategies and ones that combined both multiplication and addition.
  • Students who used diagrams generally used the same type of diagram (i.e., array or grouping) in each part of the question.

Many Figure It Out and Numeracy Develop Project resources explore multiplication.