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ESOL Online. Every child literate - a shared responsibility.
Ministry of Education.

Cart before the horse

Adapted from NZ Maths unit Cart before the horse

Adaptions for ESOL students: Pat Boyle

Year: 10
Level: 5
Duration: 1 week

Achievement objectives being assessed

Number, level 5:

  • convert numbers expressed in standard form to ordinary form. and vice versa;
  • solve practical problems involving decimals and percentages;
  • share quantities in given ratios.

Problem Solving, Level 5:

  • pose questions for mathematical exploration.

Algebra, Level 5:

  • generate patterns from a structured situation, find a rule for the general term, and express it in words and symbols;
  • solve linear equations;
  • use equations to represent practical situations.

Measurement, Level 5:

  • interpret and use information about rates in a variety of ways, for example, graphically, numerically, or in tables.

Geometry, Level 5:

  • recognise when 2 shapes are similar, find the scale factor, and use this to find an unknown dimension.

Mathematical Processes:

Specific learning outcomes

Students will be able to:

  • use their mathematical knowledge to invent problems;
  • solve other student's problems.

Teacher background reading

The problem

Choose and open a sealed envelope. Make up a problem whose answer is the one that you have found in the envelope.

What is the problem about?

When you look at the Copymasters, you will realise that this problem could be used many times during the year in any of the curriculum Strands. Alternatively you might want to mix up the sets of answers and use the problem towards the end of the year.

To be able to make up a problem from scratch like this requires a deeper understanding of the problem than just being able to solve it.

Hence this problem will provide:

  • a means of assessing a student's knowledge of the recent maths that they have done;
  • a way of delving deeper into the subject;
  • an insight to the 'mathematics language' skill of the student;
  • an opportunity for the student to experiment with the use of the language of mathematics.

Stumbling blocks

  1. Unfamiliarity with this type of activity.
    It is likely that students who haven't been given an exercise like this before will do no better than produce a sum such as 312 + 423 = 735, where 735 is the answer required. There are two directions that you can work from here. The first is to see what other sum they can make that has the answer 735 (perhaps a subtraction, multiplication or division sum). This gives them the opportunity to explore the number 735, it helps to cement their basic number facts, and it gives them a chance to use all of the four arithmetic operations. The second direction is to get them to embed the sum in a story. So you could ask them to put the problem into an actual situation. It may be necessary to prompt by providing 'part of the story'.
  2. Weak maths skills.
    Mathematically weaker students may need to be prompted to help them produce a sum. You may need to recall for them what sums they have done so far. Try to lead them on to word problems by providing a starter, like, "I had a garage sale and got $700 for my stereo, $30 for my surfboard, and... (etc)". Draw the students' attention to the use of the past tense verb ("had"... "got") and they way the verb is understood in the following phrases ("$30 for my surfboard") if the students are not using these language patterns accurately. Read more about the benefits of joint productive activity (CREDE). They will probably only be able to mimic these problems. But even that is a start towards deeper learning and understanding.
  3. Weak language skills
    Language constraints may limit the student's response. This may be indicated by students presenting very 'basic' problems in response to the answer provided or students presenting more complex problems, but relying upon maths symbols and not language. Be aware that scenario 'a' can be confused with the 'weaker maths skills' issue. Scenario 'b' may not be noticed, as the student will often produce a satisfactory result. If language constraints are suspected as the cause of basic answers, or answers that are not phrased in a language context - Intervention (Word 45KB) may be necessary.

Advanced groups:
More able students might be extended to problems that need more than one step or might be challenged to produce a problem with an answer such as 'Hannah'.

This open problem allows the students to use their imagination. It should give them the chance to invent some interesting word problems and to put the mathematics that they have learned so far together in creative ways.

If you plan to assess a particular area of mathematics using this problem, then it should probably not be used until the students have become confident in handling that area.

Finally you might like to note that this problem is one of a series of problems that extend from Level 1 to Level 6. It might be useful for you to see how the problems develop. The lessons are You Be The Teacher, Level 1, Make Up Your Own, Level 2, Invent-A-Problem, Level 3, Create a Question, Level 4 and Working Backwards, Level 6. It may be that there are some useful ideas in the other Levels that you can use with your class even at this Level.

Teaching sequence

There are several ways to approach this problem. If you think that this problem can be tackled easily by many of your class, then adopt Method A. This way you give little in the way of hints to the class as a whole. However, you may need to help some of them individually.

Method B is for a class that needs more help and it begins with a short session to remind the class of some of the things that they have done so far.

Method A: Start with all the students together

Tell them that today they are going to do something different. The aim is to make up their own maths problems. There is only one restriction. That is, the answer has to be the one you are going to give them in a sealed envelope. The problems that they make up can be anything they like.

  1.  Let the students work on the problem together in small groups. Help the ones who are having trouble. Those students who finish quickly could be asked to make up another problem using another envelope.
  2.  When a child has finished a problem put the problem into sealed envelopes. When you have collected enough problems the envelopes could be given to other students to solve either straight away or later on.
  3.  You could also give small prizes for the best problem, the funniest problem, and so on. The students could vote for the problems they like best.

Method B: Start with all the students together

  1.  Tell them that you would like them to make up some problems of their own today. Ask them what problems they can remember working on.
  2.  Ask the students to give you a number. Then try to get them to make up a problem using that number as the answer.
  3.  Now say that you have some answers in sealed envelopes. You want them to make up a problem that has that number as the answer.
  4.  Let them work in groups to come up with some problems of their own. If they can only produce a sum rather than a problem then you could get them to find other sums that make up that number or help them to produce a word problem.
  5.  The students' problems could go into an envelope for later use.
  6.  Those students who finish quickly might like to try to write another problem, solve someone else's problem or try the Extension problem.
  7.  Pose some of the students' problems from the sealed envelopes for the whole class to solve.
  8.  You might like to keep some of these problems to use with the class over the next few weeks. You could also give small prizes for the best problem, the funniest problem, and so on. The students could vote for the problems they like best.


The problem is still the same except that the answers are taken from set 10.


The solutions here will depend on your class. We would like to see some of them so that we could put them on this website.

Published on: 09 Jan 2018