Maths Support: Combining algebraic fractions

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Christian Lawson-Perfect

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Christian Lawson-Perfect 11 years, 6 months ago

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Question groups

Group 13 questions

1.
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Express $\displaystyle a \pm \frac{c}{x + d}$ as an algebraic single fraction.
2.
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Express $\displaystyle \frac{a}{x + b} \pm \frac{c}{x + d}$ as an algebraic single fraction over a common denominator. Contains a video in Show steps.
3.
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Express $\displaystyle b+ \frac{dx+p}{x + q}$ as an algebraic single fraction.
4.
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Express $\displaystyle ax+b+ \frac{dx+p}{x + q}$ as an algebraic single fraction.
5.
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Express $\displaystyle \frac{ax+b}{x + c} \pm \frac{dx+p}{x + q}$ as an algebraic single fraction over a common denominator.
6.
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Express $\displaystyle \frac{ax+b}{cx + d} \pm \frac{rx+s}{px + q}$ as an algebraic single fraction over a common denominator.
7.
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First part: Express $\displaystyle \frac{a}{px + b} +\frac{c}{qx + d},\;a=-c$ . Numerator is an integer. Second part: $\displaystyle \frac{a}{px + b} +\frac{c}{qx + d}+ \frac{r}{sx+t}$ as single fraction
8.
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First part: express as a single fraction: $\displaystyle \frac{a}{x + b} + \frac{c}{x + d},\; a \neq -c$ . Second part: Find $\displaystyle \frac{a}{x + b} + \frac{c}{x + d}+\frac{r}{x+t}$ as a single fraction.
9.
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First part: express as a single fraction: $\displaystyle \frac{a}{px + b} + \frac{c}{qx + d}$ . Second part: Find $\displaystyle \frac{a}{px + b} + \frac{c}{qx + d}+\frac{r}{sx+t}$ as a single fraction.
10.
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Express $\displaystyle \frac{a}{x + b} + \frac{cx+d}{x^2 +px+ q}$ as an algebraic single fraction over a common denominator.
11.
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Express $\displaystyle \frac{a}{x + b} +\frac{c}{(x + b)^2}$ as an algebraic single fraction.
12.
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Express $\displaystyle \frac{a}{x + b} +\frac{cx+d}{(x + b)^2}$ as an algebraic single fraction.
13.
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Express $\displaystyle \frac{a}{(x+r)(px + b)} + \frac{c}{(x+r)(qx + d)}$ as an algebraic single fraction over a common denominator. The question asks for a solution which has denominator $(x+r)(px+b)(qx+d)$ .

Topics

No topics have been defined.

Diagnostic algorithm

Extend built-in algorithm?

Base diagnostic algorithm

state (Produces the initial value of the state object): // should be renamed "initial_state"
    [
        "topics": map(
            [
                "topic": topic,
                "status": "unknown" // "unknown", "passed", or "failed"
            ],
            topic,
            values(topics)
        ),
        "retries": 3,
        "finished": false,
    ]

topics_by_objective (A dictionary mapping a learning objective name to a list of indices of topics):
    dict(map(
        let(
            ltopics, values(topics),
            indices, filter(lo["name"] in ltopics[j]["learning_objectives"], j, 0..len(ltopics)-1),
            [lo["name"],indices]
        ),
        lo,
        learning_objectives
    ))

unknown_topics (Which topics are still unknown?): 
    map(x["topic"],x,filter(x["status"]="unknown",x,state["topics"]))

first_topic (The first topic to pick a question on):
    unknown_topics[floor(len(unknown_topics)/2)]["name"]

first_question (The first question to show the student):
    random(topics[first_topic]["questions"])

get_dependents (An expression which gets the topics to update after answering a question):
    expression("""
        [target] + flatten(map(eval(get_dependents,["target":t,"correct":correct]),t,topics[target][if(correct,"depends_on","leads_to")]))
    """)

correct (Did the student get the current question right?):
    current_question["credit"]=1

after_answering (Update the state after the student answers a question):
    let(
        ntopics, eval(get_dependents,["target":current_topic,"correct":correct])
    ,   nstate, state + ['topics': map(
                    if(tstate["topic"]["name"] in ntopics, tstate + ["status":if(correct,"passed","failed")], tstate),
                    tstate,
                    state["topics"]
                )]
    , nstate
    )

action_retry (Use up one retry and visit the same topic again):
    [
        "label": translate("diagnostic.use retry"),
        "state": state + ["retries": state["retries"]-1],
        "next_question": random(topics[current_topic]["questions"])
    ]

action_stop (Stop the exam):
    [
        "label": translate("diagnostic.end test"),
        "state": state,
        "next_question": nothing
    ]

action_move_on (Move to the next topic, or end the exam if there are no more):
    let(
        state, after_answering,
        immediate_next_topics, topics[current_topic][if(correct, "leads_to", "depends_on")],
        unknown_topics, map(x["topic"],x,filter(x["status"]="unknown",x,state["topics"])),
        unknown_immediate_topics, filter(x["name"] in immediate_next_topics,x,unknown_topics),
        next_topics, if(len(unknown_immediate_topics), unknown_immediate_topics, unknown_topics),
        finished, len(next_topics)=0 or state["finished"],
        topic,
            if(not finished,
                next_topics[floor(len(next_topics)/2)]["name"]
            ,
                nothing
            ),
        [
            "label": translate("diagnostic.move to next topic"),
            "state": after_answering,
            "next_question": if(not finished, random(topics[topic]["questions"]), nothing)
        ]
    )

can_move_on:
    action_move_on["next_question"]<>nothing

next_actions (Actions to offer to the student when they ask to move on):
    let(
        feedback, retries_feedback+"\n\n"+translate("diagnostic.next step question")
    ,   [
            "feedback": feedback,
            "actions": if(not correct and state["retries"]>0, [action_retry], []) + if(can_move_on,[action_move_on],[action_stop])
        ]
    )

after_exam_ended (Update the state after the exam ends):
    let(
        state, after_answering,
        ntopics, map(t+["status": if(t["status"]="unknown","failed",t["status"])],t,state["topics"]),
        state+["finished": true]
    )

finished (Is the test finished? True if there are no unknown topics):
    len(unknown_topics)=0 or state["finished"]

total_progress:
    let(
        num_topics, len(state["topics"]),
        known, filter(tstate["status"]<>"unknown",tstate,state["topics"]),
        passed, filter(tstate["status"]="passed",tstate,known),
        num_known, len(known),
        num_passed, len(passed),
        [
            "name": translate("control.total"),
            "progress": if(num_topics>0,num_known/num_topics,0), 
            "credit": if(num_known>0,num_passed/num_topics,0)
        ]
    )

learning_objective_progress:
    map(
        let(
            tstates, map(state["topics"][j],j,topics_by_objective[lo["name"]]),
            known, filter(tstate["status"]<>"unknown",tstate,tstates),
            passed, filter(tstate["status"]="passed",tstate,known),
            num_topics, len(tstates),
            num_known, len(known),
            num_passed, len(passed),
            ["name": lo["name"], "progress": if(num_topics>0,num_known/num_topics,0), "credit": if(finished,num_passed/num_topics,if(num_known>0,num_passed/num_known,0))]
        ),
        lo,
        learning_objectives
    )

progress (Progress on each of the learning objectives, plus total progress):
    learning_objective_progress+
    total_progress

retries_feedback:
    translate("diagnostic.now assessing topic", ["current_topic": current_topic]) + " " +
    let(
        retries, state["retries"], 
        pluralise(retries, translate("diagnostic.one retry left"), translate("diagnostic.retries left", ["retries": retries ]))
    )
    + " " +
    let(
        p,total_progress["progress"],
        percentage, dpformat(100p, 0),
        translate("diagnostic.percentage completed", ["percentage": percentage])
    )

weak_objective_threshold (The amount of credit below which a learning objective is considered weak):
    0.6

finished_feedback:
    let(
        weak_objectives, filter(p["credit"]<weak_objective_threshold, p, learning_objective_progress),
    //
        translate("diagnostic.test is over") + " "
        +
        if(len(weak_objectives)=0,
            translate("diagnostic.passed all lo")
        ,
            translate("diagnostic.more work on lo", ["los": 
                join(map(lo["name"], lo, weak_objectives),", ")])
        )
    )

feedback:
    if(finished,
        finished_feedback,
        retries_feedback
    )

Custom diagnostic algorithm

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Enter review mode immediately on ending the exam?

	Always	In submitted mode	Only in review mode	Never
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Maths Support: Combining algebraic fractions

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Christian Lawson-Perfect 11 years, 6 months ago

Question groups

Topics

Learning objectives

Base diagnostic algorithm

Custom diagnostic algorithm

On leaving a question

On timeout

5 minutes before timeout